Hybrid Quantum Systems (Quantum Science and Technology) 9811666784, 9789811666780

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Quantum Science and Technology

Yoshiro Hirayama Koji Ishibashi Kae Nemoto   Editors

Hybrid Quantum Systems

Quantum Science and Technology Series Editors Raymond Laflamme, University of Waterloo, Waterloo, ON, Canada Daniel Lidar, University of Southern California, Los Angeles, CA, USA Arno Rauschenbeutel, Vienna University of Technology, Vienna, Austria Renato Renner, Institut für Theoretische Physik, ETH Zürich, Zürich, Switzerland Jingbo Wang, Department of Physics, University of Western Australia, Crawley, WA, Australia Yaakov S. Weinstein, Quantum Information Science Group, The MITRE Corporation, Princeton, NJ, USA H. M. Wiseman, Griffith University, Brisbane, QLD, Australia Section Editor Maximilian Schlosshauer, Department of Physics, University of Portland, Portland, OR, USA

The book series Quantum Science and Technology is dedicated to one of today’s most active and rapidly expanding fields of research and development. In particular, the series will be a showcase for the growing number of experimental implementations and practical applications of quantum systems. These will include, but are not restricted to: quantum information processing, quantum computing, and quantum simulation; quantum communication and quantum cryptography; entanglement and other quantum resources; quantum interfaces and hybrid quantum systems; quantum memories and quantum repeaters; measurement-based quantum control and quantum feedback; quantum nanomechanics, quantum optomechanics and quantum transducers; quantum sensing and quantum metrology; as well as quantum effects in biology. Last but not least, the series will include books on the theoretical and mathematical questions relevant to designing and understanding these systems and devices, as well as foundational issues concerning the quantum phenomena themselves. Written and edited by leading experts, the treatments will be designed for graduate students and other researchers already working in, or intending to enter the field of quantum science and technology.

More information about this series at https://link.springer.com/bookseries/10039

Yoshiro Hirayama · Koji Ishibashi · Kae Nemoto Editors

Hybrid Quantum Systems

Editors Yoshiro Hirayama Center for Science and Innovation in Spintronics Tohoku University Sendai, Miyagi, Japan

Koji Ishibashi Advanced Device Laboratory RIKEN Wako, Saitama, Japan

Kae Nemoto Principles of Informatics Research Division National Institute of Informatics Tokyo, Japan

Ministry of Education, Culture, Sports, Science and Technology ISSN 2364-9054 ISSN 2364-9062 (electronic) Quantum Science and Technology ISBN 978-981-16-6678-0 ISBN 978-981-16-6679-7 (eBook) https://doi.org/10.1007/978-981-16-6679-7 © Springer Nature Singapore Pte Ltd. 2021 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

Together with about 50 scientists around Japan, we pursued the basic science of hybrid quantum systems under the framework of the project “Science of Hybrid Quantum Systems” (HQS). This project was started in July 2015 as part of a Grant-inAid for Scientific Research on Innovative Areas, organized by MEXT (the Ministry of Education, Culture, Sports, Science and Technology) in Japan. The main streams of quantum science and quantum technology at that time were focused around quantum information processing, such as quantum cryptography, the realizations of qubits on a number of different physical systems such as superconductivity, electron spin, and atom and ion traps, and the integration of these qubits toward quantum computers. The project began with the finding that there was an approach distinctly different from a path of integration of the same type of qubits, and this approach would have potentials to lead us to a new landscape of quantum science. Even with a small number of qubits, we can achieve unprecedented sensor sensitivity. By developing quantum transducers with different physical quantities, we can exchange quantum information on different scales such as energy and distance. We investigated the quantum coupling between different physical quantities, such as charges, Cooper pairs, electron spins, nuclear spins, photons, and phonons, aiming to impact a wide range of fields from science and engineering to medicine. Significant effects and interesting results have been obtained over the 5 years of the project, and many collaborations began and have flourished, even continuing beyond the end of the project. We are planning to publish two books (“Hybrid Quantum Systems” and “Quantum Hybrid Electronics and Materials”) from Springer Nature to summarize our progress and to broadly discuss the importance of this hybridization work.

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This book, “Hybrid Quantum Systems”, summarizes results related to quantum transducers, quantum sensing using coherent spin operation, and theoretical studies on hybrid quantum systems. There are many stimulating chapters discussing the various hybrid systems in which charges, spins, nuclear spins, photons, and phonons are coherently coupled. We hope this book will help you understand the interest, importance, and diversity of “Hybrid Quantum Systems”. Finally, we are grateful to the members of the Editorial Office of Springer-Nature Publishing for their excellent help. We would also like to thank all the individual authors for their significant efforts. Sendai, Japan Wako, Japan Tokyo, Japan

Prof. Yoshiro Hirayama Prof. Koji Ishibashi Prof. Kae Nemoto

Contents

Control of Spin Coherence and Quantum Sensing in Diamond . . . . . . . . . Norikazu Mizuochi

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Wide-Field Imaging Using Ensembles of NV Centers in Diamond . . . . . . Shintaro Nomura

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Collective Effects in Hybrid Quantum Systems . . . . . . . . . . . . . . . . . . . . . . . William John Munro, Josephine Dias, and Kae Nemoto

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Rare Earth Non-spin-bath Crystals for Hybrid Quantum Systems . . . . . Takehiko Tawara

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Electron Spin Resonance Detected by Superconducting Circuits . . . . . . . Rangga P. Budoyo, Hiraku Toida, and Shiro Saito

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Hybrid Quantum Systems with Spins in Diamond Crystals and Superconducting Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Yuimaru Kubo High-Temperature Spin Qubit in Silicon Tunnel Field-Effect Transistors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Keiji Ono Ge/Si Core–Shell Nanowires for Hybrid Quantum Systems . . . . . . . . . . . . 165 Rui Wang, Jian Sun, Russell S. Deacon, and Koji Ishibashi Photonic Quantum Interfaces Among Different Physical Systems . . . . . . 197 Toshiki Kobayashi, Motoki Asano, Rikizo Ikuta, Sahin K. Ozdemir, and Takashi Yamamoto Hybrid Quantum System of Fermionic Neutral Atoms in a Tunable Optical Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 Hideki Ozawa, Shintaro Taie, Yosuke Takasu, and Yoshiro Takahashi

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Phonon-Electron-Nuclear Spin Hybrid Systems in an Electromechanical Resonator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 Yuma Okazaki and Hiroshi Yamaguchi Cavity Quantum Electrodynamics with Laser-Cooled Atoms and Optical Nanofibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 Takao Aoki Robust Quantum Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 Yuichiro Matsuzaki Transferring Quantum Information in Hybrid Quantum Systems Consisting of a Quantum System with Limited Control and a Quantum Computer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 Ryosuke Sakai, Akihito Soeda, and Mio Murao

Control of Spin Coherence and Quantum Sensing in Diamond Norikazu Mizuochi

Abstract Quantum superposition, namely coherence, is fundamentally important in quantum science. It is not easy to maintain it for a long time because it is very sensitive to environmental noises. Quantum sensing utilizes this character to realize high sensitivity. Therefore, control of coherence is a central issue in quantum technology. In this chapter, we mainly present recent our researches about the extension of coherence times of NV centers in diamond (1.1), electrical control and detection of spin coherence in diamond (1.2), and quantum hybrid sensors in diamond (1.3). In section (1.1), researches of the extension of coherence times of the NV centers by utilizing n-type diamond and a quantum hybrid system of dressed states are presented. Recently, we realized the longest inhomogeneous spin-dephasing time (T 2 * ≈ 1.5 ms) and Hahn-echo spin-coherence time (T 2 ≈ 2.4 ms) of single electron spins in NV centres, ever observed in room-temperature solid-state systems. In section (1.2), an electrical control for extension of coherence times and electrically detected magnetic resonance are presented. The latter is the first demonstration of room-temperature electrical detection of nuclear spin coherence in diamond and any other materials. In section (1.3), quantum hybrid sensors in diamonds are presented. Furthermore, the investigation of the magnetic resonance spectrum of a high-density ensemble of the NV centers, which is required for ultra-high sensitivity, is presented by introducing an appropriate model. By adopting this model, we estimated the optimal temperature sensitivity with the appropriate concentration of the NV centers. Keywords Diamond · NV center · Quantum sensor · Dressed state

1 Extension of Coherence Times of NV Centers in Diamond Solid-state spins are a leading contender in quantum technology [1]. Enhancing the inhomogeneous spin-dephasing time (T 2 * ) and the Hahn-echo spin-coherence time (T 2 ) is a central issue. The electron spin plays a significant role for quantum N. Mizuochi (B) Institute for Chemical Research, Kyoto University, Kyoto, Japan e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 Y. Hirayama et al. (eds.), Hybrid Quantum Systems, Quantum Science and Technology, https://doi.org/10.1007/978-981-16-6679-7_1

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sensing and for coherent connectivity with other qubits such as photons [2–4], nuclear spins [5–7], and superconducting qubits [8, 9]. Therefore, the coherence times define the physical behavior of the quantum device, and their improvement allows new perspectives for quantum applications. For example, their increase directly improves direct current (DC) and alternating current (AC) sensitivities of nitrogen-vacancy (NV) sensors [10, 11] quantum-gate fidelity [12, 13], and quantum-memory times [10, 12, 14, 15]. The extension of the spin coherence time has been demonstrated by suppressing noises with various techniques such as tailored dynamical decoupling [16–18], measurements at low temperature [19], measurements at high magnetic field [20], and decoupling by fast charge-state changes with high power laser irradiation [21] in addition to removal of noise sources by growth techniques [10, 14, 22, 23]. From the viewpoint of material science, enhancements of T 2 * and T 2 have been realized by the development of growth techniques to suppress paramagnetic defects [24], isotope engineering to reduce nuclear spins [10, 12, 22], and annealing techniques to remove defects [25]. Regarding diamond crystals, there has been remarkable progress in the quality of single crystal diamond grown by chemical vapour deposition (CVD). By suppressing impurities and defects, the T 2 of NV centres has been enhanced. For example, T 2 = 0.7 ms is reported, which is the longest T 2 at room temperature amongst diamonds which contain a natural abundance (1.1%) of 13 C [22, 26]. Furthermore, by depleting the 13 C isotope, the electron spin showed significantly long room-temperature spin-dephasing times (T 2 * = 100 μs [27], 470 μs [21]) and spin-coherence times (T 2 = 1.8 ms in single-crystal [10] and 2.0 ms in poly-crystal diamond [28]). In principle, T 2 can be extended closer to T 1 (~6 ms [5]), however, this has not been reached yet. The reason considered is the effect of residual paramagnetic defects and/or impurities in the bulk. During the CVD growth [29] and ion-implantation [30], it is known that many vacancies are generated. Once they combine, complexes or defect clusters are formed. It is very hard to remove them by annealing techniques because they are very stable. Therefore, the longest T 2 * and T 2 of single NV centres have been realized by native NV centres created during CVD growth, while those of NV centres created by ion-implantation and a high annealing temperature have not exceeded the longest ones.

1.1 Long Coherence Times and High Magnetic Sensitivity with NV Centres in Phosphorus-Doped n-type Diamond Extending coherence times is rather challenging. Although enrichment of the spinzero 12 C and 28 Si isotopes greatly reduces spin-bath decoherence in diamond and silicon, the solid-state environment provides undesired interactions between the electron spin and the remaining spins of its surrounding. Here we demonstrate,

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contrary to widespread belief, that an impurity-doped (phosphorus) n-type singlecrystal diamond realizes remarkably long spin-coherence times [14]. Single electron spins show the longest inhomogeneous spin-dephasing time (T 2 * ≈ 1.5 ms) and Hahn-echo spin-coherence time (T 2 ≈ 2.4 ms) ever observed in room-temperature solid-state systems, leading to the best sensitivities. The extension of coherence times in diamond semiconductors may allow for new applications in quantum technology. Recently, charging of vacancies suppressed the formation of vacancy complexes by confining implantation defects into a space-charge layer of free carriers created by a thin sacrificial boron-doped p-type diamond layer [30]. After removing this layer, the T 2 of the NV centres at the shallow surface region was improved only. In their research, the intrinsic diamond was used, but n-type conductivity is crucially important because of the stabilization of the negatively charged state of the NV centres (NV− ) [31, 32], and because of the electrical controls used in diamond quantum devices.

1.1.1

Sample and Doping Concentration Comparison

The n-type diamond samples A ~ H were grown onto Ib-type (111)-oriented diamond substrates by microwave plasma-assisted CVD with enriched C (99.998%) and with phosphorus concentrations ranging from 3 × 1015 to 1 × 1017 atoms/cm3 . For all samples, the growth conditions are the same except for the PH3 /CH4 gas ratio to change the phosphorus concentration. We address individual NV centres with a homebuilt confocal microscope. All experiments were conducted at room temperature. In order to compare the doping concentrations, two parameters were measured for a number of NV centres in every sample: the population of the NV− state, and T 2 . The NV− population is determined by single-shot charge-state measurements, and T2 with a Hahn-echo measurement. The results of samples A ~ H are plotted in Fig. 1.1. In the highly doped diamond samples C ~ H, a charged state of NV− near 100% is realized, while, regardless of the paramagnetic dopant (electron spin 1/2), T2 can surpass 2 ms. The different local surroundings of each NV centre are considered to be the reason for the scatter of T2 for each NV− population. NV centres with a higher NV− population show a longer maximum T2 . Moreover, sample C with [P] = 6 × 1016 atoms/cm3 seems to produce NV centres with longer T2 than diamond with both smaller and larger phosphorus concentrations, with an average over the first 21 measured NV centres of = 1.8 ms and almost 40% has T2 > 2.0 ms. This might indicate a competition between a positive effect of phosphorus doping and the negative effect of magnetic noise stemming from phosphorus, and therefore an optimum concentration could exist.

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Fig. 1.1 T 2 vs NV population for eight samples A ~ H with different doping levels: 3 × 1015 (crosses), 1 × 1016 (up-pointing triangles), 5 × 1016 (left and right-pointing triangles), 6 × 1016 (squares and diamonds), and 1 × 1017 atoms/cm3 (circles and pluses). The error bars indicate standard errors. The top-right of the graph is enlarged

1.1.2

Coherence Times

Owing to the pure NV− state and long T 2 , sample C with [P] = 6 × 1016 atoms/cm3 was investigated more in-depth. At first, T 2 * was studied, measured via the exponential decay of a free induction decay (FID) measurement. Since it is hard to measure T 2 * due to the strong effect of the environment, the measurement time is decreased by using both short and long delays in one measurement sequence, resulting in T 2 * = 1.54 ms. Compared with the previously reported long T 2 * of phosphor in an isotopically engineered 28 Si crystal (270 μs) and of NV centres in diamond (470 μs), this is the longest T 2 * for an electron spin ever observed in solid-state systems. The DC magnetic field sensitivity of the single NV centre can be derived to be ~6 nT/(Hz)1/2 . To find NV centres with a long T 2 , Hahn-echo measurements were conducted for the ones with a long T 2 * . Measurements for the MS = 0 state and the MS = 1 state were subtracted and normalised to reject common-mode noise, and the result was fitted to the exponential exp (−(Delay/T 2 * )). Figure 1.2c shows the longest T 2 consistently measured (2.43 ms), which is the longest T 2 for an electron spin ever observed in solid-state systems at room temperature. In these references, as opposed to our results, the measurements are performed without common-mode noise rejection, the results are rather noisy, and n was fixed at 2 for the fits. Hence, the uncertainty in T 2 decreases (since T 2 and n appear in the same exponent only), and the fitted T 2 increases (since for long T 2 , generally n < 2; for our measurement, forcing n = 2 gives T2 = 2.93 ms). Moreover, as visible in Fig. 1.2b, even the average = 1.8 ms of our measured NV centres rivals with the longest T 2 measured in single-crystal diamond (T 2 = 1.8 ms), and almost 40% are longer than the longest T 2 measured in poly-crystal diamond (T 2 = 2.0 ms), while both references only show their best measurement.

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Fig. 1.2 T 2 * and T 2 in sample C. a Result of FID measurement (data with crosses, sinusoidal exponential-decay fit with line, T 2 * = 1.54 ms). Please note the breaks on the horizontal axis; all data are fitted with a single function. b Results for the S|0> measurement (crosses) and for the S|1> measurement (circles). The top and bottom dashed lines correspond to a maximum population of the |0> and |1> states respectively. The middle dashed line is when both states are populated equally. c Echo signal derived from subtracting (b)’s S|0> (data with crosses, exponential-decay fit with line, T 2 = 2.43 ms). The dashed line at 0 indicates when the states are populated equally

1.1.3

Sensitivity

Since the AC magnetic field sensitivity is proportional to 1/(T 2 )1/2 , the NV centre with the longest T 2 was examined. The concept for the measurement is based on the oscillation of the population of the spin state with the magnetic field amplitude (BAC ). Therefore, to measure BAC , a working point of maximum gradient is chosen (the dashed lines in Fig. 1.3a), and then the Hahn-echo intensity is measured. Via the gradient, this intensity relates directly to BAC . The sensitivity is η = δBmin (T meas )1/2 with δBmin the minimum detectable magnetic field amplitude, and T meas the measurement time. δBmin relates directly to the uncertainty of the measurement, and is given by δBmin = σ1 /grad with σ1 the uncertainty of a single Hahn-echo measurement and grad the gradient in the working point. Since σ1 depends on the noise in the system, which generally scales over measurement time with 1/(T meas )1/2 , the sensitivity is independent of the measurement time, hence its usefulness for comparing systems. Also, please be aware that the sensitivity of a measurement should be determined in the same way the measurement itself is conducted, thus a technique such as used here is the only correct way to obtain a sensitivity. To obtain the optimum sensitivity, first, the T 2 data are used to estimate the optimum magnetic field frequency for the measurement, which is f AC = 833 Hz. Next, the gradient in the working point was determined. Since this is a constant given the measurement parameters and environment, a relatively long time averaging measurement can be used. An average result is shown in Fig. 1.3a, which gives grad

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Fig. 1.3 AC magnetic field measurement. a Single measurement to find the working points which have the maximum gradient (100 000 iterations, data with crosses, sinusoidal fi with line). The working point indicated with a circle is an example, the dashed arrows show how a measured intensity translates to a magnetic field amplitude. b Logarithmic plot of δBmin versus T meas

= 1.56 × 107 intensity T −1 . Next, the uncertainty of a measurement in the working point is extracted by measuring this point 100 times for a number of measurement times T meas ranging from 13 s to 11 min. In Fig. 1.3b, the resulting δBmin is fitted to η/(T meas )1/2 , from which follows the sensitivity η = 9.1 nT/(Hz)1/2 .

1.1.4

Discussion

The AC magnetic field sensitivity is improved by almost a factor of two compared to previous results. The reasons for this improvement are the longer T2 , the larger Rabi contrast and the slightly higher photon count due to n-type diamond. That the doping of phosphorus extends the spin-coherence times and gives better magnetic field sensitivities is against intuition, because phosphorus is paramagnetic at room temperature in diamond due to a large activation energy of 0.57 eV, and thus causes magnetic noise. The time extensions in n-type diamond are considered to be due to charging of the vacancies, which suppresses the formation of paramagnetic vacancy complexes during growth. Potentially, this mechanism is similar to

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the recently reported interpretation of suppression of vacancy creation during ionimplantation through a sacrificial boron-doped p-type layer. During the CVD growth, it is known that many vacancies are generated, which causes generation of thermally stable impurity-vacancy and multi-vacancy complexes. However, their generation can be suppressed by Coulomb repulsion of charged vacancies in n-type diamond, where an accepter level of the vacancy is located at 2.6 eV below the conduction band, which is lower than the donor level of phosphorus (0.57 eV). For a two-level system, T 2 is known to be ultimately limited by the longitudinal spin-relaxation time T 1 . We applied Carr-Purcell- Meiboom-Gill (CPMG) dynamic-decoupling sequences with common-mode noise rejection, and T 2dd = 3.3 ms was derived. Although T 2dd is longer than T 2 , it is still shorter than T 1 (6~7.5 ms). To obtain information about the sources of decoherence, we carried out noise spectroscopy and T 1 measurements.[Dav] . The noise spectrum of deep NV centres can be analyzed by a single Lorentzian. For our data (see Fig. 1.4), since the cut-off frequency eludes us, we fixed it as the highest probed frequency (thus τc < 0.25 μs), and fitted to find the minimum for  ( > 0.1 MHz). The minimum density of the paramagnetic impurities/defects npara was derived from  under the assumption that the noise source originates from dipolar interaction between these and the NV centre only (npara ~ /α with α = 3.3 × 10–13 cm3 /s), giving 3 × 1017 cm−3 , which is much larger than doped P. For a Lorentzian bath, in the limit of very short correlation times (τ c  T 2 ), the dynamical-decoupling sequence is inefficient and there is no improvement with the number of pulses. In addition, there is a limitation to the π-pulse duration, and their spacing restricts the maximum CPMG filter frequency. We consider that these explain why the decoupling technique is not very effective for the extension of T 2 . As for the contribution of nuclear spins, from the applied external magnetic field (1.8 mT), the Larmor frequencies of the nuclear spins of 14 N, 15 N, 13 C, 31 P, and 1 H were calculated and they are indicated in Fig. 1.4. They were not detected in the noise spectrum, which indicates as well that the contribution of the nuclear spins to the decoherence is small.

Fig. 1.4 Noise spectrum extracted from dynamic-decoupling measurements (with 2n pulses with n = 1 ~ 9, data with dots, Lorentzian fit with line, correlation time τc fixed to represent the highest probed frequency). The Larmor frequencies of several nuclear spins are indicated (dashed lines); B = 1.8 mT

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1.2 Extension of the Coherence Time by Generating MW Dressed States in a Single NV Centre in Diamond Here, we focus on the microwave (MW) dressed state based on Autler-Townes splitting (ATS). Using ATS, a large number of the dressed states can be generated. It has also been reported that T 2 of the MW dressed states is longer than that of undressed states. In this research [33], we experimentally demonstrate the generation of the MW dressed states in the single NV centre in diamond by ATS at ambient conditions in order to analyse fundamental phenomena. We show the extension of T 2 under the generation of the dressed states. Mechanism of Autler-Townes Splitting Figure 1.5a shows the energy level of the NV electron spin coupled with the 14 N nuclear spin of the NV centre, where |ms , mI  is defined as the electron and the 14 N nuclear spin of the NV centre, respectively. After laser illumination, the NV centre is equally polarised in |0,0, |0,1, and |0,−1 depicted by the open circles under the application of a static magnetic field (B0 ). Figure 1.5a also depicts the irradiation of an unperturbed drive field whose frequency is close to a resonant frequency of a transition between |0,1 and |−1,1. When the drive field is considered as a classical mw mode, the NV centre can be coupled to the mode of the drive field. Then, each |0,1 and |−1,1 is split into two levels described in Fig. 1.5b. As an example, Fig. 1.5b depicts the generation of four dressed states of |0,1|1, |−1,1|0, |0,1|2, and |−1,1|1 in the presence of coupling between the NV centre (|0,1 and |−1,1) and the mode of the drive field (|0, |1, and |2). This phenomenon is called (weak) ATS. Figure 1.5b also shows the energy levels of the dressed states which are characterised by the Rabi frequency of an NV electron spin () and frequency of the drive field (ω). Fig. 1.5 a Energy diagram of the NV centre under irradiation of a weak drive field. b Dressed energy level coupling with a mode of the drive field

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Results and Discussion The sample is a type IIa (111) diamond. In our experiment, we chose an NV centre that is weakly coupled to other nuclear spins (e.g., 13 C nuclear spin). We measured the optically detected magnetic resonance (ODMR) spectrum with a 1-μs pulsed laser by sweeping the frequency of a 5.5-μs pulsed probe MW (Pmw) pulse (π pulse) depicted top of Fig. 1.6. In Fig. 1.6a, the ODMR spectrum has three dips with 2.1 MHz splitting, which corresponds to the hyperfine splitting of the 14 N nuclear spin of the NV centre. Experimental Generation of Dressed States by ATS First, we measured the change in the dressed-state resonant frequencies by changing the power of continuous drive MW (Dmw ) with pulse sequence depicted in the top of Fig. 1.6. These experiments use the three Dmw frequencies of 2834.75 MHz (Dmw 1), 2837.05 MHz (Dmw 2), and 2839.18 MHz (Dmw 3) to generate dressed states. The results are shown in Fig. 1.6b. The signals for each Dmw frequency split into three above ~10 μT. Fig. 1.6 a ODMR spectrum without any drive fields. b Resonant frequencies as a function of the strength of the drive field (Bdrive ). Black, red, and blue plots show the changes in resonant frequencies under the irradiation of Dmw frequencies of 2834.75 MHz (Dmw 1), 2837.05 MHz (Dmw 2), and 2839.18 MHz (Dmw 3), respectively. Solid lines are fitted for each resonant frequency. c ODMR spectrum under a Dmw at a frequency of 2834.75 MHz and a Dmw power of 33 μT. We can observe the Mollow triplet, which we call ATS

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Here we focus on the 2834.75 MHz of Dmw 1. It should be noted that all three Dmw frequencies have the same dependences on the power of the continuous Dmw . The ODMR spectrum under continuous irradiation at Dmw 1 with the power of 33 μT is shown in Fig. 1.6c. Figure 1.6c shows the ODMR spectrum at ~2834.75 MHz splits into three peaks under the irradiation of the Dmw . Figure 1.6b shows that the resonant frequencies of the dressed states as a function of Bdrive . The solid lines show the linear fitting for each observed data. The absolute values of these slopes in Fig. 1.6b agree well with the gyromagnetic ratio of the NV electron spin (γNV ), so that means the resonant frequencies of side peaks are linearly proportional to the Rabi frequencies of the NV electron spin. These our results are consistent with the theoretical prediction in ATS, demonstrating the generation of more than four dressed states by the ATS [33]. Coherence Time of Dressed States First, in order to investigate the magnetic moments of the dressed and undressed states, we measured Rabi oscillations of the dressed states and the NV electron spin as described in Ref. [33]. The result indicates the Rabi frequency of the dressed spin states is the same with that of undressed spin states [33]. Consequently, the magnetic moment of the NV electron spin and the dressed states are the same with each other. Next, we measured coherence time of the dressed states (T 2ρ ) and T 2 of a single NV centrein a 12 C enriched diamond. The top of Fig. 1.7 shows the pulse sequence for the T 2ρ and T 2 measurements. It is noted that while T 2ρ was measured with the continuous Dmw irradiation, T 2 was measured without continuous Dmw irradiation. Since the pulsed laser and continuous Dmw were simultaneously irradiated to the NV centre during the T 2ρ measurements, we kept the pulse sequence time (T seq ) constant adjusting interval between the final π/2 Pmw pulse and the readout laser pulse depicted in Fig. 1.7. Then, the dressed spin states can be initialised by simultaneous irradiation of the pulse laser and the continuous Dmw in the T 2ρ measurements. Additionally, Fig. 1.7 (Top) Pulse sequence to observe T 2ρ and T 2 with applying a phase cycle to the final π/2 pulse. (Bottom) Black and red plots show the results of T 2ρ and T 2 measurements, respectively. They are fitted by exponential decay curves described by black and red solid lines

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a phase cycling technique was applied to T 2ρ measurements in order to remove common-mode noise from laser fluctuations. In the case of a Pmw and Dmw strength of ~0.43 MHz and ~1.2 MHz, respectively, the result of T 2ρ (black plots) and T 2 (red plots) measurements fitted with exponential decay curves are shown in Fig. 1.7. The results show that we observed a coherence time of T 2ρ ~ 1.5 ms of the dressed states, which is more than two orders of magnitude longer than T 2 ~ 4.2 μs of the undressed states. While such an extension can also be demonstrated by a dynamical decoupling technique, e.g., a CPMG sequence in the NV centres, an extension of two orders of magnitude by the ATS is much larger than the extension of about one order of T 2 in the dynamical decoupling techniques. The extended T 2 by the ATS is also close to the longest T 2 of a single NV centre in a 12 C enriched diamond. Related Research About Dressed State: Experimental Demonstration of TwoPhoton Magnetic Resonances in a Single-Spin-System of a Solid In the research described at previous section, the dressed states among the electron spin with microwave were generated. In addition, we investigated dressed states including nuclear spins by introducing a radio frequency (rf). In this experimental demonstration, we used the electron spin and the 14 N nuclear spin of the NV center to generate the dressed states [34]. While the manipulation of quantum systems is significantly developed so far, achieving a single-source multi-use system for quantum-information processing and networks is still challenging. A virtual state, a so-called “dressed state,” is a potential host for quantum hybridizations of quantum physical systems with various operational ranges. We present an experimental demonstration of a dressed state generated by two-photon magnetic resonances using a single spin in a single NV center in diamond. The two-photon magnetic resonances occur under the application of microwave and radio-frequency fields, with different operational ranges. The experimental results reveal the behavior of two-photon magnetic transitions in a single defect spin in a solid, thus presenting new potential quantum and semiclassical hybrid systems with different operational ranges using superconductivity and spintronics devices. We discussed the strategy for generating a hybrid system with a generated dressed state. Trifunovic et al. [35] have proposed a hybrid strategy between magnetic materials and electron spin in NV centers in diamond using dipole coupling for a fusion of classical and quantum information processing. In this case, our demonstration of two-photon magnetic resonance (TPMR) indicates that we can hybridize these spins with wide-range operational fields through an NV center because we can optimize the parameters for coupling between both spins in both the mw and rf operational fields. We clearly observed the generated dressed states using TPMRs via the single-defect spin in diamond. We showed that the generated dressed states can be converted reversibly by driving both fields. These results indicate the potential for hosting hybridizations of physical systems with different operational ranges. Thus, our results pave the way for a new fusion of physical systems with single qubits and wide operational ranges.

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2 Electrical Control and Detection of Spin Coherence in Diamond 2.1 Electrical Control for Extension of Spin Coherence Times of NV Centers in Diamond The extension of the spin coherence times has been demonstrated by suppressing noises with various techniques such as, dynamical decoupling, measurements at low temperature, measurements at high magnetic field, and decoupling by fast chargestate changes with high power laser irradiation in addition to removal of noise sources by growth techniques. On the other hand, realization of the suppression by an electric field is important because the electric field can be locally operated in individual onchip nanoscale devices such as scalable quantum device in a dense array and quantum nanoscale sensing device. Furthermore, it does not need huge energy consumption facilities for operation and rare materials synthesized by isotopes without a nuclear spin. The electrical control is challenging because the electric fields do not couple directly to the spin unlike the magnetic field. Previously, the static electric field dependence on magnetic resonance frequencies of the NV center was reported [36]. By using the dependence, application of the NV center to the nanoscale electric field sensors was demonstrated [37, 38]. In the previous research [37], the analysis of the electronic structure of the NV center reveals how an applied magnetic field influences the electric-field-sensing properties. In this study [39], coherence times, which were estimated from a free-induction-decay (T 2 FID ) and a Hahn-echo decay (T 2 echo ), were measured under the externally applied static electric fields. We report the increase of T 2 FID and T 2 echo under the electric field and discuss the mechanism. The basic idea is as follows. The Hamiltonian is expressed as Hgs

   2 1 1 || = 2 Dgs + dgs E z Sz − S(S + 1)  3  1 ⊥ + 2 dgs E x (S y2 − Sx2 ) + E y (Sx S y + S y Sx )  1 + μ B ge S · B 

(1)

where , μB , and ge are the reduced Planck constant, the Bohr magneton, and the g factor of the electron spin, respectively. S is a spin operator and the spin quantum ⊥ /h number of the NV center is 1 (S = 1). d gs || /h = 0.35 ± 0.02 kHz·cm/kV and dgs = 17 ± 3 kHz·cm/kV [36] are the measured axial and non-axial components of the ground triplet state electric dipole moment, respectively. The following Eq. (1) is derived from the spin Hamiltonian described above for the change ω± of the resonance frequency due to the magnetic field B and the electric field E. However, it is limited to the region where the amount of change in energy is sufficiently smaller

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⊥ than zero magnetic field splitting Dgs = 2870 MHz (h Dgs  μ B ge B, dgs E ⊥ ). || Ez ± ω± = dgs

 ⊥2 E 2 μ2B ge2 Bz2 + dgs ⊥

(2)

Next, the mechanism of increasing coherence time is explained. Since spin decoherence is caused by fluctuations in the resonance frequency, Eq. (1), which expresses the change in the resonance frequency due to the external field, is important. Looking at the second term of Eq. (1), which is a square root of the sum of the squares of the magnetic field term and the electric field term. In this case, when the contribution of either the electric field or the magnetic field is larger than that of the other, the influence on the resonance frequency of the other becomes smaller. Furthermore, when one is much larger than the other, the resonance frequency hardly changes even if the one with the smaller contribution changes a little. Figure 1.8 shows this relation. When the fluctuation of the resonance frequency due to magnetic noise is the cause of decoherence, it can be expected that the fluctuation of the resonance frequency is suppressed and the coherence time is increased by applying the electric field. In this research [39], we showed the electrical control for extension of the spin coherence times of 40 nm-deep ion-implanted single NV center spins in diamond by suppressing magnetic noises. NV centers were created by ion implantation and subsequent annealing on a IIa (100) single-crystalline diamond substrate. 14 N with a natural abundance concentration (99.6%) was implanted at a depth of about 40 nm by an ion-implantation with a kinetic energy of 30 keV. Electrodes and an antenna to apply an electric field and microwaves, respectively, were formed on the substrate (Fig. 1.9a). From the current–voltage property of the structure to apply an electric field (Fig. 1.9b), it was supposed that the voltage at the interface between the diamond and the electrode was zero and the electric field strength in diamond bulk was linearly proportional to the applied electric field. We applied 120 V DC across two contacts spaced by 10 μm. The spin coherence times, estimated from a free-induction-decay and a Hahn-echo decay, were increased up to about 10 times as shown in Fig. 1.10 and 1.4 times (reaching 150 microseconds), respectively [39].

Fig. 1.8 Comparison of the magnitude of resonance frequency fluctuation ω when the electric field strength is different. The solid line in the figure shows how the energy level of each spin sublevel changes depending on the magnetic field in the z direction. (Left) Under zero electric field. (Right) When electric field large, for example, in the case of E ~ 100 kV/cm

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Fig. 1.9 a Schematic image of the measurement setup. b Current–voltage property between the electrodes on the sample. Linear I-V property is observed. c Distribution of the electric field strength in the case where 100 V between the electrodes. Dots marked as NV1, NV2, and NV3 indicate the positions of the NV centers of NV1, NV2, and NV3, respectively. d Coordinate system of the NV centers

Fig. 1.10 a Ramsey pulse sequence to measure T 2 FID . b Decay curves of the Hahn echo signal. Fit function of f echo (2τ ) = y0 + A · exp(−(2τ/T2echo )3 ) is employed. c Electric field dependence of T 2 FID . Horizontal axis, which represents the electric field strength, is normalized by the magnetic field along the z-axis. Dashed curve represents the fitted curve for NV1. Chained curve represents the fitted curve for NV2

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The electric field dependence of T 2 FID and T 2 echo at a range up to ~100 kV/cm were quantitatively analyzed based on the spin Hamiltonian. Assuming only a magnetic field fluctuation, the behavior of T 2 FID is basically elucidated. The behavior of T 2 echo is well reproduced assuming fluctuations in both the electric and magnetic fields. Although the magnetic field fluctuation is the dominant decoherence source for T 2 FID in the entire range of the electric field in our experiment and for T 2 echo in the low electric field region, the dominant decoherence source for T 2 echo in the high electric field region is the electric field fluctuation. The difference in the dominant decoherence source under the electric field is due to the difference in the amplitude– frequency characteristics, corresponding to the difference in the correlation time of the magnetic field fluctuation and the electric field fluctuation in our experiment. The enhancement of the coherence times by the electric field can contribute to the improvement of the sensitivity in thermometry, pressure and AC electric field sensing, although it is not effective to the magnetic field sensitivity [27]. The present technique can be utilized for not only for the NV center in diamond but also the highspin centers (S > 1) with the zero-field splitting, such as promising centers in silicon carbide [40, 41]. Our study opens up the new technique of the electrical decoupling of the spin coherence in solid from the magnetic noises.

2.2 Room Temperature Electrically Detected Nuclear Spin Coherence of NV Centres in Diamond We demonstrate electrical detection of the 14 N nuclear spin coherence of NV centres at room temperature. Nuclear spins are candidates for quantum memories in quantuminformation devices and quantum sensors, and hence the electrical detection of nuclear spin coherence is essential to develop and integrate such quantum devices. In the later section of (1.3.1), hybrid quantum magnetic-field sensor with an electron spin and a nuclear spin to enhance a sensitivity is presented. In the present study, we used a pulsed electrically detected electron-nuclear double resonance technique to measure the Rabi oscillations and coherence time (T 2 ) of 14 N nuclear spins in NV centres at room temperature. We observed T2 ≈ 0.9 ms at room temperature. Our results will pave the way for the development of novel electron- and nuclear-spin-based diamond quantum devices. Nuclear spins in a semiconductor have a long coherence time (T 2 ) due to the good isolation from environmental noise. Therefore, they are candidates for quantum memories in quantum-information devices and quantum sensors [42–44]. Using nuclear spins (e.g., nitrogen and carbon) in diamond for quantum memories, highly sensitive magnetic sensors, quantum repeaters, quantum registers, etc., have been demonstrated at room temperature. In these demonstrations, the detection of nuclear spin coherence is essential. Nuclear spin coherence can be detected via the electron spins of NV centres, which also have a long T 2 at room temperature. NV electron

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spins can be detected by optical techniques and electrical techniques. The electrical technique is an important technology for developing and integrating quantum devices. Furthermore, a theoretical model predicts that its detection sensitivity is approximately three times higher than that of the optical technique [60]. While the photoelectrical detection of the electron spin coherence of an ensemble of NV centres and photoelectrical coherent spin-state readout of single NV centres at room temperature has been demonstrated, the direct electrical detection of nuclear spin coherence remains challenging. Thus, we focus on an electrically detected electronnuclear double resonance (EDENDOR) technique to demonstrate room-temperature electrical detection of nuclear spin coherence. Electrical Detection of Nuclear Spin The first EDENDOR measurement was demonstrated by Stich and collaborators for phosphorus (P) donors in silicon at 4.2 K [45]. After this demonstration, two other groups independently demonstrated pulse EDENDOR measurements of P-donors in silicon [46, 47]. They measured Rabi oscillations and T 2 of P-donor nuclear spins from 3.5 to 5 K. Furthermore, pulsed EDENDOR measurements of proton nuclear spins in organic semiconductors have been demonstrated [48]. Proton nuclear spin resonances have been measured at room temperature, but Rabi and T2 measurements of proton nuclear spins have not been reported yet. To the best of our knowledge, there have been no demonstrations of room-temperature electrical detection of nuclear spin coherence in diamond or any other materials. The EDENDOR signals of 14 N nuclear-spin coherence are observed by measuring the change in the electrically detected electron-spin echo intensity of the NV centres. The electrically detected magnetic resonance (EDMR) of the NV centres measures the photocurrent change due to electron-spin resonances of the NV centres. The photocurrent can be generated under illumination by a 532-nm laser via a two-photon ionisation process, depicted in Fig. 1.11a. The Fig. shows that the |±1 NV electron spin at the 3 E state has a transition probability to the long-lived (~220 ns) metastable 1 E states, where |ms  describes an NV electron spin. This causes the photocurrent to decrease due to a magnetic resonance transition from |0 to |±1 after the optical

Fig. 1.11 (Colour online) a Schematic of the EDENDOR measurements of NV centres in diamond. b Process for the EDENDOR measurements

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initialisation to |0. Based on this mechanism, this study measures the electron and nuclear spin coherences with the pulse sequence depicted in Fig. 1.11b. To measure the nuclear magnetic resonance of 14 N nuclear spins, we used the Davies ENDOR technique [49]. First, an MW π-pulse is applied to the transition between |0 to |+1 after illumination by a pulsed laser. This π-pulse can generate hyperfine coupling between NV electron spins and 14 N nuclear spins and the polarisation between |+1,0 and |+1,+1, where |ms ,mI  are electron and nuclear spins, respectively. Then, the RF pulse is applied. Finally, Q was measured by applying a Hahn echo sequence and the following laser pulse. We observed EDENDOR spectra by measuring Q as a function of the irradiated RF frequency. Setting the MW frequency to 2916 MHz, the input MW power to 5 W, and the input RF power to 5 W, we observed the spectrum shown in Fig. 1.12. The observed data can be fitted with the Gaussian function shown by the solid red line in Fig. 1.12. We can estimate that the observed resonance frequency corresponds to the transition between |+1,0 and |+1,+1 of the 14 N nuclear spins. Rabi Oscillations of

14 N

Nuclear Spins

The top of Fig. 1.13 shows the pulse sequence for the measurements of the 14 N nuclear-spin Rabi oscillations. The sequence shows that we measured Q as a function of the length of the RF pulse. Here, we set the MW frequency to 2916 MHz, the input MW power to 5 W, and the RF frequency to 3.5 MHz. The results of the electrically detected nuclear-spin Rabi measurements with four different input RF powers are shown in the bottom of Fig. 1.13. The red, black, blue, and green points correspond to the results with input RF powers of ~2.5, 5, 10, and 20 W, respectively. The observed oscillations are fitted by sinusoidal curves, which are shown as solid lines in the bottom of Fig. 1.13. The results of the curve fittings showed that all Fig. 1.12 Pulse sequence (top) and the result (bottom) of an electrically detected ENDOR spectrum. ±x indicate the phase of MW pulse

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Fig. 1.13 Pulse sequence (top) and the result (bottom) of an electrically detected nuclear Rabi oscillation. ±x indicate the phase of MW pulse. (Inset) Rabi frequencies as a function of the square root of RF power

Rabi oscillations have the same phase offset which depends on the polarisation of 14 N nuclear spin after applying the MW π pulse [49]. Furthermore, the oscillation frequencies observed by the curve fittings are plotted as a function of the square root of the input RF power in the inset of Fig. 1.13. The plots are fitted well by a linear function with the intercept at zero, as shown by the solid line, which certifies that the observed oscillations correspond to Rabi oscillations between |+1,0 and |+1,+1. Hence, the Rabi oscillations of the 14 N nuclear spins can be observed with EDENDOR at room temperature. Echo Decay of

14 N

Nuclear Spins

In echo decay measurement of 14 N nuclear spins, a nuclear-spin Hahn echo sequence was added between the first MW π- and second MW π/2-pulse, allowing the nuclear spin echo intensity to be measured via the change in the ESE intensity. We set the input RF power to 10 W and the other experimental conditions were the same as those for electrically detected nuclear spin Rabi oscillations. Then, we measured Q as a change of the ESE intensity as a function of the freely evolving time of 2τ. From the experimentally observed decay and the fitting with an exponential curve by fixing the echo amplitude at half the Rabi amplitude, consequently, 14 N nuclear spin coherence time T2 (n) ≈ 0.9 (5) ms was estimated [49]. Hence, we successfully observed T2 (n) with the EDENDOR technique at room temperature.

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2.3 Ferromagnetic-Resonance Induced Electromotive Forces in Ni81 Fe19 |p-Type Diamond For an electrical control of spin in diamond, we have an interest in the spin current injection to diamond. We report on direct-current (DC) electromotive forces (emfs) in a nickel–iron alloy (Ni81 Fe19 )|p-type diamond under the ferromagnetic resonance of the Ni81 Fe19 layer at room temperature [50]. The observed DC emfs take its maximum around the ferromagnetic resonant frequency of the Ni81 Fe19 , and their signs are reversed by reversing the direction of an externally-applied magnetic field; it shows that the observed DC emfs are spin-related emfs. The origin of the Lorentzian emfs can be inverse spin-Hall effect in the Ni81 Fe19 |ptype diamond; however, this is still unclear. Additional experiments are required to clarify its origin, e.g., spinpumping-induced spin transport measurements.

3 Quantum Hybrid Sensors 3.1 Hybrid Quantum Magnetic-Field Sensor with an Electron Spin and a Nuclear Spin in Diamond In this research [51], we propose a scheme to improve the sensitivity of magnetic-field sensors by using a hybrid system of an electron spin and nuclear spin in diamond. The concept is that the electron spin has a strong coupling with the magnetic fields, and so we use this to accumulate the phase from the fields. On the other hand, since the nuclear spin has a longer coherence time than the electron spin, we can store the phase information in the nuclear spin (Fig. 1.14a). By repeating this procedure in Fig. 1.14 a Schemetic image of the method. b A pulse sequence to detect magnetic fields with an electron spin and a nuclear spin in the NV center

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this hybrid system, we showed that it is possible to detect static magnetic fields with the sensitivity far beyond that of previous sensors using just a single electron spin. Since the nuclear spin is coupled with an electron spin via a hyperfine coupling, it is possible to transfer the information attained by the electron spin to the nuclear spin for the storage. Actually, a controlled-NOT (C-NOT) gate between the electron spin and the nuclear spin has been already demonstrated in many previous researches [5]. Thus, we can construct an efficient hybrid magnetic-field sensor to combine the preferable properties of these two different systems (Fig. 1.14b). However, if we consider the effect of decoherence, there is of course a difficulty with this simple picture, namely, a propagation of the error from the electron spin to the nuclear spin. Due to the dephasing effect of the electron spin, the non- diagonal term of the entangled state decreases as quickly as that of the electron spin does. This dephasing error might be accumulated in the nuclear spins, which could destroy the phase information obtained from the target magnetic field. Especially, if the dephasing noise is Markovian, the sensitivity of the hybrid field sensor is as small as that of the conventional one, due to the error propagation. In the Markovian dephasing model, the nondiagonal term of the density matrix decays exponentially. On the other hand, in the case that the relevant dephasing in the NV center is induced by low-frequency noise [22] which is not Markovian, we can suppress the error accumulation. Under the effect of low-frequency noise, the nondiagonal term of the density matrix decays quadratically. Due to this property, the initial decay of the non-Markovian noise is slower than that of the Markovian noise. We numerically evaluated the method and concluded that the sensitivity of our sensor is one order of magnitude better than the conventional one if the gate error is below 0.1% as shown in Fig. 1.15. It should be noted that this can be realized in non-Markovian dephasing model, so it is applicable to low concentration NV center. Note that our scheme can be interpreted as an application of quantum Zeno effect (QZE) [52, 53]. It is known that QZE occurs in a system to show a quadratic decay while QZE cannot be observed for exponential decay process. We consider that this research is important in the point that it shows the advantage to utilise the quantum hybrid system for quantum sensing in some conditions compared with the conventional method. Fig. 1.15 Sensitivity r with respect to the error ε. If the gate error is below 0.1%, the sensitivity of our sensor is one order of magnitude better than the conventional one

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Fig. 1.16 ODMR with zero applied magnetic fields. The red line denotes a numerical simulation and blue dots denote the experimental results

3.2 ODMR of High-Density Ensemble of NV− Centers in Diamond The ODMR is a way to characterize the NV− centers. For ultra-high sensitivity, highdensity ensemble of NV− centers are required, therefore, elucidation of a magnetic resonance spectrum of a high-density ensemble of the NV centers are very important. Particularly, for high magnetic field sensitivity, elucidation of it at zero magnetic field is essential. In previous many researches, it is reported that the ODMR spectrum exhibited small splitting with several MHz at zero magnetic field. In most of them, it is simply interpreted that it is due to the strain induced by imperfection of the crystallinity. However, if it is true, most of the NV centers must have a similar strain splitting with each other. In our research, we showed that the small splitting should be interpreted not by the single strain splitting but by strain distributions, inhomogeneous magnetic fields, and homogeneous broadening width [54]. Recently, a remarkably sharp dip was observed in the ODMR with a high-density ensemble of NV centers, and this was reproduced by the previously reported theoretical model [55], showing that the dip is a consequence of the spin-1 properties of the NV− centers. We showed much more details of analysis to show how this model can be applied to investigate the properties of the NV− centers. By using our model, we have reproduced the ODMR with and without applied magnetic fields. Also, we theoretically investigate how the ODMR is affected by the typical parameters of the ensemble NV− centers such as strain distributions, inhomogeneous magnetic fields, and homogeneous broadening width. Our model could provide a way to estimate these parameters from the ODMR, which would be crucial to realize diamond-based quantum information processing. In addition, by fitting the Hahn echo decay curve with a theoretical model, we showed that both the amplitude and correlation time of the environmental noise have a clear dependence on the spin concentration [56]. These results are essential for optimizing the NV center concentration in high-performance quantum devices, particularly quantum sensors.

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3.3 Optimization of Temperature Sensitivity Using the ODMR Spectrum of a NV Center Ensemble By adopting the model in the above Sect. 1.3.2, we estimated the optimal temperature sensitivity with the appropriate concentration of the NV centers [57]. Temperature sensing with NV centers using quantum techniques is very promising and further development is expected [58, 59]. Recently, the ODMR spectrum of a high-density ensemble of the NV centers was reproduced with noise parameters sycg as inhomogeneous magnetic field, inhomogeneous strain (electric field) distribution, and homogeneous broadening of the NV center ensemble [54]. In our study [57], we use ODMR to estimate the noise parameters of the NV centers in several diamonds. These parameters strongly depend on the spin concentration. This knowledge is then applied to theoretically predict the temperature sensitivity. Using the diffractionlimited volume of 0.1 μm3 , which is the typical limit in confocal microscopy, the optimal sensitivity is estimated to be around 0.76 mK/(Hz)1/2 with an NV center concentration of 5.0 × 1017 /cm3 . This sensitivity is much higher than previously reported sensitivities, demonstrating the excellent potential of temperature sensing with NV centers. We investigated the dependence of the noise parameters of high-density NV center ensembles. Inhomogeneous magnetic fields and homogeneous broadenings nearly linearly depend on the spin concentrations, whereas the inhomogeneous strain (electric field) distribution nearly linearly depends on the NV center concentration. Additionally, to illustrate the importance of such parameter dependences when optimizing a quantum device, we theoretically estimated the influence of spin concentration on the performance of the NV center ensemble as a temperature sensor. Based on the theoretical calculations, the sharp dip structure in the ODMR spectrum is suitable for a temperature sensor. The optimal sensitivity is predicted to occur around 0.76 mK/(Hz)1/2 with an NV center concentration of 5.0 × 1017 /cm3 and a spatial resolution of ~ 0.1 μm3 , which is the diffraction-limited volume in a typical confocal microscope. This estimated value is better than the previous experimental results [58, 59]. Our results are essential to control and use NV center ensembles as high-performance quantum devices, particularly as temperature sensors.

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Wide-Field Imaging Using Ensembles of NV Centers in Diamond Shintaro Nomura

Abstract Imaging methods exploiting quantum coherence are attracting much interest in a variety of platforms for their high sensitivity. Among them, wide-field imaging using ensembles of NV centers in diamond provides advantages in high stabilityand high resolution-imagings, and has already been exhibiting its potential in various applications. In this chapter, we present wide-field imaging using ensembles of NV centers, which is expected to find new applications for quantum sensing. Keywords Quantum sensing · Diamond · NV centers · Microwave imaging · Wide-field microscopy

1 Introduction Imaging has been a key technology in physical and life sciences. Recent advances in high-sensitivity measurement exploiting quantum coherence are further extending the application fields of imaging. The imaging utilizing an electron spin resonance of nitrogen-vacancy (NV) centers in diamond [1–4] has a number of advantages. First, NV centers show high charge stability under ambient conditions against photoexcitations [5] and temperature change because NV centers are embedded in a rigid and stable crystal. Second, the temperature range of the operation as a sensor is extremely broad ranging from mK to 700 K [6]. The NV spin states can be actively initialized to the ground state by optical excitation instead of passive equilibration with a cold bath. Moreover, the inhomogeneous spin coherence time (T2∗ ) was found to be temperature independent up to at least 625 K [6]. Third, the high sensitivity is achieved by utilizing elaborate RF/microwave pulse sequences originally developed for high-resolution nuclear magnetic resonance (NMR) spectroscopy, ion-trap, or superconducting qubits, such as a periodic [7] and a concatenated [8] dynamical decoupling techniques for suppressing decoherence. Fourth, the spatial resolution S. Nomura (B) University of Tsukuba, Tsukuba, Japan e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 Y. Hirayama et al. (eds.), Hybrid Quantum Systems, Quantum Science and Technology, https://doi.org/10.1007/978-981-16-6679-7_2

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S. Nomura

Fig. 1 Schematics of a a confocal microscope for scanning probe microscopy and b a wide-field microscope for the imaging utilizing an electron spin resonance of NV centers in diamond. APD: avalanche photodiode, BS: beam splitter

is practically not limited by the size of the sensor because a single NV− center constitutes a single sensor with the N-V distance of 1.478 A [9]. Scanning probe microscopy has been developed to image magnetic field at high spatial resolution [10–12]. The spatial resolution is defined by the distance between the sample and the apex of a diamond probe tip in which an NV center is embedded as schematically shown in Fig. 1a. Whereas the scanning probe microscopy offers the best spatial resolution, preparation of a probe tip with an NV center at an appropriate position requires high manual skills. As a result, the applications of this technique are still limited. The other drawback of scanning probe microscopy is that a long measurement time is usually required to get an image. This restricts the application of the scanning probe microscopy to life science, where the condition of a specimen may change with the lapse of time. The throughput of measurement for acquiring an image is markedly increased by applying wide-field microscopy [13–17]. Photoluminescence from an ensemble of NV centers in a large field of view, typically µm2 –mm2 , is collected and imaged using multi-pixel photodetectors, such as a charge-coupled device or a scientific cMOS camera as schematically shown in Fig. 1b. The measurement set-up is relatively simple, in particular, an active feedback mechanism to stabilize the position of the laser beam spot is usually not necessary. A drawback of this method is a shorter dephasing time T2∗ . To increase the photoluminescence intensity, a diamond chip with a higher concentration of nitrogen is usually used as compared to the method

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using a single NV center. This results in a decrease in T2∗ due to the increase in the spin-spin scatterings with P1 centers [18], divacancies [19], and NV-NV interactions. The other drawback is limited spatial resolution. The spatial resolution of standard far-field optics is limited by r|| = 0.6098 λ/NA, where λ is the wavelength of the light and NA is the numerical aperture of an objective lens. However, this Abbe’s diffraction limit may be circumvented by using super-resolution techniques, such as stochastic optical reconstruction microscopy (STORM) [20] and stimulated emission depletion (STED) microscopy [21]. Since the super-resolution techniques have been developing rapidly, it is expected that the spatial resolution below 20 nm may be achieved. In this chapter, we will review some of the important developments in the widefield imaging using ensembles of NV centers in diamond for sensing utilizing their long spin coherence time. We describe on detection of static magnetic field and microwave intensity by using ensembles of NV centers in diamond.

2 Experimental Techniques for Wide-Field Imaging 2.1 Diamond Samples Used for Wide-Field Imaging Since the spatial resolution is limited by the distance between the surface of a diamond and the layer of the NV centers, accurate control of the position of the layer of NV centers is essential. A thin layer of NV centers is prepared in close proximity to a surface of a diamond. Two methods are commonly used for preparation of a thin layer of NV centers: shallow implantation of nitrogen ions [22–26], and growth of a high quality thin N-doped layer on a surface of a diamond by microwave plasma-assisted chemical vapor deposition (CVD) [27–30]. In the former method, low energy (0.4– 20 keV) N+ or N+ 2 ions are implanted to a single crystal of ultrapure diamond. The areal density of the nitrogen is controlled by the fluence of the ion beam, typically 109 –1014 N/cm2 . The depth of the NV layer is controlled by the acceleration energy of the ion beam, typically 0.4–20 keV, corresponding to the average depth of 2–20 nm, as estimated by using SRIM simulation [23, 25]. 15 N ions are often used to discriminate the implanted nitrogen from the native nitrogen in the diamond [22]. After implantation of nitrogen ions, the diamond sample is usually annealed at 800– 1000 ◦ C to induce diffusion of the vacancies. The diffusion of nitrogen atoms is small because of the high activation energy [25]. Then the sample is treated with a mixture of sulfuric, nitric, and perchloric acid to remove any residue on the surface and to prepare an oxygen-terminated surface. The production efficiency of NV centers was estimated to be 0.8% for the case of implantation of 5 keV-+ N ions [23]. In the latter method, the N-rich layer with a thickness of 5 nm—several µm is grown on a diamond [27–30]. The formation of NV centers occurs during the growth, and hence ion implantation or high energy electron irradiation is not necessary to generate NV centers [30].

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Fig. 2 Schematics of a pulse sequence to measure pulsed ODMR using an sCMOS camera

We employed the shallow implantation of nitrogen ions. A (100)-oriented CVDgrown Type IIa ultra pure single crystal diamond chip with dimensions of 2.0 × 2.0 × 0.5 mm3 (Element 6) was used. N concentration was less than 5 ppb. 15 N+ 2 ions were implanted at 10 keV with fluence of 2 × 1012 –2 × 1013 cm2 . The implanted diamond chips were annealed at 800 ◦ C and cleaned by acid [17, 31].

2.2 Experimental Technique Schematics of a pulse sequence to measure pulsed ODMR using a camera is shown in Fig. 2. A green laser pulses with a duration of typically τ p = 1 µs is irradiated to NV ensembles to prepare the spins in the |0 state. The system reaches equilibrium during the 1 µs pulse [32]. After a τ0 waiting time for the relaxation of the electrons in the metastable state to the |0 state, a microwave pulse sequence is applied. The spin state is read-out by applying a second laser pulse with a duration of τ p , which simultaneously prepares the spins in the |0 state. This pulse sequence is repeated by n cycles, where n is tens of thousands of cycles, for a total time equal to the integration time of the camera. The signal is normalized by a reference acquired by the same sequence with the microwave pulse off. By repeating the above cycles with a short repetition time of 10–100 ms, fluctuation in the normalized PL image due to the fluctuations in the laser power is suppressed. Spontaneous emission from the NV ensembles is detected to measure the spin states. The emission of photons occurs randomly in time, and follows the Poisson distribution, giving the shot noise. For the number of detected photons Nph , the stan dard deviation of the photon number is given by σ = Nph . Hence the noise due to fluctuation in the photon number is √1 . Other sources of noise to be examined Nph

are the read noise and the dark current of cameras. Read noise of scientific CMOS (sCMOS) cameras (∼1 e− ) are smaller than that of CCD cameras (∼10 e− ) primarily because each pixel in the sCMOS camera is equipped with an on-chip amplifier. The read noise of an electron-multiplying CCD is suppressed by the electron multiplication to be smaller than 1 e− . Both sCMOS and CCD cameras have small dark

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current ( 1/ N we are operating in a nonlinear regime with enhanced decay (superradiant decay).

3 Quantum Effects In the previous section we showed that superradiance was expected in a hybrid system composed of an electron spin ensemble coupled to a microwave resonator. In fact it was demonstrated in experiment. Does this mean it is quantum? Superradiance by itself can not be considered a truly quantum phenomena—instead it is one associated with coherent effects. However it is straightforward in these hybrid quantum systems as we could have two ensembles coupled with our resonator [55] which would open up the possibility of entanglement between them. To investigate this situation we are going to extend our ensemble/resonator model to now include two ensembles as we depict in Fig. 5. Our previous Hamiltonian (14) can simply be extended to [56, 57] A B     σzi A + σi 2 i=1 2 i=1 z B

N

H = ωa † a + 2i(a † − a) cos ωt + + ig

N

NA NB    i   i  σ+ A a − a † σ−i A + ig σ+ B a − a † σ−i B i=1

(28)

i=1

Fig. 5 Schematic illustration of a hybrid quantum system formed from a microwave resonator coupled to two spatially separated ensembles

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where we immediately observe that there are no direct coupling terms between the two spatially separated ensembles. However there is an interesting question related to this in terms of whether the shared resonator is able to distinguish which ensemble the emitted photons came from. A particularly interesting case is when it can not. We can write the master equation with the resonator adiabatically eliminated as [56] ρ˙ = −i

 NA  2

i=1

NB i γ NA γ NB σzi A + 2 i=1 σz B , ρ + 2 i=1 L(σ−i A ρ) + 2 i=1 L(σ−i B ρ) NB NB 2 + γ2⊥ i=1 L(σzi A ρ) + γ2⊥ i=1 L(σzi B ρ) + 2 κg L(ζρ) (29)

where we have the collective lower operator ζ is given by ζ =

NA 

σ−i A +

i=1

NB 

σ−i B

(30)

i=1

The dynamics is governed by the time scales associated with γ  γ⊥  2g 2 /κ. Operating on this fastest time scale in a rotating frame, we can simply (29) to [56] ρ˙ =

2 g2 L(ζρ) κ

(31)

which corresponds to a relaxation operation. That operation however does not act ensemble A and B independently. Instead it jointly acts and so we expect a little different behaviour from it. As a simple example consider what our evolution would be if we started with an initial state of the | ↑ A ↓ B . It is straightforward to show that 1 − e−2 g t/κ e−2 g t/κ | ↓ A ↓ B ↓ A ↓ B | + | ↑ A ↓ B + ↓ A ↑ B ↑ A ↓ B + ↓ A ↑ B | 2 4 1 + | ↑ A ↓ B − ↓ A ↑ B ↑ A ↓ B − ↓ A ↑ B | (32) 4 2

ρ(t) =

2

In this limit as t → ∞ we have the steady state solution [56] ρ(t) =

1 1 | ↓ A ↓ B ↓ A ↓ B | + | ↑ A ↓ B − ↓ A ↑ B ↑ A ↓ B − ↓ A ↑ B | 2 4

(33)

where we immediately notice that system B under this relaxation behaviour has actually gained population and is now 1/4 excited. The steady state has the form of a maximally entangled mixed state [58] and is in fact entangled. The above example was for a system where superradiance would not be seen as it only involved a single spin in each separated region. What happens if we have multiple spins. Consider N A spins in ensemble A and N B spins in ensemble B. What behaviour we observe depends on our initial state. If all the spins in ensemble A and B are excited, the combined system undergoes superradiant decay to the collective ground state (all spin down). On the other hand if the spins in ensemble A are initially

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A

B

Fig. 6 Plot of the magnetization (2 Sz /N) versus scaled time for our dual ensemble configuration with N A = 106 , N B = 102 . All spins in N A are initially excited while N B spins are in the ground state. While ensemble A is shown to super radiantly decay, ensemble B undergoes super absorption. The rate of maximum decay/absorption occurs when 2 Sz  /N = 0

excited while those in B are in the ground state, a different behaviour is seen (as shown in Fig. 6). In this case the spins in ensemble A undergo the usual super radiant decay while those in ensemble B undergo super absorption. For N A N B the population of ensemble A tends to its ground state while that of B tends to its excited state. This is true even in the steady state where our dynamics is governed only by the collective damping at zero temperature. In fact for N B = 1 we can show that [56, 57] N A (N A + 1)2 − 2 2 2 (N A + 1)2 (N A − 1)2 − 2 Sz B  ∼ 2 (N A + 1)2 Sz A  ∼ −

(34) (35)

In the limit of N A 1 the magnetization of ensemble A goes to Sz  → −N A /2 while B goes Sz  → 1/2. This means ensemble B is approaching the fully excited state. We call this a negative temperature as our system is in thermal equilibrium. A similar behaviour is seen (as observed in Fig. 6) even when N B = 100. The interesting question is whether we have any quantum correlations between A and B. It is straightforward to show for arbitrary N A with N B = 1 that the logarithmic negativity (an entanglement monotone) given by [57] ⎡ E Log Neg = log2 ⎣

N A (N A + 1) +



4N A3 + (N A + 1)2

(N A + 1)2

⎤ ⎦

(36)

is strictly greater than zero (but tends to zero as N A → ∞). In Fig. 7 we plot E Log Neg versus N A (with N B = 1) where we clearly observe a peak near N A ∼ 5. However even with N A = 106 we still have E Log Neg ∼ 0.03. The detection of entanglement

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Fig. 7 Plot of the degree of entanglement as measured by the logarithmic negativity E Log Neg as a function of ensemble A size N A with N B = 1. Initially ensemble A is fully excited while B is in its ground state

Fig. 8 Plot of the decay time as a function of ensemble A size N A with N B = 1. With all spins in A initially excited, and the single spin in B in the ground state, ensemble A decays superradiantly. The decay time is measured as the steepest negative gradient of Sz A 

in this two ensemble approach clearly indicates we are operating in the quantum regime. It is also interesting to examine the behaviour of the decay time with two ensembles present. With ensemble A and B initially in the excited and ground states respectively, we can examine the time it takes for ensemble A to decay. In general, this can be found via the steepest negative gradient of Sz A  (see Fig. 6). For N A N B , the behaviour of the decay time is well approximated by (25). However, this is not exact for cases where N A N B is not true. As an example, we take N B = 1 and calculate the decay time (via the gradient of Sz A ) for the same range of N A as in Fig. 7. This is shown in the plot of Fig. 8. Interestingly, we observe the longest decay time occurs

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for an ensemble of N A = 5, the same ensemble size that maximises entanglement between the two domains.

3.1 Applications The example above where we have one system super radiantly decay while super absorption occurs on the second system is quite interesting from an application point of view. System B gains energy much faster than γ . This nonlinear effect is known as a quantum battery [59–61] and has recently been a topic of significant interest in the community. It enables the charging of energy into a system with a quantum advantage. In the case outlined above, this hybrid quantum systems approach gives both superextensive capacity and charging [61]. The example above where we have one system super radiantly decay while super absorption occurs on the second system is quite interesting from both a fundamental physics point of view but also an application one. • First and foremost, we are observing the rapid transfer of energy from one system to another (at a rate much greater than the decay rate γ of the qubits in the ensemble). Further in the long-time level one of our ensembles which started in it ground state becomes at completely excited. It effectively has that excited state as its thermodynamics steady state. As such we are relaxing to a negative temperature state. • Second when one explored the dynamics of what is occurring in system B, we find it gains energy in a very nonlinear fashion (it is the reverse of the superradiant decay—known as superabsorption). Its maximum rate of energy gain scales as the square of the number of qubits in the ensemble. This effect has recently become known as a quantum battery [61] and shows other technologies beyond quantum computing, communication, sensing & imaging are possible. Quantum batteries are part of the newly emerging field of quantum thermodynamics [62–64]. Such batteries utilize the unique properties of quantum thermodynamics to design batteries that are superior to their classical counterparts (they give a quantum advantage). Using the hybrid quantum systems with two ensembles one should be able to realize quantum batteries with superextensive capacity and charging [61].

4 Summary and Perspectives In this chapter we have shown how hybrid quantum systems can be used in the nonlinear regime to explore effects such as amplitude bistability and superradiance. We further showed how these semiclassical phenomena in our hybrid system can be adapted to give truly quantum phenomena with no classical analog. As part of these we showed how a hybrid system composed of two ensembles coupled to one

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resonator can exhibit both super radiant and super absorptive properties at the same time. Further we could show entanglement between the two ensembles meaning they had not classical analog. Finally we mentioned how this behaviour may provide a quantum advantage for the charging of batteries. Acknowledgements W.J.M, J.D. and K.N. thank Yusuka Hama, Emi Yukawa, James Quach for useful discussions and Andreas Angerer, Stefan Putz, Johannes Majer, Jorg Schmiedmayer, Hamzah Fauzi, Yoshiro Hirayama for their experimental insights. This research was partly supported by the MEXT KAKENHI Grant-in-Aid for Scientific Research on Innovative Areas Science of Hybrid Quantum Systems grant no.15H05870 and the JSPS KAKENHI grant no. 19H00662.

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Rare Earth Non-spin-bath Crystals for Hybrid Quantum Systems Takehiko Tawara

Abstract Optical coherent manipulation of the quantum electronic superposition states in solid-state materials is essential for the realization of quantum information networks. Rare-earth doped crystals are good candidates as the platforms for hybrid quantum system because they have a deterministic quantum energy state with long population and coherence lifetimes for optical coherent manipulation. However, the practical coherence time of rare-earth doped crystals is much shorter than the prospective theoretically predicted value due to fluctuation of magnetic moment induced by spins of crystal constituent elements. In this chapter, I will discuss the creation and optical characterization of rare earth non-spin-bath crystals produced by the isotopic purification of rare-earth guest ions and nuclear-spin engineering of the host crystal. Keywords Rare earth · Coherence · MBE · Epitaxy · Waveguide

1 Rare-Earth Doped Crystals as Platform for Hybrid Quantum Systems The anticipated quantum information communication network will be composed of a quantum channel configured to transmit quantum information and quantum nodes configured with a quantum optical memory and quantum media converters [1]. Because photons will be used as a quantum information transmission medium due to their excellent quantum properties, a 1.5 µm telecom-band optical fiber network with low optical loss is used for the quantum channel. Since the quantum nodes will be linked to this quantum channel, it is easy to imagine that solid-state memories and converters operating at 1.5 µm photons are indispensable. T. Tawara (B) College of Engineering, Nihon University, 1 Nakagawara, Tokusada, Tamura, Koriyama, Fukushima 963-8642, Japan e-mail: [email protected] NTT Basic Research Laboratories, 3-1 Morinosato-Wakamiya, Atsugi, Kanagawa 243-0198, Japan © Springer Nature Singapore Pte Ltd. 2021 Y. Hirayama et al. (eds.), Hybrid Quantum Systems, Quantum Science and Technology, https://doi.org/10.1007/978-981-16-6679-7_4

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The basic operating principle of a quantum node is based on coherent quantum coupling between photons and matter (electron charge or spin). That is, it is based on a reversible and coherent transfer of quantum information between the photon state and the wave function of material. In recent years, operation verification of quantum memories using this coherent coupling have been reported in various material systems, including semiconductor quantum dots [2], donors in Si [3], and color centers in diamond [4]. However, in each case, the interaction wavelength ranges from visible to 1 µm, and linking with the quantum channel requires wavelength conversion. Furthermore, a recent interesting topic is quantum frequency conversion in superconducting qubits through coherent quantum coupling between microwave and optical photons [5, 6]. The superconducting qubits are being intensively explored as a platform for quantum information processing as discussed elsewhere in this book, but microwave qubits are not suitable for long distance quantum-communications because thier propagation loss is higher than that of telecom-band photons. If quantum frequency conversion can be achieved, it will allow for the long-distance distribution of quantum states through optical fibers and, as a result, allow for the use of optical quantum memories. When a solid material is considered as a platform of such a hybrid quantum system, one of the candidates is a rare-earth (RE) doped crystal [7], which has long been studied as a laser material. Erbium ions (Er3+ ), in particular, have an optical transition at 1.5 µm that corresponds to the telecom-band wavelength, which allows for direct storage and retrieval of telecom-band photons [1, 8–12]. The intra-4 f orbitals characterizing RE ions doped in host crystals are shielded by outer electron orbitals from electromagnetic disturbances. Consequently, the electronic states in the weakly perturbed intra-4 f orbitals exhibit long energy relaxation time T1 (up to milliseconds) and atomic-like discrete energy levels. On the other hand, the coherence time T2 is generally on the order of microseconds, which is much shorter than 2T1 defined as the theoretical limit of T2 . The T2 of the optically coupled energy levels determines the performance of quantum manipulations and therefore needs to be longer. Here, the spectrum homogeneous width (Γh ) and T2 are simply expressed by the following relations. 1 = Γ ph + Γg−g + Γg−h + Γn π T2 1 Γn = 2π T1

Γh =

(1) (2)

Γh directly reflects T2 and is composed of the sum of the phonon broadening (Γ ph ), guest-guest (Γg−g ) and guest-host (Γg−h ) interactions, and the natural linewidth (Γn ), where guest and host refer to doped RE ions and the constituent elements of host crystal, respectively. Γn is specific to each element, and Γ ph can be strongly suppressed by lowering the crystal temperature. From this, it can be seen that suppressing Γg−g and Γg−h is essential for enhancement of T2 . The major source of Γg−g and Γg−h is the fluctuation of the magnetic moment around the target two-level system [13–15].

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This fluctuation is mainly induced by the electron and nuclear spins in the constituent elements of the crystal. For instance, the magnetic moment of Y2 SiO5 crystal, which is the most commonly used host crystal for RE dopants, is smaller than that of other hosts, but it is still 3.8 × 1020 μN /cm3 . This will affect T2 of RE 4 f orbital electrons due to its random fluctuation. Since quantum nodes based on coherent quantum coupling have to achieve T2 comparable to 2T1 , it is essential to explore non-spin-bath crystals that eliminates magnetic fluctuation from the elements. In this chapter, I focus on Er-doped crystals operating with telecom-band photons as a platform for hybrid quantum systems, and show T2 can be increased by purifying the isotope of Er ions to suppress Γg−g . In addition, I will discuss the epitaxial growth of nuclear-spin-free host crystals on Si substrates to suppress Γg−h and the fabrication of planar photonic structures on those thin films.

2 Magnetic Purification of Guest Ions The intra-4 f orbitals in Er ion are weakly perturbed by the crystalline environment and their energy levels are split through several interactions as shown in Fig. 1. As a result, natural (non-purified) Er ions show Stark energy levels, which are formed by the crystal field. Erbium is a Kramers ion, so the double degeneracy of electron spins due to Kramers degeneracy, resulting in a Stark splitting becomes J + 1/2. Therefore, to obtain longer T2 , a magnetic field must be applied to quench the electron-spin contribution [16]. However, only isotope 167 Er3+ , whose natural abundance is 23%, has a nuclear spin, which is I = 7/2, and the Stark level splits into 16 hyperfine sublevels even in the absence of any external magnetic field. Thus, homogeneous broadening Γh in hyperfine sublevels should be narrower than that in the Stark levels. One suitable host material for Er ions is the Y2 SiO5 (YSO) crystal as its constituent elements either have relatively weak magnetic moments (−0.137 μ N for 89 Y) or a

Fig. 1 Energy structure of Er ions

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small natural abundance of magnetic isotopes (4.7% with −0.554 μ N for 29 Si and 0.04% with −1.89 μ N for 17 O). This ensures low perturbation by the host material at low temperatures, which increases coherence times of the optical transitions. The wavelength corresponding to the optical transition between the ground (4 I 15/2 ) and the first excited state (4 I 13/2 ) depends on the crystallographic sites of the Er ions in the YSO host crystal. The Er ions substitute a Y atom at two nonequivalent lowsymmetry C1 sites surrounded by either six or seven neighboring O atoms. In the absence of an externally applied magnetic field, zero-phonon optical transitions at sites 1 and 2 have vacuum wavelength of 1536.48 and 1538.90 nm, respectively. This section will discuss the effect of Er3+ isotopic purification in the YSO host crystal on the homogeneous linewidth (Γh ∝ 1/T2 ) for the suppression of Γg−g and also the fundamental optical coherent manipulation in 167 Er3+ ions.

2.1 Growth of 167 Er3+ Doped Y2 SiO5 Isotopically purified 167 Er3+ doped-YSO bulk single crystals were grown by means of the Czochralski (Cz) method [15]. Prior to the Cz growth, a commercially available 167 3+ Er isotope (Isoflex Russia) was mixed with high purity powders of SiO2 and Y2 O3 resulting in a normal dopant concentration of 0.001 at%. The crystal showed no detectable abundance of isotopes other than 167 Er3+ in a secondary ion mass spectroscopy (SIMS) measurement. The grown ingot was 5 cm in diameter and 15 cm in length as shown in Fig. 2. Samples with a surface area of 5 × 5 mm2 and a length of 6 mm along the b-axis were cut from a grown ingot. The surfaces were polished and coated with an anti-reflection layer. As a reference, we used a nonpurified 0.005% Er3+ doped-YSO single crystal grown by the Cz method at the Scientific Materials Corporation. This crystal has 0.001% 167 Er3+ (23% abundance in natural Er3+ ) in the YSO; therefore, we can quantitatively compare the optical characteristics of the samples, where the only parameter is the difference in the kind of the ions surrounding the 167 Er3+ ions. To characterize the Stark energy levels, the photoluminescence (PL) spectrum of the grown samples were measured at various excitation wavelengths [photoluminescence excitation (PLE)]. Figure 3 shows PLE color plots of the purified 167 Er3+ -doped and non-purified Er3+ -doped samples measured at 4 K. If the excitation wavelength

Fig. 2 Isotopically purified 167 Er3+ doped YSO ingot grown by Cz method (Figure from Ref. [15]. Copyright (2017) The Japan Society of Applied Physics)

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Fig. 3 PLE color plots of a purified 167 Er3+ and b non-purified Er3+ in YSO measured at 4 K (Figure from Ref. [15]. Copyright (2017) The Japan Society of Applied Physics)

corresponds to the Er3+ energy level, the excitation photons are absorbed by electrons with the transition from the ground state of the 4 I 15/2 manifold to the excited state of the 4 I 13/2 manifold. After a certain lifetime, these excited electrons relax to the ground state with photon emission. When the energy level does not agree with the excitation wavelength, since there is no photon absorption, luminescence does not appear either. In Fig. 3, we can see many PL emission peaks originating from the optical transitions between various Stark levels of 4 I 13/2 and 4 I 15/2 manifolds. The PL profile (linewidth and peak positions) of purified 167 Er3+ -doped YSO shows exactly the same state as the reference non-purified sample, which is consistent with the profiles reported in the literature [17, 18]. Moreover, the PL decay lifetime of the zero-phonon transition estimated by single exponential fitting is 11 ms in both samples [15]. These results indicate that the two samples have nearly the same crystal quality. Hence, quantitative comparisons of their optical properties are meaningful.

2.2 Spectral Hole Burning Spectral hole burning (SHB) is a nonlinear spectroscopic technique that was first introduced in nuclear magnetic resonance (MNR) spectroscopy [19]. In optical SHB, a narrow-linewidth pump laser saturates the absorption of some subset of ions through selective excitation, which results in increased transmission at the pump frequency. This leads to the formation of a hole in the absorption spectrum. The spectral-hole width Γhole depends on the pump intensity and coherence time, and is expressed as  1 2 (1 + 1 + 2 T1 T2 ) 2π T1  = Γh (1 + 1 + 2 T1 T2 )

Γhole =

(3) (4)

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Fig. 4 Transmittance spectrum of optical transition between hyperfine sublevels of 167 Er3+ at 2.7 K. The horizontal axis shows the detuning from the master laser frequency. The arrow shows the pump frequency for SHB measurements

where  is the normalized pump intensity and Γh is the homogeneous linewidth [20]. If  is weak enough, such as 2 T1 T2  1, Γhole equals 2Γh . On the contrary, when 2 T1 T2  1, Γhole increases in proportion to Γhole = Γh (T1 T2 )1/2 . In the SHB experiments, we should therefore use a pump laser whose power is low enough for evaluation of the coherence time T2 . Here we focus on the zero-phonon optical transitions between hyperfine sublevels of the ground (4 I 15/2 ) and first excited state (4 I 13/2 ) at site1. Figure 4 shows the transmission spectrum (without pump) of the ensemble transition between hyperfine sublevels of 167 Er3+ in the non-purified sample at 2.7 K. In the SHB experiments, the frequency of the master laser was fixed (∼195 THz), and the frequencies of the pump and probe were independently detuned by electro-optic modulators (EOMs). Due to convolution of the inhomogeneous broadening with each hyperfine sublevel, we can not distinguish the individual transitions between the various hyperfine sublevels. Figure 5 shows the spectral hole profiles of purified and non-purified samples with/without the pump, where detuning means the frequency difference between the pump (about 195.1157 THz) and probe. The input intensities of the pump and probe light was set to 2.5 mW/cm2 and 3.2 × 10−3 mW/cm2 , respectively, because the saturation pump intensity (2 T1 T2 ≈ 1) was about 6.9 mW/cm2 (see inset of Fig. 5a). The values of the spectral hole width Γhole of the 167 Er3+ hyperfine sublevel are 0.43 MHz in the purified 167 Er3+ -doped YSO and 1.62 MHz in the reference. Half of these widths correspond to the values of homogeneous linewidth Γhole , which are 220 kHz in the purified 167 Er3+ -doped YSO and 810 kHz in the reference. Now, referring to Eqs. (1) and (2), it is clear that we can not control Γn because it depends on T1 , which is the nature of the element. The lifetime of the excited 4 I 13/2 manifold (11 ms) contributes Γn = 14.5 Hz to homogeneous linewidth Γh . In this experiment, it is apparent that the contributions of Γ ph to Γh were not different for the two samples at the same temperature. This is also true of Γg−h in the same host crystal. Therefore, Γg−g was the only parameter necessary for producing the difference in homogeneous linewidth Γh .

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Fig. 5 SHB profiles of a isotopically purified 167 Er3+ and b non-purified Er3+ doped in YSO, shown by black curves. The horizontal axis shows the detuning from the pump laser frequency. The inset in a shows the pump power dependence of the spectral hole width (Figure from Ref. [15]. Copyright (2017) The Japan Society of Applied Physics)

The Kramers 167 Er3+ ion has large total angular momentum J = 15/2, which is the sum of the total orbital angular momentum L and total spin angular momentum S. This means that Er3+ ions themselves produce a larger electronic magnetic moment μ = g J [J (J + 1)]1/2 (g J : Landé g-factor) than that of non-Kramers RE ions. When we focus on the 167 Er3+ isotopes, fluctuation of the local magnetic field induced by electron spins of the surrounding non-purified Er3+ ions shortens the coherence time of the optical transition in the 167 Er3+ isotopes even if the Er3+ concentration is only 0.005 at%. In the above experiments, removal of this fluctuation is interpreted as being the main factor for the spectral narrowing in the purified 167 Er3+ -doped YSO. The degree of its fluctuation as Γg−g is enormous and becomes about 590 kHz quantitatively. This implies that isotopic purification is essential for prolonging T2 in RE ions in solid-state materials. T2 estimated from the measured Γh by using Eq. (4) shows a remarkable enhancement from 0.39 to 1.45 µs due to the isotopic purification.

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2.3 Coherent Transients The previous section discussed the effect of purified guest Er ions on Γh (∝ 1/T2 ) in the frequency domain. However, experimentally estimated Γh includes temporal fluctuations in the frequency of the pump laser, and evaluation in the time domain is required to obtain more accurate Γh . The coherence of an atomic system is a transient that exists only immediately after excitation. Coherent transients reflect the dynamics of the motion of the atomic system, including decoherence, as well as the temporal change of the population distribution at each energy level. This section discusses optical coherent transients such as Rabi oscillation and photon echo between the ground and optically excited hyperfine sublevels in 167 Er3+ ions [21]. Here, the coherent transients in -like three level systems will be considered. This is because optical coherent manipulation such as photon storage protocols generally requires three or more quantum energy levels. In 2010, two -like three level systems in 167 Er3+ doped in YSO were identified through SHB spectroscopy [22]. The separation between the two ground levels in the 4 I 15/2 manifold coupled to an excited level in the 4 I 13/2 manifold was found to be 880 MHz for system 1 and 740 MHz for system 2. Such three-level schemes are required for photon storage protocols in ensembles of atoms such as electromagnetically induced transparency (EIT) [23, 24], controlled reversible inhomogeneous broadening (CRIB) [25] and stimulated Raman adiabatic passage (STIRAP) [26]. Now, we are concerned with the characteristics of system 1 (Fig. 6a). More precisely, this -like system consists of two zero-phonon optical transitions between hyperfine sublevels of the lowest Stark level at 4 I 15/2 and 4 I 13/2 with a frequency separation of 880 MHz. To identify two transitions (∼195.11 THz) with the same excited state in the -like system, we performed SHB spectroscopy again. Whereas an optical pump

Fig. 6 a Schematic of a -like three-level system in the hyperfine structure of 167 Er3+ . b Spectral holes (sharp peaks) observed on the inhomogeneously broadened spectrum that corresponds to the 4I 4 15/2 ↔ I 13/2 hyperfine transitions. Positive and negative peaks indicate the main spectral hole and antihole, respectively (Reprinted with permission from Ref. [21] © The Optical Society)

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Fig. 7 Rabi oscillation of the integrated photoluminescence versus pump width t (Reprinted with permission from Ref. [21] © The Optical Society)

beam has a fixed optical frequency at which the optical transition from state |1 to state |2 occurs, the probe beam has a continuously swept frequency. This sweep is a means to monitor in the spectral domain the population distribution induced by the optical pump beam, thus making it possible to examine the population distribution in the -like system during excitation. In this -like system, we can identify two optical transitions by measuring the probe transmission (absorption) versus the swept probe frequency (Fig. 6b). The frequency at which the transmission increases or decreases sharply suggests the existence of the energy level. The increase in transmission around the detuning of 10.3 GHz suggests that the optical pump beam causes a population deficiency in state |1, and the decrease in it around 11.2 GHz suggests the existence of the population in state |2, which are known as the spectral hole and antihole, respectively. To explore the pulse area conditions in this -like system, Rabi oscillations were measured at 2.2 K. Figure 7 shows the integrated PL intensity as a function of the excitation pulse width, which is proportional to pulse area. Square optical pulses were applied with a constant intensity of 45 mW and a mode-field diameter of 900 µm. The electric dipole interaction between states |1 and |2 induces periodic (Rabi) oscillations of the population in the electronic states. The occupancy of state |2 is directly reflected by the intensity of the PL, which shows a Rabi oscillation overlapped with an incoherent background. Two Rabi cycles can be seen with a time period of 1.2 µs, which gives a Rabi frequency  R of 2π × 810 kHz. The transition dipole moment μ = 7.6 × 10−32 C m can be deduced from the definition of the Rabi frequency,  R = |μ/|, where μ and  represent the transition dipole moment and the estimated electric field amplitude respectively. The calculated dipole moment is in good agreement with previously reported values for Er3+ transitions at 1.5 µm, which were obtained by using different methods [27, 28]. The obtained dipole moment and Rabi frequency are important parameters for optical coherent manipulation of the electronic states. The π and π /2 pulse areas estimated from the Rabi oscillation measurements in Fig. 7 were used to perform a two-pulse photon echo measurement at 2.2 K. In

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Fig. 8 Photon-echo signals observed at each delay τ . The inset is the measured echo envelope (black circles) and single exponential fit (red line) for the slow decay component (Reprinted with permission from Ref. [21] © The Optical Society)

this experiment, after time τ of the first pulse (π /2 pulse: pulse width 250 ns), the second pulse (π pulse: pulse width 500 ns) was irradiated to the sample, and an echo signal was observed at a time delayed by τ from the second pulse. As shown in Fig. 8, the two-pulse photon echo signals were observed at each delay τ . The echo intensity follows a double exponential decay, of which the slow decay, occurring opt after 3–10 µs, corresponds to the T 2 in this -like system. A single exponential opt opt fit by exp(−4τ /T 2 ) in this region yields T 2 = 12 µs. This is about an order of magnitude longer than the value in previous reports for Er3+ at zero magnetic field [29, 30]. The origins of this temporal extension include a low abundance of non-zero nuclear spin isotopes in the host crystal (YSO), resulting in reduction of nuclear spin flips; the effect of isotopic purification of 167 Er3+ [15]; and the lowering of the concentrations of guest ions (167 Er3+ here) compared with previous reports, resulting in the reduction of Er3+ -Er3+ interactions [18]. The fast decay appears in the time domain of 0–2 µs, whereas the slow decay dominates after 2 µs. The fast decay is likely to originate from the interaction between an electronic spin in the rareearth guest ion (167 Er3+ here) and a nuclear spin in the host material (89 Y3+ ), which is known as the superhyperfine interaction [11, 31, 32]. This interaction modulates the echo envelope and appears as fast decay. opt The parameters obtained in this experiment, such as  R and T 2 , which determine the performance of optical quantum manipulation, illustrate the potential of using 167 3+ Er :Y2 SiO5 at zero magnetic field for quantum information processing applications. In particular, the STIRAP technique requires  R  Γ , where Γ is given by Γ = 1/T2 [33]. For solid state materials, experimental conditions do not always satisfy the criteria. However, in this case, the estimated coherence time is so long that they are satisfied even under the limitations of maximum optical intensity. Since the obtained parameters satisfy this condition, the -like system in 167 Er3+ :YSO is a suitable platform for executing coherent manipulation protocols such as STIRAP without external magnetic fields.

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3 Magnetic Purification of Host Crystal In the previous section, we discussed the effect of guest-guest interaction Γg−g on T2 and its suppression by Er isotope purification. Another artificially manipulable factor for increasing T2 is the guest-host interaction Γg−h . A major source of population decoherence in the two-level system is a fluctuation of the magnetic moment around the guest ions [13–15]. This fluctuation is mainly induced by nuclear spin moments of the constituent elements of the host crystal. As described in the previous section, the nuclear spin abundance ratio of each element constituting YSO is quite low, and the magnetic disturbance to the electronic state of the doped Er ion (substituted with Y) is small. However, since bulk YSO crystal grown by the Cz method has a 6 space group in Schönflies notation), it is not suitable monoclinic crystal structure (C2h for epitaxial growth on typical semiconductors with cubic diamond or zincblende structures such as Si. In other words, it is difficult to fabricate optical confinement nanostructures such as waveguides and cavities in Er-doped YSO and integrate them on the same substrate. Therefore, quantum coherent coupling with high efficiency using the cavity quantum electrodynamics (c-QED) effect, and seamless connection to the other optical devices are virtually impossible. This is a serious problem in realizing quantum nodes. The above problems point to the need create a host crystal that has a small magnetic moment and is capable of being epitaxially grown on a semiconductor substrate for realizing a highly efficient hybrid quantum coupling platform. Figure 9 shows the natural abundance of isotopes with nuclear spin in each element. Scandium-45 (45 Sc) has nuclear spin I = 7/2 like 89 Y does (I = 1/2), as shown in Fig. 9, but the cubic phase for scandium oxide is very stable. Therefore, it is convenient for examining basic optical properties of doped Er ions. In contrast, cerium (136,138,140,142 Ce) is the only

Fig. 9 Natural abundance of isotopes with nuclear spin in each element

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RE element with no nuclear spin. Thus, cerium oxide thin films are very interesting as non-spin-bath crystal on semiconductor substrates. Here, the epitaxial growth of magnetically purified host crystals on Si substrates and the optical characteristics of doped Er ions in those hosts will be discussed. Moreover, fabrication of photonic nanostructures on those RE epitaxial layers will be demonstrated.

3.1 (ErSc)2 O3 Grown on Si Substrates 3.1.1

Crystal Growth of (ErSc)2 O3

First, the fundamental condition for host crystal growth on Si substrate and the population dynamics of doped Er ions in the host are investigated. It is known that RE sesquioxides (R2 O3 , where R is rare earth) can form a cubic bixbyite structure as shown in Fig. 10a, and, interestingly, the lattice constants of all of them are generally twice that of the Si plane (a = 5.43 Å). Consequently, they can be grown epitaxially on a Si surface as a host for Er ions, and we can obtain structural flexibility and functionality by fabricating optical structures like waveguides or cavities by nanoprocessing or selective area growth. Here, erbium oxide (Er2 O3 ) is used as a starting material for the crystal growth [36], and the distance between Er3+ is controlled by replacing some of the Er3+ ions with scandium (Sc3+ ) ions. A unit cell of sesquioxide Er2 O3 and one of Sc2 O3 have the

Fig. 10 a Unit cell of bixbyite Er-doped Sc2 O3 crystal. In this schematic, red and green atoms indicate RE ion at C2 and C3i sites, respectively. b Energy diagram of Er3+ ions in Y2 O3 [34] (Reprinted with permission from Ref. [35] © The Optical Society)

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lattice constants of a = 10.54 Å and 9.855 Å, respectively, and contain 32 RE ions, 24 at sites with C2 (non-inversion) symmetry and eight at sites with C3i (inversion) symmetry. In Er2 O3 , the nearest neighbor distances of Er3+ ions between the C2 -C2 , C2 -C3i , and C3i -C3i sites are 1.945, 2.656, and 5.268 Å, respectively. The transition from the first excited state (4 I 13/2 manifold with Stark levels Yi( ) , i = 1–7) to the 3+ ions exhibits ground state (4 I 15/2 manifold with Stark levels Z( ) j , j = 1–8) in Er photon emission at around 1.5 µm as shown in Fig. 10b [34]. Here, (forced) electricdipole transitions between the Stark levels are allowed for Er3+ in C2 sites (Yi -Z j ), and only magnetic-dipole transitions are possible for Er3+ in C3i sites (Yi -Z j ). In contrast, Sc2 O3 is completely transparent to photons in the visible-to-telecom-band range [35, 37]. We grew 50-nm-thick (Erx Sc1−x )2 O3 with Er concentration x from 1.000 (Er2 O3 ) to 0.012 on Si(111) surfaces with a 7 × 7 reconstruction by molecular-beam epitaxy (MBE) at a growth temperature of 715 ◦ C. The streak pattern of reflection high-energy electron diffraction (RHEED) was maintained during the growth. The composition of the grown films was determined by Rutherford backscattering. An ω-2θ scan of the X-ray diffraction (XRD) measurements after growth showed that single-crystal (ErSc)2 O3 layers were grown as shown in Fig. 11a, and crystal quality was approximately equivalent for all Er compositions. Moreover, it was found that the Er concentration dependence of the lattice constant of (Erx Sc1−x )2 O3 satisfied Vegard’s law. Thus, we can assume macroscopic uniformity of the Er3+ distribution in the grown samples. A cross-sectional image obtained with a transmission electron microscope (TEM) also proved that the (ErSc)2 O3 was epitaxially grown on the Si(111) surface (Fig. 11b). Figure 12 shows the PL spectra of various Er concentrations under the resonant excitation of the third level from the bottom of the first excited Stark manifold at the

Fig. 11 a XRD ω-2θ scan of grown sample with x = 0.0027. b Cross-sectional TEM image of grown Er-doped Sc2 O3 (Reprinted with permission from Ref. [35] © The Optical Society)

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Fig. 12 PL spectra for various Er concentrations under the resonant excitation of the assigned Y 1 level at the C3i site. Inset shows PL peak intensity and FWHM of the Y 1 -Z 1 transition as a function of Er concentration. The dotted curves are guides for the eye (Reprinted with permission from Ref. [35] © The Optical Society) Table 1 Summary of the number of Er3+ ions per unit cell, the Er3+ density, and the average distance between Er ions for grown (Erx Sc1−x )2 O3 samples Er conc. (at%) No. of Er (cell) Er density (cm−3 ) Avg. Er distance (Å) 100 27.1 5.4 2.7 1.2

32 8.7 1.6 0.9 0.4

2.7 × 1022 7.4 × 1021 1.4 × 1021 7.7 × 1020 3.4 × 1020

3 5 9 11 14

C3i site (Y 3 level in 4 I 13/2 ). Table 1 summarizes the number of Er3+ ions per unit cell, the Er3+ -ion density, and average distance for the grown samples. The strongest PL peak at a wavelength of 1548 nm with x = 1.000 corresponds to the transition between bottom Stark energy levels in the first excited and ground states at the C3i site (from Y 1 in 4 I 13/2 to Z 1 in 4 I 15/2 ) [38]. The transition energy of this peak shifts to 1551 nm because the crystal field surrounding Er3+ is changed by decreasing the Er composition [39, 40]. The peak intensity per Er3+ ion normalized by the Er3+ density and layer thickness and the PL fullwidth at halfmaximum (FWHM) of the main Y 1 -Z 1 transition are summarized as a function of Er composition in the inset of Fig. 12. The efficiency of optical transitions in Er3+ decreases with increasing Er3+

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Fig. 13 Time-resolved PL of (Erx Sc1−x )2 O3 with x = 1.000 (solid circles) and 0.054 (open circles), detected at the Y 1 -Z 1 transition. Gray curves show the results calculated by Eqs. (6)–(9) with the parameters in Table 2. Inset shows Er composition dependence of PL decay time (Reprinted with permission from Ref. [35] © The Optical Society)

density up to x = 0.270. This is known as concentration quenching, which is caused by the enhancement of the energy migration among the Er ions and its capture by the non-radiative centers (crystal defects, etc.) during the migration [41–43]. As the non-radiative relaxation process becomes dominant, the emission lifetime generally decreases rapidly. However, the measured lifetimes of the main Y 1 -Z 1 transition measured under the same excitation condition as shown in Fig. 12 are almost a constant values of 2 ms below x = 0.270 as shown in Fig. 13. Here, all the emission decay curves are fitted by using single exponential function I P L ∝ exp(τ/τT r ). This behavior suggests that the concentration quenching in epitaxial (Erx Sc1−x )2 O3 is not mainly caused by an increase in the population-capturing probability at non-radiative centers with higher Er compositions.

3.1.2

Energy Transfer Between Er Ions

To analyze these population dynamics in (ErSc)2 O3 , we focus on the effect of the energy transfer up-conversion (ETUC) in Er3+ . The ETUC means that the cooperative up-conversion (an Auger-like process with de-excitation and excitation) between Er3+ occurs, and, as the result, photons with a wavelength shorter than the excitation one are emitted [44]. Figure 14 shows PLE color plots of an Er2 O3 sample under 4 I 13/2 manifold excitation (1520–1550 nm) and the PLE spectra detected at the energy levels of 4 S 3/2 (around 550 nm) and 4 I 13/2 manifolds. Strong PL caused by the ETUC from the higher states in the Er3+ 4 f orbital is observed only when the excitation wavelength corresponds to the Stark energy level of the 4 I 13/2 manifold. The UC PL was observed in all (ErSc)2 O3 grown samples, but its intensity was greatly reduced by decreasing the Er composition as shown in Fig. 15. The PL peaks from the various UC states (UCSs: 2 H 9/2 , 4 S 3/2 , 4 F 9/2 , 4 I 9/2 , 4 I 11/2 ) correspond to the predicted transitions between the Stark levels and the ground state (4 I 15/2 )

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Fig. 14 PLE color plots of (ErSc)2 O3 detected at a 4 S 3/2 and b 4 I 13/2 manifolds under 4 I 13/2 manifold excitation. The noise in the center part in the PLE color plots at the 4 I 13/2 manifold is caused by scattering from the excitation laser (Reprinted with permission from Ref. [35] © The Optical Society)

Fig. 15 UC spectra of (Erx Sc1−x )2 O3 with a x = 1.000, b 0.054 and c 0.012, respectively. The spectrum intensity of b and c is multiplied by 6 and 35, respectively (Reprinted with permission from Ref. [35] © The Optical Society)

at the C2 site, while optical transitions above the 4 I 13/2 state at the C3i site are not observed [34]. To estimate the ETUC rates from the first excited state, we investigate the excitation-power dependence of the PL intensity under the resonant excitation of the Y1 state for samples with x = 1.000, 0.054 and 0.012. Figure 16 shows the integrated PL intensity of the transition from the 4 I 13/2 manifold and UCSs to the

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Fig. 16 a–c Integrated PL intensity of the transition from the 4 I 13/2 manifold and UCSs to the ground energy level of the 4 I 15/2 manifold. Red, black and blue filled circles indicate the transition from the 4 I 13/2 manifold at the C2 and C3i sites and UCSs to the 4 I 15/2 manifold, respectively. d Calculation model for the rate equation analysis (Reprinted with permission from Ref. [35] © The Optical Society)

ground energy level of the 4 I 15/2 manifold. Here each PL intensity is not normalized by the Er3+ density, and the PL spectra from various UCSs are integrated in the 500 (4 S 3/2 ) to 1020 nm (4 I 11/2 ) wavelength region to treat them as originating from a single UCS. To reproduce this excitation power dependence, we consider the energy transfer model (Fig. 16d) [45, 46] and solve the steady-state rate equations based on this model as follows:

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dn 0 dt dn 1 dt dn 2 dt dn 3 dt dn 4 dt

= −B31 n 3 n 0 + B13 n 1 n 2 + A1 n 1

(5)

= B31 n 3 n 0 − B13 n 1 n 2 − A1 n 1

(6)

= −Γ + B34 n 23 + B31 n 3 n 0 − B13 n 1 n 2 + A3 n 3 + A4 n 4

(7)

= Γ + 2B34 n 23 − B31 n 3 n 0 + B13 n 1 n 2 − A3 n 3

(8)

= B34 n 23 − A4 n 4

(9)

where Γ is the pumping rate in consideration of the Y1 level saturation, n i (Ai ) is the population probability (measured linear decay rate) of the i-th level, and Bi j is the energy transfer rate from the i to j states (fitting parameters). The population probability satisfies the relation of Γ dt = n 1 + n 2 + n 3 + n 4 . The initial population in the ground state (n 0 and n 2 ) is assumed to have a ratio of n 0 /n 2 = 1/3, which reflects the abundance of each site in the unit cell. The transfer process assumed a cross-relaxation type for the energy transfer between the C2 and C3i sites and an Auger type for the ETUC. Note that the non-radiative relaxation rate of the excited populations is neglected in this calculation, because, as we have previously revealed, its contribution in the epitaxial (ErSc)2 O3 layers at the measurement temperature of 4 K should be very small [47]. We also neglect the energy transfer between the same sites (C2 -C2 and C3i -C3i ) because we can not distinguish them from the spectrum without the energy transfer. Using this model, we can accurately fit the excitation power dependence with various Er compositions as shown by the solid curves in Fig. 16a–c, and the transfer rates we obtained are listed in Table 2. These calculations reproduce very well the experimental results-not only the intensity correlation among the transitions but also the slope of the intensity as a function of the excitation power, although the non-radiative relaxation is ignored in this model. The estimated energy transfer rate (Bi j ) is greatly enhanced with increasing Er composition, although the linear decay rates A1 and A3 for x = 0.012 and 0.054 are almost constant. In particular, the ETUC rate (B34 ) for the higher Er composition is much faster than the linear decay rate. Here, we consider the population dynamics in (ErSc)2 O3 on the basis of the above experimental results. We observed concentration quenching in (ErSc)2 O3 with a constant emission lifetime with single exponential decay as shown in Figs. 12 and 13. Moreover, we measured the UC luminescence for all Er compositions (Fig. 15) and found that the ETUC rate is greatly enhanced with increasing Er composition (Fig. 16 and Table 2). These results indicate that the concentration quenching in epitaxial (ErSc)2 O3 mainly originates from the population that escapes from the excited states to the UCS by ETUC (UC quenching in intra-ion) and that the contribution of the population transfer from Er3+ sites to the non-radiative centers (inter-ion migration quenching) is very small. In the steady-state rate equation analysis, all of the energy transfer rates Bi j are 1/10 or less than the linear decay rates Ai at x < 0.054. Therefore, the weight of these Bi j to the lifetime (dn 1or 3 /dt) will be insignificant. In fact,

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Table 2 Summary of the measured values (Ai ) and obtained transfer rates (Bi j ) in ms−1 units x

A1

A3

A4

B13

B31

B34

1.000 0.054 0.012

7.14 0.45 0.54

10.0 0.63 0.77

250 31.3 2.45

1.00 0.002 0.0005

125 0.033 0.006

2000 0.01 0.002

the calculation results, which were obtained by the time evolution analysis using the same rate equation [Eqs. (6)–(9)] with the parameters in Table 2, also reproduce well the measured PL lifetime as shown by gray curves in Fig. 13. Therefore, we conclude that the mechanism of the concentration quenching in epitaxial (Erx Sc1−x )2 O3 mainly originates from the ETUC quenching due to the enhancement of the UC rate B34 below the Er composition x ∼ 0.3. In contrast, the average distance between Er3+ in (ErSc)2 O3 with x > 0.3 becomes less than 5 Å, and this will induce wavefunction overlapping [48, 49] and an increase in the coherence volume of the Er3+ ions.

3.2 Er-Doped CeO2 Grown on Si Substrates As mentioned above, among many RE elements, Ce is the only one with no nuclear spin (Fig. 9). In other words, Ce has excellent characteristics that have no magnetic fluctuation and do not affect the quantum state of Er ions. As for oxygen, only 17 O (abundance of 0.04%) has a nuclear spin moment (−1.894 μ N ). Therefore cerium oxide is the most promising Er-doped host crystal on a Si substrate. Cerium oxide has several competing phases, such as fluorite CeO2 , bixbyite and hexagonal Ce2 O3 . Among them, fluorite CeO2 is the most stable phase and its lattice constant (a = 5.411 Å) matches that of Si (a = 5.43 Å). In this section, the growth and optical characteristics of Er-doped CeO2 (Er:CeO2 ) as an authentic non-spin-bath crystal are discussed [50]. First, to achieve epitaxial and stoichiometric Er:CeO2 layers, i.e., (Er+Ce)/O = 1/2, on Si (111) substrates, the growth was carried out with various Ce/O2 flux ratios. Here, the growth temperature, layer thickness, Er deposition rate, and O2 flow rate were fixed at 640 ◦ C, 30 nm, 0.01 Å/s, and 0.1 sccm, respectively. Figure 17 shows the RHEED patterns for the 30-nm-thick Er:CeO2 films grown with various Ce deposition rates (0.03 to 0.82 Å/s). The incident electron beam was parallel to the [112] axis of the Si substrate ([112]Si). As the Ce deposition rate increases, the pattern changes from halo-like (Fig. 17a) to streaky (Fig. 17c). Further increases in the Ce rate leads to a spotty ring pattern (Fig. 17e); structures of the films are amorphous, singlecrystalline, and polycrystalline for excessive, optimal and insufficient oxidations, respectively. The stoichiometric composition of the film grown with the Ce rate of 0.33 Å/s is further confirmed by Rutherford back scattering (RBS) results as described later.

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Fig. 17 RHEED patterns after Er:CeO2 growth: Ce rate dependence. Er fluxes were supplied at a constant rate (0.01 Å/s). a–e O* + O2 supplied with a fixed O2 flow rate of 0.1 sccm. f Ce supplied at a rate of Å/s, which is three times higher than that in c. Oxygen flow rate was also increased so that the Ce/O2 ratio becomes equal to that in c (Reprinted with permission from Ref. [50] © The Optical Society)

As long as the appropriate Ce/O2 ratio was used, films with identical quality were achieved when we grew them at a three times higher rate (Fig. 17f). Specifically, the Ce and O2 flow rates were increased to 1.00 Å/s and 0.3 sccm, respectively, while the Er rate (0.01 Å/s) was kept constant. This means that a higher growth rate is advantageous for achieving a more dilute doping of Er, and equivalently, a less pronounced Er–Er interaction. Although epitaxial films were prepared, there remains a possibility that some competing phases (e.g., hexagonal or bixbyite Ce2 O3 ) were formed other than the fluorite Er:CeO2 [51]. To confirm the formation of Er:CeO2 , we investigated the film structure by XRD. Figure 18a shows a 2θ − ω scan XRD pattern for an Er:CeO2 specimen. The d value calculated from the peak position is 3.11 Å, which agrees well with the spacing between the fluorite CeO2 (111) planes [52]. The thickness of the Er:CeO2 layer estimated from the intervals of the satellite peaks in the XRD pattern is 26.4 nm. In addition, to determine the valence of Ce (Ce4+ for CeO2 or Ce3+ for Ce2 O3 ), we measured the X-ray photoelectron spectroscopy (XPS) spectrum for the Ce 3d core level (Fig. 18b). All the peaks in the Ce 3d spectrum can be assigned to tetravalent Ce [53]. These XRD and XPS results indicate that the epitaxial layer is composed of Er:CeO2 and amount of the Ce2 O3 (and/or Er:Ce2 O3 ) phases, if any, is negligibly small. Measurements of RBS depth profiles for the Er:CeO2 layer for the composition analysis showed that a uniform composition was accomplished in the entire layer except for the interface region. Remarkably, uniform doping of 1% Er was achieved for the film grown when the rates of Er and Ce were 0.01 and 1.00 Å/s, respectively, and the O2 flow rate was 0.3 sccm. Cross-sectional TEM and high-angle annular

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Fig. 18 a 2θ − ω scanned XRD pattern an d b XPS spectrum for Er:Ce2 O3 /Si(111) (Reprinted with permission from Ref. [50] © The Optical Society)

dark field scanning TEM (HHAADF-STEM) observations also confirmed the high crystallinity of the Er:CeO2 layer with a very flat surface (Fig. 19). They also indicate that the Er:CeO2 layer is free from any unidentified phases, orientations, or stacking sequences. However, in the HAADF-STEM image shown in Fig. 19, an amorphous layer can be seen at the interface between the Er:CeO2 layer and Si substrate. The formation of this interface layer, which is peculiar to the growth of CeO2 on Si substrates, is widely known, and so far, two models to explain it have been proposed and discussed [54–58]. One is that solid state reaction already occurs at the very early stages of the growth [56–58]. The other is that a redox reaction takes place at the CeO2 /Si interface after the growth, leading to the formation of an amorphous layer consisting of CeO2−x and SiO2 [54, 55]. Interface engineering for controlling such solid-state or redox reactions is crucial for electronic devices. In contrast, for waveguide-type optical devices without current injection, such an amorphous layer formed at the interface may not deteriorate device performance. Instead, it may positively serve as a layer blocking energy transfer from the Er ions in CeO2 to the Si substrates [43]. Subsequently, optical characteristics of the grown Er:CeO2 layer with various Er concentration were investigated. Again, the Er concentration was varied by changing the Ce and O2 supply rates while the Ce/O2 flux ratio and Er deposition rate (0.01 Å/s) were kept constant. Figure 20 show PLE color plots of Er concentration dependence. The Er concentrations (1, 2, and 4%) were estimated from RBS measurements. An optical emission in the Er ions, which originates from the transition between the Stark-level manifolds of 4 I 13/2 and 4 I 15/2 formed by the crystal field splitting, is observed at a wavelength of around 1.53 µm for all the Er concentrations. Moreover, the PL peak and absorption positions completely agree with each other irrespective of the Er concentration. The excitation spectrum is detected at 1.533 µm for the 1% sample, which is equivalent to an absorption spectrum. In the measured wavelength range, the most intense emission and absorption appear at 1.533 and 1.512 µm,

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Fig. 19 Cross-sectional images of grown Er:Ce2 O3 /Si(111) measured by TEM (upper) and HAADF-STEM (lower) (Reprinted with permission from Ref. [50] © The Optical Society)

Fig. 20 PLE color plots for Er:CeO2 /Si(111) with Er concentrations of a 4%, b 2%, and c 1%. The measurements were performed at 4K (Reprinted with permission from Ref. [50] © The Optical Society)

respectively, for all the samples. This indicates that the Er ions are certainly located at the Ce sites in the CeO2 lattice. Figure 21 shows the PL spectra, which correspond to slices of Fig. 20 at the resonant excitation condition (λexc = 1.5122 µm). The emission intensity of the PL peak at around 1.533 µm clearly increases with decreasing Er concentration. This is owing to an enhancement of the quantum efficiency of the PL emission for the lower Er concentration. Specifically, the reduction of the Er concentration makes each Er ion more completely isolated and suppresses Er–Er interactions, leading to less energy transfer. Moreover, the degree of inhomogeneous broadening of the spectra is reduced with decreasing Er concentration, and the spectra become narrowed. The emission lifetime with decreasing Er concentration (inset of Fig. 21) is prolonged by the same mechanism. The lifetime of the 1% Er:CeO2 sample was 1.5 ms. This value is comparable to the lifetime of the 1% Er in the epitaxial Sc2 O3 host crystal as shown in the previous section [35, 36]. As we have also reported, Er ions diluted by three

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Fig. 21 PL spectra under the resonant excitation for Er:CeO2 /Si(111) with various Er concentrations. Inset shows radiative lifetime as a function of Er concentration. Solid circles and squares indicate Er:CeO2 and Er:Sc2 O3 [35, 36], respectively. The black dotted line shows the intrinsic lifetime of Er [15] and the red dotted curve is the guide to the eye (Reprinted with permission from Ref. [50] © The Optical Society)

orders of magnitude (∼0.001 at%) and doped into a YSO single crystal show the intrinsic emission lifetime of about 11 ms [15], longer by a factor of five or one order magnitude. Nevertheless, the lifetime is monotonically prolonged with decreasing Er concentration in the 1% range, and hence, the present Er:CeO2 /Si system provides a promising route to achieve the intrinsic lifetime on Si chips by further reducing the Er concentration.

3.3 Photonic Structures on Epitaxial RE Oxide As described above, we have succeeded in epitaxial growth of Er doped RE oxide (REO) host crystal on a Si(111) substrate by the MBE method. When these epitaxial REO layers are used, we can expect not only the function of controlling the distance between RE ions by using an alloy, but also the addition of a light confinement function, such as that provided by a heterostructure or photonic structure. In this section, the fabrication and properties of photonic structures using these REO will be discussed. It is well known that REO is a very hard material and difficult to process by both dry and wet etching [59]. Therefore, the usual top-down process is not suitable for fabricating REO nanostructures, and we first consider a bottom-up process, namely selective area growth. Figure 22 depicts the fabrication process for photonic

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Fig. 22 Schematic of PhC fabrication process with selective area growth

Fig. 23 SEM images of PhC fabricated by using selective area growth

crystal (PhC) structures with selective area growth on a silicon-on-insulator (SOI) substrate. This SOI substrate consists of a 200-nm-thick Si(111) device layer, 1-µmthick buried-oxide (BOX) layer, and Si (100) handle wafer. Before the REO growth, PhC patterns are transferred to the Si (111) top surface by EB lithography and dry etching. Then the BOX layer is removed with a hydrogen fluoride (HF) solution to make a Si (111) PhC membrane. An REO epitaxial layer, here Er2 O3 , with a thickness of approximately 20 nm is grown on the patterned Si (111) surfaces with 7 × 7 reconstruction. Cross-sectional SEM images indicate a fine PhC structure with vertical sidewalls of air holes is maintained even after Er2 O3 growth as shown in Fig. 23. A photonic lattice was designed to induce two photonic band gaps (PBGs) matching the excited states of 4 I 9/2 (∼1.45 eV) and 4 S 3/2 (∼2.20 eV) of Er3+ intra-4 f transitions. The designed values of the photonic lattice constant and air hole radius are 290 and 100 nm, respectively. PL spectra of the fabricated Er2 O3 /Si PhCs were obtained under resonance excitation with the lowest excited state of 4 I 13/2 (1536 nm) measured at 4 K. Figure 24 shows the PL transition from 4 I 13/2 , 4 I 9/2 , and 4 S 3/2

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Fig. 24 PL spectra from Er2 O3 /Si PhC membrane (red) and un-patterned area (black) from a 4 S 3/2 , b 4I 4 9/2 , and c I 13/2 manifolds

excited states to 4 I 15/2 ground states. Note that the PL from 4 I 9/2 and 4 S 3/2 states, which have higher energy than the excitation energy, is caused by the ETUC in Er3+ ions [44]. In an Er2 O3 PhC, the PL intensity from both excited states 4 I 9/2 and 4 S 3/2 remarkably reduces compared with that in an unprocessed sample. This is because the PL transition from these states is forbidden by the PBG of the Er2 O3 /Si PhC membrane, and the population on those states should relax to the lower 4 I 13/2 state. As evidence of this scenario, the PL intensity from the first excited state of 4 I 13/2 is enhanced at the same time. Applying such a PhC structure to an REO thin film on a Si substrate is expected to provide optical device structures such as PhC nanocavities or optical waveguides. Other structural possibilities include optical confinement by the refractive index (n) contrast of heterostructures [60]. Normally, an electromagnetic wave is confined and propagated in a layer with a high refractive index. In the case of an REO/Si heterostructure, however, due to the low refractive indices of REOs compared with Si, most of the electromagnetic field will be confined in Si, rather than in the oxide, which will significantly reduce the interaction strength between Er and the optical field.

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Fig. 25 a Distribution of electromagnetic field intensity of the fundamental TM mode in a horizontal slot waveguide. b Cross-section TEM image of Gd2 O3 -based horizontal slot waveguide. c and d SEM images of fabricated grating coupler and microring resonators, respectively (Reprinted with permission from Ref. [60] © The Optical Society)

To solve this problem, a horizontal slot waveguide structure is applied to the REO/Si material system. An epitaxial REO layer with n ∼ 1.9 is sandwiched between two Si layers with n = 3.4. With this configuration, the electromagnetic field energy for TM polarization can be well confined in the low-refractive-index REO layer [61]. Furthermore, a ridge configuration with only the top Si layer etched is used to provide lateral optical confinement, which simplifies the fabrication process by eliminating the necessity for etching of REO. Figure 25a shows the simulated electromagnetic field intensity distribution of the fundamental TM mode of a horizontal slot waveguide with about a 150-nm-thick bottom Si layer and 50-nm-thick REO (here Gd2 O3 is assumed), and width of 1 µm. It can be clearly seen that most of the electromagnetic energy is well confined in the thin REO layer. Based on this simulation, we grew an REO/Si heterostructure with the same structural parameters on a SOI substrate, as shown in Fig. 25b. To obtain a flat REO/Si interface, an amorphous layer is used for the upper Si layer because the surface energy of REO is much lower than that of Si. This slot waveguide also has grating couplers (GCs) for coupling with single-mode fibers, and a microring resonator for characterization of waveguide propagation loss (Fig. 25c, d). Figure 26 shows TM-polarized transmission spectrum of a microring resonator with a radius of 50 µm. Periodic resonance can be clearly seen in this spectrum. We

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Fig. 26 Transmission spectra of waveguide-coupled microring resonators for TM polarizations. The widths of the bus waveguide and microring are both 0.8 µm and the ring radius is 50 µm. The resonances around 1535 nm, together with its fitted curve, is also shown in the inset (Reprinted with permission from Ref. [60] © The Optical Society)

cam extract the propagation loss of the ring waveguide by fitting the resonant dips to the theoretic transmission formula of microring resonators [62], T =

a 2 − 2ra cos(φ) + r 2 1 − 2ra cos(φ) + (ra)2

(10)

where a = exp(−αL) is the single path amplitude transmission coefficient, α is the power attenuation coefficient of the ring waveguide, φ = β L is the single path phase shift, β is the propagation constant of the ring waveguide, and r is the selfcoupling coefficient of the coupling waveguide. Using this relation, the propagation loss for TM polarization was estimated to be 284 dB/cm, which is a rather large value compared with that of state-of-the-art Si waveguides. This large optical loss would mainly come from the optical absorption of the top amorphous Si and REO layers. To reduce the propagation loss further, dedicated optimizations of growth and processing conditions are necessary, and is a subject needing further investigation.

4 Summary This chapter discussed the details of the growth and optical characteristics of the Er-doped non-spin-bath REO crystals operating with telecom-band photons as a platform for hybrid quantum systems. The suppression of Γg−g by purifying the isotope of Er ions resulted in an increase in optical coherence time T2 from 0.39 to 1.45 µs in the frequency domain, and to 12 µs in the time domain. According to these results, the isotope purification of Er ions caused a remarkable reduction of

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Γg−g by strongly suppressing interactions among the 167 Er3+ ions and other isotopes. It was very effective for suppressing the decoherence of the populations. Moreover, coherent transients, such as Rabi oscillation and two-pulse photon echo, which are fundamental principles of coherent population manipulation, were demonstrated in a -like 167 Er3+ three-level system at the operation wavelength of 1536 nm without a magnetic field. In addition, to suppress Γg−h , the epitaxial growth of non-spin-bath REO host crystals on Si substrates was demonstrated. Various defect-free REO epitaxial thin films were successfully grown by using the MBE method. The investigation of the dynamics of Er populations doped into (ErSc)2 O3 revealed that the ETUC is the dominant mechanism of population dissipation from the target energy level. To make this disappear, it was found that the Er concentration has to be less than 30%. In addition, CeO2 epitaxial host crystals, which has quite a small magnetic moment, were successfully grown on Si substrate. The Er doped into them showed well-defined optical transitions at the wavelength of 1533 nm, and the efficiency was effectively enhanced by diluting the Er concentration to less than 1%. Methods for fabricating photonic nanostructures, which are required to evaluate the coherence characteristics of Er doped into these REO epitaxial hosts and to integrate optical circuit on Si chip, were proposed. The fabrication of a PhC membrane by selective area growth of REO and a horizontal slot waveguide based on an REO/Si heterostructure was demonstrated. With the slot waveguide structures, the optical transition probability can be controlled by PBG, and the REO layer showed strong optical confinement of TM-polarized light. As described above, three important technologies-isotope purification of guest ions, magnetic-moment purification of host crystals, and optical nanostructure fabrication-have been established for realization of the hybrid quantum systems. Utilizing these technologies, we aim to achieve on-chip coherent quantum coupling in the telecom-band. Acknowledgements I would like to thank all the collaborators, T. Inaba, X. Xu, H. Omi, E. Kuramochi, K. Shimizu, H. Yamamoto, H. Gotoh (NTT BRL), R. Kaji and S. Adachi (Hokkaido Univ.) for their dedicated experimental cooperation and helpful discussions throughout the research project. I am also grateful to my students, T. Hozumi, Y. Kawakami, T. McManus, A. Llenas, G. Mariani, G. Nakamura, M. IJspeert, W. Szuba, V. Fili and M. Hiraishi for their enthusiastic research support. This manuscript summarizes the results obtained from the contributions of all the people above.

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Electron Spin Resonance Detected by Superconducting Circuits Rangga P. Budoyo, Hiraku Toida, and Shiro Saito

Abstract Superconducting circuits are promising components for future quantum information technologies such as quantum computing, quantum simulation, and quantum sensing. Superconducting quantum interference devices (SQUIDs) and superconducting flux qubits can be used as a sensitive magnetic field sensor to detect local electron spin resonance (ESR). These types of sensor are characterized by good sensitivity and high spatial √ resolution for electron spins. In fact, we have achieved a sensitivity of 20 spins/ Hz with a sensing volume of 6 fl combining a flux qubit and qubit readout using a Josephson bifurcation amplifier. As a demonstration, we measured the ESR spectrum of erbium electron spins in a Y2 SiO5 crystal and nitrogen vacancy centers in diamond. In both cases, we obtained reasonable electron spin parameters by fitting a simulation to the spectrum. In this way, we proved superconducting circuits to be promising magnetic field sensors for evaluating the properties of solid materials. Keywords Quantum sensing · Superconducting qubits · Electron spin resonance

1 Introduction Quantum science and technology have developed rapidly towards the realization of useful quantum computers. Among the many potential physical systems, superconducting quantum circuits are especially promising solid-state devices for quantum R. P. Budoyo · H. Toida · S. Saito (B) NTT Basic Research Laboratories, NTT Corporation, 3-1 Morinosato-Wakamiya, Atsugi, Kanagawa 243-0198, Japan e-mail: [email protected] R. P. Budoyo e-mail: [email protected] H. Toida e-mail: [email protected] R. P. Budoyo Centre for Quantum Technologies, Singapore, Singapore © Springer Nature Singapore Pte Ltd. 2021 Y. Hirayama et al. (eds.), Hybrid Quantum Systems, Quantum Science and Technology, https://doi.org/10.1007/978-981-16-6679-7_5

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state manipulations. The coherence time of superconducting qubits has improved from 1 ns to 0.1 ms in the last 20 years [1–3]. Furthermore, gate fidelities above 99.4% have been achieved by many groups [4–6]. Superconducting qubit-based technologies have matured to the stage where they can be used to make small-scale systems. In fact, many research groups have started scaling up superconducting quantum computers, and Google has recently reported that it had achieved “quantum supremacy” with a 53-qubit superconducting chip [7]. These technologies can be used for new applications, especially quantum sensing. The reason why the coherence time of superconducting qubits was short in the past was because they were fragile against external noise. As such, superconducting qubits should therefore be a good sensor for external fields. In particular, superconducting flux qubits √ were demonstrated in 2012 to be sensitive to an external magnetic field of 3.3 pT/ Hz [8]. In addition, the flux qubit has good spatial resolution; the size of the qubit loop is a few micrometers. These features make it suitable for local detection of electron spins. In this chapter, we will review our experimental results on using superconducting circuits to perform local electron spin resonance measurements.

1.1 Superconducting Quantum Circuits Two Josephson junctions in a superconducting loop form a device known as a dc superconducting quantum interference device (dc-SQUID), which is a sensitive detector of magnetic flux (see Fig. 1c) [9]. Figure 1c shows the superconducting current flowing through the dc-SQUID, which is expressed as I = I1 + I2 = IC sin φ1 + IC sin φ2     ext ext sin φ1 + π . = 2IC cos π 0 0

(1) (2)

The second line of this equation contains the fluxoid quantization relation among an external flux through the SQUID loop ext and the phase differences across the junctions φ1 , φ2 . IC is the critical current of each junction. 0 = h/2e is the magnetic flux quantum, h is Planck’s constant, and e is the elementary charge. The maximum value of I , namely the switching current of the dc-SQUID, is written as Isw

    ext  . = 2IC cos π  

(3)

0

This magnetic flux dependence of Isw (see Fig. 1c) makes the dc-SQUID a sensitive magnetic flux sensor. A superconducting flux qubit consists of a superconducting loop with three Josephson junctions [10, 11]. A schematic of the flux qubit is shown in Fig. 1b.

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Fig. 1 a Schematic diagram of a dc superconducting quantum interference device (dc-SQUID). Cross marks represent Josephson junctions. b Schematic diagram of a superconducting flux qubit. c Switching current of dc-SQUID as a function of external magnetic flux. d Energy levels of the ground and first excited states of superconducting flux qubit. The blue (red) curves represent the ground (first excited) states

One of the junctions is designed to have α times smaller area than the other two. The sum of the Josephson energies of the junctions is expressed as U = (1 − cos φ1 ) + (1 − cos φ2 ) + α(1 − cos φ3 ) EJ   ext . = 2 + α − cos φ1 − cos φ2 − α cos φ1 − φ2 + 2π 0

(4) (5)

Here, E J is the Josephson energy of the nominal junctions, while φi and ext represent phase differences across the i-th junction and the external flux through the qubit loop, respectively. The energy U plays the role of potential energy of a mass in a phase space spanned by φ1 and φ2 . When ext ≈ 0 /2, the potential energy forms a double-well potential, in which the localized state in the left (right) well corresponds to a clockwise (counter clockwise) circulating current state. At the optimal point ext = 0 /2, both states have the same energy. However, the degeneracy can be lifted by tunneling of the mass between the two wells, which can be controlled by the value of α. Thanks to the double-well potential, the lowest two energy levels are well separated from the higher levels. Therefore we can utilize them as a qubit without exciting higher levels. Figure 1d shows the qubit energy levels as a function of the external flux. The important point for the application to a magnetic flux sensor is that the range of the flux is much narrower than that of the dc-SQUID shown in

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Fig. 1c. This means that the flux qubit has higher sensitivity to an external flux than the dc-SQUID has as long as the same methods of signal readout are used for both cases.

1.2 Local Electron Spin Resonance Electron spin resonance (ESR) spectroscopy is a powerful means to obtain information about unpaired electrons. It has been used for medical and pharmaceutical analysis and in materials science and solid-state physics. Conventional ESR spectroscopy is performed by measuring microwaves transmitted through or reflected in an X-band 3D cavity with a length of about few centimeters (see Fig. 2a). The sensing volume is large (of the order of a ml) and the poor signal-to-noise ratio means that a large number of spins (∼1013 spins) are required to obtain a strong enough signal. Recently, local ESR measurements have been performed using superconducting 2D resonators composed of a narrow inductor and interdigital capacitors (Fig. 2b) [12–14]. In this configuration, the coupling between the resonator and an electron spin becomes much stronger than in the conventional case and electrons in a small √ region just below the narrow inductor can be detected. A sensitivity of 12 spins/ Hz has been demonstrated in a sensing volume of 6 fl [15]. Here, the ESR microwave frequency is set to be close to the cavity or resonator resonance because of the

a

SuperconducƟng resonator

b Out

In

0.7 mm C/2 L C/2 Silicon substrate including electron spins

c

Spin

Qubit

+

Few cm Electron spins

3D cavity

Readout circuit

5 µm Microwave line SQUID Qubit

Spin sample Silicon substrate

Fig. 2 a Setup of conventional electron spin resonance (ESR) experiment. b Local ESR detected by a 2D superconducting resonator [12]. c Local ESR detected by a superconducting flux qubit or by a dc-SQUID

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narrow resonator bandwidth. Consequently, the magnetic field applied to the target spin sample has to be swept to obtain the ESR spectrum. On the other hand, in our setup, both the ESR microwave frequency and the magnetic field can be swept because we directly measure the magnetic flux created by the electron spins using a dc-SQUID or a flux qubit instead of a resonator. Hence, we can measure the samples in a wider parameter range and get more information. Figure 2c shows √ a schematic diagram of our local ESR setup, which has a sensitivity of 20 spins/ Hz and a sensing volume of 6 fl.

2 ESR Using dc-SQUID In order to validate the concept of spin detection using superconducting magnetometers, the properties of paramagnetic spins in solid state materials were investigated using a dc-SQUID. This section is an adaptation of a previous publication [16].

2.1 Experimental Setup The experimental setup for spin detection using a dc-SQUID is shown in Fig. 3a. The electron spin ensemble is directly attached to a silicon chip on which a dc-SQUID is fabricated. A magnetic field oriented perpendicular to the chip B⊥ is used to generate magnetic flux bias in the dc-SQUID. An in-plane magnetic field B is applied to polarize the electron spins. The dc-SQUID is connected to a readout circuit; a pulsed readout signal is applied to it. A microwave signal is radiated through an on-chip microwave line to excite the spin ensemble.

Fig. 3 a Experimental setup. This figure is adapted (the part related the flux qubit has been deleted) from [17] under a Creative Commons Attribution 4.0 International License [18]. b Principle of detecting the magnetization of spins by using a dc-SQUID. The magenta modulation curve shifts to the blue curve in the presence of the polarized electron spin ensemble

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Additional magnetic flux applied to the loop of the dc-SQUID  is converted into a shift in the switching current ISW by fixing the operation point of the dcSQUID as shown in Fig. 3b. In this case, the additional magnetic flux is generated by the magnetization of the spin ensemble. Thus, if the magnetization is modulated by external factors (e.g. temperature, magnetic field, or microwave signal), the magnetometer generates a signal corresponding to the stimulation. The dc-SQUID only detects the magnetic flux oriented perpendicular to it. Because the applied magnetic field for the spin polarization is oriented to the inplane direction, a large magnetization signal would not be observed. However, a finite magnetization would appear if there is anisotropy or asymmetry in the setup. In the magnetometry discussed in the next section, we use a material with an anisotropic g-factor tensor to convert the in-plane magnetic field to a magnetization oriented perpendicular to the dc-SQUID. Asymmetry in the microwave spin excitation is naturally introduced, because the placement of the microwave line is asymmetric about the dc-SQUID as shown in Fig. 3a. This asymmetry causes a gradation of the microwave power in the spin ensemble, and thus, a finite signal is generated for the magnetometer. In addition to these methods, we can generate a finite signal by careful placement of the spin ensemble, e.g. by placing the sample such that it partially covers the dc-SQUID. However, this method may require micro- or nano-manipulation of the spin ensemble and is out of the scope of this chapter.

2.2 Magnetometry Here, we describe magnetometry performed on erbium-doped yttrium orthosilicate (Y2 SiO5 /YSO, 200 ppm) as a function of temperature T and in-plane magnetic field B . The in-plane magnetic field is oriented parallel to the D1 axis in the crystal reference frame.1 The modulation curves of the dc-SQUID are depicted in Fig. 4a. In this experiment, the spin polarization ratio is controlled by varying the temperature of the sample, while the in-plane magnetic field is fixed at 5 mT. A clear shift in the modulation curve is observed. Note that a shift does not occur when there is no applied magnetic field. Thus, the shift is caused by the combined effect of the temperature and the magnetic field. The intensity of the in-plane magnetic field is varied up to 40 mT to investigate the relationship between the detected magnetic flux and the parameters. The shift in the modulation curves, which directly translates into magnetic flux, is summarized as a function of the ratio between temperature T and in-plane magnetic field B in Fig. 4b. This dependence of the detected magnetic flux  is well reproduced by the following equation:

1

YSO crystal is typically described in the D1 -D2 -b orthogonal coordinate system, where the D1 and D2 axes are the optical extinction axes and the b axis is one of the crystal symmetry axes of the monoclinic YSO crystal [19].

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Fig. 4 a Switching current of the dc-SQUID as a function of applied magnetic flux bias under in-plane magnetic field of 5 mT. b Detected magnetic flux as a function of the ratio between the temperature and the in-plane magnetic field. Symbols indicate shifts in the modulation curves. The solid line is the fit to the data points using Eq. (6). The dashed line is a linear fitting using the slope at zero in-plane magnetic field. The dotted line is the level of the saturation magnetic flux estimated from the fitting. Adapted from [16]

 (B , T ) = sat tanh

μ B geff B 2k B T

 ,

(6)

where sat is the saturation magnetic flux for a completely polarized spin ensemble; μ B is the Bohr magneton, geff is the effective g-factor, and k B is the Boltzmann constant. From the fitting of the experimental result to Eq. (6), the effective g-factor and saturation flux are estimated to be 5.9 ± 1.4 and 1.75 ± 0.21 0 . The value of the effective g-factor is consistent with the value calculated using the parameters from the literature [20] within the range of errorbar.

2.3 ESR Spectroscopy The results of ESR spectroscopy of a type-Ib diamond crystal are shown in Fig. 5. In this experiment, the in-plane magnetic field is along the [100] direction. Because type-Ib diamond has a lot of nitrogen impurities, P1 centers, which are point defects in diamond that substitute a carbon atom with a nitrogen atom, are the focus of the observation. Raw ESR spectra are plotted for several in-plane magnetic fields in Fig. 5a. In this measurement, the switching probability, which is the average of many pulsed measurements to determine whether the dc-SQUID switched to the voltage state or not, is determined by setting the magnetic flux bias of the dc-SQUID as in the case of readout of a flux qubit [21]. The spectra have three peaks with splittings of ≈100 MHz. This result is consistent with the properties of the P1 center: because ≈99.6 % of nitrogen atoms in nature have a nuclear spin of one, the resulting ESR spectrum splits into three peaks due to the hyperfine interaction.

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Fig. 5 ESR spectroscopy of P1 centers in diamond. a Raw ESR spectrum for various in-plane magnetic fields. The intensity of the in-plane magnetic fields is −50, −30, −20, −10, 40, 45, 50, and 55 mT from bottom to top. Lines have offsets of 4B %/mT for clarity. b ESR peak positions as a function of in-plane magnetic field. The dots indicate the positions of the peaks in (a), and the lines are fittings using Eq. (7). Adapted from [16]

The peak positions are plotted as a function of the in-plane magnetic field in Fig. 5b. The following spin Hamiltonian fits the results: HP1 = μ B gB · S + I · A · S,

(7)

where g is the g-factor of the P1 center, B = B + B⊥ , and S (I) are electron (nuclear) spin vector operators. Here, the hyperfine tensor A is diagonal: ⎛

⎞ Ax 0 0 A = ⎝ 0 Ay 0 ⎠ . 0 0 Az

(8)

The hyperfine parameters used in the numerical calculation are A x / h = A y / h = 57.9 MHz and A z / h = 114.4 MHz [22], and it is assumed that there is no alignment error between the crystal axis and the orientation of the in-plane magnetic field. The g-factor is estimated from the fitting parameter to be 2.11.2 This value is not consistent with the g-factor of the P1 centers (g = 2.0027 [22]), possibly because of the enhancement of the magnetic field near the superconductor due to the Meissner effect or error in the calibration constant of the in-plane magnet. It is necessary to take ESR spectra in several conditions of in-plane magnetic field orientation to determine the hyperfine coupling constants A x , A y , and A z from experiments. While in principle it is possible to change the orientation of the in-plane magnetic field (e.g. by using a vector magnet), determining the hyperfine parameters is beyond the scope of the current research. Instead, a comparison is made of the level spacings in a regime where the Paschen-Back effect is strong (i.e. the hyperfine splitting is much smaller than the Zeeman splitting); the value of ≈100 MHz is consistent with the literature [22]. This value is slightly different from our previous result [16] (g = 2.12), because the fitting here is performed using the spin Hamiltonian instead of the linear function in [16].

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2.4 Estimation of Sensing Volume and Sensitivity The sensing volume and sensitivity are figures of merit of this sensing technique. The sensing volume can be estimated by multiplying the detection area and the effective sensing height. Because a dc-SQUID is only sensitive to the magnetic flux through its loop, the sensing area of the device can be calculated from the shape of the dc-SQUID. In our device, it is ≈101 µm2 . Here, we assume the effective sensing height to be the height at which the coupling strength between the persistent current of the dc-SQUID and the spin starts to decrease rapidly. This height is estimated to be ≈1 µm from the literature [23]. These numbers give a sensing volume of ν ≈ 100 µm3 = 0.1 pl. The sensitivity is estimated by converting the flux sensitivity into the corresponding number of spins. The flux sensitivity is defined as the minimum √ distinguishable shift in the modulation curve. It is estimated to be δ 3.5 m0 / Hz by using the intensity of noise in the switching probability described in the literature [17]. The sensitivity is converted into the number of spins by using the magnetometry results discussed in Sect. 2.2, sat = 1.750 . This value corresponds to the magnetic flux generated by completely polarized spins in the sensing volume ν. Considering the concentration of spins in the crystal n = 3.7 × 1018 spins/cm3 and the sensing volume ν, the saturation flux is estimated to be generated by N = nν = 3.7 × 108 spins. Finally, the sensitivity δS is estimated as δS =

√ δ N  106 spins/ Hz. sat

(9)

3 ESR Using a Josephson Bifurcation Amplifier The dc-SQUIDs described in the previous section are read out by monitoring their switching to the voltage state. This process heats the device, which imposes a significant waiting time (100 µs) between individual measurements. This limits the repetition rate and in turn the ESR measurement sensitivity. On the other hand, readout operation of a Josephson bifurcation amplifier (JBA) does not involve heating. As a result, ESR spectroscopy using non-switching readout of a JBA offers improved sensitivity compared to using switching readout of a dcSQUID. This section is an adaptation of a previous publication [24].

3.1 Josephson Bifurcation Amplifiers A Josephson bifurcation amplifier is simply a superconducting resonator with a strong nonlinear element. As is typical in superconducting quantum circuits, Josephson

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Fig. 6 a Schematic diagram of a JBA consisting of a coplanar waveguide resonator and a single Josephson junction. b Illustration of the bifurcation behavior on the resonance of the JBA. c Schematic diagram of a tunable JBA, where a single Josephson junction is replaced by a dc-SQUID. d Resonance frequency of a tunable JBA as a function of the magnetic flux through the dc-SQUID

junctions are the sources of the nonlinearity. Figure 6a shows a circuit diagram of a JBA consisting of a λ/2 coplanar waveguide resonator with a Josephson junction. The term “bifurcation amplifier” comes from the fact that the resonator possesses two distinct oscillation states when driven strongly, a low amplitude state and a high amplitude state, as shown in Fig. 6b [25, 26]. When the JBA is dispersively coupled to a superconducting qubit, the JBA can latch to one of the two states depending on the state of the qubit. This allows for fast and high-contrast readout of the superconducting qubits [27, 28]. When a dc-SQUID is embedded in a JBA, as shown for example in Fig. 6c, the resonant frequency of the JBA depends on the magnetic flux through the dc-SQUID loop, as shown in Fig. 6d. By monitoring the shift in the resonance frequency, the JBA can be used as a magnetometer and, by extension, an ESR spectrometer.

3.2 ESR Spectroscopy Setup Figure 7a shows the setup for ESR spectroscopy using a JBA. Typically a λ/2 coplanar waveguide resonator with a dc-SQUID embedded in the middle of the resonator is used. The spins of interest are attached to the chip substrate, covering the dc-SQUID. Two separate magnetic fields are applied using superconducting magnets: One magnetic field is oriented parallel to the chip surface B and has a strength up to tens of mT to polarize the spins. The other, much smaller, magnetic field B⊥ (typically of order 10 µT) is oriented perpendicular to the chip to adjust the flux through the SQUID loop. The JBA is driven using a capacitively-coupled microwave input line, and the transmitted signal from the capacitively-coupled output line is monitored. The microwave drive for ESR is supplied using a separate inductively-coupled microwave line or the same capacitively-coupled input line driving the JBA. For optimum sensitivity, the

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Fig. 7 a Schematic diagram of ESR spectroscopy using a JBA. b Probability of switching to JBA high amplitude state as a function of flux bias (switching curve) swept near JBA resonance. Adapted from [24]

JBA drive frequency ωJBA is chosen to be at a bias point where the bifurcation is large and the frequency is very sensitive to change in flux, ∂ωJBA /∂  0. In typical operations of our ESR measurements, B is set fixed and the ESR drive frequency ωESR is stepped within the range of interest. Setting ωESR fixed and stepping B is also possible. In both cases, ωESR should not be too close to ωJBA (|ωJBA − ωESR |/2π  500 MHz) as it will distort the JBA transmission profile. Except for the range around ωJBA , ESR spectroscopy can be performed at all ωESR within the working frequency range of the ESR microwave input line (e.g. up to 18 GHz for typical connectors). For a given ωESR , the microwave for spin excitation is applied continuously to the spin sample while the microwave for the JBA drive is pulsed, using the same pulsing sequence as that used in qubit readout [27, 28]. Typically, measurements are made for approximately 1000 JBA pulses and the probability of the JBA to be in the high amplitude state Psw (“switching probability”, due to the similarity to the SQUID switching probability) is obtained from the average value of these measurements. When the flux bias (B⊥ ) is swept near the JBA resonance point, Psw will rapidly change from 0 to 1, for example as shown in Fig. 7b (called the “switching curve”). When the ESR signal is expected to be small, B⊥ is set to a value at which Psw = 0.5 when there is no microwave for spin excitation. In this case, the ESR spectrum is obtained by plotting Psw as a function of ωESR . When the expected ESR signal is large, B⊥ is instead swept to find the bias point where Psw ≈ 0.5. The ESR spectrum in this case is given by the change in B⊥ versus ωESR . This latter method is slower than the former one (the speed can be increased by implementing an effective active feedback for finding the correct B⊥ value), but it has a much larger detection range. The latter method was used to obtain the results discussed in the following section.

3.3 ESR Spectroscopy of Er:YSO Using a JBA This section describes the results of ESR spectroscopy of an erbium-doped yttrium orthosilicate (Y2 SiO5 /YSO) crystal using a JBA. Er3+ dopants in YSO have attracted

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interest for use as solid-state spin qubits due to the presence of optical transitions at 1.5 µm (a standard wavelength used in telecommunications) [29] and the predicted presence of zero first-order Zeeman (ZEFOZ) transitions with enhanced coherence [30].

3.3.1

Er:YSO Spin Hamiltonian

Er3+ dopants (electron spin S = 1/2) consist of even isotopes with nuclear spin I = 0 (77% natural abundance) and 167 Er isotopes with I = 7/2 (23% natural abundance). For even isotopes, the spin Hamiltonian is Heven = μ B B · g · S,

(10)

where B = B + B⊥ and g is the electron g-factor tensor. The transition frequencies are given by (11) ω = μ B geff |B|, where geff is the effective g-factor determined by the orientation of B. For 167 Er, the electron spin Hamiltonian also has hyperfine and quadrupole interaction terms: H167 = μ B B · g · S + I · A · S + I · Q · I − μn gn B · I,

(12)

where μn is the nuclear magneton, A is the hyperfine tensor, Q is the quadrupole tensor, and gn is the nuclear g-factor. There are 16 states and 120 transitions; typically, the Hamiltonian must be fully diagonalized in order to find the transition frequencies. In the case of Er:YSO, the g, A, and Q tensors are highly anharmonic, and thus, the transition frequencies will depend strongly on the direction of B. Furthermore, the Er dopants are at two different crystallographic sites with different sets of spin tensors, and each site also contains two magnetic inequivalent subclasses, related by rotation. For these reasons, Er:YSO is a prime example of a material that would benefit from characterization using superconducting loop-based devices instead of conventional single-frequency ESR. In fact, recent reports using tunable cavity ESR spectroscopy [31] and optical Raman heterodyne spectroscopy [32] have shown that the Er:YSO spin tensors obtained using conventional 9.5 GHz ESR spectroscopy [20] do not extrapolate well to the low-magnetic field and low-frequency range.

3.3.2

Spectroscopy Results

In the experiment, YSO crystal doped with 200 ppm Er3+ (in natural isotope abundances) was glued on top of a JBA chip. Relative to the crystal reference frame, the in-plane magnetic field was oriented parallel to the D1 -D2 plane, about 9◦ from the D1 axis. This orientation was subsequently confirmed by electron backscatter

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Fig. 8 a ESR spectra of Er:YSO sample measured using a JBA with B oriented parallel to the D1 -D2 plane, 9◦ from D1 axis. b Simulated ESR spectra for the same parameters. Adapted from [24]

diffraction spectroscopy of the crystal. For this B orientation, the two magnetic subclasses for each site are equivalent. The spectroscopy was performed at B between 0.27 and 6.50 mT and ωESR /2π between 0.1 and 5.2 GHz (upper limit set by ωJBA /2π = 5.93 GHz), with the dilution refrigerator temperature set at 200 mK. Figure 8a shows the resulting spectra. For comparison, Fig. 8b shows simulated spectra [33] for the measurement values created from the spin tensor values obtained from previous spectroscopy measurements of Er:YSO [31, 32]. Overall, the frequencies of the measured transitions agree very well with the simulations. However, there are small deviations between the two spectra, for example in the several spacings between parallel transitions, which mainly depend on the quadrupole transition strengths. These deviations can be used for further optimization of the spin tensor parameters.

3.4 Measurement Sensitivity The sensitivity and other figures of merit for ESR spectroscopy using a JBA are estimated similarly to when using a dc-SQUID, as discussed in the previous section. The parameters in the experiment were the same as those for the device used in the magnetometry of Er:YSO in the previous section.

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Fig. 9 Calculated ESR measurement sensitivity for the JBA sample used in spectroscopy of Er:YSO as a function of number of measurements N p in 1 s, for several values of temperature T and magnetic field B . The dashed curves are fits of the sensitivity values to Eq. 14. Points in the shaded region are not included in the fit. Adapted from [24]

The dimensions of the dc-SQUID loop were 12 µm × 13 µm. As discussed in the previous section, the effective thickness is estimated to be about 1 µm. This gives a sensing volume of about ν ≈ 150 µm3 = 0.15 pl. The 200 ppm Er:YSO used in the experiment had a spin concentration of n = 3.7 × 1018 spins/cm3 , meaning about N = nν = 5.5 × 108 spins were located within the sensing volume. From spin magnetization measurements (similar to the measurement described in Sect. 2.2, but using the JBA), it is estimated that saturation of N spins corresponds to a shift of sat ≈ 3.40 in the flux detected by the JBA relative to all spins in the ground state. The final parameter needed for the sensitivity estimation is the smallest distinguishable flux shift. This value can be obtained by repeatedly measuring the JBA switching curve (e.g. √ Fig. 7b). Under typical measurement conditions, this value is about δ = 90 μ0 / Hz. The measurement sensitivity can then be estimated as δS =

√ δ N = 1.5 × 104 spins/ Hz. sat

(13)

To investigate the limiting factors of the sensitivity, Fig. 9 shows the estimated sensitivity for several T and B values as a function of number of measurements N p within 1 s (i.e. repetition rate is N p /1 s). For large N p  105 , the sensitivity apparently becomes worse. This drop in sensitivity is due to the reduced bifurcation effect, which is likely due to incomplete JBA relaxation between pulses.3 The sensitivity values for N p < 105 can be fit to the expression, δS(N p ) =

α + β, Np

(14)

This means the JBA still offers a minimum repetition time of about 10 µs, an order of magnitude shorter than the minimum repetition time for the dc-SQUIDs discussed in the previous section.

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where α and β are fitting parameters. The first term within the square-root is related to the averaging of the binomial distribution of the switching probability Psw ; in particular, α is related to the characteristics of the JBA. The fitted values of α appear to increase with increasing temperature and are independent of B . This is consistent with the observation that the JBA resonance broadens at higher T , while it remains relatively unchanged when B is varied, at least within the range of B in the experiment. √ The β term corresponds to additional noise sources. The sensitivity of 15,000 spins/ Hz applies√when B = 0. When B = 0, the apparent sensitivity improves to about 8000 spins/ Hz. This suggests that a strong coupled flux noise is generated by the B superconducting magnet when B = 0. The fact that β = 0 even when B = 0 suggests the presence of additional noise sources, including flux noise generated by the B⊥ magnet and, as will be shown in the following sections, the intrinsic 1/ f flux noise of the SQUID.

4 ESR Detected by a Flux Qubit with Switching Readout The previous section showed that the sensitivity can be improved by changing the readout method of the dc-SQUID magnetometers. In this section, we describe how sensitivity can be enhanced by replacing the magnetization detector by a flux qubit. Here, the method for reading out the flux qubit is based on the switching readout of the dc-SQUID described in Sect. 2. This section is an adaptation of a previous publication [17].

4.1 Experimental Setup The experimental setup for spin detection using a flux qubit (Fig. 10a) is similar to the one described in Sect. 2.1. One microwave generator and a microwave combiner are added to the setup to perform spectroscopy of the flux qubit. In this setup, the magnetization of a spin ensemble is converted into information of qubit frequency. Here, a dc-SQUID is a readout device for the flux qubit. The dc-SQUID generates two different signals reflecting the state of the flux qubit: a finite voltage drop between two terminals of the dc-SQUID (switching to voltage state) or a zero voltage drop (remaining superconducting state). The voltage signals are binarized with some threshold and recorded several thousand times, and the state of the flux qubit is estimated. The probability of obtaining a voltage state is defined as the switching probability PSW . The principle of detecting spin polarization by using a flux qubit is illustrated in Fig. 10b. The energy spectrum of the flux qubit f q () has a hyperbolic shape [21]. The qubit spectrum shifts depending on the magnetization of the spin ensemble (magenta or blue line), because it can be treated as an additional magnetic flux bias.

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Fig. 10 a Experimental setup. b Principle of detecting the magnetization of a spin ensemble using a flux qubit. The blue qubit spectrum (dotted line) is shifted to the magenta curve (solid line) by the difference in the polarization ratio of the spin ensemble. These figures are adapted (some labels are modified) from [17] under a Creative Commons Attribution 4.0 International License [18]

The change in the magnetic flux  is converted into a change in the resonance frequency of the flux qubit  f q , if the operation point (magnetic flux bias to the flux qubit) is fixed at a point in the qubit spectrum having a finite slope.

4.2 Magnetometry Magnetometry of Er:Y2 SiO5 (50 ppm) is performed as a function of the temperature T and the in-plane magnetic field B  D1 without applying a microwave signal for spin excitation. Here, because of the limitation of the applicable in-plane field to the flux qubit, the maximum in-plane magnetic field is 10 mT. As shown in Fig. 11a, there is a clear shift in the qubit spectrum depending on temperature under the in-plane field of 4 mT. Figure 11b summarizes the temperature and in-plane magnetic field dependence of the shift in qubit spectrum. The figure shows a linear increase in the detected magnetic flux. This can be interpreted as the linear rising part of the hyperbolic tangent dependence [Eq. (6)], because the Zeeman energy (corresponding to the inplane magnetic field) is much smaller than the thermal energy here. Note that this trend in the magnetization corresponds to the small field region in Fig. 4b. Because Fig. 11b does not show saturation behaviour due to the small in-plane field unlike in the case of Sect. 2.2, it is hard to derive both an effective g-factor and saturation flux from the experimental results. Instead, to derive the saturation flux, the effective g-factor can be calculated from the existing parameters [20]. The effective g-factor turns out to be 6.2 [16] for this experimental setup. By combining this number and the slope derived from the fitting (Fig. 11b), the saturation flux is estimated to be 0.180 . This is consistent with the previous observation using a dc-SQUID (Sect. 2.2) considering the difference in the concentration of erbium impurities and the ratio of the detector area.

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Fig. 11 Magnetometry of Er:Y2 SiO5 . a Temperature dependence of the qubit spectrum. The inplane magnetic field is 4 mT. b Summary of the temperature ([50, 100, 200] mK) and in-plane magnetic field dependence of shift in the qubit spectrum. Symbols are experimental results, and the solid line is a linear fitting. These figures are adapted (color schemes, symbols, and labels have been modified) from [17] under a Creative Commons Attribution 4.0 International License [18]

4.3 ESR Spectroscopy ESR spectroscopy of a type-Ib diamond crystal is performed at the base temperature of the dilution refrigerator. Here, the target spin ensemble is NV− centers in diamond, because a high spin polarization ratio is expected even under a small magnetic field due to its large zero field splitting (≈2.88 GHz). The detected magnetic flux as a function of the spin excitation frequency is depicted in Fig. 12a. Two peaks are visible around ≈2.8 and ≈3.0 GHz. These peaks arise from the three-level nature of NV− centers; namely they are spin-one systems. In addition to the two large peaks, small split peaks are also visible. These can be explained by misalignment between the in-plane magnetic field and the crystal axis. Because NV− centers have four possible axes in the crystal, the ESR peak can split into eight in a magnetic field. For this particular experiment, the in-plane magnetic field is almost along the [100] direction, and the component perpendicular to the qubit chip plane is kept small for qubit operation. Thus, the main alignment error is considered to be due to an in-plane rotation that splits the peaks into four. From these tiny splittings, the misalignment can be estimated to be ≈3◦ . The in-plane magnetic field dependence of the ESR peak frequencies is summarized in Fig. 12b. The peak positions are well fitted by the following spin Hamiltonian: 

HNV = μ B gB · S + h DSz2 + h E S y2 − Sx2 ,

(15)

where g is the g-factor of the NV− center, B = B + B⊥ , D is the zero field splitting, and E is strain. Here, E = 5 MHz [34]. g = 1.996 and D = 2.88071 GHz are derived from the fitting. The g-factor is slightly different from the reported value [35], possibly due to the Meissner effect or error in the calibration constant of the superconducting magnet.

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Fig. 12 ESR spectroscopy of NV− centers in diamond. a Detected magnetic flux as a function of spin excitation frequency. The in-plane magnetic field is 5.8 mT. b Spin resonance frequencies as a function of in-plane magnetic field. Symbols are experimental results, and the lines are the fitting curves calculated using the spin Hamiltonian for NV− centers, i.e. Eq. (15). These figures are adapted (color scheme, and labels are modified) from published work [17] under a Creative Commons Attribution 4.0 International License [18]

4.4 Estimation of Sensitivity The figures of merit of spin sensing using a flux qubit are the sensing volume and sensitivity. The sensing volume is estimated by simply multiplying the sensing area and the effective sensing height, as previously shown in Sect. 2.4. For this particular device, the sensing area is calculated to be 47.2 µm2 from the design, and the effective sensing height is assumed to be ≈1 µm [23]. These numbers give a sensing volume of ν ≈ 50 µm3 = 50 fl. The sensitivity is estimated by converting the noise in the qubit measurement into the number of spins by deriving the number of spins corresponding to a unity signal-to-noise ratio. To estimate the sensitivity from the experiment, the noise in the qubit measurements δ PSW is measured as a function of the number of repetitions N p (Fig. 13). Here, noise in switching probability δ PSW is defined as the standard deviation of the switching probabilities PSW for N p measurements. The measurement interval (i.e. repetition time) is 200 µs in the experiment; thus, the corresponding number of repetitions per unit time is N p = 5000. Next, the noise in the switching probability δ PSW is converted into the number of spins δS by using the following equation: δS = δ PSW

∂ f q ∂ δ N , ∂ PSW ∂ f q δ

(16)

−1 where ∂ f q /∂ PSW is the slope derived from the line shape of the flux qubit res−1

is the slope derived from the qubit spectrum onance peak [PSW ( f q )], ∂/∂ f q

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Fig. 13 Noise in switching probability. Symbols indicate experimental results, and the solid line is the theoretical curve derived by assuming a binary process. The arrow indicates the number of measurements during the integration time of one second. This figure is adapted (the color scheme and labels have been modified) from [17] under a Creative Commons Attribution 4.0 International License [18]

[ f q ()], and δ N /δ is a conversion constant for converting the flux into the number of spins derived from magnetometry, for example. From the experimental parameters, √ the sensitivity is derived to be ≈400 spins/ Hz. It is worth mentioning that the noise in the switching probability δ PSW does not follow the theoretical curve in the experiment. The slope of the noise in Fig. 13 is expected to be −1/2 because a single qubit measurement should obey a binary process in theory. However, deviation from the ideal curve occurs especially when the integration time becomes long, as shown in Fig. 13. This is mainly caused by 1/ f type flux noise, which is reported to be a limiting factor of the sensitivity of SQUIDs [36–38], or the dephasing time of flux qubits [39–42]. This 1/ f noise will be discussed in the next section.

5 ESR Detected by a Flux Qubit with JBA Readout The previous sensitivity improvement had by changing a readout method from switching readout of a dc-SQUID (Sect. 2) to non-switching readout of a JBA (Sect. 3), also can be applied to the state readout of a flux qubit using a dc-SQUID (Sect. 4). As before, the repetition rate can be improved by using a JBA instead of a dc-SQUID. As this section describes, ESR spectroscopy using a flux qubit with JBA readout also offers improved sensitivity compared to using a flux qubit with dc-SQUID readout. This section is an adapted from a previous publication [43].

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5.1 ESR Spectroscopy Setup As discussed previously, when a JBA is dispersively coupled to a flux qubit, at the correct bias conditions, it can latch to one of two JBA states (a ‘low’ or ‘high’ amplitude state) depending on the state of the qubit (|0 or |1 ). This characteristic can be used for fast and high-contrast readout of superconducting qubits using a JBA [27, 28]. A flux qubit is typically inductively coupled to a tunable JBA including a dc-SQUID, either by having the flux qubit and JBA SQUID loop adjacent to each other [28] or by having the flux qubit embedded inside the JBA SQUID loop [44]. ESR spectroscopy using a flux qubit with JBA readout follows the same principle as ESR spectroscopy using a flux qubit with dc-SQUID switching readout. Figure 14a illustrates the measurement scheme. A flux qubit is embedded inside a dc-SQUID loop positioned at the center of a λ/2 superconducting coplanar waveguide resonator. The spin ensemble of interest is positioned on top of the flux qubit loop. As with the other measurement schemes discussed in this chapter, two magnetic fields are applied: a magnetic field parallel to the chip surface B for spin polarization, and a magnetic field perpendicular to the chip surface B⊥ to bias the flux qubit. The microwave signals for the flux qubit and for the ESR drive share the same line, which is inductively coupled to the flux qubit, while the JBA input and output lines are capacitively coupled to the JBA. Figure 14b shows the pulse sequence. For typical spin ensembles and ESR microwave powers, the ESR drive is turned off just before the qubit readout. The drive is turned off because a strong ESR drive affects the visibility of the qubit. In principle, when the ESR drive power is sufficiently low, it can be kept continuous. Qubit readout consists of a qubit drive followed by a JBA readout drive. Typically, a spectroscopy sequence is chosen, where the qubit drive is longer than its coherence time, and the frequency is stepped to obtain the qubit spectrum (the switching probability Psw vs. qubit drive frequency f q ). For small expected signals, the qubit frequency can be fixed at the point at which the slope (∂ Psw /∂ f q ) is steepest, i.e. the bias point for optimum sensitivity (see Eq. 16).

Fig. 14 a Schematic diagram of ESR spectroscopy using a flux qubit with JBA switching readout. b Typical pulse sequence. Adapted from [43]

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5.2 ESR Spectroscopy of Er:YSO ESR spectroscopy of an Er:YSO crystal was performed using a flux qubit with JBA readout. The flux qubit shared an edge with the JBA (see Fig. 14a). The Er:YSO crystal (see Sect. 3.3 for description) had 50 ppm Er3+ dopants which were isotopically purified such that the 167 Er isotope had approximately 92% abundance. This setup used a 2d vector magnet such that B can in principle be oriented in any direction parallel to the surface of the chip, i.e., in the D2 -b plane of the crystal. As an example, Fig. 15a shows the ESR spectrum for B = 1.7 mT, with the temperature set at 200 mK. For direct comparison, Fig. 15b shows the simulated spectra [33] for the same magnetic field and temperature, computed using the spin Hamiltonian of Eq. 12 and the most recent 167 Er:YSO spin tensor parameters [31, 32]. Furthermore, since the same flux qubit chip also allows ESR spectroscopy measurements using the JBA (as in Sect. 3), a direct comparison can be made between ESR spectroscopy measurements using the JBA and those using the flux qubit. Figure 15c shows the JBA-based spectra for the same magnetic field and temperature. Considering that the sensitivity of using the qubit is expected to be more than an order of magnitude higher than using the JBA (about 3 orders of magnitude higher, as shown in the following section), the qubit-based spectra Fig. 15a should be able to resolve

Fig. 15 a ESR spectrum of Er:YSO at B = 1.7 mT oriented along D2 axis, measured using a flux qubit with JBA readout at 200 mK. b Simulated ESR spectrum for the same magnetic field and temperature. The red (green) dashed curve corresponds to the spectrum for Er site 1 (2), and the blue solid curve corresponds to the overall spectrum. c ESR spectrum measured using a JBA for the same magnetic field and temperature. Adapted from [43]

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more spin transitions. However, it is also important to note the differences in the ESR microwave drive between the two spectroscopy setup: 1. In JBA-based spectroscopy, the ESR drive is not turned off during the JBA pulses. 2. The ESR microwave drive powers are different. The JBA-based spectroscopy uses higher power, which results in broadened transitions and different relative heights between peaks.

5.3 Measurement Sensitivity and 1/ f Flux Noise The parameters described here for the sensitivity estimation are for the device described in Sect. 5.2. The expression of the sensitivity for ESR using a flux qubit with spectroscopy-type pulse sequence (Eq. 16) was derived in Sect. 4.4. The same expression can be used in this case. The flux qubit had a loop area of about 6 µm2 . Similar to the other devices discussed in this chapter, an effective thickness of 1 µm gives a sensing volume of about ν ≈ 6 µm3 = 6 fl. The 50 ppm Er:YSO crystal used in the experiment had a concentration of n = 9.3 × 1017 spins/cm3 , meaning about N = nν = 6 × 106 spins were located within the sensing volume. The other parameters in the sensitivity calculation were obtained from the spectroscopy of the qubit, the spin magnetization measurement of the Er:YSO, and the standard deviation of the experimental switching probability. Using typical bias values in spectroscopy and choosing the optimum parameters, Eq. 16 gives the sensitivity value, δS = δ Psw

√ ∂ f q ∂ δ N ≈ 20 spins/ Hz. ∂ Psw ∂ f q δ

(17)

This value is about an order of magnitude better than the sensitivity of ESR spectroscopy using a flux qubit with dc-SQUID switching readout. Considering that the JBA allows a repetition √ time about 20 times shorter compared with using dc-SQUIDs, only a factor of ≈ 20 ≈ 4.5 sensitivity improvement comes from the faster repetition. The other improvements come from the higher visibility of the qubit spectrum and the narrower linewidth of the qubit resonance peak (both resulting in lower ∂ f q /∂ Psw ). Figure 16a shows the standard deviation of the switching probability as a function of the number of measurements N p in 1 s, with the JBA bias chosen such that Psw ≈ 0.5, a typical baseline for spectroscopy. At low N p , the standard deviation follows the expected binomial standard deviation Psw (1 − Psw )/N p ; however, it eventually levels off to ≈0.5% for N p  104 . The switching probability noise in the sensitivity calculation corresponds to this value, since typical JBA readout repetition time is 10 µs (N p = 105 for 1 s). Similar to the discussion in Sect. 3.4, this constant value suggests the presence of an additional noise source. However, unlike the previous results, this value does not appear to depend significantly on the magnetic field

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Fig. 16 a Standard deviation of switching probability σ Psw as a function of number of measurements N p . The shaded region corresponds the typical number of measurements within 1 s. b Welch spectral density of switching probability S Psw ( f ) as a function of frequency f . Adapted from [43]

strength, which suggests that the leading noise source in this case is not coupled noise from the magnets but something else. To characterize this noise source, Psw can be measured continuously for a long duration (approximately 7 h in the reported experiment). Figure 16b shows the resulting Welch power spectral density extracted from the Psw data, which follows a 1/ f type (‘flicker’) noise spectrum given by S Psw =

A2Psw ( f /1 Hz)α

(18)

where A2Psw is the noise at 1 Hz and the exponent α ≈ 0.93 is obtained from the fit (line in Fig. 16b). In SQUIDs and superconducting qubits, the flux √ noise commonly contains a 1/ f α noise component with typical A ∼ (1μ0 )/ Hz and 0.5  α  1.0 [36–42, 45].4 Assuming the noise in Psw comes from flux noise, the flux noise level can be calculated using A = A Psw (∂/∂ Psw ), while ∂ Psw /∂ can be calculated from the JBA switching curve as a function of . This gives a flux noise level of √ A ≈ (5μ0 )/ Hz, which is within the typical range of values for 1/ f flux noise in SQUIDs and qubits. Thus, it can be inferred that the sensitivity of the device is limited by 1/ f flux noise.

4

Even more than 30 years after it was first observed [36], the source of this noise is still an open question; surface spins on the SQUIDs are the most likely source [37]. Other proposed sources include adsorbed oxygen molecules [46] and even cosmological fluctuations [47].

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6 Summary and Perspectives This chapter has described local magnetization measurements and local ESR measurements using superconducting quantum circuits. In particular, it described measurements of the ESR spectrum of Er electron spins in a Y2 SiO5 crystal and of NV− centers or P1 centers in diamond over a wide range of magnetic field strengths and microwave frequencies. The experimental results are well reproduced by simulations incorporating the previously reported parameters. This proves that local ESR measurements using superconducting circuits are useful for studying solid-state materials. The sensitivity to the electron spins was improved by three orders of magnitude by replacing a dc-SQUID sensor with a superconducting flux qubit. We also changed the readout method from switching readout to readout using a JBA. The former method requires a long repetition time for quasiparticle relaxation to occur because the dcSQUID switches to a voltage state after each measurement. On the other hand, we can repeat measurements frequently in the latter case because no quasiparticles are created in the JBA. This enables us to average more signals per unit time, resulting in about an order of magnitude improvement in sensitivity. We also increased the spatial resolution; namely, we reduced the sensing volume by making loop of the detector smaller. Figure 17 summarizes the sensitivities and sensing volumes of local

Fig. 17 Sensitivity and sensing volume of local ESR detected by superconducting circuits. SQ and FQ represent sensors, a SQUID and a superconducting flux qubit, respectively. SW and JBA represent readout methods, a switching readout and a JBA readout, respectively. All resonators are 2D lumped-element resonators, which show better sensitivity and sensing volume compared with coplanar waveguide resonators [12–15]

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ESR measurements using superconducting circuits. Interestingly, superconducting flux qubits and superconducting resonators arrive at similar properties, although their starting points are different. With the goal of improving the sensitivity even further, we are developing a capacitively shunted flux qubit in a 3D cavity to increase the qubit’s coherence time. Together with dispersive qubit readout using a Josephson parametric amplifier, we expect to achieve single electron spin detection. This may lead to observation of coherent coupling between the flux qubit and the spin of a single electron, which can be utilized for quantum transducers between microwave and optical frequencies for quantum networking. To take advantage of the high spatial resolution, we are going to develop a sensor array to take ESR images. Bio-materials are an interesting target in this project, because the size of a cell is about 10 µm, which is about the size of the qubit loop. We are trying to detect irons in neurons by using a flux qubit. Although the measurement is performed at mK temperatures, our detector can be used to diagnose lesions. Superconducting flux qubits as magnetic field sensors will present new possibilities for medical diagnosis, materials science, and solid-state physics. Acknowledgements The authors thank Yuichiro Matsuzaki, Kosuke Kakuyanagi, Xiaobo Zhu, William J. Munro, Kae Nemoto and Hiroshi Yamaguchi for valuable discussions and support. This work was supported by CREST (JPMJCR1774), JST.

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Hybrid Quantum Systems with Spins in Diamond Crystals and Superconducting Circuits Yuimaru Kubo

Abstract Hybrid quantum systems with spins and superconducting circuits are discussed. Recent studies have shown that impurity spins in solid crystals, such as diamond or silicon, are very promising for a spin-ensemble quantum memory for microwave photons. Keywords Hybrid quantum systems · Quantum memory · Superconducting circuits · NV centers in diamond · Spin qubits

1 Introduction Spins are one of the most fundamental and natural microscopic quantum systems as a choice for quantum technology resources. Recent studies have shown that impurity spins in solid crystals possess very long decoherence times [1–6]. Moreover, they turn out to be compatible with other quantum systems, composing spin-based hybrid quantum systems [7, 8]. In particular, hybrid quantum systems with superconducting circuits are very promising (Fig. 1). In this chapter, we discuss the spin-superconductor hybrid quantum systems. First, in Sect. 2, the spin systems focused on in this chapter will be discussed. Next, in Sect. 3, the fundamental concepts of superconducting circuits will be discussed. In Sect. 4, a typical example of spin-based hybrid quantum systems with superconducting quantum circuits will be discussed. In particular, a spin ensemble quantum memory for microwave photons will be focused on.

Y. Kubo (B) Science and Technology Group, Okinawa Institute of Science and Technology Graduate University, 1919-1 Tancha, Onna, Okinawa 904-0495, Japan e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 Y. Hirayama et al. (eds.), Hybrid Quantum Systems, Quantum Science and Technology, https://doi.org/10.1007/978-981-16-6679-7_6

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Superconducting qubits e

Spin

u rcond Supe tor a reson

s

cting

Hybrid quantum information device Quantum technology

• Quantum RAM (spins) and processor (supercond. qubits)

• Ultra sensitive electron spin resonance spectrometer

• microwave

optical photon

quantum state transducer

Fig. 1 Concept of the spin-based hybrid quantum systems discussed in this chapter

2 Impurity Spins in Diamond In this section, we discuss the basics of impurity spins in diamond, especially a color center called the nitrogen-vacancy (NV) center. NV centers are also discussed in detail in Chap. 1 and 2.

2.1 NV Centers in Diamond Figure 2a represents a crystallographic structure of an NV center. It consists of a substitutional nitrogen (N) with an adjacent vacancy. In a negatively charged NV center, the electronic ground state is a spin triplet S = 1 with a zero-field splitting D/2π = 2.88 GHz between the spin states m s = 0 and m s = ±1. At sufficiently low magnetic field, the spin quantization axis is along the N-V bond. The NV center’s spin Hamiltonian writes   HN V / = DSz2 − γe BNV S + E Sx2 − S y2 + SAN I + P Iz2 ,

(1)

where γe (−2π×28 MHz/mT) is the gyromagnetic ratio of the electron spin, S (I) is the spin operator of the electron (14 N nuclear) spin S = 1 (I = 1), and E is the straininduced coupling. AN (AN⊥ = −2π × 2.7 MHz, AN = −2π × 2.1 MHz) and P = −2π × 5MHz are the hyperfine interaction tensor and the nuclear quadrupole moment with the 14 N nuclear spin, respectively. The transverse spin operators Sx and S y are coupled to the local electric field and strain, which significantly modify the spin eigen states m S = ±1 at low magnetic fields [9, 10].

Hybrid Quantum Systems with Spins in Diamond Crystals …

(a)

(b)

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mS +1

E -1 N V

I=1

mI

0 -1 +1 0 +1 -1

D

S=1

C

ms=0

0 ±1

Fig. 2 Nitrogen-vacancy (NV) center in diamond. a Schematic of an NV center. b Energy diagram. D and E are the zero-field and strain splittings

Because of its spin-dependent fluorescence and relaxation from the orbital excited state, one can readout the electron spin state of an NV center and polarize to m s = 0 by shining green laser even at room temperature [11, 12]. This optical reset and readout enable one to perform quantum control on the electric and nuclear spins, and also make NV centers extremely attractive for a magnetic sensor [13]. Moreover, NV center spins possess very good coherence [4, 5, 14].

3 Circuit Quantum Electrodynamics Among the solid-state implementations of quantum devices envisioned up to now, the one based on superconducting circuits in the circuit quantum electrodynamics (QED) architecture [15–17] has been best advanced in the quantum computer development race. They are currently the state-of-the-art quantum system for faulttolerant quantum computing [18–20]. They also offer a unique platform to perform various quantum physics and quantum optics experiments in unprecedented parameter regimes. In this section, we summarize the basics of the superconducting qubits and the circuit-QED architecture.

3.1 Superconducting Resonators: Quantum LC Oscillators The simplest passive superconducting quantum circuit widely implemented in the circuit-QED architecture is a resonant circuit consisting of an inductor in parallel with a capacitor, i.e., an LC resonator (Fig. 3). Noting V as the voltage across the capacitor and I as the electric current associated with the voltage, the whole circuit

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(a)

(b)

I

Φ V

+Q L

-Q

ωLC

C

Fig. 3 A resonant inductor-capacitor circuit (LC resonator). a Circuit diagram of an LC resonator. The voltage V and current I at the node connecting the inductor and capacitor are given by V = ˙ b The harmonic potential and energy levels ˙ I = Q. ,

ˆ ˆ and Q, can be quantized [21–23] in terms of two canonical conjugate operators  where the former is the flux through the inductor and the latter the charge stored on the capacitor, respectively. The two variables satisfy the canonical commutation relation   ˆ Qˆ = i , (2) and the Hamiltonian is written as ˆ2  Qˆ 2 + , Hˆ LC = 2C 2L

(3)

which has the same form as of a harmonic oscillator. Therefore, the Hamiltonian can be rewritten as   1 † ˆ HLC = ωLC aˆ aˆ + , (4) 2 √ −1 where ωLC = LC , aˆ † and aˆ are the resonance frequency of the resonator, the creation and annihilation operators,1 respectively. Namely, a quantized LC linear resonator has equally-spaced energy levels separated by ωLC . In the above discussion, we have reduced a complex macroscopic circuit that includes an enormous number of electrons (1010 ) to a system with only two collective degrees of freedom, the charge Q and flux . This simplification looks unrealistic for such a macroscopic electric circuit. However, it is valid under a unique circumstance, i.e., superconductivity. In the superconducting state, all the electrons pair up and form so-called Cooper pairs, which condensate into a special ground state with an identical phase of the macroscopic wave function. The system has a finite excitation gap, the energy needed to break a Cooper pair, in which no excitations except Cooper pairs are allowed to exist. For conventional metal superconductors, the energy gap is typically on the order of ∼K; thus, superconducting electrons are protected by any disturbance on that energy scale.

1

1 Here, aˆ = +i √2Cω

LC

Qˆ +

√

1 ˆ aˆ † , 2LωLC

1 = −i √2Cω

LC

Qˆ +

√

1 ˆ . 2LωLC

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3.2 Superconducting Qubits In the previous subsection, we have seen that a linear quantum system can be engineered by a superconducting LC-circuit. However, in order to implement a quantum bit (qubit), a nonlinear quantum element is required, such that the energy levels are non-uniformly separated. Hopefully, the nonlinear element is non-dissipative. There is an astonishing superconducting element that is very nonlinear, even at the singlephoton level, and non-dissipative. That is a Josephson junction, a superconducting tunnel junction.

3.2.1

Josephson Junctions and SQUID

A schematic of a Josephson tunnel junction is shown in Fig. 4a. It consists of two superconducting electrodes separated by a thin insulating barrier. A supercurrent I J , associated with the tunneling of Cooper-pairs across the insulating layer, can , where Ic the flow according to the Josephson relations I J = Ic sin ϕ and V = ϕ0 ∂ϕ ∂t critical current of the Josephson junction, ϕ0 = /2e is the reduced superconducting flux quantum, and ϕ = ϕ1 − ϕ2 the phase difference of the macroscopic phase of the Cooper-pairs’ wave function. When I J  Ic , the Josephson junction behaves as a point-like inductance L J = ϕ0 /Ic , and the energy associated with the phase difference ϕ across the junction is given by E J (1 − cos ϕ), where E J = Ic ϕ0 is called the Josephson energy. In short, the inductance and energy of a Josephson junction are entirely determined by the critical current Ic , which one can engineer by tuning the junction’s size and the insulating layer thickness. Apart from the inductive perspective, a Josephson junction also intrinsically possesses a capacitive perspective arisen by the junction geometry. Therefore, it also has a charging energy E C = Q 2 /2C, which leads to the Hamiltonian   Qˆ 2 + E J 1 − cos ϕˆ . Hˆ JJ = 2C

(5)

The nonlinearity in the Josephson inductance term is the key to engineering nonlinear quantum systems, as we will see in the following. There is an essential element composed of Josephson junctions, called superconducting quantum interference device (SQUID). It is a superconducting loop interrupted by two Josephson junctions, as drawn in Fig. 4d. Because of the flux quantization, an externally applied flux  through the loop and the phase differences δ1 , δ2 of the two Josephson junctions are linked as δ1 − δ2 = 2π

 0

(6)

Assuming the two Josephson junctions have an identical Ic , the bias current I is given by

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(b)

Superconductor

(c)

Insulator Superconductor

e e

I

(d)

δ1

Φ

200 nm

Cooper pair

(e)

LJ(Φ)

δ2

Fig. 4 The Josephson junction and SQUID (superconducting quantum interference device). a Cartoon of a Josephson junction. b The circuit symbol of a Josephson junction. c Scanning electron micrograph of a Josephson junction used in the experiments in Sect. 4. The two superconducting electrodes made out of evaporated aluminum film are separated by a thin insulating layer of aluminum oxide. d Circuit diagram of a SQUID, and e its equivalence: tunable inductance L J ()

   I = Ic (sin δ1 + sin δ2 ) = 2Ic cos π × sin (δ1 + δ2 ) . 0

(7)

What (7) implies is that a SQUID can be regarded as a Josephson junction whose supercurrent is tunable by flux; that is, a SQUID is a flux-tunable inductance: L J () =

ϕ0 , Ic ()

(8)

  where Ic = 2Ic cos π 0 . This tunability of SQUIDs has enabled to change the frequencies of superconducting qubits [24] or resonators [25, 26]. Moreover, L J () can rapidly be changed in the time scale of ∼ns if one is ever able to implement a means of rapid flux-tuning, e.g., fast on-chip flux lines. Rapid tuning of the Josephson inductance L J enabled to switch the interaction between qubits and a resonator on and off non-adiabatically [18, 27].

3.2.2

The Transmon Qubit

The superconducting qubit most commonly used is the “transmon” type invented by the group of R. Shoelkopf at Yale [28]. Since the invention, transmon qubits keep being improved [29] and developed for the integrated superconducting quantum computers [30]. This qubit is an advanced version of the Cooper-Pair-Box, or the so-called charge qubit, invented by the group at CEA-Saclay. The charge qubit is the first solid-state qubit that demonstrated a coherent superposition of states by the group of NEC [24]. Charge qubits are extremely sensitive to charge noise in the

Hybrid Quantum Systems with Spins in Diamond Crystals … (a)

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(b)

LJ

C

e

ωge g

Fig. 5 The transmon qubit. a Circuit diagram of a transmon qubit, a Josephson inductance L J in parallel with a capacitor C. b The non-equally spaced energy levels of a transmon qubit in an unharmonic cosine potential

environment, which is extremely challenging to eliminate. As a result, the coherence times of charge qubits cannot be made to be better than ∼1 µs. Transmon qubits mitigate the impact from the charge noise by shunting the nonlinear Josephson inductance by a large capacitance (Fig. 5a) such that the Josephson energy is much bigger than the charging energy, E J  E C = Q 2 /2C (c.f. E J  E C for charge qubits). In other words, one can regard transmon qubits as an anharmonic LC-resonator with a non-linear Josephson inductance, as shown in Fig. 5. The low√ est two states |g , |e have a transition frequency ωge ≈ 8E J E C [28], which is different from that of between other states. Therefore, operating at sufficiently low temperature, let’s say at 10 mK in a dilution refrigerator, enables transmon qubits to be polarized in the lowest energy state |g and to selectively operate as an effective two-level system. Most importantly, neither the charge Qˆ nor the phase ϕˆ is well defined in transmon qubits; thus qubit state readout cannot be performed by simply measuring either of them. Instead, in order to perform qubit readout, one needs a resonator coupled to the transmon qubit. Qubit state can be deduced through a resonator in the so-called dispersive (off-resonant) regime, where the qubit and resonator frequencies are detuned much more than the interaction g. This is one of the fundamental concepts of the cavity-QED (quantum electrodynamics) [31, 32], which we will see in the next 3.2.3. See also Chap. 12.

3.2.3

Circuit-QED: Qubit Coupled to a Resonator

Here we focus on only the Hamiltonian and interaction terms that play essential roles in the circuit-QED. The details of cavity-QED can also be found in Chap. 12. The most fundamental and essential concept in the cavity QED is the Jaynes-Cummings Hamiltonian. In the circuit-QED case, we can write down it as     ωge 1 + σz + g aˆ σˆ + + aˆ † σˆ − , Hˆ / = ωLC aˆ † aˆ + 2 2

(9)

where σˆ z , σˆ +(−) are the Pauli spin operator, and raising (lowering) operators of the qubit, respectively. The third term is the qubit-oscillator interaction Hamiltonian

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in the rotating wave approximation,2 which represents the exchange of a single microwave photon from the LC resonator to the qubit, and vice versa, with a rate g. Importantly, in order to make this exchange of a single microwave photon possible, g must overwhelm other decays, such as the decay rates of the resonator and the qubit. The rate g is called coupling constant and defined as the product of the vacuum fluctuation of the resonator field and the dipole matrix element associated with the transition between |g and |e [23, 28, 32, 33]. The circuit-QED architecture has played an essential role in the superconducting circuit-based quantum information research, which led to the integrated quantum computer Research and Development.

4 Spin Ensemble Quantum Memories for Microwave Photons As we have seen in Sect. 3, the circuit-QED architecture has been proved to be one of the essential building blocks of superconducting quantum computers [18, 19, 34]. Moreover, its fundamental idea, namely coupling a linear quantum system (resonator) to non-linear ones (qubits), has also been extended to other quantum degrees of freedom. In the past decade, a number of breakthrough demonstrations of hybrid quantum systems based on circuit-QED architecture have been reported. In this section, we focus on a hybrid quantum system with spin ensembles and superconducting quantum circuits.

4.1 Strong Coupling of a Spin Ensemble to a Superconducting Resonator In order to coherently exchange quantum information among different quantum systems, the coupling strength g must exceed any other decays. This sounds particularly challenging for electron and nuclear spins, where the excitations are magnetic dipole transitions. As the magnetic dipole energy is generally much smaller than that of an electric dipole, g turns to be smaller by the same order of magnitude. For example, when a single electron spin is placed at the magnetic anti-node of a coplanar waveguide resonator (such one in Fig. 6), the magnetic coupling constant is obtained to be g ∼ 1−10 Hz, which is far below the reach of the strong coupling regime.

     Originally, the coupling Hamiltonian is given by g aˆ † + aˆ σˆ + + σˆ − = g aˆ σˆ + + aˆ † σˆ − +   † g aˆ σˆ − + aˆ σˆ + . The latter is a rapidly oscillating term, which can be dropped if the interaction strength g is pertabative, i.e., much smaller than ωLC and ωge .

2

Hybrid Quantum Systems with Spins in Diamond Crystals … ~ cm

127

1 mm ~ 0.

~ μ m

Fig. 6 Schematic of circuit QED. A superconducting transmon qubit is coupled to a onedimensional coplanar waveguide resonator

Coupling an ensemble of N electron spins √ to a resonator can mitigate this problem, as the coupling constant is enhanced by N [35]. The system can be described by the Tavis-Cummings Hamiltonian Hˆ TC / = ωr aˆ † aˆ +



   ω j /2 σˆ z, j + g j aˆ σˆ +, j + aˆ † σˆ −, j , j

(10)

j

where ωr is the resonator frequency, and ω j , σˆ +(−), j , and σˆ z, j are the frequency, raising (lowering), and Pauli operators associated with spin j. In the low excitation regime where the number of excited spins is much smaller than N , the spin operators can be replaced sˆ j . With the “bright” mode operator  operators  by bosonic  2 ˆb = 1 g j sˆ j , where gens = g j , the Hamiltonian is rewritten as gens

Hˆ TC / = ωr aˆ † aˆ +



 ω j sˆ †j sˆ j + gens aˆ bˆ † + aˆ † bˆ .

(11)

j

The Quantronics group at CEA-Saclay demonstrated strong coupling of an ensemble of NV-centers in diamond and a superconducting resonator [36]. Figure 7 shows the experiment. A chunk of diamond crystal is glued on top of a superconducting coplanar waveguide resonator. The diamond contains about 10 ppm of NV-centers, yielding the number of NV center spins coupled to the resonator mode volume N ∼ 1012 . As a result, this makes the ensemble coupling constant gens /2π enhanced to 10 MHz, which exceeds the linewidth of a typical superconducting resonator κ/2π  MHz and that of an inhomogeneously broadened electron spin ensemble /2π ∼ MHz. In this experiment, the resonator frequency ωr is tuned by changing the flux  threading the SQUID loops inserted in the transmission line of the resonator (Fig. 7a). The two polaritonic peaks manifest the strong coupling in the resonator transmission spectroscopy at around 0.3 0 shown as a solid white curve in Fig. 7c. In this experiment, gens ≈ 2π × 11.6 MHz was obtained. Similar experiments with an ensembles of NV centers [37], P1 (substituted nitrogen) centers in diamond [38, 39], chromium ions in sapphire (ruby) [38], rare-earth ions in optical crystals [40, 41], and a magnon mode in a spherical ferrimagnetic insulator YIG (yttrium ion garnet) crystal [42, 43] have been reported. For further details of the measurement setups, see Refs. [36, 44, 45].

128

Y. Kubo (c) 2.95

3 mm

10 µm

(b) Br

GND

2.90

2.85 10-5 2.80

BNV

|S21|2 40 mK BNV = 0.99 mT

Φ (0,0,1)

-32 dB

Transmission |S21| (dB)

Diamond

Frequency ω / 2π (GHz)

(a)

2.75 0.1

0.2

-70 dB 0.3

0.4

Magnetic Flux Φ / Φ

0

Fig. 7 Strong coupling of an NV center spin ensemble and a superconducting resonator. a Photograph of the device (top) and scanning electron microscope image (bottom) of the SQUID array placed in the dashed box on the circuit. b Schematic of the device seen from the top. The SQUID array is inserted in the middle of the transmission line of the coplanar waveguide resonator. A local flux  is applied to the SQUID array by feeding a DC current through an on-chip flux bias line. c Transmission spectra of the resonator as a function of local flux . The new eigenfrequencies of the coupled system are drawn in red solid curves, whereas the original (uncoupled) eigenfrequencies are drawn in yellow-dashed curves

4.2 Coherent Coupling of a Superconducting Qubit and a Spin Ensemble The demonstration of strong coupling between a spin ensemble and a superconducting resonator tells us that a coherent exchange of quantum information between the spin ensemble and a qubit via the resonator would be possible. In other words, the spin ensemble possibly works as a quantum memory of microwave photons prepared by a superconducting qubit. In this subsection, we review one example demonstrated by the Saclay group [46].

4.2.1

Device: The Spin-Superconductor Hybrid Quantum Circuit

Figure 8a is the spin-superconductor hybrid quantum circuit. Similar to Fig. 7a, a chunk of diamond crystal is glued at the center of a superconducting circuit chip. The diamond crystal contains about 2.5 ppm of NV centers (NV, magenta), which are magnetically coupled to the microwave photons in the coplanar waveguide resonator (quantum bus B, orange). The bus resonator’s frequency B is made tunable by a local flux generated by passing a current through an on-chip flux bias line (F). The quantum bus B is also coupled to a superconducting transmon qubit (Q, red). There is another non-linear resonator R, through which the qubit Q is driven and readout.

Hybrid Quantum Systems with Spins in Diamond Crystals … a

129

b

1 mm

Q drive

B drive

F R

|S21|

Readout Pe

Qubit (Q) Diamond (NV) Quantum bus (B)

gQ

gR

R

Q

B

NV g

F

Φ

BNV

Flux bias line (F)

Flux

Readout resonator (R)

Fig. 8 The hybrid quantum circuit. a Photograph (top) and cartoon (bottom) of the quantum memory experiment using the hybrid quantum circuit realised in Ref. [46]. A transmon qubit (red) is coupled to an ensemble of nitrogen-vacancy (NV) centers electron spins (pink) via a frequency tunable quantum bus resonator (orange)

A Josephson junction is inserted at the middle of the resonator R, which makes the resonator non-linear. This non-linearity enables one to perform a single-shot readout of the qubit state with high-fidelity [47] by means of Josephson bifurcation amplification (see also Chap. 5). Qubit state readout is performed by measuring the phase of a reflected microwave pulse on R, which depends on the qubit state [28, 47].

4.2.2

Quantum Memory Demonstration

Spectroscopy of hybrid quantum circuit Transmission |S21 | versus probe frequency ω and local flux  are color-plotted in Fig. 9b. A static magnetic field BNV = 1.1 mT is applied in parallel to the chip surface. The crystallographic orientation of the diamond with respect to BNV is schematically represented in Fig. 9a. One of the four bonding orientations labeled as I (red) is aligned parallel to BNV . Because the transition frequencies of NV centers are sensitive to the projection of BNV to the N-V bond axis, the orientation of BNV makes the ESR frequencies of the group I ω±I are lifted further than those of the other three bonds (group III, blue), ω±III , which are degenerated. As a result, under this experimental configuration, two sets of transition frequencies ω±I and ω±III emerge, as seen in the transmission spectra in Fig. 8a. Analyzing the avoided-level crossings with the NV center transitions at ω±I and ω±III in the spectra using the coupled oscillator model [36, 46], the ensemble coupling constants g±I /2π = 2.9 MHz and g±III /2π = 3.9 MHz are obtained. As seen in the qubit spectra on the bottom right in Fig. 9, the quantum bus resonator B is also strongly coupled to the transmon qubit Q. With the qubit-resonator Jaynes-Cummings Hamiltonian, the coupling constant g Q ≈ 2.3 MHz is obtained.

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Y. Kubo (b) 2.95

(a)

-40

ms = ±1

I

+1

+I +III -III -I

-1 2.88 GHz

ω+ΙΙΙ

ω+Ι

Frequency, ω/2π (GHz)

BNV = 1.1 mT

2.90

ω+III ω-III

2.85

-70

ω-I

2.62

0.5

ωQ

Pe

ω−ΙΙΙ ω−Ι

0

2.57

ms = 0

|S21| (dB)

ω+I III

0.30

0.35

Flux, Φ/Φ0

0.42

Fig. 9 Spectroscopy of the hybrid quantum circuit. a Top: schematic representation of the static magnetic field BNV with respect to the diamond’s crystallographic orientations in the experiment. Since BNV  [1 1 1], the NV centers are divided into two sub-ensembles: the ones that have the N-V bond orientation parallel to BNV (labeled as I in red), and the other degenerated three (III in blue). Bottom: energy level diagram with the four associated transitions under this experimental configuration. b Transmission |S21 | of the bus resonator B is color-plotted as a function of the probe frequency ω and the flux  applied to the SQUID through the flux bias line F. Left insets: zooms on the two avoided crossings, where sub-structures due to the hyperfine interaction with 14 N are visible in-between the two main polaritonic peaks. Bottom right inset: The qubit excited state probability Pe as a function of ω and  is color-plotted

Storage and retrieval of a single microwave photon Figure 9b manifests the quantum bus B is strongly coupled to both the qubit Q and the spin ensemble N V in the hybrid quantum circuit. Therefore, a dynamical quantum-state transfer from Q to N V through B is now possible as discussed below. The pulse sequence used for the experiments is shown in Fig. 10a, where the energies (frequencies) of each quantum system are schematically represented versus time. First, the frequency of the quantum bus resonator ω B is set to lower than those of the qubit Q and the lowest NV center transition ω−I . After a sufficiently long wait time ≈10 µs, much longer than the qubit relaxation time, the resonator relaxation time, and the free-induction-decay time of the NV center ensemble N V , the qubit Q is excited by a microwave π pulse, which brings the qubit from the ground state |g to the excited state |e . Immediately after the π pulse, the bus resonator frequency ω B is swept across ω Q slowly enough such that the quantum state of the qubit | Q = α|g + β|e is adiabatically swapped to the bus resonator states | B = α|0 B + β|1 B , which is called “aSWAP” here. More precisely, this operation makes the system evolve from | Q ⊗ |0 B to |g ⊗ | B . Therefore, with

Hybrid Quantum Systems with Spins in Diamond Crystals …

(a)

NV

π

(b)

ωB

τ

Q B

Excited state probability, Pe

Fig. 10 Single microwave photon swap demonstration. a Pulse sequence. Energies of each quantum system are schematically represented as a function of time. b Single microwave photon swap oscillations. Qubit excited state probability Pe versus the interaction time τ is plotted. The single microwave photon is fully transferred into the NV center ensemble at the time τs = 97 ns, which comes back at τr = 146 ns

131

R

aSWAP

ω-I 0.4

τs 0.2

0

τr

200

400

600

Interaction time, τ (ns)

the qubit Q in the excited state |e , the single quantum of excitation is transferred from the qubit Q to the bus resonator B by the aSWAP; namely, B is in the single-photon Fock state |1 B .3 The bus resonator frequency ω B is then non-adiabatically brought to resonance in the lowest NV center ensemble ω−I for an interaction time τ , which induces single microwave photon swap oscillations, or vacuum Rabi oscillations, between the bus resonator B and the NV spin ensemble N V . At the end of the swap oscillations after time τ , the bus resonator frequency is brought back close to the qubit frequency ω Q . Then another aSWAP operation is performed such that the state of the bus resonator | B at the end would be swapped back into the qubit state | Q . Finally, the qubit excited state probability Pe was measured by repeating the same pulse sequence at each τ by twenty thousand times. Figure 10b displays the result Pe (τ ). A few cycles of oscillation of Pe are observed, revealing a storage in the spin ensemble of the single quantum of excitation initially in the qubit at τ S = 97 ns, and a retrieval back into the qubit at τr = 146 ns. Similar results are also obtained with the other NV center ensembles, at ω+I and ω±I I I , with the period of oscillations depending on the coupling g to the bus resonator. The fidelity defined by Pe (τr )/Pe (0) was obtained to be 0.07 (0.14) for the group ±I (±III). These low values are not mainly due to the short spin dephasing times, but rather to an interference effect caused by the hyperfine interactions with the nuclear spin of 14 N [46]. Indeed, sub-gap structures arising from the hyperfine interaction are seen in the spectroscopy in the inset of The quantum SWAP gate operation could also be realized by non-adiabatically tuning ω B in resonance with ω Q for a duration π/2g Q . In this experiment, aSWAP was chosen to the resonant as it is more immune to flux noise in the SQUID loop of the tunable bus resonator B [45, 46].

3

132

Y. Kubo

Fig. 9b. A similar demonstration with a flux qubit has also been realized by the NTT group [48]. For further details of the measurement setups, see Refs. [45, 46]. Storage and retrieval of a coherent superposition of state In order to fully demonstrate quantum memory operation out of the NV center ensemble, one has to test if an arbitrary quantum state, i.e., a coherent superposition of states, can be transferred to the spin ensemble and retrieved. To this end, instead of |e , the qubit is now prepared in √12 (|g + |e ) by a microwave π/2 pulse. The coherent superposition of states is then swapped to the bus resonator B by applying the aSWAP gate. The B state is now in √12 (|0 B + |1 B ), which was non-adiabatically brought in resonance with the ensemble −I at ω−I . Similarly to the single-photon swap, the resonator-spin ensemble system is then let evolve during a time τ , followed by another non-adiabatic change of ω B and aSWAP gate, which swaps the state | B (τ ) back into | Q . At the end of the sequence, the qubit’s Bloch vector was reconstructed by performing a quantum state tomography to test if the phase of the initial superposition of state is well preserved and retrieved. The quantum state tomography consists of applying another rotation by a π/2 pulse on the qubit about either x axis (X ), y-axis (Y ), or no rotation (I ). This enables one to measure the mean values of Pauli matrices σ X , σY , or σ Z of the qubit, which are converted to the excited state probability Pe . The obtained off-diagonal matrix element ρge is plotted as a function of the interaction time τ in Fig. 11c. No coherence is left in the qubit at τ = τs , as expected for full storage of the initial state into the ensemble. Then, coherence is retrieved at τ = τr , although with an amplitude ∼4 times smaller than that at τ = 0 (i.e., without interacting with the NV centers). The results of Fig. 11b and c demonstrate that a superposition of the two qubit states (α|g + β|e ) can be stored and retrieved in a spin ensemble, although with limited fidelity, and thus represent a proof-of-concept of a spin ensemble quantum memory for superconducting qubits.4

4.3 Towards a Spin-Ensemble Quantum RAM What was discussed in the previous section is the storage protocol (the “write step”) in the quantum memory operation based on a spin ensemble [50]. For a full quantum RAM operation, one also needs to retrieve the stored information at will (the “read step” ). This section reviews the multimode storage and retrieval of weak coherent microwave signals [50–52], which are inspired by the optical quantum memory

4

A similar demonstration with a superconducting flux qubit has also been reported [49].

Hybrid Quantum Systems with Spins in Diamond Crystals …

(a)

NV

X(π/2)

Q B

aSWAP

0.4

Q

(b) Coherence 2|ρge|

ωB

τ

133

I, X,Y

R

NV

+

0 +1

τS

0.2

0

100

200

300

Interaction time,τ (ns) Fig. 11 Quantum memory demonstration. a Pulse sequence. In the same way as Fig. 10a, energies of each quantum system are schematically represented as a function of time. √ b Swap oscillations of a coherent superposition of quantum states. A superposition (|g + |e )/ 2, initially prepared in the qubit (Q), is sent to the spin ensemble (NV) via the bus resonator B. The coherence (modulus of the off-diagonal matrix element ρge ) is plotted versus the interaction time τ . A few storage (dashed arrows) and retrieval (solid arrows) cycles are seen

proposals [53, 54]. These are the important proof-of-concept demonstrations of the retrieval protocol (the “read step”) towards quantum RAM operations for microwave photons.

4.3.1

Operating Principle of Spin Ensemble Multimode Quantum RAM

As seen in Figs. 10 and 11, even without the interference effect caused by the 14 N nuclear spins, the stored quantum information is damped in the time scale of T2∗ ∼ 100 ns because of the inhomogeneous broadening of the spin ensemble. Here T2∗ is the free-induction decay (FID) time, defined by T2∗ =  −1 , where  is the spins’ inhomogeneous linewidth. However, this does not mean that the stored quantum information is lost, but it rather leaks towards the N − 1 other uncoupled single-excitation states, called the “dark-modes” [44, 55–57]. What’s interesting and appealing is that these N − 1 dark modes are orthogonal to each other. Namely, one can store another quantum information into the same spin ensemble after a time of the order of a few T2∗ . Moreover, this can be repeated as long as the total storage protocol time is shorter than the decoherence time T2 of the spin ensemble. The stored quantum information can then be retrieved by applying a strong refocusing

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pulse (ideally a π-pulse) to the memory spin ensemble [50].5 This quantum memory retrieval process has been experimentally demonstrated [51, 52, 58], convincing that a multimode quantum RAM operation can be implemented with a spin ensemble.

4.3.2

Device for Multimode RAM Operation

One obstacle to implementing such a multimode quantum RAM operation would be the long spin relaxation times at millikelvin temperatures, which can be as long as ∼10 h [59]. Indeed, in the quantum memory operation, strong refocusing pulses need to be periodically applied to the memory spin ensemble that has always to be polarized into a particular quantum state at the beginning. Therefore, after a strong refocusing pulse is applied to the spin ensemble quantum memory, which is inverted or saturated then, one needs to actively reset it in a reasonable time scale. Fortunately, as seen in Sect. 2, NV centers’ electron spins can be optically initialized into |m s = 0 by a green laser, which is implemented in the device used in the experiment below. Figure 12a is a schematic of the device that consists of a planar superconducting lumped-element LC microwave resonator, with a frequency ωr = 2.88 GHz, on which a chunk of diamond crystal containing about 2 ppm of NV centers is placed. The NV centers close to the inductor L are coupled to the resonator. Microwave tones are sent to the resonator through a coupling capacitor Cc , which was designed to yield a quality factor Q = 80. An optical fiber is glued on the top surface of the diamond right above the inductor L so that the coupled NV centers can be optically repumped into |m s = 0 . With this device, the NV centers can be repumped into |m s = 0 with an efficiency of 90 % by a green laser pulse of 1.5 mW for 1 s [45, 51]. The drawback of this optical reset is an elevation of the cryostat temperature from 10 up to 400 mK.

4.3.3

Multimode Storage and Retrieval of Weak Microwave Fields

Figure 12c and d shows the pulse sequence of the multimode storage and retrieval experiment and the results, respectively. After resetting the NV center ensemble by a green laser pulse, a train of six storage microwave pulses (of ∼104 microwave photons) is stored in the NV ensemble. A strong refocusing pulse R (∼109 photons) 6 is then applied 10 µs after the sixth pulse to excite the entire spin ensemble, which then starts refocusing from the equivalent waiting time of 10 µs after R and emitting In the real quantum RAM protocol [53, 54], two π-pulses are used during the refocusing process in order to avoid echo emission on top of excess noise generated by the inverted spin ensemble, which is excited by the first π-pulse, combining with a rapid detuning of either the resonator or the spin resonance frequencies such that the spin echo is “silenced”. 6 Note that the refocusing pulse R is not a well-defined π-pulse for the entire spin ensemble due to the inhomogeneity of the spacial microwave magnetic field mode generated by the resonator; the spins experience different Rabi frequencies for a given input power depending on their position with respect to the meandering inductor, which reduces the echo amplitude. 5

Hybrid Quantum Systems with Spins in Diamond Crystals … Green laser (c)

(a)

Cc

Laser reset

C

L gens Spins memory

RETRIEVAL

R

θ6

e2 e1

e6

Time

Amplitude (V)

L

θ1 θ2 2

(d)

IQ quadratures (V)

C

Storage, Refocus

STORAGE

Laser reset

Cc

(b)

135

θi

1 R 0

ei x 5

−π/4

θ1 θ2 θ4

1

ei x 10 4

2 1

0 6 5

-1

θ3 θ5θ6 π/4

0

10

3

R

20 Time (μs)

30

40

Fig. 12 Multimode storage and retrieval of coherent microwave photons in a spin ensemble with a refocusing protocol. a Cartoon of the device. A chunk of diamond crystal is glued on a planar superconducting lumped-element LC resonator consisting of a meandering inductor L and an inter-digitated capacitor C in parallel. Green laser pulses are sent through an optical fiber, glued on top of L. b Diagram of the experiment. The NV center ensemble close to L is coupled to the superconducting resonator. c Pulse sequence. Before starting each microwave pulse sequence, the NV center ensemble is reset to |m s = 0 by a green laser pulse. After the reset, a train of six microwave pulses θi , of about 104 photons in the resonator, are sent and stored into the spin ensemble, followed by a strong refocusing pulse R (∼109 photons) that triggers the retrieval as spin echoes ei in reverse order. d Retrieved echoes. Upper panel: amplitude of reflected microwave signal. Six spin echoes ei (enlarged vertically by a factor of 5) are retrieved in reverse order. Although R appears to be comparable with the storage pulses in this graph, in reality, it is truncated due to saturation of the cold microwave amplifier placed at 4 K. Lower panel: in-phase (I, cyan) and quadrature-phase (Q, magenta) homodyne signal. The phases of the storage pulses θ1 , θ2 , and θ4 are set to be −π/4, whereas those of θ3 , θ5 , and θ6 are π/4

spin echo signals ei in reverse order. After repeating the sequence by 104 times and averaging the signals, six echoes are observed with an amplitude of about five photons as shown in the upper panel of Fig. 12d. The lower panel of Fig. 12d displays that the phases of each storage pulse θi are also retrieved in the corresponding ei , on top of a constant phase offset −π/2 arising from the phase of the local oscillator for the homodyne detection. The obtained efficiency is about 2 × 10−4 , which can be well reproduced by a numerical simulation that takes into account the inhomogeneous broadening of the NV spin ensemble, its 14 N hyperfine structure, the spatial inhomogeneity of microwave field, and the measured Hahn-echo spin decoherence time T2 ≈ 8 µs [51]. The simulation also identifies that the main limitation of the low efficiency is the shorter T2 . Indeed, in the next experiment [52], where a diamond crystal isotopically

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purified to 12 C is used, the retrieval efficiency reaches ∼10−3 after 100 µs. This one order of magnitude improvement on both the storage time and the retrieval efficiency was made possible thanks to the longer decoherence time T2 ≈ 84 µs of the NV center ensemble in the 12 C-purified diamond crystal where the magnetic noise generated by the nuclear spins is suppressed by reducing [13 C] down to 0.03 % (cf. [13 C] = 1.1 % in the natural abundance).

5 Summary and Perspective Hybrid quantum systems with NV center ensembles and superconducting circuits have been discussed. Two important proof-of-concept demonstrations towards a spinensemble quantum memory, i.e., the “write” and “read” steps, have been realized. The advantageous multimode nature of the spin ensemble quantum memory has also been confirmed, although the efficiency is still as low as ∼10−3 . There are a couple of rooms for improvement. First, the microwave magnetic field inhomogeneity can be mitigated either by engineering microwave resonator geometry [60, 61] or by implanting nitrogen ions at well-defined positions in a diamond crystal (followed by a high temperature anneal to creates NV centers). Either means will result in a homogeneous microwave magnetic field over the entire spin ensemble, which will enable one to apply microwave pulses, such as a π or π/2 rotation, with high fidelity. Apart from the resonator geometry, it may also be interesting to combine fast adiabatic passage pulses, which have indeed been successfully implemented to drive donor spins via a superconducting microwave resonator on a silicon substrate [62]. Second, further extension of spin decoherence time T2 would also be possible by reducing the NV center concentration in a diamond crystal isotopically purified to 12 C. Alternatively, other spin species that possess long T2 , such as bismuth [1] or phosphor [63, 64] donor spins in purified 28 Si substrates, or rare-earth ions in an optical crystal [65, 66]. Third, a more efficient way of resetting the quantum memory spin ensemble would eventually be necessary in order to repeat the experimental sequence faster. To this end, cavity-enhanced spin cooling [67–69] may be useful. Another promising direction with the spin-superconductor hybrid quantum system is the ultra-sensitive electron spin resonance spectroscopy. Recent experiments show that exploiting the technologies developed in the superconducting quantum circuits research leads to improving the sensitivity of electron spin resonance, i.e., using Josephson parametric amplifiers [70–74], a Josephson bifurcation amplifier [75], SQUIDs [76], or qubits [77, 78]. Besides, the spin-based quantum transducer [79– 82], which converts microwave and optical photons bidirectionally, would also be a challenging but essential objective for future quantum networks.

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High-Temperature Spin Qubit in Silicon Tunnel Field-Effect Transistors Keiji Ono

Abstract We have improved the operating temperature of silicon qubits by adopting deep silicon impurities and tunnel field-effect transistor. The quantum interference effect of a single qubit was observed by high-speed modulation of qubit energy by device gate voltage. Keywords Silicon · Spin · Qubit · Deep impurity · Tunnel field-effect transistor

1 Background and Core Technology Qubits are expected to be useful not only as building blocks for quantum computers but also for sensing and other related applications. The construction of qubits using either superconductor circuits, atoms, and ions has been studied thus far. Electron spin based qubits making use of a single localized electron spin in crystalline silicon is another one of these construction methods. Silicon qubits are highly compatible with existing silicon technologies such as those used in the microfabrication industry, and can thus be expected to coexist with established silicon integrated circuits. This makes it possible for vast amounts of knowledge accrued in the fabrication of the existing technologies, relating to materials, design, processing, inspection, etc. to be leveraged for those based on Silicon qubits. Good quantum coherence would then be realized by isotope control materials. Previous research on silicon qubits has been conducted with greater emphasis placed on the improvement of operation fidelity and large-scale integration based on qubit-to-qubit coupling. However, the operating temperature for all these remains a low temperature of 0.1 K (K) or less. In order to open up new possibilities of silicon qubits, we overlooked the above important issues, and focused on improving the operating temperature of qubits as priority. If a silicon qubit capable of operating at room temperature can be produced, it can be mass-produced with existing silicon technology at low cost, and a new applications fused with conventional silicon K. Ono (B) Advanced Device Laboratory, RIKEN, 2-1 Hirosawa, Wako 351-0198, Saitama, Japan e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 Y. Hirayama et al. (eds.), Hybrid Quantum Systems, Quantum Science and Technology, https://doi.org/10.1007/978-981-16-6679-7_7

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electronics can be expected. Even if the operating temperature does not reach room temperature, but some moderate temperature between 0.1 K and room temperature, it will still be a useful qubit because it can be evaluated with a cooling device that is small, inexpensive, and easily replaceable. Such an outcome could help accelerate research and development by virtue of saving space and time. A silicon qubit uses the spin state of localized electrons in silicon. Previous studies have used electrons confined in quantum dot structures or electrons trapped in shallow impurities, such as phosphorus, as localized electron spins [1–10]. The quantum dot device has a structure in which source and drain electrodes are attached to a tiny space-dot-through a tunnel barrier, and various properties of electrons in the dot can be examined by electric conduction characteristics. However, the confinement energy of dots produced by general microfabrication technology (about several tens of nanometers) is small, and its operating temperature is as low as 0.1 K or less. Qubits operating at high temperatures require more strongly localized electrons that are immune to thermal noise. To attain such strongly bound electrons, we used deep impurity electrons in silicon [11–13]. Compared with conventional shallow impurities that determine the polarity of N-type and P-type, such as phosphorus and boron, deep impurity create their impurity levels deep in the band gap (Fig. 1). Deep impurities can be introduced into silicon by conventional silicon technology such as ion implantation. In addition to deep impurities composed of a single element, an impurity pair formed by a combination of aluminum and nitrogen impurities also forms deep levels. As described in later sections of this paper, this aluminum-nitrogen impurity pair was used as a deep impurity for this study. A tunnel field-effect transistor (TFET) device structure was adopted as a method for electrical access to deep impurity levels. A tunnel field effect transistor (TFET) includes an N-type source electrode and a P-type drain electrode, and can be regarded as a gate-tunable PIN structure. It is attracting attention as a next-generation low-power-consumption device as it can switch more rapidly than conventional metal–oxide–semiconductor field-effect transistors (MOSFETs) [14–17]. As will be Fig. 1 Schematic of Shallow and deep impurity levels in Si

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described later, when deep impurities are introduced into the I layer of this PIN structure, that is, the channel of the TFET, a tunnel current can flow between the source and drain electrodes via the deep impurity level, if the TFET has a sufficiently short channel length. In such a device, single electron tunneling transport through deep impurities can be expected. Single electron tunneling is characteristic in tunnel conduction between a source and a drain through localized levels (impurities source electrode → localized level → drain electrode). Since the impurity level is strongly localized, a strong Coulomb repulsive force acts between two or more electrons passing through the impurity level to suppress electrical conduction. By adjusting the gate voltage, this Coulomb repulsion can be overcome. However, only a single electron can pass the impurity at one time as two or more electrons cannot enter the impurity. This phenomenon only occurs at temperatures below the Coulomb repulsive energy (Fig. 2a–c).

Fig. 2 a Schematic single quantum dot device and b its schematic potential energy landscape. Single electron tunneling is blocked due to strong Coulomb repulsion in the dot. Coulomb repulsion prohibits the electron number N to change from, e.g., 1–2. c This Coulomb blockade is lifted for an adequate source/drain voltage or appropriate gate voltage where the Coulomb repulsion is overcome by the change of the dot potential energy. d Schematic of double quantum dot device and e its schematic potential energy landscape for double dot in spin blockade condition, where the spin up electron in dot 1 cannot move to dot 2 because another spin up electron already occupies the dot 2, and the dot 2 only accepts a spin down (not up) electron due to Pauli exclusion. f This spin blockade is lifted if the spin up electron in dot 1 is flopped to spin down by means of electron spin resonance

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Spin blockade often occurs in tunneling transport between source and drain via two localized levels [18]. Depending on a specific condition, when a tunnel current flows through two impurity levels between the source and drain electrodes (source electrode → impurity level 1 → impurity level 2 → drain electrode), a spin dependent tunneling appears. The spin dependence is due to Pauli’s exclusion principle. When the spin state of each impurity is the same, the tunnel between the impurities is blocked and the current between the source and drain electrodes is suppressed. When either spin state is changed, this blockade is lifted and a current flows (Fig. 2d–f). Since it can be realized with a relatively simple device structure, it is one of the standard methods for reading out spin qubits. It works even under thermal energy much larger than the energy difference between the two-level systems that form qubits, making it suitable for reading qubits at high temperatures. The existence of deep impurities and defect levels such as dangling bond defects has been of interest from the viewpoint of MOSFET switching stability [19]. These tasks can also be viewed as studies accessing many deep impurities using PIN or PN structures. For example, the current flowing from the electrode of the MOSFET to the substrate includes a tunneling current via a dangling bond defect at the MOS interface, and the electron spin of the defect is detected by electrical detection magnetic resonance [20–26]. In addition, it has been reported that switching characteristics are improved by introducing many deep impurities into silicon TFETs7-9. It should be stressed that both these tasks were conducted at room temperature, which suggests that electrical access to deep impurities and their spins is possible even at room conditions. Thus, the key issue is how to access a single deep impurity.

2 Electrically Accessing a Deep Impurity In a fine MOSFET with a channel length of several tens of nanometers, the transistor on-current does not flow when the device gate voltage is set near the threshold (immediately before the transistor is turned on). However, if there is only one shallow impurity in the channel, the situation changes, and electrical conduction occurs with tunneling transport, i.e., source → impurity level → drain via the shallow impurity [27–31]. This is the same as the above-described tunneling transport in the quantum dot device. If the shallow impurities are replaced with deep impurities, one may expect that high temperature operation may be possible, but this is not the case. In order to measure the above-mentioned quantum dot electrical conduction, the impurity level and the Fermi level of the source/drain electrodes need to be closely aligned. Also, with an increase in depth of the impurity level, the tunnel barrier between the electrode and the impurity widens in proportion (Fig. 3d, e). As a result, the tunnel current sufficient for the electric conduction measurement cannot be realized. This problem may be avoided if the channel length of the MOSFET can be reduced extensively, but that is not possible with a MOSFET having a channel length of at least several tens of nanometers.

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Fig. 3 Schematic of a NMOSFET (n-type metal–oxide–semiconductor) field-effect transistor. b PMOSFET, and c TFET (Tunnel FET). d Potential profile in the source/drain direction for NMOS. When there is a deep impurity level deep in the band gap at the channel, the deeper the level, the thicker the tunnel barrier between the deep impurity and the source or drain electrodes, which results in it becoming more difficult to obtain a large enough tunnel current for electrical conduction evaluation. This situation is the same in e PMOS (p-type MOS), but the situation is different in f PIN (p-type intrinsic n-type) structure. g In this system, the thickness of the charge depletion layer (i.e. intensity of the internal electric field) depends on the source/drain voltage and the carrier concentration of the electrode, so that a tunnel current flows via the deep impurity level no matter how deep the level is. h When the charge depletion layer is thin enough, band-to-band tunneling occurs even without impurity

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This problem can be solved by adopting TFET (PIN structure) instead of MOSFET. In a TFET with a short channel length (I-layer of PIN structure), tunneling through deep impurities in the channel is possible if the I-layer thickness is set slightly before an interband tunneling occurs. Unlike MOSFETs, even the deepest impurity level, located in the middle of the band gap, can provide sufficient tunneling current for electrical conduction measurements (Fig. 3f–h). As will be described later, such electric conduction can be realized with a TFET that can be fabricated by general microfabrication techniques such as those with channel length measuring several 10th’s of 1 nm.

3 Device and Measurement Devices based on TFETs are manufactured by almost the same process as conventional MOSFETs. Using a 100 nm thick Si-on-insulator (SOI) wafer, the source and drain electrodes are formed by shallow donor and acceptor ion implantation respectively and activated by high temperature rapid thermal annealing. Furthermore, Al and N ions are implanted over the entire region consisting of the source, channel, and drain, and annealing is performed at a relatively low temperature for a lengthy period of time to form Al-N impurity pairs [32–35]. A high k/metal gate technology is thus formed (Fig. 4). TFET was fabricated using AIST’s 100 mm manufacturing facility. Several types of TFETs were fabricated on each wafer simultaneously, including TFETs with different gate lengths and widths, with gate lengths varying from 10 µm to 60 nm. Long channel TFETs with gate lengths longer than 100 nm have been successfully operated as conventional TFETs, with all transistors working and the fluctuation effectively suppressed. On the other hand, short-channel TFETs with gate lengths shorter than 90 nm exhibited short-channel effects with increased off-current because drain bias strongly affected channel potential. Short channel TFETs with gate lengths shorter than 80 nm behaved similar to quantum dot-like devices described below. The electrical conduction characteristics of these devices were evaluated at temperatures ranging from 1.5 K to 300 K. Drain current I D (or source current I S ) was measured while applying gate voltage V G and source voltage V S (or drain voltage V D ). A magnetic field was applied in parallel to the source-drain current, and an alternating magnetic field was generated by applying a microwave current in the vicinity of the device, which was used for the single electron spin resonance described later. Among devices with various channel lengths from 60 nm to several microns, quantum dot-like electron transport properties were observed.

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Fig. 4 Device overview a Whole wafer photo. b Transmission electron microscope image of crosssectional element of TFET device with schematic of ion-implanted deep impurity, and c Schematic diagram of Al-N impurity pair forming a deep impurity level. d Schematic sectional view of the device fabrication method. After forming N and P electrodes by ion implantation of shallow donor impurities and acceptor impurities, deep impurities (Al-N impurity pairs) are formed in front of the device by ion implantation and low-temperature annealing, and finally a gate electrode is produced and completed [11]

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4 Single Electron Transport at Room Temperature Figure 5c shows the V G dependence of conductance G (differential conductance dI D /dV S at V S = 0) for one such short channel element (channel length 60 nm, referred to as device A). Clear peaks are observed at low temperatures, one of which even remains at room temperature. This indicates that the device functioned as a single electron transistor operating at room temperature [36–42]. dID/dVS (nS) 㻌0.01

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Fig. 5 Characteristics of device A (channel length 60 nm). a dI D /dV S color map at a temperature of 40 K. Coulomb diamond exhibits maximum widths at V G = 0.35 V and −0.55 V, corresponding to single electron charging energies of 0.3 eV and 0.1 eV, respectively. The high dI D /dV S seen in all four corners of the figure is also observed in normal (with large channel length) TFETs, with positive V S due to interband tunneling of the PIN structure and negative V S due to diffusion current. b I D –V S characteristics at V G c V G dependency of G. The peak position shift at high temperature is due to the change in MOS capacitance due to thermally excited carriers d Schematic band diagram of single-dot electron transport with deep impurities [11]

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Figure 5a is a color intensity map of dI D /dV S at a temperature of 40 K. Quantumdot-like conduction characteristics (Coulomb diamond characteristics) with large single-electron charging energy are shown. These characteristics are typical of single quantum dot devices, and the single-electron charging energy estimated from the width of these diamond-type patterns was 0.1–0.3 eV, which is considered adequate for room temperature operation. Interestingly, this intensity map is asymmetric with respect to V s. That is, dI D /dV S in the positive V s range is much larger than that in the negative V s range (Fig. 5b). These features indicate that “quantum dots” are located in the channel of the TFET, that is, since the width of the space charge region of the PIN structure depends on V S , the tunnel coupling between the “quantum dot” and the electrode depends on V S . Therefore, a negative V S will result in weak tunnel coupling. This is a prominent feature of quantum dot conduction in PIN structures. Such asymmetry is not observed in the conventional quantum dot element or quantum dot electric conduction in MOSFET. The room temperature single-electron transport, single-electron charging energy, and V S asymmetry described above, all reveal the deep impurities in the observed “quantum dot” in our device, which are shown schematically in Fig. 5d. It is shown that quantum-dot-like transport through a single deep impurity level is realized. In addition, such a quantum dot-like electric conduction characteristic was not observed in the TFET device without the formation process of the Al-N impurity pair, which suggests that the observed deep levels are Al-N impurity pairs. However, it is noted that only a limited number of deep impurities contribute to the transport. The contributing impurities are located approximately at the midpoint between the source and drain when the absolute value of V G is relatively small. In case the deep impurities are located much closer to the source than the drain, the impurity level becomes significantly lower than the Fermi energy of the electrode and electrons are trapped at the impurity level as a result, unable to move to the electrode. Conversely, when a deep impurity is located near the drain side, the impurity level becomes significantly higher than the Fermi energy of the electrode, and the electrons of the electrode cannot occupy the impurity level. Therefore, either case does not contribute to electron transport (Fig. 5d).

5 Double-Quantum-Dot Transport Transport properties that appear when two “quantum dots” are connected in series between the source and drain [43, 44] have been observed in several other devices, especially those with slightly longer channel lengths. Figure 6a shows the color intensity map of the differential conductance obtained from such an element (referred to as element B). Coulomb diamond is almost closed at V G = 0.25 V, but there is a finite gap (~0.02 V) in terms of source-drain voltage. In addition, as can be clearly seen in the vicinity of (V D , V G ) = (0.1 V, 0.1 V), a zigzag shape is seen in the Coulomb diamond.

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Fig. 6 Characteristics of device B (channel length 70 nm). Unless otherwise noted, all measurement temperatures are 1.5 K. a Linear scale dI S /dV D colormap. b I S –V D characteristic at V G = 0.253 V. The inset shows the I S color map when microwaves with a frequency of 32.7 GHz are irradiated at various intensities near the sharp peak of V D = −0.03 V. c Color plot of I S at (V D , V G ) = (0.055 V, 0.253 V) as a function of applied magnetic field B and irradiation microwave frequency f . The diagonal line in the figure shows the increase in I S due to electron spin resonance (ESR). Similar ESR was observed in the enclosed area marked by the dotted line in (a). d Temperature dependence of the ESR peak observed at a magnetic field of 1.255 T at (V D , V G ) = (0.06 V, 0.25 V). The measured curve is displayed without an artificial base current offset. Therefore, the base current increases as the temperature increases. e Schematic band diagram of double-dot electron transport including deep impurities and acceptor shallow impurities [11]

Sharp current peaks were observed in the I S –V D characteristic (Fig. 6b). Furthermore, when microwave voltage was applied to the device, the current peak showed a clear Landau-Zener-Stückelberg-Majorana (LZSM) interference pattern (Fig. 6b inset) [45]. The LZSM interference are observed at frequencies f ranging from 10 to 70 GHz. The dependence of the peak height on the power of microwave agrees with the theoretical square Bessel function expression. LZSM interference is observable

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even if the energy of one photon hf , is smaller than the measurement temperature (1.5 K). The periods of the interference peaks measured in V D depends linearly on the frequency between 10 and 70 GHz. Using this linear relation, we estimated the conversion factor between the V D and energy difference between two dots as 0.17 meV/mV (17%). Using this factor, the I S peak width in I S –V D characteristic (Fig. 6b) was converted into energy of 0.06 meV, which is less than that corresponding to the measurement temperature (0.12 meV, corresponds to 1.5 K). This relationship indicates that the electron transport is limited by the lifetime of the single-particle states in the dots and is independent of temperature. It also provides evidence of resonant tunneling in the series-coupled double quantum dot [46]. All of these are observed with double quantum dot devices. The “double quantum dots” observed from the size of Coulomb diamond suggest that they are formed of deep impurities with a confinement energy of 0.2 eV or more, and shallow impurities with a relatively weak confinement energy (~0.01 eV). (Fig. 6e).

6 Spin Blockade and Qubit The spin blockade phenomenon was observed in multiple double quantum dot devices including device B. It is known that the weak leakage current that flows in the spinblocked state is proportional to the spin flip events in the dot. Under single-electron spin resonance (ESR) conditions in a dot, if one of the spins is reversed by ESR in a static magnetic field, the spin blockade is lifted and the leakage current increases [47–51]. Similar ESR responses were observed in all (V G , V D ) regions surrounded by the dotted line in Fig. 6a, strongly suggesting that spin blockade occurred in this region. The ESR response was observed at temperatures up to 12 K (Fig. 6d). This is because the orbital state of shallow impurities is excited at this upper temperature limit, and the Pauli exclusion is no longer effective due to double occupancy of the same orbital state in the shallow impurity. It should be noted that single electron ESR can be observed even at a temperature much higher than the spin Zeeman energy (ESR frequency). This is one of the advantages of spin state readout using spin closure. In contrast to other techniques commonly used in spin qubit readout, single shot readout with spin-to-charge conversion [52] require temperatures well below the Zeeman energy. The leakage current in the spin blockade region limits the mean stay time of electrons in the dot. In device B, a leakage current of about 10 pA was observed, which corresponds to a mean stay time of about 10 ns. This time is consistent with an ESR linewidth of approximately 0.1 GHz at low microwave power. Additionally, similar spin blockade was observed in another device (device C) exhibiting double quantum dot characteristics, although the spin blockade leakage current in this device was about 1 pA, thus corresponding to a mean stay time of about 100 ns. Reflecting this relatively long mean stay time, the width of ESR is as narrow as 4 MHz (Fig. 7a) and is close to the spin coherence time (T 2 *) of natural Si [53].

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Fig. 7 Characteristics of device C (channel length 80 nm). All measurements were made at 1.5 K unless otherwise noted. a Color plot of dI D /dB at (V S , V G ) = (0.33 V, −0.36 V). as a function of applied magnetic field B and irradiation microwave frequency. Two diagonal lines in the figure show the increase in I S due to electron spin resonance (ESR). b ESR line shape observed at magnetic field B = 0.276 T. c Steady state change I D of the drain current as a function of the pulse length of the pulse modulated microwave. Pulse repetition period is 1 µs. The plot is displayed offset. The increase in background current with increasing pulse length is due to the static shift of effective V G caused by microwaves. d Dependence of Rabi frequency on microwave power PMW . e Microwave detuning f dependence of Rabi oscillation at temperatures of 1.5 K, 5 K, and 10 K [11]

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Rabi oscillation of spin qubit can be observed by applying pulse-modulated microwaves instead of the conventional continuous microwaves. The mechanism is as follows. First, two spins (a deep impurity spin and a shallow impurity spin) are set to a parallel state (one of spin triplet states) by spin blockade, which is an initial state. Here, when a microwave pulse with ESR frequency is applied, spin rotation according to the pulse length occurs. As a result, the two spins change to a superposition of parallel (triplet state) and antiparallel (singlet state), and the spin blockade is lifted with a probability proportional to the antiparallel component. By repeating this pulse at regular intervals, spin initialization, rotation, and readout are repeated, and the spin blockade leakage current increases as a time ensemble average. Rabi oscillation is observed as the pulse length increases (Fig. 7b). The amplitude of Rabi oscillation (~0.1 pA) is almost the same as the pulse repetition rate (1 µs). It was also observed that the frequency of the Rabbi oscillation changed in proportion to the square root of the microwave power (Fig. 7c). The color map of Fig. 7d shows the dependence of the Rabi oscillation on the detuning of the microwave frequency from the resonant frequency f , showing well-known patterns used as evidence of the qubit operation. Rabi oscillations were clearly observable at temperatures ranging from 5 to 10 K. Therefore, the operating temperature of the qubit will be about two orders of magnitude higher than that of other silicon qubits (about 0.1 K.) The cause of the invisibility of the Rabi oscillations at 10 K is the decrease in Rabi amplitude and background noise level. This is due to the thermal excitation of the shallow impurity levels described above.

7 Quantum Interference In addition to the above-described coherent operation using pulsed microwaves, there is a method of observing the interference effect of spin qubit. The ESR spectrum of continuous microwaves, as in Fig. 7a, is observed in a wide range of V D , V G , and indicating spin blockade region. Within this region, it is found that the g-factor of the spin depends on V G due to the Stark effect on the localized electron spin [54]. The g-factor can be changed by about 1% by changing V G within the spin blockade region (Fig. 8b). In contrast to general modulation of Zeeman energy by a static magnetic field B, g-factor modulation by V G enables high-speed modulation of the Zeeman energy. Figure 9 shows the measurement and calculated results when square wave modulation (Fig. 9a) is applied to the gate of the device B. When the modulation frequency is much lower than (T 2 *)−1 = 4 MHz, two ESR peaks appear reflecting the two gfactors at each stage of modulation, i.e., high and low of the square waves (Fig. 9d). This is because the time constant of I SD measurement is as slow as about 0.3 s, and the peak height is about 1/2 of that without modulation. As the modulation frequency of this square wave is increased, these two obvious ESR peaks undergo a complex interference pattern (Fig. 9c), and then at a higher modulation frequency (>>(T 2 *)−1 ) it shows a single ESR peak (Fig. 9b). Figure 9e–g presents respective

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Fig. 8 a Measurement setup including square wave modulation of gate voltage. b I SD of the device versus the frequency f for various magnetic field detuning B away from B = 0.2755 T. c I SD of the device versus the frequency f for various gate voltage detuning V G away from V G = −0.36 V. All the upper curves are shifted vertically for clarity [12]

theoretical calculations of the qubit upper-level occupation. Changing modulation amplitude shows similar behavior with similar crossover frequency. Similar pattern was observed for latching modulation of an energy of a superconducting qubit [55]. This indicates that an electron spin modulated faster than its coherent time experiences an external field with an average value of modulation, which is a universal phenomenon known as motional averaging in conventional magnetic resonance. While conventional magnetic resonance experiments have investigated the behavior of the ensemble average of a large number of spins, such as 1012 spins, it can be said that this experiment showed that motional averaging was also observed for a single electron spin.

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1 0.5 MHz

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We explored the motional averaging not only for the symmetric square modulation, but also the asymmetric square modulation, where dwelling in one state is longer than in another. The results were obtained in the same way as when the shape of the square wave is asymmetrical, and are shown in Fig. 10. Here we can see that the electron spin experiences the external field of the weighted average value of g-factor modulation. We repeated similar measurements with various high/low ratios of the square modulations (from 20 to 80%). In Fig. 10h we plotted heights of the two ESR peaks I L , I H , at lowest modulation frequency. For each duty ratio we also plotted distances between above two peak positions and the motional averaged main peak position, f L , f H , at highest modulation frequency (Fig. 9i). Both of the peak heights and distances reflect the duty ratios. The asymmetric square modulation places the qubit in one of the two positions, and the low-modulation frequency characteristics reflect the weighted time spent in those two states, while for high modulation frequency the main ESR peak is situated in-between the two qubit states. The ratio of the peak heights and the frequency distances are plotted in Fig. 10j, and show the motional-averaged main peaks, which indeed appear at the weighted

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Fig. 10 a Waveform of asymmetric square modulation added to the gate voltage. The ratio of the high and low stages of the square wave is fixed at 4: 1 (duty ratio of 20%), and the modulation frequency is changed from 0.25 MHz to 25 MHz. b ESR spectrum at a modulation frequency of 25 MHz. c Modulation frequency dependence of ESR spectrum. In order to emphasize the ESR peak position, the derivative of I SD against f was color plotted. Shift of the peak frequencies, f L , f H are indicated d ESR spectrum at a modulation frequency of 0.25 MHz. Peak heights, I L , I H are indicated. e–g Numerical calculation results corresponding to (b–d). h Plot of I L , I H , i f L , f H , and j their ratio for various different high/low stage of asymmetric square waves [12]

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average frequency. The deviations from linear dependencies in these plots, especially for the duty ratio of 20%, are due to the gate-voltage dependence of the ESR peak height. That is also seen in Fig. 8b. Weighted motional averaging can be regarded as analog calculation of weighted average value with only one qubit. That is, for the main peak position f in the fast modulation, the two peak positions f 1 , f 2 in the slow modulation, and the weight factors W 1 , W 2 , f = W1 f 1 , +W2 f 2 For Fig. 10c, 9.021 = 0.2 · 8.992 + 0.8 · 9.028 was calculated. The calculation procedure is (A) (B) (C) (D)

Set input and weight. Check the input valued are installed correctly by the ESR peak positions and their height at slow modulation. Maintain the modulation waveform and increase the frequency to fast. Interference peak position is output.

Obviously, this is a simple calculation that can be done even with a calculator, but it should be noted that the calculation method is very “quantum” because what is being done in stage (C) are; • Multiple candidates for solution appear due to quantum interference effects, • The probability of a plausible solution increases, These two are common in many well-known and complex quantum algorithms. It is interesting to see a glimpse of the features of quantum computation in this simple system.

8 Other Devices To date, 41 devices with channel lengths suitable for single electron conduction (60, 70, and 80 nm) have been evaluated, 37 of which exhibit single or multiple quantum dot transport, having a large single electron charging energy. At present, only three (devices A, B, and another device D shown in Fig. 11a, b) exhibited single-electron transport at room temperature. Device E (whose characteristics are illustrated in Fig. 11c) operated at temperatures up to 140 K, although the charging energy was as high as those of the three aforementioned devices. At room temperature the single electron transport current in the order of 1 pA has become difficult to see due to the large gate leakage current. The characteristics of three additional devices are provided in Fig. 11d–f as examples of multiple-dot-like transport at 1.6 K.

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Fig. 11 Characteristics of five Al–N-implanted short-channel TFETs. a dI D /dV S intensity map measured at 60 K for device D, which had a channel length of 70 nm. b G–V G curve measured at 300 K for device D. c dI D /dV S intensity map measured at 140 K for device E, which had a channel length of 70 nm. d–f dI D /dV S intensity maps measured at 1.6 K for device F, which had a channel length of 70 nm (d); device G, which had a channel length of 60 nm (e); and device H, which had a channel length of 60 nm (f) [11]

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9 Outlook In the future, we would like to realize silicon qubits that operate at room temperature and are produced with sufficient yield. Spin blockade requires a pair of deep donor and deep acceptor levels. In the work described, device B and C used the pair made of a deep impurity and a shallow impurity, although by introducing a pair in which both are deep impurities, operation at a higher temperature is possible. It is known that there are various deep impurities in silicon, and there is a vast accumulation of knowledge about the level depth and its spin state. Utilizing these assets will be the key to future development. Moreover, if the position of deep impurities will be controlled by the progress of single ion implantation technology [56, 57], further yield improvement can be expected. Even if a silicon qubit that operates at a high temperature with deep impurities can be produced, the current device can only be a single isolated qubit. Although one qubit can be applied to measurement and security, it cannot be a component of a quantum computer. At the end of this article, I would like to comment on the technology required to build a silicon quantum computer with deep impurity qubits. The qubits required for the implementation of the gated quantum computers are required to have high fidelity control/reading out, and large-scale integration enabled by scalable qubit-qubit couplings. The most well-known qubit-qubit coupling method controls the direct exchange interaction between two electron spins with a gate electrode [58], which can be applied to weakly localized shallow impurities, but is not suitable for strongly localized deep impurities. On the other hand, long-range qubit coupling methods such as spin chain [59–64], dipole–dipole interaction [65], or microwave resonator [66, 67] have been proposed. These technologies may be applicable to deep impurities. The qubit readout method employed in this work, based on spinblocked leakage current, is basically a time ensemble measurement of an electron spin qubit, and cannot perform quantum-non-demolition measurement required for gated quantum computers. However, if a deep impurity atom has a non-zero nuclear spin that is coupled to an electron spin by hyperfine interaction, this nuclear spin qubit can be readout quantum non-demolitionally through spin-blocked leakage currents [68, 69]. This is the case even though there are many hurdles that should be overcome. Beyond these problems, there lies the dream of realizing a room-temperature silicon quantum computer. We believe that a future with people carrying a quantum computer in their pockets, like they do smartphones now, is a possibility. Acknowledgements We thank T. Mori, S. Moriyama, S. N. Shevchenko, and F. Nori for fruitful discussion and help.

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Ge/Si Core–Shell Nanowires for Hybrid Quantum Systems Rui Wang, Jian Sun, Russell S. Deacon, and Koji Ishibashi

Abstract Ge/Si core–shell nanowires are an attractive material to control and manipulate spins through the strong spin–orbit interaction for holes which are accumulated in the Ge core. They may find application as a spin-qubit coupled with microwave photons or for engineering of helical states that are an important component for realization of Majorana Fermions in combination with superconductivity. In this chapter, we describe magnetoresistance measurements in gated nanowire devices to study the spin–orbit interaction and the possible signatures of helical states in the nanowire. Toward the quantum mechanical spin-photon coupling, we also describe a charge qubit (double quantum dot) coupled with photons in a superconducting microwave resonator. Keywords Ge/Si nanowire · Spin–orbit interaction · Hole charge or spin qubit · Light–matter interaction

Present Address: R. Wang Department of Physics, Tokyo University of Science, 1–3 Kagurazaka, Shinjuku 162-0825, Tokyo, Japan e-mail: [email protected] Present Address: J. Sun Hunan Key Laboratory of Super Micro-Structure and Ultrafast Process, School of Physics and Electronics, Central South University, Changsha 410083, China e-mail: [email protected] R. Wang · J. Sun · R. S. Deacon · K. Ishibashi (B) Advanced Device Laboratory, RIKEN, Wako 351-0198, Saitama, Japan e-mail: [email protected] R. S. Deacon e-mail: [email protected] R. S. Deacon · K. Ishibashi Center for Emergent Matter Science (CEMS), RIKEN, Wako 351-0198, Saitama, Japan © Springer Nature Singapore Pte Ltd. 2021 Y. Hirayama et al. (eds.), Hybrid Quantum Systems, Quantum Science and Technology, https://doi.org/10.1007/978-981-16-6679-7_8

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1 Introduction Quantum nanostructures where quantum states are coherently manipulated have been attracting much attention in terms of quantum computing and sensing. Regarding the spin-qubit application, the group-IV nature and the state-of-the-art isotopic purification of Si and Ge imply that Ge/Si is a promising platform for spin information devices with long coherence due to the absence of nuclear spin scattering [1]. The spin–orbit interaction (SOI) can play an important role in the manipulation of individual spins because it can mediate interactions between the spin and electric field [2]. With a similar mechanism, the SOI is useful to couple individual spins with a photon in a microwave resonator which works as a quantum bus through which the quantum information is exchanged between distant qubits [3]. The SOI is also essential to realize helical edge states in topological insulators [4]. The InAs and InSb nanowire based artificial helical states coupled with the superconductors are currently being actively studied to search for Majorana zero modes which could be used for topological quantum computation [5–8]. Ge/Si core–shell nanowires are an attractive alternative to these systems, also possessing a large SOI. The Ge/Si core–shell nanowires used in this study form a p-type one-dimensional (1D) heterostructure with a 2–3 nm Si shell surrounding a 10–20 nm wide Ge core (Fig. 1a). The nanowires were epitaxially synthesized by a two-step vapor–liquid– solid method [9] and are dopant free. Under transmission electron microscope (TEM) the Si shell and Ge core lattice and interface are clearly identified as shown in Fig. 1b. An ~0.5 eV valance band offset between Ge and Si makes holes naturally reside within the Ge core, confined as a one-dimensional hole gas (1DHG). The wires exhibit a high mobility with mean free paths up to ~500 nm having been reported [10]. In this chapter, we demonstrate that the Ge/Si core–shell nanowire can be an important building block for quantum devices that make use of the SOI as described above. First, the SOI in the nanowire is described and experimentally demonstrated through study of the weak anti-localization [11]. Then, the possible formation of a helical state in the nanowire geometry is described [12]. Finally the charge-photon interaction is demonstrated using quantum dots embedded in a microwave resonator [13].

2 Evaluation of the Strength of Spin–Orbit Interaction 2.1 Spin–Orbit Interaction in a Ge/Si Nanowire A key property of the Ge/Si nanowire is the predicted strong Rashba type spin– orbit interaction (referred to henceforth in this work as direct Rashba SOI, DRSOI), originating from the combination of quasi-degenerate low energy orbital levels of holes and the strong spin–orbit coupling at the atomic level [14]. The spin splitting

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Fig. 1 Germanium/silicon (Ge/Si) core/shell nanowire. a Schematic drawing and b transmission electron micrographic (TEM) image of a Ge/Si nanowire. Single crystalline structures of Si shell and Ge core are distinguished in b. Carrier momentum k is confined in z-direction along the wire axis. Electric field E (in y-direction) is applied transversely to the wire and the induced effective spin–orbit field BSO is perpendicular to both Ey and k. c Horizontal splitting of the low-energy hole spectrum induced by Ey in the E−k diagram. The spin splitting energy E DRSOI is indicated on different orbital states in c

of the hole band can be easily introduced with a moderate external electrical field breaking the structural inversion symmetry. A direct dipolar coupling to an external transverse electrical field ensures the DRSOI scaling linearly with the core diameter R, and that SOI is predicted to be one or two orders of magnitude larger than the conventional Rashba SOI (RSOI), which is inversely proportional to R. This large and tunable SOI makes the Ge/Si nanowire especially suitable as a platform for spin qubits [15]. Figure 1c shows the numerically calculated hole spectrum as a function of electrical field Ey (with E in the y-direction) with the external magnetic field B = 0,

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according to the theoretical model described in Ref. [14, 15]. Without considering the strain at the core–shell interface, the lowest two orbital levels at k z = 0 are quasidegenerated and depart at large k z (short dash lines), where k z is the carrier wave number along the wire axis. Upon applying a transverse electrical field Ey , the spin degeneracy is lifted with each band splitting into two branches. Like the conventional Rashba SOI, the spin splitting energy, E DRSOI , (the cross point with respect to the bottom of each orbital band) is linearly proportional to the electrical field strength, but is significantly larger in the case of the RSOI. For instance, with a moderate electric field strength of 1 V/µm (which can be easily achieved with electrical gating), the E DRSOI is about 2 meV. An applied magnetic field can lift the time reversal degeneracy. Figure 2 presents the calculated eigenenergy and spin projection (in x, y, z direction) of the lowest

Fig. 2 Splitting of the lowest valence band with a fixed external electric field strength E y = 6 V/µm and a varied external magnetic field B applied to a Ge/Si NW with RGe = 5 nm and a strain induced energy gap  = 20 meV. B S O ∝ E y × k z is linearly proportional to the strength of the applied electric field. The relative orientation angle θbetween external magnetic field B and B S O determines the valance band dispersion relation, Zeeman splitting and spin  state, as compared by the numeric calculations of eigen energies and expected spin projection Sx, y, z with a B = 0, b B = 1 T, θ = 90◦ , c B = 1 T, θ = 45◦ , and d B = 1 T, θ = 0◦ . As E y and B are both perpendicular to the wire, Sz  is always zero throughout

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hole state as a function of a magnetic field B = (Bcos(θ ), Bsin(θ ), 0) applied in the x–y plane with a fixed E y = 6 V/µm. The case for B = 0 is shown as a reference (Fig. 2a) where the spectrum is symmetric in the E–k diagram. The spin projection along the x direction shows an antisymmetric ground/excited state as a function k z , which is consistent with the principle of Rashba type SOI that the effective SOI field  is expressed as B S O, x = α0 E y × k z /gμ B with α 0 the Rashba constant and the spin orientation locked to the momentum. An external magnetic field B perpendicular to BSO will induce a Zeeman splitting (B in y direction of Fig. 2b) and hence lift the Kramer degeneracy. The Zeeman energy gap is linearly proportional to the strength of B. Under this circumstance, the helical state will be observed in the transport measurement as the appearance of a reentrant dip in conductance when the Fermi level enters the Zeeman gap (See Sect. 3 for further details). The position of the re-entrance is determined by the strength of Ey . The spin projection of eigenstates is also affected by the external field B (right panels in Fig. 2). Once the magnetic field B deviates from the y direction, as shown in Fig. 2c with an angle θ to BSO , it leads to an asymmetric spectrum. The Zeeman splitting shows an angle dependent gap as E Z = gμ B B sin θ . The Zeeman gap varnishes when θ = 0 (Fig. 2d). With the asymmetric E–k spectrum the motion of carriers confines in a particular direction at the lowest ground state with a well-defined spin polarization. The linear dependence of E DRSOI on Ey and the angular dependence of E Z on B determines the respective position and depth of the re-entrance in the conductance trace as a function of gate. These crucial criteria for the search of nanowire helical states are extensively discussed in Sect. 3.

2.2 Weak (Anti-)localization The SOI has been observed in various materials with different dimensionality. For instance, its strength has been extracted from the beat pattern of Shubnikovde Haas oscillations [16, 17] and angle-resolved photoemission spectroscopy in two-dimensional systems, or from the avoid crossing of the orbital states in the magnetic field dependent spectroscopy in quantum dots [18, 19]. A typical method to extract the SOI strength in extended diffusive nanowires is to investigate the weakantilocalization (WAL) or weak-localization (WL) through magnetoconductance (MC) measurements [20]. The WL and WAL in electrical transport are interpreted to originate from the quantum interference correction to the classical conductance in a diffusive system. Wavefunctions of a time-reversed pair of closed loop trajectories that are induced by random defect scattering interfere destructively (constructively) according to the presence (absence) of the SOI. An external magnetic field destroys the time-reversal symmetry, and a carrier acquires an additional phase dependence on the path due to the vector potential, which lifts the WAL or WL. Experimentally, with a sweeping magnetic field, an enhanced or suppressed conductance at a zero magnetic field is observed as a signature of WAL or WL, respectively.

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Through analysis of the WL/WAL the strength of SOI has been extracted in various 1D nanostructures, i.e. InAs and InSb nanowires, and etched InGaAs quantum strips [21–23]. Among those studies, the WAL has always been investigated in devices with one global gate where the carrier density and electric field across the NWs are tuned simultaneously. Sweeping the gate voltage will alter several carrier characteristic lengths at the same time, such as mean free path, dephasing length and spin orbit length, hence making the analysis complicated. The Ge/Si nanowire considered in this report exhibits a much narrower diameter (typically ~10–20 nm) than the abovereported group III–V 1D structures, so the conductance is less sensitive to the external magnetic field. To extract a reliable value of the SOI and demonstrate the electrical field tunability of spin–orbit coupling, some practical issues have to be considered in the magneto-transport. 1.

2.

3.

Due to the small cross-section of the Ge/Si nanowire, the specular scattering from the nanowire boundary significantly affects the transport behavior. We adopt a complete theoretical model to consider both the flux and spin precession cancellation, ensuring a reliable Rashba spin–orbit length. Besides the strength, the tunability of the SOI by electrical modulation is of equal importance for controlled spin qubit manipulation and on/off switching of spin transistor or valves. A dual gated device architecture allows the exclusive investigation of the evolution of the spin–orbit length with electrical field while other physical characteristics can be maintained at a constant. The universal conductance fluctuation (UCF) is always superimposed upon other features in the magnetoconductance, and is likely to merge with and obscure the feature of the WAL. By superimposing a small AC signal on the DC gate voltage, the UCF is largely removed from MC traces, enabling a good fitting to WAL theory models.

We quantitatively analyzed the WAL with a one-dimensional model in a “pure” regime, taking into account the nanowire boundary scatterings [22, 24, 25]: ⎡  − 21 − 21  2e2 ⎣ 3 1 4 1 3 1 4 1 1 × + 2+ 2 − + 2+ 2+ 2 G(B) = G ∞ − hL 2 l 2ph 3ls 2 l 2ph 3ls le lB lB ⎤ − 21 − 21   1 1 1 1 1 1 1 ⎦, − + 2 + + 2+ 2 2 l 2ph 2 l 2ph le lB lB (1) l 2B =

C1le lm4 C2 le2 lm2 + , W3 W2

(2)

C3le l 4R . W3

(3)

ls2 =

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Here, G ∞ is the classical conductance without quantum correction. L and W are length and width of the wire with h being the Plank constant, l ph the carrier dephasing length, ls the spin relaxation length, and l R the Rashba spin–orbit length. The magnetic dephasing length l B is obtained according to the Beenakker and van Houten model in Eq. 2 considering flux cancellation, where lm = (/eB)1/2 is a magnetic length and constants C1 = 9.5(4π ) and C2 = 4.8(3) are geometrical coefficients for the case of specular (diffusive) boundary scattering [24]. Equation 3 describes the relation between the spin relaxation length ls and the Rashba spin–orbit length l R , where C3 = 130 is a geometrical constant. This model was first proposed by Kettemann to elucidate the cancellation of WAL when wire width is close to or smaller than the spin-orbit length [25]. The characteristic lengths are also related to the corresponding times: l 2ph,s,B = Dτ ph,s,B , where D = le υ F /2 is the diffusive coefficient, le is mean free path and υ F is Fermi velocity. Parameters τ ph , τs and τ B are the phase decoherence, spin life and magnetic dephasing time, respectively. Before discussing the measurement results it is worthwhile looking into the effect of geometrical confinement on the magneto-transport. Figure 3a, b show the analytically derived magnetoconductance (MC) as a function of B and l R /l ph (or ls /l ph ) using Eq. 1–3, without and with consideration of the cancellation of spin precession,

Fig. 3 Weak-antilocalization in a Ge/Si nanowire without (a) and with (b) the consideration of geometrical confinement on spin relaxation. The evolution of calculated magnetoconductance (MC) as a function of a l S /l ph and b l R /l ph using a one-dimensional model are compared. The NW radius RGe is 10 nm and length L = 1 µm. The mean free path le = 20 nm and l ph = 100 nm are assumed for these calculation

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respectively. Without the inclusion of geometrical confinement effects (W  l R , and assuming ls = l R ), ls should be shorter than l ph to observe the WAL. For instance, when ls ∼ l ph , the crossover of WAL and WL is observed in Fig. 3a. In the case of W ≤ l R the spin relaxation rate will decrease according to the Kettemann model, and the WAL is more difficult to detect. As shown in Fig. 3b when l R ≤ 0.1l ph , a clear WAL is observed as a negative magneto-conductance. It is instructive for us to extract a reliable Rashba spin–orbit length from the experimental results in a Ge/Si nanowire with the typical width of 10–20 nm. Similar suppression of WAL due to boundary scattering has been observed and discussed in InGaAs etched quantum wires and InSb nanowires [22, 23]. To study the validity of the Kettemann model in the nanowire with the set of characteristic length (W, le , l ph , l S , l R ), we refer readers to Ref. [22] for more discussion.

2.3 Dual Gated Device Ge/Si core/shell nanowires were dry transferred onto a target substrate using a homemade mechanical manipulator with micrometer precision [11, 26]. In Fig. 4a a schematic drawing of a dual gated device is shown. Simply speaking, the carrier density and the asymmetric electric field across the transport channel of the nanowire can be independently controlled by the top and back gates (denoted as TG and BG, respectively). Figure 4b shows a false color scanning electron microscopic (SEM) image of a fabricated device with the surrounding circuitry. The magnetic field was applied perpendicular to the NW and substrate surface. The measurements are conducted at a temperature of 1.5 K. To ensure transparent contacts the gate voltages for the contact gates (CG) were always kept negative. For detailed device information, we refer readers to Ref. [11]. To remove the Universal Conductance Fluctuations (UCF) a small AC voltage was superimposed on the gate allowing them to be averaged out. The time domain magnetoconductance (MC), G, was measured by a standard lock-in technique. Each data point in the MC traces is an average over a two-seconds span with gate AC excitations as shown in Fig. 4c. Figure 4d shows the evolution of the MC for a 1 µm long device with increasing VBG, AC /VT G, AC and a swept magnetic field up to ±9 T. The DC gates are set to maintain G(0) ∼ 0.3e2 / h, where G(0) is the conductance at zero magnetic field. With sufficient AC amplitude (usually < 10% of the whole DC scan range) the UCF can be significantly suppressed (blue line in Fig. 4d), ensuring a good fitting with the theoretical models previously discussed.

2.4 Electrical Modulation of Spin–Orbit Interaction The electrical transport is first investigated at B = 0 using a standard lock-in technique with a 20 µV source-drain AC excitation and a frequency of 137 Hz. Only

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Fig. 4 Device architecture and measurement method. a A schematic drawing of a dual electrical gated device. b False color scanning electron micrograph (SEM) with circuit diagram. c Real time signals of TG (red), BG (blue) AC voltages and the measured differential conductance G (black) in a two second span. The recorded G of each data point in the magnetic transport is the average of all signal collected in two seconds. f T G,AC = 10 Hz and f BG,AC = 1Hz. d Comparison of G as a function of magnetic field with different VTG, AC and VBG, AC . By averaging the MC with enough AC gate voltage amplitude, the universal conduction fluctuation (UCF) can be dramatically suppressed. Reprinted with permission from Ref. [11]. Copyright 2017 IOP Publishing

DC voltage is applied onto the gates to independently modulate the carrier density (n T G and n BG ). Upon electrical gating, the net carrier density is the summation of both gating effects, n = n T G + n BG . The difference between the top and bottom gating breaks the structural inversion symmetry. The induced built-in electric field across the wire is derived as, E ≈ (n T G − n BG )/W C Q (W is nanowire width, C Q is the quantum capacitance of the nanowire). The electrical conductance of a 1 µm long wire as a function of VBG, DC and VT G, DC is presented in Fig. 5a. The lever arm of TG and BG (C T G and C BG ) is evaluated as γ = C T G /C BG ∼ 5.5, with C T G ∼ 730 aF and C BG ∼ 130 aF [11]. The magnetoconductance (MC) traces as a function of VT G and VBG are plotted in Fig. 5b. The device could be completely pinched off with a sufficiently large positive gate voltage, showing a typical p-type transport. This is consistent with the observation that with increasing VBG the pinch off voltage is shifted to a more negative VT G . The hole mobility of the device is extracted with Drude’s model, μ = G/ne = dG/d VT G ×(L 2 /C T G ), where μ is the mobility and e is the elementary charge. The mobility is evaluated in a range of 200–500 cm2 V−1 s−1 , which is

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Fig. 5 Characterization of the electrical and magnetic conductance. a Differential conductance G of a 1µm long Ge/Si device as a function of the TG and BG DC voltage offset, where the red and black dots indicate the gate voltage setting with G being kept constant at 0.5 and 0.25e2 / h, respectively. b line cuts of G in a as a function of VTG, DC with VBG, DC varied from −4 to 4 V with 0.5 V per step. UCF is observed on each G trace while the AC gate voltages are not applied. The inset shows the gate dependence of the extracted mobility (black circles) and mean free path (blue circles). The red solid line is the smooth to guide the eye. c MC traces with VTG, DC varied from 0.5 to −1.5 V with 0.2 V per step and VBG, DC fixed at −4 V. Black circles represent the experimental data, red solid lines are the fits using the input parameters l e and μ extracted from the inset of b. Reprinted with permission from Ref. [11]. Copyright 2017 IOP Publishing

consistent with other reports [10]. The mean free path le = ν F τe = m ∗ ν F μ/e is evaluated in a range of 10–20 nm, wherem ∗ = 0.28 m e is the effective mass of heavy holes in the Ge/Si NW,m e is the free electron mass, ν F is the Fermi velocity and τe is the elastic scattering time. The mean free path le ∼ W L indicates a quasi-1D diffusive transport regime in the NW. The mean free path le and a built-in electrical field are used as parameters for the fitting. The measured MC traces in the full carrier density range are presented in Fig. 5c. All the MC traces show a clear WAL feature. By fitting with Eq. 1, we find that carrier dephasing length l ph is around 100 nm and spin relaxation length ls is around 50 nm, implying a strong SOI in the Ge/Si nanowire. For each n, ls is smaller than l ph , as expected as WAL is always observed.

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Considering the geometry confinement in Eq. 3, we extract the Rashba spin–orbit length l R in the range 4–10 nm, almost one order of magnitude shorter than those of InAs and InSb nanowires [21, 22]. The electric field control of the SOI strength was further implemented in the Ge/Si NW devices. By changing the TG and BG voltages with the relation VBG ≈ −γ VT G (as indicated by the dots in Fig. 5a), the conductance was kept constant but the built-in electric field across the wire is varied. The built-in electric field (E) is proportional to an asymmetry of the carrier distribution, E ∝ (n T G − n BG ). In Fig. 6, we present the spin relaxation length ls , Rashba spin–orbit length l R and the Rashba coefficient α as a function of gate voltage (and electric field of E), where α = 2 /m ∗l R . At four different carrier density conditions, a broad peak-like variation in ls is always observed when sweeping VT G and VBG . Fitting with the Kettemann model, the Rashba spin–orbit length l R is about 8 times shorter than ls and a 20– 50% change of l R is achieved through the gating. The maximum values of ls and l R always appear around the positions where the system is most balanced, i.e. where n T G = n BG . Correspondingly, the Rashba SOI coefficient α is smallest when E inside the NW is minimized (Fig. 6c). These results hold the promise that the SOI in the Ge/Si nanowire can be efficiently tuned by electrical field. The Rashba constant is evaluated as α0 = α/eE ≈ 5 nm2 , almost one order of magnitude larger than the conventional bulk Rashba constant ∼ 0.4 nm2 , a straightforward proof for the existence of the peculiar DRSOI in the Ge/Si NWs as theory predicts [14]. The spin splitting energy is tuned in a range E D R S O I = 2 /2m ∗l 2R =1.5–4 meV, implying that the Ge/Si nanowire is a good platform for searching helical state and Majorana Fermions.

Fig. 6 Electrical modulation of spin–orbit interaction. a spin scattering length ls , b spin orbit length (or coherent precession) length l R , and c Rashba coefficient α as a function of the asymmetric gating. In c the equivalent electrical field across the nanowire is plotted on the inset top axis. Reprinted with permission from Ref. [11]. Copyright 2017 IOP Publishing

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3 Detection of Helical Spin State in Ge/Si Core/Shell Nanowire 3.1 Principle We have discussed the helical state in a general picture in Sect. 2. In this section we will focus on the experimental measurements of the helical spin states in the one-dimensional (1D) ballistic systems, e.g. the quantum wires or the quantum point contact (QPC) devices. A typical measurement scheme involves the detection of quantized conductance in the 1D system, which reflects the band dispersion. A gate voltage is applied to control the chemical potential and scan the Fermi energy crossing the energy modes. Summing over all modes, one obtains the conductance quantized at G = n × 2e2 / h. The step size is twice the conductance quantum e2 / h as the spin-up and spin-down modes are degenerate. Strong Rashba type SOI can lift the spin degeneracy in momentum space at zero magnetic field. However in this case the Kramer’s degeneracy is still preserved, which holds the conductance at an integral multiple of 2e2 / h. Hence, the quantized conductance is measured exactly in the same way as the system without SOI. The presence of the substrate and gate imposes an out of plane electric field E, resulting in an in-plane pseudo-magnetic field, i.e. Rashba spin–orbit field, B S O ∝ k × E, ideally oriented perpendicular to the wire axis, where k is momentum (Fig. 7). Applying a magnetic field (B) perpendicular to B S O opens a helical gap of E Z = gμ B B at the band touching points, with g and μ B being the Land´e g-factor and Bohr magneton, respectively. Inside the gap the conductance is e2 / h lower than outside. This gives the distinct transport signature of the helical state in the 1-D system as a re-entrant conductance feature at the n × 2e2 / h conductance plateau. An optimal situation is realized when the helical state and the SOI energy (E S O = m ∗ ×α 2 /22 with m ∗ and α being effective mass and Rashba coefficient, respectively) are comparable. Under such circumstance, the widths of the 2e2 / h plateau and e2 / h re-entrant dip are both sizable and maximally visible [27]. Increasing the external magnetic field linearly enlarges this re-entrant conductance pattern associated with the helical gap.   According to the Hamiltonian of the Rashba SOI, Hˆ R = α (z × k) · σ , where z is a unit vector along electric field E, and σ is the vector of Pauli spin matrices, rotating the applied magnetic field towards B S O by the angle of θ closes the helical gap sinusoidally since the helical gap is correlated to the B component perpendicular to B S O as E z · sinθ . When B is aligned with B S O , the spin mixing vanishes, and a quenched helical gap is expected. Additionally, when rotating B, two subbands are Zeeman split by an additional energy of E z · cosθ . This angle dependency is a unique feature of the SOI and can be used to confirm its origin for the re-entrant conductance feature.

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Fig. 7 Transport characteristics with helical state. a An illustration showing a nanowire device with a Rashba spin–orbit field B S O perpendicular to the wave vector k and the electric field E; Energy dispersion of two lowest spin sub-bands with SOI and the corresponding conductance measurement b without magnetic field, c in a B-field perpendicular to SOI field B S O , d in a B-field with an angle to B S O

3.2 Experimental Considerations In actual experiments the aforementioned re-entrant conductance feature that allows the identification of the helical state is fragile. It is preserved only in almost ballistic 1D wires. Importantly, the materials selected should have relatively large g-factor. Subsequently, the opened helical gap E Z = gμ B B in the finite magnetic field of a few tesla should be in a detectable range of a few meV. Besides the selection of the material, several realistic and technical issues have to be considered during experiments.

3.2.1

Wire

In a 1D system, ballistic transport can only be realized and observed when the mean free path of the carriers is much longer than the channel length. The typical mean free path in the high quality III–V semiconductor nanowires and Ge/Si core/shell nanowires is a few hundreds of nanometers. Rainis and Loss demonstrated theoretically that in a disordered nanowire with a mean free path of ~300 nm, the re-entrant conductance feature can be completely smeared out when the channel length is much longer than the mean free path [27]. When channel length is close to that of the mean free path, the re-entrant feature is recognizable at an extremely low temperature. This

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points out that even with high quality nanowires, a nanodevice with its channel length much lower than few hundreds of nanometers and low measurement temperature of a few kelvins are necessary in order to see clear 1D transport and the re-entrant conductance.

3.2.2

Contacts

Ideally, the metal contacts to the semiconducting channel should be ohmic with sufficient transparency in order to measure the quantized conductance and the helical state induced re-entrant conductance feature. As a circuit in series, the output signal of a quantum wire or a QPC device are the summation of the channel resistance and contact resistance at the metal–semiconductor interfaces. The first conductance plateau of 2e2 / h has a resistance of ~12.9 k . Considering the poor contacts inducing a contact resistance of orders of magnitude larger than this value, the signal of the quantized conductance and the re-entrant conductance feature residing on it are heavily masked by the large background and become less distinguishable. Now we consider a nanowire in a real device, which inevitably hosts unintentional charge disorders and has imperfect contacts. Electrons and holes therefore can be reflected at the metal contacts or at the potential barriers formed at charged disorders. Subsequently, even with the channel length of the same order or shorter than the mean free path, the reflections can lead to a periodic oscillating signal, the so-called Fabry– Pérot (FP) interference, which superposes on the conductance curve as a function of gate voltage. The re-entrant conductance feature associated with the helical gap is strongly masked by the superimposed FP oscillations, therefore easily causing the helical state to be indistinguishable in a realistic measurement [27].

3.2.3

Geometry

Due to the strong screening from the metallic contacts the gate voltage induces a non-uniform chemical potential distribution along the nanowire as shown in Fig. 8a. The two segments of the wire near the contacts are therefore less electrostatically controlled with a so-called onset potential length λ. Due to the onset potential profile the effective channel length L e f f is shorter than the designed channel length L ch , where L e f f = L ch − 2λ. As described by Rainis et al., in order to identify the reentrant conductance of the helical state in the quantized conductance measurement an adiabatic potential profile in the channel must be fulfilled. The realistic onset potential length λ should be close to an optimal value λ∗ = 2v F /E z , with v F = α/ being the Fermi velocity at zero field [27]. For either too small or too large λ masking effects stemming from the different mechanisms previously discussed are significant, making the re-entrant conductance feature difficult or impossible to observe (Fig. 8b).

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Fig. 8 a Schematic of a nanowire with two contacts. An underlying gate is utilized to tune the chemical potential in the center of the wire. The chemical potential varies from the contacts to the center of the wire by an onset potential length of λ. b Simulated conductance behavior in a nanowire as a function of the chemical potential considering the superimposed Fabry–Pérot interferences with varied geometries and gate onset potential length in units of the optimum optimal value λ∗ = 2v F /E z . The potential difference between the contacts and wire is fixed at 0.5 meV. Reprinted with permission from Ref. [27]. Copyright 2014 American Physical Society

3.3 Helical State Studies in III–V Nanowires 3.3.1

GaAs Nanowire

Quay et al. presented the first observation of the hole helical state in the cleaved-edge overgrowth 1D hole wire realized in a carbon-doped AlGaAs/GaAs/AlGaAs hole quantum well [28]. The re-entrant conductance feature as the signature of the helical state was observed on the measured conductance plateau of the second conduction mode (Fig. 9a). The helical gap in the first conduction mode possessed too small an energy scale to be measured in the experiment. They additionally demonstrated the Zeeman expansion of the helical gap with the application of a 9 tesla magnetic field, which suggested an effective g-factor of 2/9 in their system along the wire direction.

3.3.2

InAs Nanowire

Heedt et al. reported the experimental observation of the re-entrant conductance feature in the lowest 1D subband associated with the electron helical state in an indium arsenide (InAs) nanowire [29]. The magnetic field dependence measurements clearly revealed the linear expansion of the helical gap with the application of the magnetic field normal to the SOI pseudo-field (Fig. 9b). Surprisingly, the re-entrant feature was still prominent even in the absence of magnetic field. They explained that the exchange interactions are enough to open the pseudo-helical gap by spin-flipping two-particle backscattering. Consequently, the transport signature of the helical state can be realized from all-electric origins without the application of magnetic field.

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Fig. 9 Signatures of helical state observed in a variety of group III–V semiconductors. a Quantized conductance measurement in the GaAs hole nanowire at 300 mK with the re-entrant feature residing on the second conductance plateau in the magnetic field applied along the wire as shown by inset. b Expanded re-entrant conductance feature measured at 6.1 K in an InAs nanowire in the increased magnetic fields applied perpendicular to the wire as shown by inset. c Conductance measured in the InSb nanowire at ∼ 20 mK as a function of gate voltage and an external magnetic field applied with an angle to B S O as shown by inset. Dashed green lines give the eye guide to the linear expansion of the re-entrant conductance feature. d Conductance measured in the same InSb nanowire device of c in the rotated magnetic field of 3.6 T from B S O by θ as shown by inset. The nonlinear evolution of re-entrant conductance feature is marked by the dashed lines. Reprinted with permission from a Ref. [28]. Copyright 2010 Springer Nature. b Ref. [29] Copyright 2017 Springer Nature. c, d Reproduced from [30] by Kammhuber et al. licensed under CC BY 4.0

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Their measurements lead to the extraction of the spin–orbit energy of 2.4 meV and g-factor of 7 for the hosting material.

3.3.3

InSb Nanowire

Kammhuber et al. detected the electron helical state in the lowest conduction mode of a 1D indium antimonide (InSb) nanowire [30]. The linear expansion of the re-entrant conductance width was observed with the increased magnetic field titled from the SOI field B S O (Fig. 9c). Furthermore, they demonstrated the angular dependence of magnetic field for the first time, where the applied field was rotated with respect to the SOI field by small angles (Fig. 9d). Under such circumstance, the width of the helical gap was controlled following the sinusoidal law. They extracted a spin–orbit energy of 6.5 meV and a g-factor of 38 from the helical state measurements for the InSb nanowire.

3.4 Hole Helical State Detection in Ge/Si Nanowires Ge/Si nanowires have been predicted to have a considerable Landé g-factor of ~4 and strong dipole-Rashba spin–orbit interaction with an energy >1 meV, making the system promising for the study of the helical state of a hole system [14, 31]. However, in Ge/Si nanowires with core diameter of ~10 nm, the hole subband separation is only several meV. Subsequently, detection of the helical state as a re-entrant conductance is more difficult than in the narrow gap III–V nanowires. In the following, the experimental measurement of the re-entrant conductance features associated with helical states are presented for the Ge/Si core/shell hole nanowires [12].

3.4.1

Device

The device used to detect the helical state in the Ge/Si nanowire is shown in Fig. 10a, which consists of a predefined metal gate on a SiO2 /Si substrate, a hexagonal boron nitride (h-BN) flake of ~30 nm thickness as a dielectric layer, a nanowire of 15 nm in diameter, and electrical contacts made from a titanium/palladium stack (0.5 nm/80 nm). The h-BN flake and nanowires were dry-transferred using a mechanical manipulator with a viscoelastic membrane (Gelfilm, Gelpak) in sequence. Electrical contacts were defined using electron beam lithography and evaporation. To ensure the optimized contact condition, a short dip in buffered hydrofluoric acid strips the surface oxide of the nanowire before metal deposition.

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Fig. 10 Signatures of helical state observed in a Ge/Si nanowire. a False color SEM of the Ge/Si device for helical state detection. A Ge/Si NW (red) with Ti/Pd contacts (yellow) is located on a gold bottom gate (orange) with h-BN (blue) as the dielectric layer. Rashba spin–orbit field B S O is perpendicular to k and E. The scale bar is 2 µm. b, c Conductance traces with no B-field application measured at 7.5K without DC bias, and at 12K with 1 mV DC bias, respectively. The red arrows indicate the re-entrant conductance features. Insets show the zoomed-in details of the red boxes. d Voltage bias spectroscopy of conductance measured without magnetic field. The blue dashed lines indicate the perimeter of the helical gap region. The red dashed lines indicate the perimeter of the e2 / h plateau at zero field. Reprinted with permission from Ref. [12]. Copyright 2018 American Chemical Society

3.4.2

Quasi-1D Transport and Re-entrant Conductance

Differential conductance G was measured in the vicinity of the first conduction mode as a function of gate voltage in the device with 280 nm long Ge/Si junction at 7.5 K (Fig. 10b, c). Quantized conductance plateaus at integer multiples of 2e2 / h were observed, indicating a quasi-ballistic 1D transport dominating in the nanowire. The plateau of e2 / h reveals the nanowire possibly possessing a strong electron–electron interaction. More interestingly, on the 2e2 / h plateaus the pronounced conductance re-entrant dips were observed without the application of magnetic field, as indicated

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by the red arrows in Fig. 10b. A similar phenomenon has been reported previously by Heedt et al. in which the feature was referred to as a pseudo-helical gap introduced by the emergence of correlated two-particle backscattering and the Rashba SOI [26]. The conductance measurement at an elevated temperature of 12 K and with a finite DC bias of 1 mV suppresses the conductance fluctuations originating from e.g. Fabry–Pérot oscillations, as shown in Fig. 10c. Moreover, under such circumstances, the zero-field pseudo-gap vanishes giving a clearer presentation of the flat quantized conductance plateaus. Quantized conductance plateaus and the re-entrant conductance can also be observed clearly as “diamonds” in the voltage bias spectroscopy (Fig. 10d). In our studies the quantized conductance and re-entrant conductance features were searched for in various devices with different geometries. The re-entrant conductance features were only found in a few cases, while quantized conductance features can be observed in a number of devices. For instance, severe fluctuations appeared in the conductance of a device with shorter junction lengths of 250 nm, which heavily masked the quantized plateaus. In these devices, the onset potential length is possibly far from the optimal λ∗ , or the contacts may have insufficient transparency. The reentrant conductance feature was also found in a longer junction of 300 nm. Further details maybe found in Ref. [12].

3.4.3

Magnetic Field Dependence

To verify the re-entrant conductance opened in the Ge/Si nanowire by magnetic field as well as its helical nature, the feature should exhibit the two aforementioned magnetic field dependences. Firstly, the re-entrant conductance feature residing on the quantized conductance plateaus is found to expand linearly with increasing magnetic field applied perpendicularly to the SOI field (Fig. 11a), as expected for the helical state associated with SOI. Some weak conductance fluctuations contributed by Fabry–Pérot oscillations or other phenomenon are also found. Fortunately, these can be easily distinguished from the re-entrant conductance feature as they possess negligible magnetic response and present as vertical “strips” in Fig. 11a. The expanded helical gap is also noticed in the voltage bias spectrum by comparing that measured without field in Fig. 10d and measured at 8 T in Fig. 11b. Another important test before confirming the measured re-entrant conductance as a result of the transport via the helical state is the angular B-field dependence. In this experiment, the device was rotated in the magnetic field using the nanowire as the axis. Figure 11c plots the angular dependence of the re-entrant conductance at a constant field strength of B = 6T and rotating from θ = 90◦ to 0◦ . When external B-field is rotated into the substrate plane, i.e. towards the SOI field, the helical gap is decreased. The helical gap energy is extracted and plotted as a function of θ in Fig. 11d. It can be fitted well to the sinusoidal law as expected. Combining the results of both B and θ dependence, one can be convinced by the helical nature of the measured re-entrant conductance feature.

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Fig. 11 Linear and angular dependence of helical energy gap on the external magnetic field. a Quantized conductance of the first conduction mode measured in the Ge/Si nanowire as a function of B⊥B S O . The linear expansion of the re-entrant conductance is marked with blue dashed lines. b voltage bias spectroscopy of conductance measured at 8 T field applied normal to B S O . The blue dashed lines indicate the perimeter of the helical gap region. c Quantized conductance at B = 6 T as a function of tilt angle θ to B S O . The evolution of the re-entrant conductance feature is highlighted using the blue dashed lines. d Helical gap energy (gray dots) extracted from c. Red solid line is the fit using E ∝ E z · |sinθ|. Reprinted with permission from Ref. [12]. Copyright 2018 American Chemical Society

3.4.4

Extraction of Landé g-Factor and SOI

From the linear field dependence of the helical state in a normal magnetic field, the Landé g-factor can be extracted. From the voltage bias spectra in Figs. 10d and 11b, it can be seen that increasing the magnetic field from 0 to 8 T enhances the energy of the helical gap from 1.5 to 3.1 meV as a result of the enhanced E z . Subsequently, this gives a g-factor of ∼3.5. Additionally, the gate lever arm is calculated as ∼3.3 meV/V from the voltage bias spectra. Hence, a similar g-factor of ∼3.6 and a pseudo-helical gap E s ≈ 1.5 meV can also be evaluated from the magnetic field dependence measurement in Fig. 11a. The evaluated g-factor from the transport through the helical state is larger than previously reported g ≈ 2 extracted in

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the strongly confined quantum dots, but resembles the theoretical prediction for the Ge/Si nanowire in weak electric field of ~1 V/µm [31]. The strength of the SOI, E S O , is roughly equal to the energy between the bottom of the subband to the center of the helical gap at zero magnetic field, as illustrated in Fig. 7b. It is translated to be ~2.1 meV from the gate voltage span of 0.64 V with the gate lever arm of ~3.3 meV/V (see Fig. 11a). The SOI length is calculated as l S O = /2E S O m ∗ ≈ 7.9 nm, with m ∗ = 0.28m 0 being the effective mass of the heavy hole which is predominant in the low-energy regime in the Ge/Si nanowire. This result is consistent with E S O = 1.5–3 meV evaluated from the weak antilocalization measurements introduced in Sect. 2. The Rashba coefficient is, therefore, calculated √ as α =  2E S O /m ∗ ≈ 0.34 eV Å.

4 Toward Spin-Photon Coupling 4.1 Double Quantum Dot Embedded in a Superconducting Cavity The strong and tunable spin–orbit coupling in Ge/Si nanowires has been demonstrated by the investigation of the weak-anticoalition and the helical state in the magneto transport described in the Sects. 2 and 3. In this section, we first focus on the experimental implementation of Ge/Si nanowire quantum dots coupled to a superconducting resonator, which is widely utilized as a sensitive dispersive read-out probe or coherent “quantum information bus” for the scalable quantum processor. With the extracted charge-photon coupling strength gc and the spin–orbit interaction energy E S O , we will estimate the potential spin-photon coupling strength gs with the same setup. The strong coupling of an electron charge or spin with photons in an on-chip cavity has been accomplished recently [32, 33], but has yet to be extensively explored in the hole system. For this reason it is of fundamental interest to explore a hole-photon hybrid system based on Ge/Si nanowires. A typical 50 transmission line resonator is fabricated using a 100 nm MoRe superconducting thin film as shown by the optical micrograph of Fig. 12 (upper panel). Close to each open end of the 1/2−λ microwave frequency resonator a Ge/Si nanowire device was placed bridging the center pin and ground plane in order to maximize the capacitively coupling strength. The magnified SEM images of a nanowire quantum device are shown in the bottom panel of Fig. 12. Each nanowire is lying on a set of dense surface gates using an exfoliated h-BN flake as the gate dielectric layer. The nanowires are dry transferred on top of the h-BN with a homemade micro-manipulator [26]. In the present study, we focused on one nanowire coupled to the resonator, so the nanowire at the other end was always pinched off. All the measurements were conducted in a dilution refrigerator with a base temperature of ~30 mK.

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Fig. 12 Optical and SEM images of a nanowire-resonator hybrid device. Upper: An optical micrograph of a typical superconducting MoRe transmission line resonator with two nanowire quantum dot devices near both open ends. Inset is a carton illustrating a TLS-cavity interaction system. The DC wires are connected to the bonding pads through on-chip LC filters. Bottom: Magnified SEM images of a gated nanowire device on top of exfoliated h-BN substrate, further zoomed nanowire and gates, and the input/output coupler of the resonator, respectively

4.2 Model A double quantum dot (DQD) is defined by applying voltages on the bottom gates. We ignore many body effects and assume a single hole confined in the DQD. Under certain gate conditions the DQD is isolated from the source-drain electrodes and only the inter-dot tunneling is allowed. The hole locates either on the left (L) or right (R) dot. The system can be treated as a charge qubit, which is described by the two-level system (TLS) model. Considering the spin degree of freedom, one has a Hamiltonian in a magnetic field, Hqb,s = 21 (ετz + 2tc τx + Bz σz + Bso σx τz ), where τi and σi are the Pauli operators in the position {L, R} and spin {↑, ↓} space [34]. ε and tc are the energy detuning and tunneling rate between left and right dots. Bz and Bso are the external applied (for Zeeman splitting) and the effective spin–orbit magnetic field, respectively. For simplicity all the parameters are in energy units. The two orbital energy levels of a charge qubit as a function of ε and tc are calculated and shown in Fig. 13a. The excited and ground orbital states are denoted as |+ and |−, respectively, with their eigenenergy, ± 21 ε2 + (2tc )2 . The presence of tc induces a hybridization of positional states |L and |R as an avoid gap in energy space.

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Fig. 13 Simulation of lowest energy level spectrum of various quantum system. a Energy levels of a charge qubit formed in a DQD as a function detuning energy ε and tunnel coupling strength tc . Excited/ground orbital states are denoted as |±. Splitting of orbital states into a four-level system is present with an applied magnetic field of b Bz < 2tc and c Bz > 2tc . The orbitals with spin ↑, ↓ are denoted as |±, ↑ (↓). tc = 0.5, B S O = 0.3, and Bz = 0.8 in b and 1.2 in c are used for simulation. Dash line corresponds to B S O = 0. In a–c the simulation parameters are in energy unit for simplicity. As illustrated by black arrows (|−, ↓ ↔ |−, ↑, etc.) in b and c a spin embedded in a resonator couples to a photon when photon energy ωc is close to Bz . Simply speaking, the spin-photon interaction is realized as a combination of the electric-dipole coupling induced orbital transition (green arrows, |−, ↓ ↔ |+, ↓, corresponding to E 0 ↔ E 2 in b and E 0 ↔ E 1 in c at ε close to zero, respectively) and spin–orbit hybridization between the closest orbital levels (green arrows, |+, ↓↔|−, ↑, corresponding to E 1 ↔ E 2 in both b and c). For a charge qubit coupled to a resonator, the lowest eigenenergy level spectrum as a function of qubit detuning ε is simulated by a Jaynes-Cummings model under the condition d 2tc / h < f c , e 2tc / h = f c , and f 2tc / h > f c . In d–f the simulation parameters are similar to the practical values of the measured device

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An applied magnetic field Bz will lift the spin degeneracy. Figure 13b, c show the numerical calculation of four eigen energy levels E n (n = 0, . . . , 3) in the regime of 2tc > Bz and 2tc < Bz , respectively. The eigen states are represented in the {|−, ↓, |−, ↑, |+, ↓, |+, ↑} basis. The presence of spin–orbit interaction hybridizes the states of |−, ↑ and |+, ↓ when Bz is close to the charge qubit energy ε2 + (2tc )2 and induces spin state mixing. For instance, clear anticrossings of |−, ↑ and |+, ↓ are observed in Fig. 13c at ε = ± Bz2 − (2tc )2 when 2tc < Bz . The states of |−, ↑ and |+, ↓ are hybridized into two levels eigen states at energy of E   basis are denoted  1 and E 2 , the respective     as |E 1  = cos 2 |−, ↑ + sin 2 |+, ↓ and |E 2  = sin 2 |−, ↑ − cos 2 |+, ↓ with 

x . The hybridization opens the way the spin–orbit mixing angle  = arctan 2tcB−B z to realize the spin-photon coupling, especially by utilizing the strong SOI in Ge/Si nanowire. As schematically illustrated by arrows in Fig. 13b, c a spin embedded in a resonator will couple to a photon by repeatedly absorbing/emitting the photon when photon energy ωc ∼ Bz (black arrows). The  to the -type  whole process maps   transitions of |−, ↓ ↔ |+, ↓ ↔ |−, ↑ at  (2tc )2 + ε2 − Bz  ∼ Bx (indicated by green arrows). The electric-charge dipole coupling drives orbital transition of |−, ↓ ↔ |+, ↓ and spin–orbit interaction mixes |+, ↓ ↔ |−, ↑. For instance, with 2tc < Bz in (b) |−, ↑ and |+, ↓ mainly contribute to the respective states of |E 1  and |E 2  and vice versa in (c). The electrical dipole coupling primarily gives rise to the orbital transition of E 0 ↔ E 1 (or E 0 ↔ E 2 ) in (b) (or (c)). In combination of the hybridization of |E 1  and |E 2  (in both (b) and (c)), the spin up/down transition is eventually realized in the manner of spin-photon coupling. Our experiment focuses on a charge qubit coupled to a single mode harmonic oscillator, which is well described by a Jaynes-Cummings (JC) model with Hamiltonian     1 1 + ωqb σ˜ z + ge f f a σ˜ + + a † σ˜ − . (4) H J C = ωc a † a + 2 2 The first and second terms describe the bare resonator and qubit, respectively. The third term indicates the qubit-photon interaction, where ge f f = gc ×2tc / ε2 + (2tc )2 is the effective coupling rate. a † and a are the photon creation and annihilation operators in the resonator. The mean photon number in the resonator is n = a † a. σ˜ +,− are the charge qubit raising and lowering operators in the orbital basis {|+, |−} and the Pauli operator σ˜ z =|++| − |−−|. The qubit energy is E qb = ωqb = ε2 + (2tc )2 , and resonator photon energy is ωc where ωc = 2π f c is the angular frequency. Figure 13d–f show the calculated quantum level spectrum of a closed Jaynes-Cummings system for (d) 2tc / h < f c , (e) 2tc / h = f c , and (f) 2tc / h > f c where external drive and system decay are not considered. The qubit-resonator energy detuning is defined as  = h( f qb − f c ). The simulation parameters are close to the experiment for straightforward comparison (refer to Ref. [13] for more parameter details). A splitting feature is always observed whenever the qubit energy is swept across the photon level (Fig. 13d), indicating the charge-photon coupling. The gap

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is determined by parameter ge f f . The qubit-photon hybridization is encoded √ √ by the superposition states as (|−, 1 − |+, 0)/ 2 and (|−, 1 + |+, 0)/ 2, where the qubit is repeatedly absorbing/emitting the photon trapped in the resonator. The splitting reaches a maximum, equal to 2gc , if f qb = 2tc / h = f c . Once the qubit is largely detuned from the photon level (Fig. 13f), the resonator mode obtains a dispersive shift by as much as ±ge2f f / depending on the qubit state. The dispersive shift of the resonance is utilized to probe the charge state of the DQD throughout the present work.

4.3 Charge Stability in a Double Quantum Dot The charge number in the DQD is defined by the voltages of each finger gate (Fig. 14a), denoted as VS B , VL , VB , VR , and VD B , respectively. The response of resonator transmission is utilized to infer the charge state as the susceptibility of the DQD to microwave photons varies with gating. Figure 14b shows the resonator transmission spectrum when the DQD is in the deep blockade (blue) and on the resonant tunneling condition between left and right dots (red). The lineshape shows a significant variation between these two conditions. The fundamental mode of the resonator is extracted when the DQD is in the Coulomb blockade regime, yielding the central frequency f c = ωc /2π = 5.9667 GHz and a width  f = κ/2π = 1 MHz. The lifetime of trapped photons in the cavity is 1/κ ≈ 160 ns. Using the response of the magnitude and phase of the resonator transmission, the charge stability diagram is presented in Fig. 14c. A clear honeycomb-like pattern is observed, as outlined by the white dotted lines, showing a 3 × 3 charge stability diagram of a DQD in the gate voltage sweeping range [35]. Combined with DC transport measurements [13], the charging energy of each dot is evaluated as E L = 2.6 meV and E R = 2.8 meV. The gates and source/drain capacitive lever-arms are α L = 0.079 eV/V, α R = 0.077 eV/V, and αs = 0.243 eV/V, respectively. The electrochemical potential of each dot considering the mutual capacitive coupling reads μ L(R) = α L(R) VL(R) + βm α R(L) VR(L) , where the cross capacitive  , the mutual capacitance between dots Cm ≈ 20 aF, coupling rate βm = Cm /C L(R)  and C L(R) ≈ 60 aF is the total capacitance of each dot. The energy detuning between left and right dot is determined as ε = μ L − μ R .

4.4 Tunable Charge Dipolar Coupling In the phase plot of Fig. 14c, one may notice that both negative and positive phase shift are observed as highlighted by arrows in the graph. This is a signature of charge qubit-photon energy detuning  ∼ 0. We now concentrate on one inter-dot charge transition line. The evolution of the phase signal for the same range of VL

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Fig. 14 Charge stability of a Ge/Si nanowire double quantum dot probed by the response of resonator transmission. a An image illustrating the formation of a double quantum dot working with different detuning ε conditions. b Comparison of the resonance transmission spectra with ε = 0 (resonant tunneling between dots) and ε  0 (deep Coulomb blockade), corresponding to blue and red dots in c respectively. c Magnitude and phase variation of the transmitted signal as a function of VL and V R . Indices m/n in left panel indicate the respective charge number in the left/right dot. The voltage sweeping range covers a 3 × 3 charge stability diagram. The arrows in the phase plot (right panel) highlight the coexistence of negative and positive signal. VS B = 4.3 V, VB = 7.145 V, VD B = 3.5 V, and Vsd = 0 V. Reprinted with permission from Ref. [13]. Copyright 2019 American Chemical Society

and VR is presented in Fig. 15a with VB altered in steps of 10 mV. VB determines the tunneling rate tc between left and right dots. As VB increases (corresponded to a smaller tunneling rate tc / h), the negative phase shift gradually changes to a positive shift. Line cuts of θ along the ε axis with different VB are compared in Fig. 15b. The lineshapes of the phase shift spectrum show a clear variation as tc reduces. This is attributed to the tunnel rate undergoing an evolution from 2tc / h > f c to 2tc / h < f c as VB increases.

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Fig. 15 Tuning of the charge qubit energy and the qubit-resonator coupling by electrical gating. a Evolution of one inter-dot charge transition line for the same range of VL and V R at zero bias with different middle barrier voltages VB . μ is the mean electrochemical potential and ε is the energy detuning. b Comparison of phase shift θ as a function of  with different V B . c Comparison of experiment and numerical simulation of full transmission spectrum as a function of f d and ε with VB = 7.155 V. Reprinted with permission from Ref. [13]. Copyright 2019 American Chemical Society

To quantitatively interpret the dynamical response of the resonance transmission, a complete quantum model is needed where the drive and leakage of resonator as well as qubit relaxation and decoherence are taken into account in addition to the JC model. By solving the Markovian master equation of Eq. 4, we can numerically calculate the steady state of the qubit and resonator transmission using the quantum toolbox in python, QuTip [36]. With input–output theory [34], the resonator transmission dependence on qubit state can also be analytically solved as: t=

2π ( f c − f d ) − iκ/2 +

−iκc  2 ge f f σ˜ z /(−2π f qb

 , − f d + iγ /2)

(5)

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where κc is the input/output coupling rate of the resonator. Parameter γ = γ1 /2 + γ is the qubit decoherence rate, which is the summation of the energy relaxation rate γ1 and the pure dephasing rate γ . σ˜ z  ≈ −1 is taken as the qubit being kept in the ground state with a weak drive. The phase variation in Fig. 15b is fitted by Eq. (5). The result shows that the inter-dot tunneling rate varies from 2tc / h = 11 GHz > f c to 2tc / h = 3.84 GHz < f c , in good agreement with the experimental observation. The coupling strength gc is in the range 2π × 35–55 MHz, which is close to reports on similar devices in other semiconductor QD systems [32, 33, 37, 38]. However, the decoherence rate of the qubit varies in a range 2π × 4.5–6.5 GHz, which is much larger than the coupling strength. As a result we do not observe the strong chargephoton coupling regime, which requires gc > γ , κ. To further verify the validity of fitting, we perform numerical calculations of the whole system with Eq. 4 using the fitted-out parameters as shown in Fig. 15c. The simulation replicates most of the main features from the measurements. Switching to spin-photon coupling may be a promising way to realize a strong coupling condition as spin is less prone to charge noise. It is of interest to estimate the potential spin-photon coupling via the SOI for the Ge/Si nanowire based on our current setup. When a DQD is in a deep blockade regime, the DQD is reduced to a single QD. The spin-photon coupling strength gs ≈ gc (E Z /E)(l/lso ) ∼ 2π × 2.5 MHz, with a typical single dot energy spacing E = 0.5 meV, Zeeman splitting E z = h f c = 24 µeV, single dot ground state size l = √m∗ E ∼ 20 nm, and a moderate spin-orbit length l S O = 20 nm. If inter-dot detuning ε is zero, the charge dipole becomes larger and we assume a charge qubit energy E qb = 2tc = h × 10 GHz = 40 µeV. The half inter-dot distance L = 50nm. The spin-photon coupling

2 strength is then estimated as gs ≈ 2gc E E Z /E qb (L/lso )η ∼ 2π × 14 MHz, √ −(L/l)2 2 η = s/ 1 − s and s = L|R = e relates to the left and right dot wave function overlap [39]. Recent reports claim the coherence time of a hole spin qubit based on a p-type Ge quantum well reaches a few µs [40], while that of electron spin in isotopic enriched Si quantum dot has been elevated to ~100 µs [1, 41]. Further improvement of the coherence of hole spin qubit in Si or Ge materials is anticipated with isotopic purification. The reasonable large spin-photon coupling strength promises Ge/Si nanowire as an attractive platform for the coherent spin–photon interface.

5 Summary The coherence times of spin qubits based on quantum dots has prolonged by four to five orders of magnitude over the past two decades, approaching the single spin operation fidelity threshold for an error-correction quantum computer. Two qubit logic gates have hence been implemented with high fidelity as evident by various benchmarking methods, bringing the universal quantum computation close to reality. Moreover, recent demonstration of single electron and spin coherently coupled to a superconducting resonator, makes the spin qubit-based quantum processor truly extensible.

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The impressive evolution is always accompanied with the progressive innovation of host semiconductor materials in combination with advanced control methods. The center of attention has been focused on electron based devices for decades, the hole materials are rarely explored but should not necessarily be ignored. We have attempted to investigate a novel p-type group IV Ge/Si core/shell heterostructure nanowire, by studying the peculiarly strong spin–orbit interaction and implementing a nanowire based quantum dot coupled to a superconducting resonator, to present the potential of the Ge/Si nanowire as a platform for spin qubits. The strength of spin– orbit interaction is evaluated as a few milli-eV and can be regulated with modest electrical field as evident from the weak-antilocalization feature observed in magnetotransport measurements. In the presence of strong spin–orbit interaction, the electrical and magnetic field dependence of a helical state is witnessed, indicating that the system may be useful for study of engineered topological states for Majorana Fermions. Finally a charge qubit is established in the Ge/Si nanowire double quantum dot. Coupling the charge qubit to a resonator, the charge dipole coupling strength is extracted close to 100 MHz, and hence the potential spin-photon coupling strength through spin–orbit interaction is estimated around 20 MHz. Acknowledgements The authors gratefully acknowledge Charles M. Lieber and Jun Yao for providing high quality Ge/Si nanowire as well as numerous fruitful discussions with Junsaku Nitta, Peter Stano and Daniel Loss.

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Photonic Quantum Interfaces Among Different Physical Systems Toshiki Kobayashi, Motoki Asano, Rikizo Ikuta, Sahin K. Ozdemir, and Takashi Yamamoto

Abstract Photonics provides an ideal platform for carrying and distributing quantum information because photons are individually addressable; they experience little or no decoherence due to extremely low thermal noise even at room temperature; and they are the fastest information carriers available. As such, photons are perfect candidates for interfacing qubits in different physical systems with different operating frequencies (e.g., photons, ions, atoms, and superconductors). The ability to build such interfaces opens newopportunities to access many different physical systems and build hybrid quantum technologies. This chapter summarizes recent progress on quantum interfaces among diverse media, such as photons, atoms, ions, and mechanical oscillators, which are pivotal technologies for building large-scale distributed quantum computing and communication networks, as well as for understanding fundamental quantum phenomena related with light-matter interactions.

T. Kobayashi · R. Ikuta · T. Yamamoto (B) Graduate School of Engineering Science/QIQB, Osaka University, Osaka, Japan e-mail: [email protected] T. Kobayashi e-mail: [email protected] R. Ikuta e-mail: [email protected] M. Asano NTT Basic Research Laboratories, NTT Corporation, Kanagawa, Japan e-mail: [email protected] S. K. Ozdemir Department of Engineering Science and Mechanics, The Pennsylvania State University, University Park, Pennsylvania, 16802, USA e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 Y. Hirayama et al. (eds.), Hybrid Quantum Systems, Quantum Science and Technology, https://doi.org/10.1007/978-981-16-6679-7_9

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1 Introduction Quantum systems realized with diverse matters [1], such as atoms, ions, semiconductors, superconducting circuits, and so on, are manipulated on their sites via interactions with their neighbors. Their hybridization with photons allows for distributing quantum states and/or processes among spatially separated systems operating at different frequencies through photon transmission. Matter based quantum systems need to be isolated from their environment by well-controlled ultra-low temperature instruments and/or by an ultra-high vacuum chamber. Thus, it is hard to move these systems from their locations and have them communicate with other systems without loss of the quantum properties. On the other hand, photonic quantum systems are robust against environmental effects even at room temperature and photonic quantum states survive and travel over distances up to 100 km as has been demonstrated in many quantum key distribution (QKD) and quantum communication experiments [2, 3]. Thus matter-photon hybrid systems have attracted much interest in many aspects of quantum information processing. Hybrid systems are elemental for large-scale distributed quantum computing and communication architectures [4, 5] connecting matter qubits or small-scale matter qubits arrays. Such distributed architectures not only enable the scale up of quantum computer but also lead to many applications of networked quantum computation, such as distributed quantum computing [6] and blind quantum computing [7, 8]. Apart from quantum computation, networked quantum sensors utilizing matter qubits are expected to enhance clock synchronization [9] and longer-baseline telescopes [10, 11]. Matter-photon hybrid systems are also vital for quantum repeaters [12– 14] proposed for long distance quantum communication to dramatically reduce the photon loss effect. Success and progress in these areas will significantly contribute to efforts towards building quantum internet [15–17], which is an active research direction. Matter-photon hybrid systems are inherently created by interfacing matter and photonic oscillators with distinct eigenfrequencies. Such interfaces are studied within the context of light-matter interactions, and the physics underlying these interactions largely depends on the nature of the matter system. In the majority of the cases, the role of photons and the processes surrounding them can be considered as a scattering process and described by beamsplitter interactions. In this chapter, we will give an overview of photonic quantum interfaces and discuss similarities and differences of various interfaces built towards engineering a distributed quantum network consisting of various matter system. quantum interfaces. This chapter consists of four sections. In Sect. 2, we introduce quantum frequency conversion (QFC), a process that changes the frequency or color of photons while preserving their quantum states. In Sect. 3, we introduce atom-photon entanglement generation via Raman scattering in a cold atomic ensemble. We will see that QFC plays a significant role for entangling atoms with photons of different colors. In Sect. 4, we introduce and discuss the use of optomechanical interactions to create entanglement between photons and phonons (i.e., mechanical oscillators). Finally,

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in Sect. 5, we present challenges and opportunities in these fields and provide a perspective for future.

2 Quantum Frequency Conversion (QFC) In optical regime, interfacing oscillators with distinct frequencies is known as quantum frequency conversion (QFC) [18]. The process is, in principle, a unitary transformation and thus it can preserve the quantum properties of the respective frequency mode or help manipulate the quantum state faithfully similar to a beam splitter for spatial modes. In this section, we introduce QFC and its applications in quantum information processing.

2.1 Second-Order Nonlinear Optical Interaction We consider second-order (χ (2) ) nonlinear optical interaction among single-mode light pulses at angular frequencies ω1 , ω2 and ω3 = ω1 − ω2 . The interaction Hamiltonian of the three-wave mixing is described by H = iχa1† a2 a3 + h.c.,

(1)

where χ is a coupling constant proportional to the second-order susceptibility of the nonlinear optical medium, ai and a1† || for i = 1, 2, 3 are the annihilation and creation operators for photons at the angular frequency mode ωi , and h.c. represents the Hermitian conjugate. When a1 is assumed to be sufficiently strong and undepleted, the Hamiltonian in Eq. (1) is approximated by ∗ a2 a3 + h.c., H = igsq

(2)

where gsq = χa1 is an effective coupling constant. This Hamiltonian corresponds to the parametric down conversion process. When modes a2 and a3 are degenerate (or non-degenerate), the single-mode (or two-mode) squeezed state is generated from the vacuum state at modes a2 and a3 . When a3 in Eq. (1) is assumed to be the pump light for χ (2) interaction which is sufficiently strong and undepleted, the Hamiltonian is approximated by H = iga1† a2 + h.c.,

(3)

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(a)

(b)

1

2

1 2

p

(c)

(2) p

1,H 2,H

QFC for H pol.

1,V 2,V

QFC for V pol.

+

Fig. 1 a Energy level diagram of the relevant optical fields. b QFC on two frequency modes with specific polarization. The Hamiltonian of this process is described in Eq. (3). c QFC on four dimensional modes with two frequency modes and two polarization modes. The Hamiltonian of this process is described in Eq. (5)

where g = χa3 is the effective coupling constant. This Hamiltonian corresponds to sum (difference) frequency generation from mode a2(1) at ω2(1) to mode a1(2) at ω1(2) . We show the energy diagram of the process in Fig. 1(a) in which we relabeled the pump frequency ω3 to ω p according to the convention. The application of the process described in Eq. (3) to the cases that involves single or entangled photons is known as QFC.

2.2 Theory and Background for QFC The χ (2) -based QFC described in Eq. (3) is equivalent to the well-known beamsplitter (BS) interaction between two modes at a1 and a2 . By using the Heisenberg representation a1(2) (t) = U † a1(2) U with U = exp(−i H t/), annihilation operators a1,out and a2,out for modes at ω1 and ω2 coming from the nonlinear optical medium are described by 

a1,out a2,out



 =

a1 (τ ) a2 (τ )



 =

t −r r∗ t



 a1 , a2

(4)

where t = cos(|g|τ ), r = eiϕ sin(|g|τ ), τ is the interaction time, and ϕ is the phase of the complex amplitude of the classical pump light at mode a3 . The transmittance T = cos2 (|g|τ ) and the reflectance R = sin2 (|g|τ ) are probabilities of photons staying in the input frequency mode and being transformed into the other frequency mode, respectively. T and R can be actively changed by the pump power because |g| is proportional to the square root of the pump power. This property of QFC enables us to apply this device to many applications in the quantum information processing. For R = 1 and T = 0, the complete QFC is achieved. Even when 0 < R < 1 is valid, QFC is achieved probabilistically by postselecting the events where the input photon is converted. These QFC settings are used for an interconnect among different physical systems, which is the main topic of this section and is explained in detail later. Another application of the QFC is to use not only the photon obtained by QFC but also

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the photon which remains unconverted. For the setting 0 < R < 1 and 0 < T < 1, the partial QFC works as a frequency-domain BS acting on the two frequency modes [19]. Especially for R = T = 1/2, the QFC corresponds to the 50:50 BS in frequency domain. In Ref. [20], the frequency-domain Hong-Ou-Mandel (HOM) interference between two photons at different wavelengths 780 and 1522 nm was observed. In Ref. [21], the frequency-domain Mach–Zehnder interferometer (MZI) consisting of two χ (2) crystals has been demonstrated by using single-photon level coherent light at 795 and 1580 nm. We note that the MZI experiment employing a genuine quantum light source was performed by using third-order nonlinearity [22]. Recently, the frequency-domain BS has been applied to single-photon detectors which cannot distinguish between photons of very different wavelengths (i.e., erasure of frequency information of the detected photons) [23]. Conventional nonlinear optical medium has polarization dependency, and works on photons only with a specific polarization. For example, the PPLN used for many experiments is type 0 phase-matched for a large nonlinearity in which interaction among photons and pump light is allowed. Superposition of two QFCs working on mutually orthogonal polarization modes (e.g., horizontal (H) and vertical (V) polarization modes) can be used to construct a QFC device acting on four-dimensional Hilbert space consisting of two frequency modes and two polarization modes. The Hamiltonian is described by H = H H ⊗ I V + I H ⊗ HV ,

(5)

where Iσ for σ = H, V is the identity operator on σ -polarized mode and Hσ is the Hamiltonian of the same form as Eq. (3) with replacement of g and ai for i = 1, 2 with the coupling constant gσ and the annihilation operator ai,σ for σ polarization mode, respectively. From the Hamiltonian, similar to Eq. (4), the transition matrix for σ polarization mode is given by tσ = cos(|gσ |τ ) and rσ = eiϕσ sin(|gσ |τ ). The transmittance and the reflectance are described by Tσ = |tσ |2 and Rσ = |rσ |2 , respectively. When R H = RV = 1 is satisfied, the polarization preserving QFC process up to the constant phase shift of ϕ H = ϕV is achieved. Even if R H = RV < 1 , a polarization preserving QFC is achieved by postselection of the event where the input photon is converted. The constant phase shift can, in principle, be compensated, and thus this process can be called as the polarization insensitive QFC (PIQFC) [24]. So far, the PIQFC has been proposed or demonstrated utilizing different kinds of experimental setups [24–31]. All of the PIQFCs are constructed by two-component QFC, one for H- and one for V-polarized photons. The setups in Refs. [25–27, 30] are based on cascaded configuration of two QFCs. Refs. [26, 27, 30] use two cascaded nonlinear crystals, and Ref. [25] uses a single nonlinear crystal deploying a quarter wave plate (QWP) and a mirror to one side of the crystal for reflecting the photon with polarization flip. The setups in Refs. [24, 28, 29, 31] are based on parallel configuration of two QFCs. Ref. [29] uses two nonlinear crystals installed into two arms of the MZI with polarizing BSs (PBSs) at the input and output. Refs. [24, 28,

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31] use a single nonlinear crystal installed into the Sagnac type interferometer with a PBS for the input and the output of the interferometer. The QFC device for dual polarization modes based on the Hamiltonian in Eq. (5) has additional interesting features: (a) For Tσ = Rσ = 1/2, it corresponds to a nonpolarizing frequency-domain half BS. (b) For TH = RV = 1 and TV = R H = 0, it corresponds to a frequency-domain polarizing BS (PBS). (c) For TH = TV and 0 < Tσ < 1, it corresponds to a frequency-domain partially-polarizing BSs (PPBS) [32]. PPBSs with proper settings of the transmittance and the reflectance can be used to perform frequency-domain quantum information protocols such as entanglement distillation [33–35], probabilistic nonlinear optical gate [36, 37], quantum state estimation [38] and manipulation of multipartite entangled states [39]. In the above discussions, we implicitly assumed narrowband pump light for QFC which preserves shape of a wave packet of an input photon. By using a spectrally tailored pump light, engineering of temporal and spectral property of input photons through QFC is possible such as spectral compression [40–43] and gating/shaping operation of a pulse in/to a specific spectral broadband mode [44–46].

2.3 QFC for Optical-Fiber-Based Quantum Network QFC is also useful as a quantum interface for the quantum internet [15–17] on an optical fiber network. For the quantum internet, a quantum interface between stationary qubits of quantum matter systems (used for storing and processing quantum information) and flying photonic qubits (used for carrying the quantum information) is highly demanded. Quantum memories and processors realized by matter systems such as atoms, ions, and rare earth doped crystal have their own affinity to particular wavelengths of photons, whereas long-distance optical fiber communication only accepts near-infrared photons in telecommunication bands at 1310 and 1550 nm. Typical loss in commercially-available optical fibers in O-band around 1310 nm and S-, C- and L-bands around 1550 nm are 0.3 and 0.2 dB/km, respectively, whereas the wavelengths around 800 nm experiences a loss of 3 dB/km. This necessitates a quantum interface for filling the frequency gap of photons. Most of such quantum interfaces via QFC have been successfully demonstrated by solidstate χ (2) devices because of their applicability to a wide frequency range with an artificial quasi-phase-matching technique. The first demonstration of QFC for entanglement-preserving frequency up-conversion from the telecom wavelength to the visible wavelength was demonstrated in 2005 [47]. On the other hand, solidstate-based QFC for entanglement-preserving frequency down-conversion from the visible to the telecom wavelength was achieved using time-bin encoded qubits in 2011 [48]. The frequency down-conversion can be applied for photons emitted from or entangled with various kinds of quantum matter systems. Such technologies are crucial for the quantum repeater protocols [49]. In that context, QFC for frequency down-conversion are actively studied experimentally. In 2013, almost noiseless QFC

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Fig. 2 Experiments of QFC for frequency down-conversion to the telecom bands. The telecom bands are O-band: 1260–360 nm, S-band: 1460–1530 nm, C-band: 1530–1565 nm, L-band: 1565– 1625 nm. The red line is for ω p = ωc , and the χ (2) -based QFC on this line is impossible. The green circles are wavelengths of matter qubits for quantum memories/processors or single photon emittors. The red triangles show successful QFC demonstrations

for entanglement-preserving frequency down-conversion [50] and the HOM interference between photons after QFC [51] were demonstrated. In recent years, QFC has been applied to photons at a wide range of wavelengths emitted from several kinds of matter systems. In Fig. 2 we list examples of wavelength pairs of input and output photons used in frequency down- conversion experiments with resonant wavelengths of quantum matter systems. As we explained in the previous section, conventional χ (2) -based QFC has polarization dependency. Most of QFC experiments for preserving the input qubit information have been performed by using energy-time or time-bin encoding [47, 48, 50, 52–54]. By recent development of QFC for polarizing photons described in Eq. (5), QFCs with polarization entanglement preservation have been achieved [24, 26, 28–31]. During a QFC process, Raman scattering of the pump light and SPDC photons pumped by second harmonic light of the pump light introduce noise photons [55– 57]. In the case of Raman process, the anti-Stokes Raman scattering is smaller than the Stokes Raman scattering [56]. Therefore, a pump light with a frequency lower than that of the target converted light has been widely used to minimize Raman process induced noise. For example, under this situation, QFC with a pump light at a wavelength very close to the target wavelength were successfully demonstrated [58], in which the 780 nm photon was converted to the 1552 nm photon by the help of a 1569 nm pump light. It is generally accepted that QFC for the region of ω p > ωc , where ωc is the target frequency of photons after QFC is difficult. In fact, experiments have revealed an increased noise in this region. However, the suppression if this noise using a cascaded QFC process was proposed in Ref. [55] and demonstrated in Ref. [59]. But a subsequent frequency conversion experiment from 637–1587 nm showed the possibility of QFC for ω p > ωc by a quantitative discussion based on the experimentally-observed parameters [60]. Following this study, genuine QFC

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experiments have been successfully performed using QFC devices and by proper selection of the frequency of the pump light [53, 61].

3 Atom-Photon Quantum Interface Atom-photon hybrid quantum systems have been developed since the early stage of quantum information processing, and several methods for interfacing atoms and photons have been proposed and demonstrated. Here, we introduce a method for interfacing photons with an atomic ensemble, which is expected to be a storage for quantum states and play an important role in long distance quantum communication. This method, referred to as the hybridization method, was proposed by Duan et al. [13] for quantum repeaters, which is the well-known DLCZ protocol. So far, many experiments toward the realization of DLCZ protocol have been reported. With atomic ensemble, anti-bunched photon statistics showing a non-classical property was shown in Ref. [62], and entanglement generation between atoms and photons was reported in Ref. [63, 64]. Furthermore, utilizing the QFC described in Sect. 2, nonclassical behavior between atomic ensemble and converted telecom photons was demonstrated [65, 67–70]. In this section, we introduce such experiments with the collectively enhanced interaction between atomic ensemble and photons, and then discuss its applications.

3.1 Atomic Ensemble Quantum Memory First, we introduce a theoretical essence of atomic ensemble quantum memory. Suppose a three-level atom in -configuration as shown in Fig. 3, which consists of an excited level |e. and two ground levels |ga . and |gb ., with two radiative transitions. The energy differences between the excited and ground levels are described by ωa and ωb . We consider two off-resonance single-mode light fields, Stokes (S) and Anti-Stokes (AS), with angular frequencies ω S ≡ ωa −  and ω AS ≡ ωb − , respectively, where  represents frequency detuning. When the atoms are initially prepared in their ground levels and the detuning  is sufficiently large, the dynamics is approximately described by a two-level atomic system [71]. An atomic ensemble that consists of such two-level atoms is described by the Dicke Hamiltonian [72]. Using rotating wave approximation, this Hamiltonian can be written as   H = ω0 Sz + ω AS a †AS a AS + ω S a S† a S + g a †AS a S S+ + a AS a S† S− ,

(6)

where g is an effective coupling constant, and a S and a AS are annihilation operators. The collective spin operators are defined as

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Fig. 3 Energy-level diagram of a three-level atom in the Lambda configuration

Sz ≡

Na  j=1

Sz( j) , S± ≡

Na 

( j)

S± ,

(7)

j=1

where Na is the number of atoms, and the spin-1/2 operators of j-th atom are given by Sz( j) ≡

 1 |gb  j gb | − |ga  j ga | , 2

(8)

( j)

S+ ≡ |gb  j ga |ei (k AS −kS )·r j , ( j)

S− ≡ |ga  j gb |e−i (k AS −kS )·r j , with k AS and k S denoting wavevectors of mode AS and S, and r j representing the position vector of the j-th atom. Here we additionally introduce the HolsteinPrimakoff transformation [73] which is a mapping from spin operators to boson creation and annihilation operators. When the number of atoms is large √ enough  1), we define the annihilation and creation operators as S ≡ S− / Na , S † ≡ (Na √ S+ / Na . As a result, the Hamiltonian can be rewritten as     Na + ω AS a †AS a AS + ω S a S† a S + g Na a †AS a S S † + a AS a S† S . H = ω0 S † S − 2 (9) √ Here, the term g Na corresponds to the collective coupling strength. Thus, the coupling between atomic collective spin wave and two photonic modes is enhanced

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with increasing number of atoms in the ensemble. The interaction Hamiltonian is similar to that of the three-wave mixing process shown in Eq. (1). For example, when we inject a weak laser pulse on the mode S with wave vector kW . The effective Hamiltonian of the process is described by H = ga ˜ †AS S † + g˜ ∗ a AS S,

(10)

√ where g˜ ≡ g Na a S . The behavior of this process is similar to parametric down conversion derived in Eq. (2) such that AS photons and atomic collective spin wave simultaneously generated. The state of the pair is represented as

 are

 √ |0 AS  0a + pc |1 AS  1a + o( pc ) where |n AS  and |n a  are the number states in AS photons and atomic spin wave, and pc is the excitation probability. The pair state is an elemental resource for DLCZ type quantum repeater scheme [13]. In the repeater scheme, atomic spin wave is used as a quantum memory that keeps a quantum system entangled with AS photon. In order to retrieve the state stored in the atomic spin wave, an additional laser pulse is injected to the atomic ensemble in the mode AS with wave vector k R . In DLCZ protocol, the retrieval is done after receiving the heralding signal indicating the success of entanglement swapping. In the retrieval process, the effective Hamiltonian is described by ∗ a S S † + g˜ read a S† S, H = g˜ read

(11)

√ where g˜ read ≡ g Na a AS . The process is similar to QFC in Eq. (3) and also to the action of a beamsplitter (BS). In the write-read process, while the atomic ensemble preserves the phase information of ei (k AS −kW )·ri the direction of the retrieved S photons will be determined by the phase preservation relation k S = k AS − kW + k R . Additionally, such a BS interaction described by Eq. (11) can be used for partial retrieval of spin wave [74] when the coupling efficiency g˜ read is set to be less than a full-read-out value by adjusting the power of the readout light.

3.2 Atom-Photon Entanglement The nonclassical properties of the readout photon pair have been first observed by using an ensemble of Cs atoms [62]. Generation of entanglement between a photon and atomic spin wave has been demonstrated in several ways. In Ref. [63], Zeeman sublevels of 85 Rb are used. The ground levels correspond to 5S1/2 , F = 3 and F = 2, and the excited level corresponds to 5P1/2 , F’ = 3, as shown in Fig. 4(a). Circularly polarized write/read pulses are injected and the polarization state of scattered AS and S photons are measured for the verification of the entanglement via S-parameter of CHSH inequality. The observed S-parameter of S = 2.29 ± 0.05 clearly indicates the violation of CHSH inequality.

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Fig. 4 Entanglement generation based on a Zeeman sublevels of atomic transitions [63] and b spatial modes of scattered photons [64]

The demonstration in Ref. [64] utilizes the spatial modes of scattered photons as shown in Fig. 4(b). After the write process, the atomic ensemble preserves the phase information as the spin wave with the wave vector katom = kW − k AS . When the AS photon is postselected in a particular wave vector (path), the wave vector of the spin wave is decided by the momentum conservation. The quantum state of the postselected AS photon and the atoms is denoted by |pathx AS  and |kx atom , respectively. When the AS photon is postselected

in two different wave vectors

path− AS , the quantum state of the AS corresponding to the states path+ AS and





 

 photon and atoms is described as α path+ AS k+ atom + β path− AS k− atom , where the coefficients satisfy |α|2 + |β|2 = 1. By adjusting the excitation probabilities such that |α| = |β|, one can obtain a maximally entangled state. Using an HWP and a PBS, the path information of AS photon can be transformed into polarization  information as path± AS to |H/V AS . In the demonstration, the wave vector of the read light k R is set such that k R = −kW , leading to k S = −k AS . By using an HWP and a PBS as shown in Fig. 4(b), the path information of readout Stokes photon is transformed into polarization. Thus, the quantum state of AS and S photons is prepared in a maximally entangled polarization state  1  √ |H AS |H S  + ei(φ AS +φS ) |V AS |V S  2

(12)

where φ AS/S represents the phase coming from the path difference of AS/S photons in the interferometer. The overall relative phase is actively stabilized by a piezo mirror in the interferometer. The verification of the entanglement was done by the measurement of the S-parameter of CHSH inequality. The observed S-parameter of S = 2.60 ± 0.03 clearly violates CHSH inequality and presents an evidence of entanglement generation.

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Atom-photon entanglement applicable to telecom optical fiber network is a significant step towards the realization of quantum repeaters for arbitrary long-distance quantum communication. As described in Sec. 10.2, highly efficient and low noise QFCs are useful for interfacing atom-photon entanglement with telecom wavelengths. A pioneering work in this direction was the realization of atom-telecom photon entanglement with QFC based on four-wave mixing in Rb atoms in 2010 [65, 67]. These were followed by experiments that used PPLN based QFC for atomphoton entanglement generation [48], demonstrating nonclassical properties [68, 69] and entanglement generation [70] after the QFC process. The atom-telecom photon entanglement was demonstrated for the polarization degree of freedom of photon in Ref. [64], and PIQFC from 780–1522 nm [70]. The reconstructed density matrix (using quantum state tomography) of the generated state was used to estimate the entanglement of formation (EoF) as E = 0.37 ± 0.11 without QFC and E = 0.25 ± 0.13 with QFC, which clearly shows the realization of atom-telecom photon entanglement.

3.3 Summary In this section, we discussed theoretical and experimental issues related to atomphoton interfaces, and showed the analogy between atom-photon interfaces and QFC discussed in Sect. 2. The combination of atom-photon entanglement and QFC forms the elemental node for the realization of quantum repeaters. Long-distance entanglement between distant atomic ensembles connected via an optical fiber network over tens of kilometers was demonstrated in Ref. [75] by utilizing QFC, atom-photon entanglement generation, and entanglement swapping between atom and telecom photons.

4 Optomechanical Interface Mechanical systems, which consists of a vibrating structure in micro/nano artificial devices, have attracted significant attention and interest as a photonic interface thanks to their long coherence time, solid-state integratable architecture, and wide-bandwidth optomechanical interactions. Integrating an optical cavity with a mechanical resonator for the photonic interface based on cavity optomechanics enables to efficiently control the interaction between photons and mechanical degree of freedom (phonons) due to the strong confinement of photonic and phononic fields [76]. Such a cavity optomechanical interaction can convert information encoded on photons with PHz frequency to phonons with MHz (and GHz) phonons vice versa. Because the information restored in phonons can be readout to photonic information in microwave or optical regime, the optomechanical transduction allows us to perform wide-bandwidth information routing. The wide-bandwidth

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optomechanical routing is expected to be applicable to a variety of photonic interfaces such as a solid-state memory [77], optical wavelength converter [78], and quantum information transducer [79]. In particular, quantum information transducer between telecom (or visible) photons and microwave photons will pave the way to construct quantum internet including quantum computers with superconducting qubits connected through quantum communication channels with standard optical fiber networks [15–17]. The strongly confined optical fields also give us various ways to control mechanical vibration optically such as high-sensitive displacement measurement at the standard quantum limit (SQL) [80–82], reduction of thermal occupation with laser cooling [83, 84], and preparation of non-classical states [85, 86]. Such an ultrasensitive and functional optomechanical interaction enables us to study fundamental physics with mechanical degrees of freedom: chaos in nonlinear dynamics [87], parity-time symmetry breaking [88], and quantum gravity [89]. Moreover, mechanical modes can interact with many different systems such as photons, spins, and electrons. Thus, cavity optomechanics can be extended to further hybrid quantum systems by combining photons, phonons and other degrees of freedom and physical systems such as nitrogen-vacancy centers [90], quantum dots [91], and electromechanical nanowires [92, 93]. Recent progress in micro/nano fabrication technology allows us to develop various types of optomechanical platforms as the functional photonic interface. In this section, we present the basic optomechanical operation in cavity optomechanical systems, and review the cavity optomechanics with state-of-the-art devices with two architectures, composite cavity architecture and external optical cavity architecture. These two architectures have different advantages and usages as photonic interfaces.

4.1 Basic Optomechanical Operation Here we briefly introduce the theoretical background for basic operations in cavity optomechanics. The dynamics of cavity optomechanical systems, where an optical mode with the annihilation operator a in a cavity is modulated with a movable boundary formed by a resonant mechanical motion with the annihilation operator b (Fig. 5a), is captured by the Hamiltonian   H = ωa † a +  b† b + g0 a † a b† + b ,

(13)

where ω and are angular frequency of optical and mechanical resonator, and g0 is a vacuum optomechanical coupling constant. The vacuum optomechanical coupling x where x is a mechanical displacement and xzpf constant is defined by g0 ≡ ∂ω ∂ x zpf is its zero-point motion. The optomechanical coupling is effectively gained by a number of photons, i.e., linearization with a → α + a leads to

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Fig. 5 a Schematics of cavity optomechanical setup. b Three different cavity optomechancial operations with optical spectra and conceptual images of operations

   Hint = G a † + a b† + b ,

(14)

where α is a strong optical field, a is a residual quantum fluctuation, G ≡ g0 |α| is a net optomechanical coupling constant. This net interaction can bring the system into a strong coupling regime between photons and phonons with increasing number of cavity photons. In practice, optical and mechanical resonators are open systems such that the coherence in optical and mechanical modes are degraded through their coupling to reservoirs. The mechanical mode is occupied by incoherent thermal phonons at room temperature because the single phonon energy  is much smaller than the thermal energy quanta k B T . The optical mode, on the other hand, is affected only by the quantum vacuum noise. For instance, a mechanical mode with 1 GHz frequency contains about 7000 thermal phonons at room temperature. Decay rates of optical mode κ and mechanical mode γ become important factors to evaluate the performance of optomechanical devices and to determine what kind of photonic operations is available in the system. In this section, we focus on two parameters, mechanical f Q product and resolved sideband factor κ/ . The former given by the product of mechanical frequency f and Q factor (i.e., f Q = 2 /(2π γ )) determines how the system conserves the mechanical coherence from the thermal decoherence. The most fundamental condition for ignoring the thermal decoherence is given by k B T/ where f Q > k B T/ takes the value of 6.1 × 1012 at room temperature. The factor k/ determines what kind of optomechanical operations can be efficiently performed as discussed in detail in the following. The resolved sideband regime (i.e., κ/ < 1), where optical cavity linewidth is narrower than the mechanical frequency, allows us to efficiently induce two-types of interactions by optically pumping a red or blue sideband with a finite detuning δ = ± (Fig. 5b). The effective Hamiltonian is modified by a rotating frame approximation as follows:

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 blue    red Hint = G a† b + ab† , Hint = G a† b† + ab .

211

(15)

These effective interactions enables us to control the eigenstates of photons and phonons in the Fock-space. This can be readily understood via an analogy to nonlinear quantum optics: the red-detuned pumping corresponds to a beamsplitter interaction whereas the blue-detuned pumping leads to a photon-phonon pair production. Such interactions are the basis of SPDC, QFC and atom-photon entanglement generation, as we discussed in previous sections. This implies that sequentially integrating these operations may help realize a quantum memory operation [94]. Recently, such a quantum memory operation was demonstrated in photonic-phononic crystal cavity [95]. The unresolved sideband regime (i.e., κ/ > 1), where the linewidth of the optical mode is larger than the mechanical frequency, does not allow us to perform rotating wave approximation in contrast to the resolved sideband regime (Fig. 1c). On the other hand, the optomechanical coupling in this regime leads to an optical phase modulation with respect to the mechanical displacement. Apparently, the interaction Hamiltonian can be written as Hint = G X O X M

(16)

where X O and X M are the phase quadrature of optical and mechanical modes. In other words, this interaction provides photonic and phononic  eigenstates in the phase  space spanned by the linear quadratures X j = c + c† /2 and Y j = c − c† /2i [ j = O(c = a), M(c = b)]. Such a phase-modulated optical signal in the unresolved sideband regime is able to sense the mechanical displacement at the standard quantum limit (SQL), where the optical shot noise and the optomechancial back-action noise balances each other. The SQL displacement measurement has been demonstrated in various types of optomechancial setups towards ultrasensitive metrology [81–83].

4.2 Composite Cavity Optomechanical Architecture One of the directions for designing optomechancial systems for quantum applications is to bring the system into the resolved sideband regime (κ/ < 1) where a single phonon and a single photon can be manipulated via the basic quantum optical interactions as discussed above. For this purpose, a composite cavity optomechanical architecture, where an optical cavity and a mechanical resonator is fabricated on a single structure, has been investigated. The optical cavity used in such studies is either a whispering gallery mode (WGM) resonator or a photonic crystal cavity because of their high-Q optical properties. A sub-GHz or GHz radial breathing mechanical mode efficiently couples to an optical cavity mode in the composite cavity optomechanical architecture, and is useful to initialize the mechanical mode into its quantum ground state by installing the resonator to a commercially-available dilution refrigerator at

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10 mK. Moreover, because of the large mode overlap between light and mechanical vibrations, the composite cavity optomechanical architecture shows a relatively large optomechanical coupling constant g0 from kHz to MHz. These enable performing various quantum optomechanical experiments such as single photon-phonon pair production [95], entanglement generation between two optomechanical systems [96], and optomechancial Bell test [97]. WGM microresonators, which forms a smooth curved geometry (e.g. a sphere), have been used to investigate cavity optomechanics because of their high optical and mechanical Q factors and relatively simple fabrication process. The optomechanical coupling was observed in various geometries (microspheres [98], microdisk [99], microtoroids [100], and hollow-core microbottle [101]) and various materials (silica glass [98–101] and III-V semiconductors [102, 103]). Because mechanical properties (e.g. mechanical Q factor) strongly depends on the geometry and material, observing optomechanical coupling in resonators with different geometries and materials has been experimentally important. Optomechanical coupling in a solid-core microbottle WGM resonator fabricated on a silica glass optical fiber was investigated in Ref. [104]. The WGM microbottle resonator consists of two necks on an optical fiber which were fabricated by heat-andpull method (see Fig. 6a). The largest diameter between the two necks of the structure was 100 μm. Optomechanical coupling between an ultrahigh-Q optical mode (κ ∼ 2π × 1.9 MHz) and a radial breathing mechanical mode ( = 2π × 35.6 MHz) was achieved in the resolved sideband regime with a factor of κ/ = 0.05 1. The sideband behavior was experimentally probed by monitoring the transmission of the optical cavity supporting the mechanical oscillation (Fig. 6b). A mechanical Q factor of 1.57 × 104 was estimated by extrapolation in the parametric oscillation process where the laser light with a blue-detuning excites the mechanical vibration (Fig. 6c). The f Q product reaches to 0.56 × 1012 which is close to 6.0 × 1012 required for laser cooling towards the quantum ground state in this structure. Moreover, a different microbottle with a smaller diameter (80 μm) showed a higher f Q product of 0.85 ×

Fig. 6 a Conceptual illustration of optomechanical coupling between a WGM optical mode and radial breathing mechanical mode in a microbottle resonator. b Transmission spectrum with an optomechanical sideband. Change in the mechanical linewidth as a function of the optical power. Interpolating this data to zero pump power gives the intrinsic mechanical Q factor

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1012 because the frequency of radial breathing mode increases to 58.9 MHz with the reduction in the diameter. Similar to the state-of-the-art devices [105–107], one can engineer the internal strain and the clamping loss in microbottle resonators which can be used in singly or doubly clamped configurations, in contrast to all other geometries of WGM resonators which have a single-clamped structure. Such a flexibility in microbottle resonators allow improving and controlling optomechanical interactions and their properties towards building photonic interfaces for manipulating phonons in both classical and quantum regimes.

4.3 External Cavity Architecture In contrast to the composite cavity optomechanical architecture where both the mechanical and optical modes reside in the same device, in external cavity architectures an optical mode indirectly couples to a mechanical resonator. The most common one is the membrane-in-the-middle configuration where a thin membrane or film is placed in the middle of a Fabry–Perot type optical cavity. Since the choice of material and fabrication processes for preparing the membrane is completely independent of the performance of the optical cavity, this external optical cavity architecture provides further versatility for designing optical and mechanical resonators. In practice, the membrane in the middle system has been widely investigated with various mechanical resonators, such as a 2D membrane [108], double 2D membranes [109], and a membrane with phononic shield [83]. Optomechanical coupling through evanescent fields has also been demonstrated by integrating WGM microresonators to relatively small mechanical structures [81, 110, 111]. Decreasing the distance between an optical cavity and a mechanical resonator causes an optomechanical coupling due to increased spatial mode overlap between mechanical modes of the resonator and the evanescent field leaking out from the optical cavity. This architecture enables achieving optomechanical coupling with various micro/nano mechanical resonators such as a nano-string [112], a 2D graphene sheet [113], and a nano-beam with phononic crystal shield [114]. Evanescent field enabled optomechanical coupling between an optical resonator and a piezoelectric electromechanical resonator was demonstrated in Ref. [115]. A GaAs doubly-clamped beam resonator (150-μm-long, 20-μm-wide, and 600nm-thick) was coupled to an optical microbottle external cavity via the evanescent coupling (Fig. 7a). Because of the mechanical frequency of 280 kHz and the optical linewidth of 0.2 GHz, the system was in an unresolved sideband regime (κ/ = 7.1 × 102  1). The evanescent optomechanical coupling enabled a high-sensitive displacement measurement which allowed the observation of thermal fluctuation of √ the mechanical mode with a displacement sensitivity of 3.1 × 102 fm/ Hz [see Fig. 7b]. Integrating the optomechanical high-sensitive measurement and the piezoelectric conversion also allows for opto-electro-mechanical feedback loop to amplify or damp thermal fluctuation (see Fig. 7c) This versatile architecture may be extended

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Fig. 7 a Conceptual illustration of an evanescent optomechanical coupling between an optical microbottle resonator and a GaAs mechanical beam. b Power spectral density of thermal flucatuion observed by the optomechanical coupling. cPower spectral density of the mechancial resonator via an opto-electro-mechanical positive (amplification) and negative (damping) feedback

to high-sensitive force metrology via electromechancial, magnetomechanical and gravitomechancial forces under the SQL.

4.4 Summary In this section, we reviewed cavity optomechanical systems as photonic interfaces. The optomechanical interaction enables us to route the information of photons to phonons and vice versa in both classical and quantum regimes, and to measure the vibration at the SQL. The photon-phonon interface would be further extended to more diverse hybrid quantum systems by integrating solid-state systems such as spins, charges, and superconducting circuits.

5 Conclusions In this chapter, we have reviewed quantum interfaces between photons and diverse media, which can create matter-photon entangled systems, translate the frequency of the photons to a useful and/or standard frequency without destroying the quantum properties of the photons, and manipulate photons by phonons and phonons by photons. These technologies allow distributing entanglement among spatially separated nodes of a large area network over optical fiber links. Photons distributed among distantly located different matter quantum systems can be manipulated by using various photonic quantum circuits. For example, Bell measurement of the photons realizes quantum teleportation of the quantum state of one photon to the matter quantum system, and thus establishing entanglement between matter quantum systems. Such entangled matter systems at a distance can be used for realizing quantum repeaters. Moreover, it has been shown that a generalized Bell measurement used in all-photonic quantum repeater more efficiently creates entanglement among distantly located quantum systems [116, 117]. Thanks to QFC, even if each local

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quantum system is made of a distinct system and operating frequency, the frequency of the photons can be adjusted so that Bell measurement and entanglement swapping are performed. Thus, those photonic quantum interfaces are prerequisite to a global scale quantum communication or quantum internet [15–17, 118]. Atoms, optomechanical systems and other matter based quantum systems will play an important role for these realizations. Acknowledgements This work was supported by MEXT/JSPS KAKENHI Grant Number 18H04291; JP21H04445; JP20H01839; Moonshot R&D, JST JPMJMS2066; CREST, JST JPMJCR1671; and Air Force Office of Scientific Research (AFOSR) Multidisciplinary University Research Initiative (MURI) Award No. FA9550-21-1-0202.

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Hybrid Quantum System of Fermionic Neutral Atoms in a Tunable Optical Lattice Hideki Ozawa, Shintaro Taie, Yosuke Takasu, and Yoshiro Takahashi

Abstract Cold atoms trapped in optical lattices offer a clean and highly flexible experimental platform to investigate exotic phases of condensed matter. Recently, nonprimitive optical lattices with multiple lattice sites per unit cell have been realized, revealing nontrivial ordering and dynamics arising from the orbital degree of freedom. Here, we present a series of experiments with fermionic isotope of ytterbium (Yb) atoms in a Lieb lattice and dimerized cubic lattice geometry, which are realized by optical means. These experimental systems can be considered as hybrid quantum systems of nuclear spins and photons in the following contexts. In the dimerized lattice, we can create a 3D array of entangled atomic spin pairs with the precise control and detection by optical means. In the Lieb lattice, the sub-lattice degrees of freedom and the tunneling couplings between the sub-lattices correspond to the “effective spin” degrees of freedom and the “effective electromagnetic field” which couples the “effective spins”, respectively. Keywords Optical lattice · Fermi degeneracy · Spatial adiabatic passage · SU(N) symmetry)

1 Introduction The combination of huge number of particles and Coulomb interactions makes a detailed understanding of the many-body phenomena extremely difficult in solid state physics. For some materials, a band theory description in non-interaction regime H. Ozawa (B) RIKEN Center for Quantum Computing, 2-1, Hirosawa, Wako-shi, Saitama 351-0198, Japan e-mail: [email protected] S. Taie · Y. Takasu · Y. Takahashi Department of Physics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan e-mail: [email protected] Y. Takasu e-mail: [email protected] Y. Takahashi e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 Y. Hirayama et al. (eds.), Hybrid Quantum Systems, Quantum Science and Technology, https://doi.org/10.1007/978-981-16-6679-7_10

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is effective, which explains the distinction between metallic, semi-conducting and insulating states, but it is no longer accurate in the complex material with strong interactions where novel physical phenomena appear. Examples range from frustrated spin systems [1] to high-temperature superconductivity [2]. A problem toward understanding these phenomena is the associated many-body state emerging at low temperatures, where the strong correlations between the particles give rise to intriguing quantum phases. Typical approach in condensed-matter physics is to introduce simplified model systems for the complicated many-body problem; one attempts to formulate minimal models which include a few crucial degrees of freedom necessary to reproduce the observed physical behavior. Yet, such models are the result of several assumptions and approximations. For several materials, the formulation of a simplified model system is difficult due to non-negligible contributions of the long-range interactions, higher lattice orbital effects, coupling between electron and phonon modes in the crystal, and so on. In addition, understanding even the apparently-simple model systems is exceedingly difficult in the low-temperature regime, as they have found to be numerically intractable due to the exponentially growing number of quantum states. Therefore, the model’s ability to provide a correct description of the system’s key properties needs to be verified. Ultracold quantum gases in optical lattices [3, 4] emerged as a promising experimental platform toward further understanding of solid state systems. The basic idea is to cool a dilute atomic gas down to quantum degeneracy and introduce the gas into an optical lattice, which is created by interference of counter-propagating laser beams. Due to the atom-light interaction, the atoms feel a periodical potential. The role of the electrons moving in a crystal is then taken by the atoms, while the crystalline structure is formed by the optical lattice. This artificial solid is a defect-free environment and has high controllability of parameters in the system. The concept to study a quantum system by choosing a physically different, but fully controllable system with the same properties originates from Richard P. Feynman; quantum simulation [5]. After the possibility was pointed out in 1998 that ultracold atoms trapped in an optical lattice accurately implement the Hubbard model, the first experimental demonstration was made in 2001 with the observation of the superfluid to Mottinsulator transition for bosonic atoms [6]. This experiment ensures the capability of ultracold atoms to simulate and study quantum phases within the Bose-Hubbard model, and promoted a series of further experimental studies with interacting bosonic atoms [7–10]. Particularly, the realization of quantum gas microscope for bosonic atoms enables spatial observation of Mott-insulating shell structure on the single-site level [11–13]. Starting from the observation of the Fermi surface of non-interacting Fermi gas, a number of experiments with cold fermionic atoms in optical lattices have been also accomplished, such as mixtures with bosonic atoms [14–17], Anderson localization [18], and transport properties [19]. The precise control of inter-atomic interaction by Feshbach resonances allows the study of strongly interacting fermionic gases in optical lattices, which lead to the first realization of a fermionic Mott-insulating

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state for repulsive interactions [20, 21]. Most surprisingly, the antiferromagnetic long-range order with 6 Li atoms in two dimensional square lattice was realized very recently [22]. Thus, development in experimental technique for manipulating and detecting fermionic atoms in optical lattices enables us to explore the Fermi-Hubbard phase diagram in theoretically challenging regimes. Making good use of atom’s internal degrees of freedom also permits to expand the possibilities for quantum simulation, and stimulated a growing interest in several atomic species, instead of the widely used alkali elements. Among others, the atomic species belonging to the category of alkaline-earth-like atoms got special attention, such as calcium, strontium and ytterbium. Alkaline-earth-like elements have been primarily used in the recent past as powerful frequency standards; their atomic structure includes low-lying metastable electronic levels and the associated ultra-narrow optical transitions provides a remarkable intrinsic precision. Along with use of alkaline-earth-like atoms in quantum metrology, their potential for quantum simulation arose when the first bosonic isotope of ytterbium was cooled to degeneracy regime [23]. Since then, various isotopes of alkaline-earth-like species have been brought to quantum degeneracy (for example, see [24–29]), and even a fermionic Mott insulating state has already been realized [30]. Among them, the fermionic 87 Sr and 173 Yb are promising candidates for the quantum simulation of stronglycorrelated phases, owing to their strong interactions and nuclear spin properties. Their positive scattering lengths a = 10.55 nm for 173 Yb [31] and a = 5.09 nm for 87 Sr [32] correspond to repulsive interactions, which is the case of interest in the context of most theoretical studies. In addition, these fermionic isotopes possess high nuclear spins I = 5/2 for 173 Yb and I = 9/2 for 87 Sr, which are strongly decoupled from the electron degree of freedom due to the absence of electronic angular momentum, and permits the emergence of a high, unique symmetry of interactions. This SU(N = 2I + 1) symmetry attracts theoretical interests, as it is predicted to have drastic effect on the properties of interacting fermionic many-body state [32], and have the possibility of emergence of novel quantum phases, for example, SU(N > 2) quantum magnetism [33, 34]. Ultracold alkaline-earth-like atoms in optical lattices are suitable tool to verify such theoretical predictions, as there are (yet) no known analogue in nature. Apart from atoms internal degrees for freedom, complex lattice geometries also provide unique physics, owing to their special band structure and orbital degrees of freedom. By superimposing optical lattices with different lattice constants, or interfering more than three laser beams at different angles, a number of optical lattices with complex structure have been realized for ultracold atoms, and fascinating experiments are performed. One dimensional optical superlattice is the simplest example, where the superexchange energy is directly measured [35]. As for two dimensional lattice system, an optical honeycomb lattice [36], optical triangular lattice [37], optical checkerboard lattice [38], optical kagomé lattice [39] are so far realized. In the optical honeycomb lattice, manipulation of Dirac points is reported. A large variety of magnetic phases is simulated in the optical triangular lattice. In the optical checkerboard lattice, a superfluid in higher bands is realized. Kagome lattice has a flat band, where it is predicted that a bosonic gas in the band shows novel

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supersolidity, meaning the counter-intuitive coexistence of superfluid and crystalline order [40]. In this chapter, we present a series of experiments with fermionic isotopes of Yb atoms in a Lieb lattice and dimerized cubic lattice geometry, which are realized by optical means. These experimental systems can be considered as hybrid quantum systems of nuclear spins and photons in the following contexts. In the dimerized lattice, we can create a 3D array of entangled atomic spin pairs with the precise control and detection by optical means. In the Lieb lattice, the sub-lattice degrees of freedom and the tunneling couplings between the sub-lattices correspond to the “effective spin” degrees of freedom and the “effective electromagnetic field” which couples the “effective spins”, respectively. More details are given in the following. A Lieb lattice system has a mathematical analogy to a three-level system with Type transition. Most interestingly, a localized state in a flat band of the Lieb lattice corresponds to the dark state in the three-level system. By adiabatically changing the tunneling amplitudes in a counter-intuitive order, we coherently transfer atoms from one sublattice to another without populating the intermediate sublattice, which can be regarded as a spatial analogue of stimulated Raman adiabatic passage. We also successfully observe a matter-wave analogue of Autler-Townes doublet effect using the optical Lieb lattice [41]. 173 Yb is characterized by SU(N = 2I + 1) symmetric repulsive interaction for nuclear spin I = 5/2. For this large-spin system, Pomeranchuk cooling is enhanced; large-spin degrees of freedom can effectively cool down the system by absorbing the entropy from motional degrees of freedom. The precise control of the spin degree of freedom provided by optical pumping technique and a novel optically-induced nuclear spin singlet-triplet oscillation technique enables us a straightforward comparison between SU(2) and SU(4). Our main finding is that larger singlet-triplet imbalance is observed in a dimerized cubic lattice for SU(4) spin system compared with SU(2) as a consequence of Pomeranchuk cooling effect [42].

1.1 Outline of This Chapter This chapter is organized as follows. • Section 2 presents demonstration of spatial adiabatic passage of ultracold fermionic atoms of 171 Yb in a Lieb lattice geometry. • Section 3 reports the creation of the SU(4)-spin system of 173 Yb by optical pumping and the result of comparison with the cases of SU(2) and SU(4) in a dimerized cubic lattice. • Section 4 briefly summarizes the works in this chapter and gives future prospects in these experimental systems.

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2 Spatial Adiabatic Passage in a Lieb Lattice A Lieb lattice system has a mathematical analogy to a three-level system with type transition. Most interestingly, a localized state in a flat band of the Lieb lattice corresponds to the dark state in the three-level system. By adiabatically changing the tunneling amplitudes in a counter-intuitive order, we coherently transfer atoms from one sublattice to another without populating the intermediate sublattice, which can be regarded as a spatial analogue of stimulated Raman adiabatic passage. We also successfully observe a matter-wave analogue of Autler-Townes doublet effect using the optical Lieb lattice [41].

2.1 Three-Level System with -type Transition A three-level system is a minimal example in which quantum interference takes place. This system is considered mainly in a context of laser coupled atomic levels, and the Hamiltonian for a -type system in a rotating frame is written in the form ⎛

⎞ δ1 1 0 H = ⎝ 1 0 2 ⎠ , 0 2 δ2

(1)

where 1 (2 ) denotes a laser-induced Rabi frequency which couples basis states |1 with |2 (|2 with |3), and δ1 (δ2 ) is the detuning of the corresponding laser 1 (2). A dark state cosθ |1 − sinθ |3 (tanθ = 1 /2 ) arises as one of the eigenstate of the Hamiltonian in Eq. (1) when the Raman-resonant condition δ1 = δ2 is satisfied. By manipulating two laser pulses in counter-intuitive order so that θ changes from 0 to π/2, the dark states smoothly evolve from |1 to |3. This process is well known as STImulated Raman Adiabatic Passage (STIRAP), and has been an important technique for robust population transfer between atomic and molecular states. Electromagnetically Induced Transparency (EIT) [43] is also an important process in a three-level system. In an EIT experiment, strong optical coupling between |2 and |3 causes the splitting of the |1 → |2 transition by the Rabi frequency, known as Autler-Townes doublet [44]. As a result, the state |1 becomes transparent for laser light driving the |1 → |2 transition at a frequency region between the doublet.

2.2 Lieb Lattice A Lieb lattice is schematically illustrated in Fig. 1. It is the square lattice decorated with the additional site at each bond center. For a single particle regime, the nearestneighbor tight-binding Hamiltonian in the Lieb lattice is given by

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(a)

(b) t

t

y

A

x

E

d

C

ky

B

kx

unit cell

Fig. 1 a Lieb lattice configuration and b the band structure in the tight-binding model with  = 0. The 2nd band is a flat band

H = −t

 i, j

ci† c j +



i ci† ci ,

(2)

i

where i, j denotes the nearest-neighboring sites and t is the tunneling amplitude. − A = , and write the Hamiltonian in We set the site offset energy as  B = C = the bases of the Bloch sums |q, X  = √1N i∈X ei q·ki ci† |0 as ⎛



−2tcos(qx d/2)

⎞ 0

⎜ ⎟ ⎜ ⎟ H =⎜ − −2tcos(q y d/2) ⎟ ⎜ −2tcos(qx d/2) ⎟, ⎝ ⎠  0 −2tcos(q y d/2)

(3)

where d is the lattice constant. Note that the order of the bases is chosen as |q, B , |q, A , |q, C. By diagonalizing this Hamiltonian, we obtain the eigenvalues:

 E = , ± 2 + 4t 2 cos(qx d/2)2 + cos(q y d/2)2 . (4) The energy dispersion is plotted in the Fig. 1 for  = 0. The Lieb lattice has a dispersionless band (flat band, in other words) at zero energy, and at q = (±π/d, ±π/d) the 1st band and 3rd band touch the 2nd band. For  = 0, the eigenstate of each band can be written as |1st, q ∝ cos(qx d/2) |q, B +

cos2 (qx d/2) + cos2 (q y d/2) |q, A

+ cos(q y d/2) |q, C , |2nd, q ∝ cos(q y d/2) |q, B − cos(qx d/2) |q, C ,

|3rd, q ∝ cos(qx d/2) |q, B − cos2 (qx d/2) + cos2 (q y d/2) |q, A + cos(q y d/2) |q, C .

(5) (6) (7)

Hybrid Quantum System of Fermionic Neutral Atoms in a Tunable Optical Lattice Square long-lattice

Square short-lattice

Diagonal lattice

+

+

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Lieb lattice

=

y x

532 nm

532 nm

266 nm

Fig. 2 Construction method of our optical Lieb lattice. Three optical lattices with different lattice constants are overlapped with the relative phases meticulously controlled

The 2nd line means that a particle in 2nd band localizes at B-site and C-site regardless of q. This is because the tunneling amplitudes from B, C-site to A-site destructively interfere each other. Such localized states construct a flat band. This flat band is intrinsically different from a narrow band in a square lattice with deep potential. While a small hopping in the square lattice results in the narrow band, a flat band appears even if particles in a Lieb lattice are moving around.

2.3 Optical Lieb Lattice In this subsection, a method to construct the Lieb lattice structure by optical means is briefly explained. Our optical Lieb lattice consists of the three optical lattices with different lattice constants; one is the square lattice with lattice spacing 532 nm, another is more √ square lattice with 266 nm, and the third is one-dimensional diagonal lattice with 2 × 266 nm, as in Fig. 2. For convenience, we call them long-square lattice, short-square lattice, and diagonal lattice, respectively. The potential is written as V (x, y)

(y)

(x) cos2 (k L x) − Vlong cos2 (k L y) = −Vlong (y)

(x) −Vshort cos2 (2k L x) − Vshort cos2 (2k L y) −Vdiag cos2 (k L (x − y) + π/2),

(8)

where k L = 2π/(1064 nm) = π/(532 nm). The first, second, and third terms denote the long-square lattice, short-square lattice, and  diagonal lattice,  respectively.  In the (y) (y) (x) (x) following, we specify each lattice depth as s = sshort , sshort , slong , slong , sdiag

     (y) (y) (x) (x) = Vshort , Vshort , Vlong , Vlong , Vdiag /E R , where E R = 2 kL 2 /(2m) is the recoil energy and m is the atomic mass of 174 Yb. The relative phases between the long- and short-lattice can be adjusted by changing the frequency difference between these lattice beams. The relative phase between the long and short lattices at the position of atoms depends on the optical path lengths

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Fig. 3 Correspondence between a Lieb lattice and three-level systems

from a common retroreflection mirror. Therefore, in general, the phases along x-axis and y-axis are not equal. We shift the frequency of the long-lattice laser by an AOM on the way of y-axis to compensate the difference of the phases. The short-term stability of the relative phases between short- and long-lattices are ±0.001π according to the relative laser linewidth. The typical phase drift is ±0.01π per day. To stabilize the phase of the diagonal lattice, we constructed a Michelson interferometer along the optical path of the diagonal lattice with the off-resonant and a 507 nm laser whose frequency is off resonant and actively stabilized. The interferometer has two PZT-mounted mirrors; one is shared with the diagonal lattice laser beam for phase stabilization, and the other is used to shift the phase to a proper one. The short-term stability of the diagonal lattice is estimated to be ±0.007π . The last several optics near the vacuum chamber are out of the region actively stabilized by the interferometer. Those optics cause the gradual drift of 0.05π per hour.

2.4 Correspondence Between a Three-Level System and Lieb Lattice A Lieb lattice system has a mathematical analogy to the three-level system with -type transition. Figure 3 summaries the relations between the two systems. Couplings dependent on the momentum play a role of Rabi couplings in a three-level system, and detunings can be taken part of by energy offsets E A , E B and E C of each sublattice. In our optical Lieb lattice system, these parameters can be separately controlled by modifying the lattice depth along each direction, which enables us to realize a coherent scheme to transport atoms among these sublattices. In particular, by adiabatically changing the tunneling amplitudes in a counter-intuitive order, we can coherently transfer atoms from one sublattice to another without the intermediate sublattice occupied. This process can be regarded as a spatial analogue of STIRAP. Since this concept, named Spatial Adiabatic Passage (SAP), was introduced in the

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context of quantum dots [45, 46] and cold atoms [47], it has continuously attracted theoretical interests and large variety of its applications have been discussed [48]. High controllability and flexibility of cold atom systems in optical lattices enable the experimental demonstration of SAP.

2.5 Spatial Adiabatic Passage (SAP) In our experiment, we employ fermionic 171 Yb with a negligibly small scattering length of −0.14 nm to avoid interaction effects. The use of fermions causes a complexity arising from the finite momentum spread due to the Pauli principle. Adiabaticity of a process associated with a certain momentum is dominated by the band gaps among the corresponding eigenstates. If we adopted a Lieb lattice structure as it is, adiabaticity could not be maintained on the corner of the Brillouin zone where a Dirac cone exists. To prevent this problem, we make slight distortion on the band structure by shifting the phase of the diagonal lattice from an isotropic condition ψ = π/2. The validity of this scheme can be perceived from the potential landscape shown in Fig. 4b. The deformation reduces the inter-unit-cell tunneling, and therefore each cell becomes more like a solitary triple well. As a result, the momentum dependence of the energy dispersion diminishes, and the Dirac cone disappears. In the viewpoint of mathematics, this converts the coupling term as T AC → eiq y d/2 (t AC + δt) + e−iq y d/2 (t AC − δt) , T AB → eiqx d/2 (t AB + δt) + e−iqx d/2 (t AB − δt) , (9) where δt denotes the imbalance between inter- and intra-unit-cell tunnelings. In the case of δt = 0, T AC along the Brillouin zone boundary (q y d = π ) vanishes throughout the process. If we introduce δt = 0, the collapse of of transport along this line can be also suppressed. SAP in the Lieb lattice is to transport atoms from B sublattice to C sublattice by a counter-intuitive manipulation of tunneling amplitudes. During this process, the intermediate sublattice, namely A sublattice, is not populated because the state adiabatically follows a dark state cosθq |q, B − sinθq |q, C, with tanθq = T AB (q)/T AC (q). Our experiment begins with loading a sample of 1.2 × 104 atoms of 171 Yb at a temperature T /TF = 0.3 into the optical lattice of s = [(27.7, 0), (0, 16), 14]. In the loading stage, the potential on a B sublattice is made much deeper than those on the others so that we can ensure that only B sublattice should be occupied in the initial lattice. Then, we suddenly change the lattice depths to [(38.9, 3.8), (8, 8), 14] in 10 µs. This is a starting point of SAP process, where the tunneling t AB is much smaller than t AC . To earn a high tunneling rate, entire lattice depths are set relatively shallow, which leads to an undesired direct tunneling t BC . We reduce t BC by increasing the diagonal lattice depth beyond the equal-offset condition, i.e. E A > E B = E C . As long as E B = E C is

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(b)

z x Fig. 4 a Band energy and b potential landscape with distortion. If we shift the phase of the deep diagonal lattice, band gaps  become large and the Dirac cone disappears. Shown is the case of ψ = (1/2 + 0.11)π and slong , sshort , sdiag = [8, 8, 14]

fulfilled, the dark state is valid and SAP can be accomplished. We adiabatically sweep the lattice depths toward another limiting configuration [(3.8, 38.9), (8, 8), 14], passing through the intermediate configuration [8, 8, 14] which is equivalent to the potential shown in Fig. 4b. The time evolution during this process is measured by sublattice mapping technique [49]. The obtained TOF images suffer from the blurring around the Brillouin zone boundaries due to unavoidable non-adiabaticity arising from the band mapping procedure and a harmonic confinement of the system. For a qualitative analysis of sublattice-populations with the blurring, we take a set of basis images in which all atoms reside on a specific sublattice and determine sublattice occupancies by projecting images onto each basis. Figure 5 shows the time evolution of sublattice occupancies N A , N B and NC during the SAP process. Significant character specific to SAP can be seen clearly: initial population on the B sublattice is smoothly conveyed into C sublattice, on the other hand, the population on the A sublattice does not increase throughout the process. The efficiency of the process is estimated from the final population to be  NC (T f ) − NC (0) /N B (0) = 0.95(2), where T f is the total sweep time. What is more, it is only in the flat band state that the fermionic atoms populate at the middle point of the SAP process. A common way to occupy a certain energy band is to fill up all lower bands. The SAP process performed above provides an efficient way to realize an exotic non-equilibrium many-body state in which all fermions reside on the flat band of the Lieb lattice and the other bands are empty.

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Fig. 5 Time evolution of sublattice occupancies during a SAP process. Solid lines are the numerical calculation result with a single-particle Schrödinger equation

2.6 Autler-Townes Doublet To set a focus on a matter-wave analogue of EIT physics, we carry out a measurement similar to a pump-probe experiment. First of all, we prepare an initial state localized on the B sublattice (See Fig. 6a). Then, we introduce a weak tunneling coupling t AB and after a certain fixed time, a fraction of atoms that tunneled into A or C sublattice is measured. Figure 6b, c show the characteristic tunneling spectra. In both (x) which dominates the energy difference spectra, we scan the “detuning” via slong E B − E A (= E B − E C ). In the case of a weak coupling t AC in (6b), we can observe a single dip corresponding to B → A tunneling, whereas a clear doublet structure can be seen for a strong coupling t AC t AB in (6c). The double peaks originate from tunnelings to A + C and A − C orbitals which are separated in energy level by the amplitude of tunneling coupling t AC . The horizontal shift of the spectrum is (y) caused by the change of the short lattice depth sshort . While the short lattice generates the same potential curve for all sublattices, its effect on E A , E B and E C slightly differs depending on the configuration of other lattices. The breadth of the observed resonance peaks is broadened by the band dispersion and spatial inhomogeneity due to the harmonic confinement. In a typical EIT spectrum, even if the doublet splitting is smaller than the natural linewidth, we can observe a sharp dip. This implies occurrence of coherent population trapping (CPT) [50] in a dark eigenstate. Since our system has no loss mechanism, CPT does not happen. Therefore, a sharp

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Fig. 6 a Sequence of after a hold time of 1.8 ms, at

the experiment. b, c Tunneling spectrum

(x) (x) the lattice depths of (40, 40), (slong , 8), 9.5 in b and (40, 7), (slong , 8), 9.5 in c. Fraction of atoms which tunnel from the B sublattice during the hold time is shown. Filled solid lines are the theoretical curves based on the single-particle Schrödinger equation

EIT dip does not appear. Yet, the observed behavior exactly corresponds to a pumpprobe detection of Autler-Townes doublet which is fairly common in atomic systems.

2.7 Conclusion In conclusion, we have succeeded in demonstrating coherent tunneling processes of cold fermionic atoms in an optical Lieb lattice. Our work has revealed the fact that the three-sublattice structure of the Lieb lattice has remarkable analogy to type three level systems in quantum optics. By using this analogy and dynamical controllability of tunneling amplitudes, SAP between two sublattice eigenstates was performed. We also observed an matter-wave analogue of Autler-Townes doublet in a tunneling process from a sublattice into a strongly coupled pair of sublattices. The

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demonstrated techniques have a latent ability to generate exotic many body states in optical lattices. In particular, all atoms are located only on the flat band at the half point of the SAP process in the Lieb lattice. This might be a general scheme applicable to other lattice with flat bands. Including higher lattice orbitals is possible extension in the context of creation of angular momentum studied in Ref. [51]. In addition, recent advances in meticulous potential engineering [52] combined with single-site-resolved imaging of lattice gases [53] will greatly expand the range of application of SAP in cold atomic systems.

3 Antiferromagnetic Spin Correlation of SU(N) Fermi Gas in an Optical Dimerized Lattice In this section, we report the observation of the nearest-neighbor antiferromagnetic spin correlations of a Fermi gas with SU(N ) symmetry trapped in an optical lattice. The precise control of the spin degrees of freedom provided by an optical pumping technique enables us a straightforward comparison between the cases of SU(2) and SU(4) systems. Our important finding is that the antiferromagnetic correlation is enhanced for the SU(4)-spin system compared with SU(2) as a consequence of a Pomeranchuk cooling effect. This work is an important step towards the realization of novel SU(N > 2) quantum magnetism.

3.1 Local Entropy Redistribution The controlled system of ultracold fermionic atoms in optical lattices is regarded as a promising candidate to gain novel insights into phases driven by quantum magnetism. This approach offers experimental access to a clean and highly flexible FermiHubbard model. Progress toward entering the regime of quantum magnetism was hampered by the ultra-low temperatures and entropies required to observe exchangedriven spin ordering in optical lattices. To solve this problem, cooling schemes based on the redistribution of entropy between different regions were demonstrated recently, and made a great deal of progresses in realizing the quantum magnetism for the repulsively interacting SU(2) spin systems [22, 54, 55]. In our work, we also make the best use of the local entropy redistribution scheme within the lattice structure to reach the regime of quantum magnetism. The atoms are prepared in a dimerized cubic lattice, where the exchange energy in a dimer is much larger than that in the other links (See Fig. 7). This setup allows us to realize the short-range quantum magnetism within the dimer at the total entropy we can currently reach. Furthermore, we introduce large-spin degree of freedom by using a fermionic isotope 173 Yb, which is characterized by SU(N = 2I + 1) symmetric repulsive interaction for nuclear spin I = 5/2. The precise control of the spin degree

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Temperature T

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Super exchange J

Dimerized lattice z

y x

Jd Super exchange J

Isotropic lattice z

Temperature T

J

y x

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Jd

Fig. 7 Local entropy redistribution scheme. The temperature in the experiment is above the exchange energy J for isotropic lattices and no magnetic spin correlations can be observed. When introducing a large exchange energy Jd on links via a dimerization, the temperature is below the large exchange energy, and magnetic spin correlations emerge within the strong links

of freedom provided by optical pumping technique enables us a straightforward comparison between the cases of SU(2) and SU(4). We investigate the large-spin influence on the quantum magnetism.

3.2 SU(N ) Fermi-Hubbard Hamiltonian in a Dimerized Cubic Lattice In this work, we measure and analyze the antiferromagnetic spin correlation of SU(N = 4, 2) Fermi gas of 173 Yb in an optical dimerized cubic lattice (Fig. 8). The Hamiltonian to describe this system is the SU(N ) Fermi-Hubbard model (FHM) in a dimerized cubic lattice, which is written as Hˆ FH = Hˆ 0 + Hˆ t ,    † cˆi,σ cˆ j,σ + H.c. Hˆ 0 = −td i, jx- ,σ  U  nˆ i,σ nˆ j,σ − μ nˆ i,σ , + 2 i,σ =σ

i,σ    † cˆi,σ cˆ j,σ + H.c. Hˆ t = −t

(10)

(11)

i, jx− ,σ

−tyz

   † cˆi,σ cˆ j,σ + H.c. ,

(12)

yz

i, j− ,σ † ) is the fermionic annihilation(creation) operator for a site i and spin where cˆi,σ (cˆi,σ † σ , nˆ i,σ = cˆi,σ cˆi,σ is the number operator, µ is the chemical potential, U is the on-site

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z

tyz

x y

tyz t

td

Atom with 4 nuclear spins

266nm Singlet Triplet

Fig. 8 Schematic view of the nearest-neighbor spin correlations in a four-component mixture of fermionic atoms prepared in a dimerized cubic lattice with the strong intra-dimer tunneling td and weak inter-dimer tunnelings t, tyz

interaction energy, and td , t, tyz are the tunneling amplitudes between the nearest neighbors in the strong link i, jx− , the weak link i, jx− along the x axis, and the yz weak link i, j− along the other two axes, respectively. We reach the regime of quantum magnetism by strongly dimerizing the cubic lattice along the x direction, where the exchange interaction energy within the dimer is enhanced. Thereby, we observe an excess of spin singlets compared with spin triplets. By employing a technique for optically inducing a singlet-triplet oscillation (STO) [56] with an effectively produced spin-dependent gradient, we verify the emergence of the antiferromagnetic correlation. We investigate the difference of the spin correlation between the SU(4) and SU(2) systems over a wide range of entropy.

3.3 Optical Dimerized Lattice Potential In this subsection, our experimental setup is briefly described. An atomic sample is prepared by loading an evaporatively cooled two- or four-component Fermi gas of 173 Yb into an optical superlattice with a dimerized cubic geometry. The lattice potential is given by (x) (x) cos2 (2kL x + π/2) − Vlong cos2 (kL x) V (x, y, z) = −Vshort (y)

(z) −Vshort cos2 (2kL y) − Vshort cos2 (2kL z),

(13)

where kL = 2π/λ is a wave number of a long lattice with λ = 1064 nm. In the followy z x x , sshort ), sshort , sshort ]= ing, we use the notation for each lattice depth as sL = [(slong (y)

(x) (x) (z) [(Vlong , Vshort ), Vshort , Vshort ]/E R , where E R = 2 kL2 /(2m)2 is the recoil energy for the

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long lattice. Unless mentioned, the lattice depth is sL = [(20, 20.8), 48, 48], which is equivalent to U/ h = 3.0 kHz, td / h = 1.0 kHz, t/ h = 37 Hz, and tyz /t = 1.3 in the SU(N ) FHM.

3.4 Spin Manipulation by Optical Pumping The experimental techniques that lead to the manipulation and detection of the nuclear spin degree of freedom are of central importance in the context of quantum simulation of SU(N )-symmetric models. In this subsection, we show how the different nuclear spin states of 173 Yb can be separately imaged and how SU(4) spin mixtures can be prepared by means of optical techniques. Although removal of one or more spin states is possible by using narrow-line transitions and the Zeeman shift, associated atom losses make cooing difficult. Instead, we can make use of optical pumping by which we gather atoms into particular spin states. Figure 9 illustrates the schematics of optical pumping via the 1 S0 (F = 5/2) ↔ 3 P1 (F = 3/2, 7/2) transitions. When we create a balanced two-component mixture, we distinguish the sublevels of the 3 P1 (F = 3/2) state by applying an external magnetic field B 16 Gauss, and irradiate the π -polarized light beam with four resonant frequencies. Consequently, an initial six-component ensemble is gathered into two sublevels of m F = ±5/2 as in Fig. 9a. On the other hand, when we prepare a balanced four-component mixture, we employ both of the F = 3/2 and F = 7/2 states in 3 P1 so that we can compensate the asymmetry of Clebsh-Gordan coefficients [57]. For example, if we were  to optically pump atoms in m F = 3/2 by using the |F = 5/2, m F = 3/2 →  F = 7/2, m F = 3/2 line alone, almost all atoms would be transported to  m F = 1/2. On the contrary, if we adapted only the |F = 5/2, m F = 3/2 →  F = 3/2, m F = 3/2 line, they would be pumped into m F = 5/2 rather than m F = 1/2. Therefore, simultaneous pumping via these transition lines with modest ratio of laser intensities can generate a balanced fourcomponent mixture of m F = ±1/2 and m F = ±5/2 as in Fig. 9b. (a)

3

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Fig. 9 Optical pumping schemes to create a two- and b four-component mixtures of 173 Yb by using a π -polarized 556 nm light beam. Green and gray arrows mean excitation and decay, respectively. The number labeled on the arrows denotes the absolute square of the normalized Clebsh-Gordan coefficient, which is equivalent to the transition strength

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Fig. 10 Optical Stern-Gerlach separation of spin components after optical pumping to create a two- and b four-spin mixtures. White arrows point to the atoms in each sublevel of the 1 S0 state. The absorption image is the average of 10 independent data. The gray line is a fit to the integrated optical density by using a multi-component Gaussian function

Figure 10 shows the images of spin distributions after optical pumping. They were observed by an Optical Stern-Gerlach (OSG) technique [29], where we apply the spin-dependent gradient by an circularly polarized laser beam with a Gaussian profile. We fit the data with a multi-component Gaussian function in order to evaluate the atom number in each spin. The measured spin population is (14) pm F =−5/2 , pm F =5/2 = [0.49(1), 0.50(1)] in Fig. 10a,  p−5/2 , p−1/2 , p1/2 , p5/2 = [0.26(1), 0.24(1), 0.25(1), 0.25(1)] in Fig. 10b. (15)



In this way, we succeed in producing almost balanced two- and four-component mixtures.

3.5 Singlet-Triplet Oscillation (STO) in Dimer In this subsection, the observation of antiferromagnetic spin correlations on neighboring sites in a dimerized lattice geometry is presented. At the early stage of evaporative cooling, we apply the optical pumping mentioned in the subsection 3.4 to create balanced two- or four-component mixtures of 173 Yb. After we load the two- or four-component Fermi gas into a strongly dimerized cubic lattice, where all beams are ramped with a spline-shaped laser intensity in 150 ms at the same time, we detect an antiferromagnetic spin correlation with the procedure as shown in Fig. 11a, which is similar to Ref.[55]. In the first stage of the detection procedure, we pin the atoms in the lattice sites by applying a two-step ramp of sL = [(20, 20.8), 48, 48] → [(25, 20.8), 80, 100] → [(25, 100), 80, 100]. The first and the second ramps take 0.5 ms and 10 ms, respectively. These lattice ramps

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also get rid of the contribution of the admixture of double occupancies in the bare ground-state singlet of the SU(N ) FHM. Then, we apply a spin-dependent gradient by a fictitious magnetic field of light, which is similar to the OSG beam. This gradient produces an energy difference  for atoms with different spins on neighboring √ sites and drives coherent oscillation between √ the singlet(= (|σ1 , σ2  − |σ2 , σ1 ) / 2) and triplet (|t0  = (|σ1 , σ2  + |σ2 , σ1 ) / 2) states at a frequency of / [56], where σi (i = 1, 2) means a spin component. When we measure the spin correlation of a four-component mixture, we adopt a linearly polarized gradient beam, with which the STOs have the same frequency for the 4 spin pairs of (m F = −5/2, −1/2), (−5/2, 1/2), (5/2, −1/2), and (5/2, 1/2), but do not occur for the 2 spin pairs of (−1/2, 1/2) and (−5/2, 5/2). After a certain STO time, we switch off the gradient and merge the dimers into single sites by changing the lattice potential to sL = [(25, 0), 80, 100] in 1 ms. On account of a fermion anticommutation relation and symmetry of the two-particle wave function, the singlet state on adjoining sites transforms to a doubly occupied site with both atoms in the lowest band, on the other hand, the triplet state evolves to a state with one atom in the lowest band and the other in the first excited band. The fraction of atoms forming double occupancies in the bottom band is detected by a photoassociation (PA) technique [17, 30, 58]. The PA process enables us to change all atoms forming double occupancies in the lowest band into electronically excited molecules that immediately get out of the trap, whereas the state with one atom in the lowest band and the other in the first excited band remains unaffected due to its odd-parity of relative spatial wave functions [59]. Therefore, we can regard the loss of atoms as the number of atoms constituting the singlet state in the initial dimerized cubic√lattice. We note that the symmetric and antisymmetric states= (|σ1 , 0 ± |0, σ1 )/ 2 also exist, particularly around the trap edge, but they merely transform to the state with one atom per site after merging the dimer, which is not detected by a PA. The PA laser is detuned by -812.26 MHz from the 1 S0 ↔3 P1 (F = 7/2) transition and has intensity enough to finish eliminating double occupancies within 0.5 ms pulse time. Figure 11b shows the typical STO of SU(4) spins in a strongly dimerized cubic lattice. We can see a clear oscillation. The oscillation is damped mainly due to the spatial inhomogeneity of the fictitious magnetic gradient and the photon scattering from the gradient beam. This oscillation manifests that the number of singlet exceeds that of triplets and that an antiferromagnetic correlation on neighboring sites exists. We can also observe a similar STO signal for an SU(2) system. The STO data are fit with the function based on our experience F(tSTO ) = −a e−tSTO /τ cos (2π f tSTO ) + b,

(16)

where a, b, τ, f are fitting parameters. Along with the STO data, we measure the total atom number in the optical lattice without applying the PA laser, which is symbolized as N . This correlation is quantified by the normalized STO amplitude A and singlet fractions ps :

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Fig. 11 a Sequence to detect a singlet or a triplet in a dimer. The case of two spins (red and blue) is shown. Depending on the time when we apply the optical gradient, the two spins form the double occupancy in the lowest band (top), or the state with one spin in the lowest band and the other in the first excited band (bottom) after merging the dimer. We distinguish these states by a PA. b Singlet-triplet oscillation of SU(4) spins in a strongly dimerized cubic lattice. The red dashed line represents the total atom number in the lattice without applying the PA. The blue solid line is the fit result with Eq. (16). The gray dotted line is the STO signal assuming no damping. Error bars denote the standard deviation of four independent data

 A=

2a/N for SU(2) system 3a/N for SU(4) system

ps = 1 −

b−a . N

(17) (18)

We note that the extracted N − b − a is equivalent to the actual atom number in the triplet state |t0  for SU(2) spins, but that is not the case for SU(4) spins because a coherent oscillation does not happen for the spin pairs of (m F = −1/2, 1/2) and (−5/2, 5/2). To take this effect into account, the measured STO amplitude is multiplied by 3/2 for SU(4) case as you can see in Eq. (17).

3.6 Comparison Between SU(4) and SU(2) Systems For the purpose of revealing how the spin degrees of freedom play role in the magnetic correlations, we investigate A and ps for various initial entropies in the harmonic trap. Figure 12a, b show the comparison between SU(2) and SU(4) spins in a strongly dimerized cubic optical lattice. The initial temperature in the harmonic trap is estimated by accomplishing the Thomas-Fermi fitting to the 10 independent momentum distributions and the initial entropy sinit is calculated from the T /TF , where TF is the Fermi temperature, using the formula for a non-interacting Fermi gas. When we take the STO data, we choose the number as N = 3.2 × 104 and the trap frequencies as (ωx , ω y , ωz )/2π = (158.3, 48.6, 141.8) Hz so that the filling n, that is to say, the number of the particle per site, amounts to n = 1 around the trap center (Fig. 12d). The solid lines are the numerical calculation in atomic limit of the SU(N ) FHM in Eq.(10), assuming the local density approximation. As the initial entropy becomes higher, the normalized STO amplitude and the absolute singlet fraction decrease,

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(a)

(b)

Black solid line: 3/2 A(N =2)

Black solid line: 3/2 ps(N =2) Blue: SU(4)

Blue: SU(4)

Red: SU(2)

Red: SU(2)

(c)

(d)

SU(4) SU(2)

Red: SU(2) ln(6)/2 Blue: SU(4)

Fig. 12 a Normalized STO amplitude and b singlet fraction of SU(N = 2, 4) Fermi gases in the strongly dimerized cubic lattice with td /t = 27. The horizontal axes show the initial entropy in the harmonic trap. The solid line is a theoretical curve that assumes adiabatic loading into the lattice. The dotted line in b is the numerically calculated multiple occupancy where we exclude the double occupancy in the ground-state singlet wave function of the SU(N ) FHM. The horizontal error bars show the standard deviation of the 10 independent temperature measurements. The vertical error bars include the fitting errors in the STO measurement and the standard deviation of the total atom number N . c Temperature of SU(2,4) Fermi gases in the lattice. The empty diamonds and filled circles are the experimental data which are calculated from a, b, respectively. d Numerically calculated density (top) and entropy distribution (bottom) at the initial entropy per particle sinit /kB = 1.5 in the cases of SU(2, 4). The gray dashed line indicates the maximum singlet entropy per site ln(6)/2 for SU(4)

because triplet states become thermally populated. We can see a clear difference between SU(2) and SU(4) systems; for the same initial entropy, the antiferromagnetic correlation is larger in the SU(4) system than that in the SU(2) system. We can attribute this result to two effects. The first is the difference of the fraction of singlet configurations among all possible states. The second is the thermodynamic cooling effect concerning spin entropy. To simply explain these effects, we consider a minimal model where two atoms with SU(N ) spin symmetry are trapped in a single dimer, neglecting double occupancies. Provided that the temperature is absolute zero, the singlet probability is ps (N ) = 1 regardless of N because the singlet has the lowest energy. On the other hand, suppose that the temperature is infinite, the singlet probability is ps (N ) = W (N )/N 2 because the probability is determined by the number of the singlet configurations W (N ) =N C2 . In the cases of SU(N = 2) and SU(N = 4), the ratio of the singlet probabilities at the same temperature becomes  ps (4)/ ps (2) =

1 for T = 0 3/2 for T = ∞.

(19)

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Because this ratio monotonically decreases from 3/2 to 1 as the temperature gets lowered, ps (2) < ps (4) < 3/2 ps (2) holds as long as the temperature is finite. The normalized STO amplitudes A(4) and A(2) satisfy the same inequality, too. The black solid lines in Fig. 12a, b indicate 3/2 ps (2) and 3/2 A(2), which would give the upper limit for ps (4) and A(4) at the same temperature, if we took only the first effect into account. Interestingly, most of the observed SU(4) data are above the black lines at the same initial entropy. This manifests the second effect that the temperature of SU(4) system is lower than that of SU(2) system, which is ensured from Fig. 12c. This phenomenon can be understood as follows. Entropy per site of the singlet ground states is given by ln(W (N ))/2. In contrast to the zero-entropy ground state of SU(2) system, the SU(4) ground state has a surplus of entropy of ln(6)/2 = 0.9. Therefore, in terms of the initial temperature, the requirement for spins to form the singlet in the SU(4) system is not as strict as that in the SU(2). This has close resemblance to the Pomeranchuk effect [60] enhanced by large spin degrees of freedom, which was already demonstrated in the paramagnetic SU(6) fermionic Mott-insulator [30]. In this work, it is clearly shown that cooling with large SU(N ) spin can be applied even in the realm of quantum magnetism. We note that in a trapped system, entropy is carried by a low-density metallic state near the edge of the atomic cloud and singlet states at the trap center survive for higher total entropy as you can see the density and entropy profiles in Fig. 12d. The data in Fig. 12a, b, especially at low initial entropies, show the discrepancy with the theoretical curve. This might be caused by several reasons including some non-adiabaticity in the lattice loading or an imperfect efficiency on the PA. From mid to high initial entropies, the measured singlet fraction is slightly overestimated because the multiply occupied states except the ground-state singlet are thermally populated at the initial lattice depth and inevitably detected by PA after merging.

3.7 Conclusion In conclusion, we have studied the significant role of the spin degrees of freedom on the antiferromagnetic correlation in a strongly dimerized cubic optical lattice by comparing the SU(2) and SU(4) systems. We observed the enhanced antiferromagnetic correlation in SU(4) due to a cooling mechanism similar to the Pomeranchuk effect. Further cooling can be expected for a larger spin system such as SU(6), which 173 Yb possesses. The setback for experiments with higher spin system is the detection technique: if we tried to execute the scheme presented here to SU(6) system of 173 Yb, we would suffer from the multiple STO frequencies. We expect that combining SU(N > 2) Fermi gas with more intriguing lattice geometry like a plaquette, which has been already implemented with optical lattices [61, 62], will open up the door to the exotic magnetic order [63, 64].

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4 Summary and Outlook In this chapter, hybrid quantum systems of photons and nuclear spins of fermionic neutral atoms in a tunable optical lattice were studied. We briefly summarize the achievements in this chapter. • Demonstration of a spatial adiabatic passage in a Lieb lattice. We revealed that a Lieb lattice system has a remarkable analogy to a three-level system with -type transition. By making use of analogy and dynamical controllability of our optical Lieb lattice, we realized the spatial adiabatic passage of massive particles for the first time. This method can be used to transfer atoms into the flat band, apart from the phase imprinting method. • Probing the large-spin effect on the short-range quantum magnetism of a Fermi gas in a dimerized cubic lattice. We invented an optical pumping technique to create a balanced four-spin mixture of 173 Yb. By loading the four-spin mixture in a strongly dimerized optical lattice, we investigated the large-spin effect on the quantum magnetism within the dimer. Our important finding is that the antiferromagnetic order is enhanced in SU(4) compared to SU(2) as a consequence of Pomeranchuk cooling effect.

4.1 Outlook Combination with dissipation is a possible direction for further experiments in the abovementioned systems. While dissipation is ubiquitous in nature, it has been used as an efficient tool for the preparation and manipulation of quantum states. In the case of cold atoms in optical lattices, dissipation has already been implemented as two-body inelastic atom loss by applying a PA laser beam [65]. The strength of the inelastic collision can be easily controlled by intensity of the PA laser. If we apply strong dissipation in the Fermi-Hubbard model of 173 Yb atoms in a dimerized cubic optical lattice, a ferromagnetic spin correlation in a dimer becomes favorable compared to an antiferromagnetic one [66, 67]. This is because spin configuration can suffer from a loss in an intermediate state, while a ferromagnetic spin configuration cannot decay on account of the Pauli exclusion principle. Thanks to this dissipative spin-exchange interaction, we can investigate from an antiferromagnetic to ferromagnetic spin correlation, depending on the strength of the dissipation. This will offer a novel approach to quantum simulations of magnetism.

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Phonon-Electron-Nuclear Spin Hybrid Systems in an Electromechanical Resonator Yuma Okazaki and Hiroshi Yamaguchi

Abstract A recent progress on hybrid systems with a GaAs-based electromechanical resonator is reviewed. Fundamental experimental techniques including fabrication of an electromechanical resonator integrated with a GaAs heterostructure and transport measurements using a cryogenic amplifier are explained. The first topic is on a hybrid system composed of a gate-tunable quantum dot (QD) and quantum point contact (QPC) integrated into a piezoelectricity-based electromechanical resonator. The piezoelectric coupling between them enables us to detect milli-Kelvin phonon states via current flowing through the QD/QPC. Noise analysis based on an equivalent circuit elucidates that the displacement sensitivity is amplifier-limited and the estimated intrinsic sensitivity with a QD transducer potentially reaches the zeropoint motion of the resonator. The second topic is on quadrupole-coupling between electrically tunable phonon states in an electromechanical resonator and a nuclear spin ensemble within it. As a consequence of a rapidly oscillating strain induced by a strongly driven mechanical resonator, nuclear magnetic resonance (NMR) frequency shifts which can be regarded as mechanical analogue of ac-Stark shift known in cavity quantum electrodynamics is observed. This prototype system potentially opens up quantum state engineering for electrons, phonons, and nuclear spins such as coherent coupling between them. Keywords Electromechanical resonator · Quantum dot · Quantum point contact · Nuclear spin · Nuclear acoustic resonance

Y. Okazaki (B) National Institute of Advanced Industrial Science and Technology (AIST), National Metrology Institute of Japan (NMIJ), Tsukuba, Ibaraki 305-8563, Japan e-mail: [email protected] H. Yamaguchi NTT Basic Research Laboratories, NTT Corporation, 3-1 Morinosato-Wakamiya, Atsugi, Kanagawa 243-0198, Japan e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 Y. Hirayama et al. (eds.), Hybrid Quantum Systems, Quantum Science and Technology, https://doi.org/10.1007/978-981-16-6679-7_11

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1 Introduction Electron-phonon coupling is one of the most fundamental processes in solid-state physics. For this reason, recent progress on a hybrid system composed of an electromechanical resonator and gate-tunable electronic states defined in a semiconductor heterostructure [1–3] is fascinating. A mechanical resonator fabricated by photo and electron-beam lithographies is one of the ideal realization of a harmonic oscillator, that is a cavity of phonon or sound [4, 5]. Development of optical or electrical means to detect and manipulate their motions together with high-quality (high-Q) factor of the resonator, reaching 105 –109 range typically, enables us to access the state of phonons with unprecedent accuracy and sensitivity. A high-mobility twodimensional electron gas (2DEG) formed at GaAs/AlGaAs hetero-interface can offer unique platform to study various kinds of quantum transport phenomena of electrons including single-electron tunneling and quantum interference effects such as anti-bunching of electrons. Fine metal electrodes lithographically defined on the wafer surface can generate tunable electrostatic potential underneath the electrodes to form confinement potential into the 2DEG. This confinement potential can define low-dimensional conductors such as a quantum point contact (QPC) and a quantum dot (QD). The electron states in these structures can be electrically controlled by varying gate voltage applied to the electrodes. By integrating these structures into an electromechanical resonator, gate-tunable electronic states can interact with high-Q phonon states because of piezoelectricity of GaAs. This hybrid systems would open an opportunity to manipulate electron-phonon interactions in artificial systems. Accessing the quantum ground states of a harmonic oscillator is a milestone in the study of mechanical resonators (MRs) [6]. This had opened opportunities to test the quantum mechanics in macroscopic scales [7–9] such as a quantum entanglement between two distant macroscopic objects [10, 11] and also to develop practical quantum sensors. To reach the quantum ground states, suppressing thermal excitation is necessary to satisfy a condition kB T < h f where T ( f ) represents the system temperature (resonant frequency of a MR) with kB being the Boltzmann constant and h being the Plank constant. This condition can be realized by employing high-frequency oscillators [12–14] or by cooling down the effective temperature of the resonator by sideband cooling [8, 15–18]. One of the most successful implementation of this is cavity optomechanical systems in which superconducting microwave resonator strongly coupled to a MR is measured with a quantum-limited Josephson parametric amplifier for the microwave signal [19]. Contrary to optomechanics in which the quantum limit sensitivity is achieved, the quantum ground state of electromechanical systems such as one between the resonator and metallic single-electron transistors (SETs) [20, 21] or superconducting quantum interference devices (SQUIDs) [22] has not been detected. In these hybrid systems, many theoretical proposals such as lasing and cooling of phonon driven by a QD had been proposed [23–34]. In metallic SETs, the tunnel coupling with the lead electrodes is defined by oxidized metal interface and lacks tunability of tunnel coupling rate. On the other hand, semiconductor based low-dimensional conductors can sustain gate-tunable electronic states and thus

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can offer ideal platform to implement a versatile hybrid electromechanical system with tunability. Despite these fantastic advantages, the study on GaAs-based hybrid structures is limitedly developed [35–37], probably due to difficulty in fabricating and measuring such a device. In this chapter, we review a recent progress on a hybrid systems between a GaAs-based MR and gate-tunable electronic states at a GaAs hetero-interface [1, 2]. Another interesting topic in hybrid structures with MRs is to implement cavity quantum electrodynamics (cavity QED) type systems using an electromechanical resonator [38, 39]. The interaction of atomic-like two-level systems or spin with light is one of the most basic toy models in quantum mechanics [40, 41]. This model is not only pedagogical, but also useful to realize practical quantum measurements such as quantum non-demolition measurements of atomic states [42]. Especially, in a high-Q optical cavity, the atom-light interaction can be effectively amplified by a factor of Q. The concept of this Q-amplification can also be implemented into a phonon or sound cavity with a MR. In particular, the Q value of a MR reaches up to the order of 109 , which is hardly achieved in an optical cavity [43, 44], and thus unprecedented quantum manipulation and readout with the aid of such a high-Q resonator would be developed. In recent years, we have succeeded to observe coupling between the Zeeman split nuclear spins and an electromechanical resonator [3]. A method for detecting nuclear spins via sound waves has long been studied as nuclear acoustic resonance (NAR) since the 1960s [45, 46]. However, the interaction between nuclei and sound is weak in a bulk crystal, and thus it is hardly developed as a practical detection method. In a high-Q electromechanical resonator, the Q-amplification of this sound-nucleus coupling can enhance the sensitivity of NAR, and also possibly leads to develop a new detection scheme as well as a quantum state manipulation such as spin squeezing [47, 48] for nuclear spins. Our experiment is a prototype system to develop such fantastic detection and manipulation schemes for nuclear spins. In Sect. 2, the device structure of our GaAs-based electromechanical resonator hybrid system, fabrication by photo/electron-beam lithographies, and measurement method using cryoamp in a dilution refrigerator are outlined. In Sect. 3, a realization of displacement transducers using QPC and QD piezoelectrically coupled with a MR and the thermal motion detected via QPC/QD current are reviewed. Finally, in Sect. 4, experiments implementing the analogy of cavity-QED with the electromechanical resonator and Zeeman split nuclear spins are reviewed.

2 Experimental Methods 2.1 Fabrication of an Electromechanical Resonator Confining electrons into low-dimensional structure starts by confining electron to a two dimensional system using a semiconductor heterostructure. Modulation-doped

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heterostructure of GaAs and AlGaAs grown by molecular beam epitaxy in ultra high vacuum chamber can offer extremely clean 2DEG with a mobility greater than 106 cm2 V−1 s−1 [49]. Figure 1a depicts a cross-sectional layer structure of modulationdoped GaAs heterostructure used to fabricate the electromechanical resonator integrated with 2DEG shown in Fig. 1b. On a (001) GaAs substrate, a 3-µm-thick sacrificial layer of Al0.65 Ga0.35 As and 1-µm-thick GaAs layer that forms a beam resonator are grown. Since an aluminium-rich AlGaAs layer can be selectively wet-etched by a hydrogen fluoride solution, the GaAs layer can be released from the substrate to suspend a beam resonator. The 2DEG is formed at a hetero-interface of GaAs and Al0.3 Ga0.7 As located 90 nm below the surface. Five-nm-thick GaAs cap layer and 55-nm-thick Al0.3 Ga0.7 As layer are uniformly doped with Si. As a consequence of Coulomb attraction by positively-ionized Si dopants, the conduction band is bended to form a triangular potential which traps free electrons that forms a 2DEG as shown in the figure. The presence of 30-nm-thick undoped spacer layer of Al0.3 Ga0.7 As suppresses electron scattering by the dopant is suppressed compared to the case without a spacer layer. The resultant 2DEG can thus have a high mobility and relatively low electron density, 2 × 106 cm2 V−1 s−1 and 3 × 1011 cm−2 in this structure. These values of mobility and density are measured for a bare wafer, and are degraded after sacrificial layer etching due to damage. To laterally confine electrons into more lowdimensional structures, Cr/Au electrodes are defined on top of the heterostructure. The electric potential is generated in the 2DEG by applying negative voltages to the electrodes. Figure 1b, c are a false-color scanning electron microscope (SEM) image and a schematic of a 1-µm-thick, 6-µm-wide and 50-µm-long doubly-clamped mechanical resonator integrated with gate-tunable QPC and QD. The beam resonator is clamped at the both ends and is suspended from the GaAs substrate with a vacuum gap of 3 µm. The resonant frequency of the fundamental flexural mode of this resonator is 1.5 MHz in room-temperature and slightly increased up to ∼1.7 MHz in lowtemperature (below 4 K). An electromechanical resonator integrated with gate-defined QD and QPC is fabricated by combination of photo/electron-beam lithographies and sacrificial layer etching as shown in Fig. 2. In the actual device, based on finite element method analysis, the QPC/QD is designed to be placed at the point where the strain associated with the fundamental flexural mode is maximized in order to maximize QPC/QD-MR coupling. Basic lithographic steps to define the resonator are as follows: 1. A 200-nm-height mesa pattern is defined on the wafer by wet chemical etching using a mixture of H2 O : H2 O2 : H2 SO4 (25:1:5 in volume ratio) at 10 ◦ C for 40 s as shown in Fig. 2a. Ohmic contacts to the 2DEG are defined by photolithography and are depositioned with eutectic AuGeNi alloy. After the deposition, the wafer is annealed at 430 ◦ C under a continuous flow of pure H2 for 60 s to alloy AuGeNi and GaAs.

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Heterostructure

Cr/Au electrode

Mesa

Si doped GaAs (5 nm)

Mesa

GaAs

Si doped Al0.3Ga0.7As (55 nm)

Al0.65Ga0.35As sacrificial layer

Al0.3Ga0.7As (30 nm)

2DEG GaAs

GaAs substrate

Mesa b

GaAs 50 μm

Resonator Cr/Au Schokey electrode c

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Fig. 1 Structure of phonon-electron-nuclear spin hybrid systems in a GaAs-based electromechanical resonator. a Layer structure of the GaAs/AlGaAs heterostructure to fabricate an electromechanical resonator integrated with a 2DEG. b False-color scanning electron microscope image of a doubly-clamped beam resonator. c Simplified schematic of an electromechanical resonator

2. Fine Schottky electrodes on top of the wafer are fabricated by electron-beam lithography followed by a deposition of Cr/Au (1 nm/25 nm) (Fig. 2b). Here an adhesion layer of Cr is employed instead of typically used Ti because of its resistance to hydrogen fluoride in sacrificial etching. 3. The lateral shape (width and length) of the resonator is defined by an optical lithography and wet-etched using mixture of H2 O:H2 O2 :H2 SO4 (25:1:5 in volume ratio) for 4 min at 10 ◦ C. This deep wet-etching define the side wall of the resonator and also expose the sacrificial layer underneath the GaAs layer as shown in Fig. 2c.

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Mesa GaAs Al0.65 Ga

As Sacri ficial lay er Substrate

0.35

(a) Wet chemical etching to define mesa

(c) Deep wet-etching to expose sacrificial layer

Electrodes

(b) Lithography of top gate electrodes

(d) Sacrificial layer etching to suspend a beam resonator

Fig. 2 Basic fabrication process of an electromechanical resonator integrated with a gate tunable low-dimensional conductors. a Fabrication of mesa. b Formation of fine gate electrode. c Wet etching to define lateral structure and expose a sacrificial layer. d Sacrificial layer etching

4. After the deep etching, the resonator is released from the substrate by selectively wet-etching an aluminium-rich sacrificial layer using dilute hydrogen fluoride (5 wt%) for 20 min at room-temperature (Fig. 2d). The typical etching rate is approximately 200 nm/min with 5wt% HF. Prior to a sacrificial layer etching by HF, the side wall of the wafer piece is coated by photo resist to protect immersion of HF through the gap between the resist and the wafer. This ancilla process is effective to suppress unwanted damage on the 2DEG. Two Schottky electrodes defined at the left clamping point as shown in Fig. 1a, b are used to drive and detect the mechanical motion of the resonator. Applying ac voltage on this electrode generates piezoelectric force which can drive the resonator. The resulting mechanical motion also generate piezoelectric voltage on these electrodes. By measuring this piezoelectric voltage, the mechanical motion can be detected. The sensitivity of this piezoelectric driving and detection scheme via Schottky electrodes is not so efficient compared with those obtained by using QPC and QD, but useful to identify basic characteristics such as resonant frequency and Q-factor of a mechanical motion. At the right clamping point, fine gate electrodes to define QPC/QD are located as their enlarged SEM images shown in Fig. 4b, d. To maximize piezoelectric coupling, the beam resonator is defined along the [011] crystal axis [50, 51], and the QD/QPC are located at the clamping point where the maximum strain can be induced by the fundamental flexural mode of the resonator.

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2.2 Transport Measurements for an Electromechanical Resonator Detection of mechanical motion of the resonator via current through a QPC/QD piezoelectrically coupled to the resonator requires measurement of ac current through the QPC/QD. Figure 3a depicts a simplified circuit of the setup used to measure current spectrum sourced from the QD/QPC at the resonant frequency of the resonator, i.e. 1.7 MHz. Transport measurements of QPC/QD are performed in a dilution refrigerator with a base temperature of 80 mK to suppress unwanted thermal effects. Detection of small ac current sourced from the QD/QPC at MHz frequency range is challenging. This is because, resistance value RQPC/QD of the QPC/QD is normally higher than the quantum resistance, ∼12.9 k. This resistance and unavoidable cable capacitance Ccable connecting a low-temperature device and room-temperature equipments form RC low-pass filter whose cut-off frequency is normally several tens kHz because of Cwire ∼100 pF, typically. This low-pass filter suppresses a current signal from the QPC/QD and deteriorates the resultant signal-to-noise ratio. To minimize this low-pass filtering, a high input impedance cryogenic amplifier (cryoamp) is employed. This cryoamp is operated at 4 K stage of the dilution fridge, and thus suppress the low-pass filtering effect due to decrease in Cwire . In addition, lower noise floor of a cryoamp can improve the signal-to-noise ratio which limits the sensitivity of the displacement via a QD/QPC. In this system, small ac current sourced from the QPC/QD is fed into a load resistor RL = 1 k, and the voltage drop across the load resistor is amplified with a series of the cryo- and room-temperature(RT) amplifiers, followed by a spectrum analyzer to measure the power spectrum. Recently low-noise cryoamps are commercially available. Most of them have an input impedance of 50 . Such low input impedance cryoamp is, however, not suitable to amplify current signal from high-impedance devices such as QPC/QD, because the current signal from high-impedance devices should be strongly attenuated due to voltage division at the input stage of the amplifier. Hence we have developed a home-made high-electron mobility transistor (HEMT) based cryoamp [52–54]. We chose a commercially available HEMT, Agilent ATF-35143 a pseudomorphic HEMT. This HEMT has an excellent reported room temperature noise properties, and has been used to construct a low-noise cryogenic amplifier for measuring thermal and shot noise generated by a QPC/QD [55–57]. A design of our amplifier is depicted in Fig. 3b. This is a basic single transistor common-source amplifier [58]. To prevent unwanted self-oscillations in the circuit, we introduce a series resistor of 47  at the input. This circuit are constructed by mounting electric parts on a printed circuit board as shown in Fig. 3c. Resistors and capacitors used in this circuit are surface-mount-type thin film resistors (Susumu RG series) and multilayer ceramic capacitors (Murata GRM series, temperature compensating type), whose electric properties are almost temperature independent down to a cryogenic temperature range.

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a

RQPC/QD

VSD

4K

80 mK RL

Cwire

RT

Cryoamp

Spectrum Analyzer

V dd 510

b 22n

22n

20 mm

100n

510 11

11 Input 47

c

Output

Input

HEMT

510

Output

V dd

HEMT

Fig. 3 System for measuring small ac currents from QPC/QD using a cryogenic HEMT amplifier. a Simplified circuit diagram for measuring the current sourced from QD and QPC. b Circuit diagram of home-made cryogenic amplifier. c Image of HEMT-based cryoamp mounted on a printed circuit board

3 Displacement Sensing with Quantum Dot and Point Contact Detection of mechanical motion via electron transport through QPC and QD in the quantum limit is the first step towards future quantum control with a hybrid system between an electromechanical resonator and gate-tunable electronic states in lowdimensional conductors formed at a hetero-interface. This section reviews the recent progress in detecting thermal vibrations of milli-Kelvin phonon states (whose corresponding amplitude is the order of 100 fm) by current spectrum measurements for QPC and QD piezoelectrically coupled to the MR [1, 2]. Evaluation of the sources of noise based on an equivalent circuit elucidates that the amplifier’s noise is dominant, and the sensitivity imposed by the intrinsic noise generated by a QD itself nearly reaches the quantum limit. A recent development of a cryoamp with lower noise is also reviewed.

Phonon-Electron-Nuclear Spin Hybrid Systems …

a

QPC

b

VGU VGC

VG

VGL

c

I

e

Δ QPC/QD

QD

d

Mechanical Resonator

Potenal Modulaon ΔV

253

Δ ΔV

I N-1

N

N+1

Δ VG

ΔV

VGC

Fig. 4 a Displacement sensing via current through a QPC/QD. b, d Enlarged false-color scanning microscope image at the clamping point of the electromechanical resonator to define QPC (QD). c, e Schematic of current I through a QPC (QD) plotted as a function of gate voltage VG (VGC )

3.1 Displacement Transducers with QPC and QD Figure 4a depicts a QPC/QD-MR hybrid system used to detect a mechanical motion of the MR via current through the QPC/QD. Motion of the MR, which is illustrated as a harmonic oscillator, causes a potential modulation due to piezoelectricity of GaAs, and this potential affect the local charge states in QPC and QD. Hence the current through the QPC/QD is also affected by the mechanical motion. Figure 4b, d are enlarged false color SEM images at the clamping points showing electrodes formed on the surface of the mesa that sustains a 2DEG. By applying negative gate voltages to these electrodes, an electrostatic potential is generated in 2DEG and a gate-tunable low-dimensional conductor can be defined. Figure 4b shows a situation that negative voltages are applied to the two colored electrodes, and a narrow constriction depicted by the dotted lines is formed. As a result, quasi-onedimensional transport channels can be defined, known as a quantum point contact (QPC). When the current I flowing through the QPC is plotted as a function of VG , step-wise current plateaux are observed as schematically shown in Fig. 4c. This is the consequence of increment in the energy gap between the one-dimensional conduction channels quantized by the narrow constriction, and is a typical conduction property of the QPC [59]. The conduction channel of the QPC can be influenced by potential modulation V via piezoelectric fields associated with the motion of the resonator. This potential modulation can be effectively treated as a modulation in VG and thus the QPC current I is modulated as schematically shown in Fig. 4c. A larger value of conversion ratio gm = d I /d VG (d I /d VGC ) can realize more efficient mechanics-to-

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current conversion. Hence the displacement transducer with QPC/QD is optimized at a gate condition where gm is maximized. By applying negative voltages VGU and VGL to the side gate electrodes in addition to the electrodes used in case of QPC, an electron confinement is defined at the gate enclosed area shown by the circle in Fig. 4d, known as a quantum dot (QD). A singleelectron charging energy in the QD typically ∼1 meV allows electrons in the QD to be changed one by one by the gate voltage VGC . A plot of current I passing through the QD as a function of VGC , periodic current peaks called Coulomb oscillation appear as schematically shown in Fig. 4e. The Coulomb oscillation reflects that the number of electrons N confined in the QD changes as N + 1 → N → N − 1 as VGC is swept. This current peak corresponds to a current flowing through a singleelectron state in the QD located within the bias window defined by electrochemical potentials of the source and drain leads. Since a confinement of single electrons into a QD is more sensitive to change in its surrounding electric potential compared with a QPC, a highly efficient transducer with a larger value of gm compared with a QPC is expected to be implemented.

3.2 Thermal Motion Measurements and Displacement Sensitivity A displacement transducer using QPC and QD is constructed as described in the previous subsection, the mechanical oscillation is converted into the current flowing through the QPC/QD, which can be amplified by the cryoamp described in Fig. 3, and then measured with a spectrum analyzer. In this scheme, a relationship between the power spectrum densities S I of the QPC/QD current and Sx of the mechanical 2 Sx , where η is a responsivity with a unit of oscillation is expressed as S I = η 2 gm [V/m] and reflects the magnitude of the resulting piezoelectric fields felt by the QPC/QD. A larger value of ηgm can realize more efficient transducers. A MR placed at a non-zero temperature is disturbed by the thermal Langevin force and self-oscillates called thermal motion. The squared amplitude x 2  of a thermal motion is given by the energy equipartition theorem as kB T = K S x 2  with K S being the spring constant [4]. From measurements of temperature dependence of the thermal motion spectra, η can be determined, and then the sensitivity of the transducer can be determined by converting the noise floor of the measured spectrum into the corresponding mechanical amplitude. Figure 5a is a plot of the current power spectrum density S I for the thermal motion of the fundamental flexural mode ( f 0 = 1.7 MHz) of the resonator at T = 80 mK measured using the QPC displacement transducer piezoelectrically coupled to the resonator. The spectrum is found on a frequency-independent offset indicated by the arrow, and a single peak corresponding to the thermal motion is observed. The solid line in the figure is the fitting result by the Lorentz function. From this fitting, the peak area A which is proportional to the squared displacement x 2  of the thermal motion

Phonon-Electron-Nuclear Spin Hybrid Systems …

QPC

A / gm2 [a.u.]

Noise-limited sensitivity

A / gm2 [a.u.]

a

255

b

QPC

c

QD

Fig. 5 Thermal motion measurements. a, Current power spectrum S I through a QPC demonstrating a thermal motion spectrum associated with the fundamental mode of the resonator at f 0 ∼ 1.7 MHz. b, c Plot of A/gm as a function of the system temperature T measured through a QPC (QD), where A is the peak area of the thermal motion spectra and gm is a temperature-dependent transconductance gm determined from dc measurements

2 can be determined. Figure 5b is a plot of A/gm as a function of temperature T . Here 2 dependence of gm . The upper axis A is divided by gm to calibrate the temperature  √ 2 indicates the corresponding displacement x  = kB T /K S of the thermal motion, where K S is estimated as K S = Meff (2π f 0 )2 and the effective mass Meff = 0.73 is estimated from the total mass of the beam [4]. Note that the coefficient 0.73 is the normalization factor of the fundamental flexural mode of a doubly clamped beam resonator to a harmonic oscillator. Figure 5b shows linear temperature dependence as guided by the solid line, reflecting the linear temperature dependence of x 2  of the thermal motion. From linear fit, the value of η can be determined. The right axis in Fig. 5a shows the corresponding Sx deduced from the calibrated value of η, showing the noise limited sensitivity of Sx = 1.7 × 10−26 [m2 /Hz]. Figure 5c shows a similar plot of thermal motion measurements, where the transducer is changed from the QPC to QD. Similar to the case with the QPC, linear temperature dependence is observed. As shown in Fig. 5a, the Lorentzian-shape peak corresponding to the thermal motion is found on the frequency-independent background noise. The value of the corresponding mechanical displacement indicated by the right axis is the noiselimited sensitivity in displacement sensing. Table 1 summarizes the sensitivities of the displacement sensing using the QPC and QD transducers. The table lists the current through QD/QPC at the optimized gate bias condition, the measured value of η, the sensitivity determined from the noise floor, and the associated position

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Table 1 Amplifier- and shot-noise-limited sensitivities with QPC and QD displacement transducers Transducer Current [nA] Responsivity Sensitivity Position Dominant [A/m] [fm/Hz0.5 ] resolution noise source [fm] QPC

54

3.8

QD

5.1

7.3

128 (25) 63 (5.5)

337 (68) 166 (15)

Intrinsic RQPC/QD

Sx

Extrinsic SShot

RL

C

STh

SAmp

Amplifier (Shot noise) Amplifier (Shot noise)

Amp +

Fig. 6 Equivalent circuit model of the measurement setup

resolution for the QPC and QD transducers. The measured position resolution can be compared with the standard quantum limit (SQL), which is the principal limitation in the displacement sensing imposed by the quantum √ mechanics [6]. The value of SQL in our resonator is 2.93 fm, estimated from xSQL = h/Meff f 0 /2π [60, 61]. The measured position resolution is 337 fm (166 fm) with QPC (QD) transducer, which is 115 (57) times larger than the SQL, and thus 2 order of magnitude improvement is necessary to access the quantum regime of this resonator. It is useful to elucidate the source of noise for further improvement in the sensitivity. Figure 6 shows an equivalent circuit of the measurement setup shown in Fig. 3a. In this figure, RQPC/QD represents the resistance of the QPC/QD, and RL is the load resistor connecting the signal line and the ground, while C represents a total stray capacitance of the measurement circuit, whose dominant source is Cwire . Sx (SShot ) represents current power spectral density corresponding to the displacement signal (shot noise generated by current flowing through the QPC/QD). Additionally, STh represents current power spectral density of thermal noise mainly generated by a parallel resistance of RQPC/QD and RL . SAmp represents a power spectral density of the inputequivalent noise of the amplifier. These current sources can be classified into either internal and external parts as shown in the circuit, where the current signal/noise in the internal part are sourced from intrinsic transport properties associated with QPC/QD transducer. The value of STh can be estimated from the Johnson-Nyquist formula of thermal noise STh = 4kB Te /Reff with the effective electron temperature Te = 100 mK and Reff ∼ RL = 1 k in our setup. The value of SAmp is determined in a manner described in Refs. [55–57] to be SAmp = 2.14 × 10−25 A2 /Hz. Shot noise arises from particle nature of electrons, whose power spectral density is given by SShot = 2eFI with e being the elementary charge and I being the current flowing through QPC/QD, where the Fano factor F reflects statistical properties involved in

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257

Table 2 Comparison between HEMT- and SQUID-based cryogenic current amplifiers HEMT-based amplifier [1] SQUID-based amplifier [63] Temperature Gain [V/A] Power consumption Bandwidth Noise floor [A2 /Hz]

4.2 K 1.4 × 104 1 mW a few MHz 2.1 × 10−25

1/2 (called quadrupole nuclei) are often involved in III-V semiconductor materials such as GaAs [64]. Owing to a non-zero quadrupole moment inherent in quadrupole nuclei, their spin state can be influenced by crystal strain of the host material [45, 46, 65–67]. This interaction between nuclei and strain, known as quadrupole interaction, can be harnessed for strain-mediated manipulation of nuclear spins and has been extensively studied in strained quantum dots [68–73] and strained semiconductors [67, 74–82]. To explore the physics of nucleus-strain coupling in a high-frequency regime, the NMR spectra was measured for a bulk material irradiated by an ultrasound. However, the resulting intensity of the ultrasound is too weak to state-manipulate the nuclear spins and the physics of nuclear spins exposed to an intense oscillation strain has remained poorly understood. In high-Q mechanical resonators, the mechanical motion periodically modulates local strain and the frequency can be sufficiently higher than the nuclear spin relaxation time so that non-adiabatic

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259

Table 3 Analogy between cavity quantum electrodynamics(QED) and phonon-nucleus coupling in an electromechanical resonator Cavity QED Electromechanical resonator Atomic states Cavity

Atom Optical cavity

Coupling

Dipole interaction

Phenomenon 1 Phenomenon 2

ac-Stark shift Sideband transition

Zeeman split nuclear spin Mechanical resonator (phonon cavity) Quadrupole interaction via lattice strain NMR frequency shift Sideband NMR peak

dynamical effects can be applied. Hence harnessing such a resonator for manipulating nuclear spins is a promising approach to achieve dynamical strain-mediated controllability. Additionally recent experimental demonstration of exotic mechanical states including phonon lasing and non-classical phonon states had opened opportunities for unprecedented control of strain. The mutual interaction between phonon and nuclear spins in such a system also enables to detect nuclear spin states via mechanical states. Despite these fantastic advantages, strain-mediated manipulation of nuclear spins via an opto/electromechanical resonator remains unexploited. A system of a mechanical resonator (phonon cavity) quadrupole-coupled to Zeeman split nuclei is analogous to atom-light systems in cavity quantum electrodynamics (cavity QED) as in Table 3. In atom-light systems, atoms are dipole-coupled with electromagnetic fields in a high-Q optical cavity. Representative phenomena observed in this system are ac-Stark shift and side-band transition [39, 42]. The acStark shift is the second-order level shift in atomic states induced by intense light fields. In dispersively coupled cavity QED system (i.e. the cavity frequency is offresonant to the atomic transition frequency), the ac-Stark shift has been harnessed for quantum non-demolition measurements of atomic states. The side-band transition is associated with photon-assisted transitions in atomic states. The mechanical analogue of this system, i.e. the role of atomic states dipole-coupled with light fields in high-Q photon cavity are replaced by Zeeman split nuclear spins quadrupole-coupled with mechanical vibrations in a high-Q mechanical resonator as summarized in the table. The corresponding phenomena in a phonon-nucleus system predicted from their resemblance are frequency shift of NMR peak which can be regarded as mechanical ac-Stark shift and side-band NMR peaks under strongly driven mechanical oscillations. Recently both mechanical ac-Stark shift and side-band NMR peaks induced by an intense mechanical strain applied by an GaAs-based high-Q electromechanical resonator are observed [3].

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4.2 Experimental Observation of Mechanical ac-Stark Shift When an external magnetic field B0 is applied to the nuclei, the nuclear spins are Zeeman split to form equally separated energy levels as described by the Hamiltonian Hˆ Z = −ω0 Iˆz , where ω0 is given by ω0 /2π = γ B0 with γ being the gyromagnetic ratio of the relevant nucleus, Iˆz is the z-component of the spin operator Iˆ, and  ≡ h/2π with h being the Planck constant. In case of quadrupole nuclei with spin I = 3/2, a single-quantum process between the Zeeman-split levels allows three degenerated transitions (see Fig. 7) which can be observed as a single NMR peak. In contrast, the coupling of this nucleus to the lattice strain is described by the quadrupole Hamiltonian of the form: Hˆ Q = A1 [3 Iˆz2 − I (I + 1)]/3 + i A2 [ Iˆ+2 − Iˆ−2 ],

(1)

where A1 and A2 are the nucleus-strain coupling constant and Iˆ+ and Iˆ− are the spin ladder operators. In this Hamiltonian, we assume no inter-nucleus interaction. For the oscillatory strain associated with the harmonic mechanical motion of the resonator, the Hamiltonian takes the form Hˆ Q (t) ∝ sin(ωM t) Hˆ Q that is periodic in time. Equation (1) can be analogous to that of an ordinary cavity-QED system irradiated by intense light fields, whose Hamiltonian is of the form ∝ sin(ωL t)(σ+ + σ− ), where sin(ωL t) describes the oscillating electromagnetic fields of frequency ωL and σ+ and σ− are the spin ladder operators. In contrast to the atom-light systems with dipole coupling, the nucleus-strain coupling is quadrupole interaction as described by the terms Iˆz2 , Iˆ+2 and Iˆ−2 , which result in the unique ac-Stark shift different from ordinary one as follows. To quantify the mechanical Stark shift resulting from the quadrupole interaction, we analyze the Hamiltonian based on the Floquet perturbation theory, which is often used to quantify a level shift induced by the ordinary ac-Stark effect in atom-light systems [83, 84]. In this theory, the Hamiltonian Hˆ = Hˆ Z + x(t) Hˆ Q is analyzed by treating the second term as a time-periodic perturbation that causes a level shift. The important conclusion from the Floquet theory is that the time-independent (dc)

Bare nuclei

Static strain

Oscillating strain

Zeeman split

dc-Stark shift

ac-Stark shift

3/ 2

0 1/ 2

0 1/ 2

 

 0  2





0



0 3/ 2



0  2

2nd order transition

Fig. 7 Energy diagram of Zeeman split nuclear spin with I = 3/2 under static and rapidly oscillating mechanical strain

0  2

 0  2 

0  2



Phonon-Electron-Nuclear Spin Hybrid Systems …

261

b ΔfNMR [kHz]

x1 [μm]

a

x1 [μm] Fig. 8 Observation of mechanical ac-Stark shift of nuclear spins under a strongly driven electromechanical resonator. a Plot of displacement x1 of the resonator as a function of driving frequency f with f 0 ∼ 1.7 MHz. b Frequency shift  f NMR of the NMR peak corresponding to a transition |−1/2 ↔ |1/2 plotted as a function of x1

components of x(t) can only contribute to the first-order energy shift, while the timedependent (ac) components of x(t) has no contribution in the first order but first contribute to the second-order energy shift. Hence x(t) is decomposed into dc and ac parts as x(t) = x0 + x1 sin(ωM t) with ωM being the frequency of the mechanical oscillation. The first-order energy shift δ is derived to be δ = ±x0 A1 for any |m, where the positive (negative) sign is for |m = ±1/2 (|±3/2), as depicted in Fig. 7. Since this energy shift depends only on the static mechanical strain (∝ x0 ), it can be regarded as the mechanical dc-Stark shift of nuclear spins. Similarly, the second-order energy shift χ is calculated to be χ = ±3x12 A22 /ω0 , where the positive (negative) sign is for |−1/2 and |−3/2 (|1/2 and |3/2). Since this energy shift quadratically depends on the amplitude of the mechanical oscillation (x12 ), this can be regarded as the mechanical ac-Stark shift. As depicted in the diagram (Fig. 7), the mechanical acStark effect blue shifts only the resonant transition |1/2 ↔ |−1/2 whereas does not change the other transitions |3/2 ↔ |1/2 and |−3/2 ↔ |−1/2. This is because the second-order virtual transitions |−3/2 ↔ |1/2 and |−1/2 ↔ |3/2 depicted by the green arrows equally shift |−3/2 and |−1/2 (|3/2 and |1/2) by χ in positive (negative) direction. The above mentioned mechanical ac-Stark shift can be observed by employing GaAs-based electromechanical resonator combined with resistive-detection of NMR for local nuclei interacting with the MR. The relevant quadrupole nuclei involved in GaAs-based resonator are 75 As, 69 Ga and 71 Ga, all of which are quadrupole nucleus having spin I = 3/2. Among them, 75 As is most appropriate for the investigation of the dynamical nucleus-strain coupling, because its higher natural abundance (100%) and larger quadrupole moment than other nuclei make the experiment easier. The resonant frequency and the quality factor of the fundamental flexural motion of the resonator used in this experiment are ω M /2π ≈ 1.71 MHz and Q ≈ 5 × 104 , respectively. In order to study the nuclear spin states influenced by this motion, the NMR spectra for local nuclear spin ensemble of 75 As contained in the clamping point where the maximal mechanical strain can be induced by the fundamental motion of the resonator. To this end, we adopt resistively detected NMR method using

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a breakdown of odd-integer quantum Hall effect (the details of this unique NMR method is out of scope of this book and described elsewhere [3, 80, 85, 86]). The number of interactive nuclei are estimated to be 109 from the effective volume of this area being approximately 1 µm× 1 µm× 20 nm. Note that the observed NMR frequency is ωNMR /2π ∼ 25 MHz, which is off-resonant to the resonant frequency of the MR, so that the MR and nuclear spins are dispersively coupled. The mechanical ac-Stark shift is expected to be observed when the mechanical resonator is strongly driven. Figure 8a shows a plot of the measured amplitude of the beam resonator x1 as a function of driving frequency. Because of the strong actuation, the mechanical amplitude is deformed to sawtooth-shape reflecting the emergence of duffing nonlinearity [4, 87]. In this situation, the mechanical amplitude can be varied not by varying the driving amplitude but by sweeping the actuation frequency. Figure 8b shows a plot of frequency shift  f NMR of the NMR peak corresponding to the transition |1/2 ↔ |−1/2 as a function of x1 . As expected  f NMR blue shifts with increasing x1 . The solid line in the figure depicts the theoretical value of  f NMR numerically calculated from the Floquet perturbation theory. The experiment and the theory agrees well with each other, suggesting that the observed frequency shift is the observation of the mechanical ac-Stark shift induced by the driven mechanical oscillation. Acknowledgements The authors thank K. Chida, M. Eto, N. Lambert, I. Mahboob, Y. Matsuzaki, G. J. Milburn, W. J. Munro, R. Okuyama, M. Ono, K. Onomitsu, S. Sasaki, S. Shevchenko, and Y. Tokura for fruitful discussions. The authors acknowledge the financial support from Grant-in-Aid for Scientific Research on Innovative Areas No. JP15H05869.

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Cavity Quantum Electrodynamics with Laser-Cooled Atoms and Optical Nanofibers Takao Aoki

Abstract A cavity quantum electrodynamics (QED) system, which consists of an atom and photons confined in a cavity, is one of the most basic hybrid systems. In the strong coupling regime of cavity QED, quantum interaction between atoms and photons manifests itself, and it becomes possible to generate, manipulate, and measure quantum states of atom and light. Therefore, a cavity QED system is an ideal testbed for investigating quantum nature of atom and light. Recently, efforts have been made toward realization of a quantum network by connecting multiple cavity QED systems by optical fibers. However, it is technically challenging to connect a large number of conventional Fabry–Perot cavities with high efficiency. Here, we review novel all-fiber cavity QED systems based on optical nanofibers.

1 Introduction Cavity quantum electrodynamics (QED) is a study of interaction of atoms and photons in optical cavities (resonators) [1]. By utilizing strong confinement of light in a cavity with small mode volume, interaction between one atom and light at the quantum level (with small number of photons or vacuum) manifests itself, and it becomes possible to generate, manipulate, and measure quantum states of atom and light. Therefore, not only is a cavity QED system an ideal testbed for investigating quantum nature of atom and light, but also it is a promising platform for realizing various quantum technologies. Conventionally, single atoms trapped in free-space Fabry–Perot cavities have been utilized for cavity QED in optical frequencies [2]. There have been significant studies such as one-atom laser [3], a deterministic single-photon source [4], vacuum Rabi splitting with one atom [5], photon blockade [6], squeezed light [7], nondestructive detection of a photon [8], a quantum gate between a photon and an atom [9], and that between individual photons [10]. In order to apply these elementary demonstrations to quantum technologies, efforts are being made toward realization of a quantum T. Aoki (B) Department of Applied Physics, Waseda University, Tokyo, Japan e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 Y. Hirayama et al. (eds.), Hybrid Quantum Systems, Quantum Science and Technology, https://doi.org/10.1007/978-981-16-6679-7_12

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Fig. 1 Schematic of a cavity QED system

network by connecting multiple cavity QED systems by optical fibers [11, 12]. However, it is technically challenging to connect a large number of conventional free-space cavities with high efficiency. In order to overcome the poor scalability of free-space cavities, novel cavity QED systems based on nanophotonic devices have been developed. Here, we review novel all-fiber cavity QED systems based on optical nanofibers.

2 Cavity QED 2.1 Jaynes–Cummings Model The basic model of cavity QED system [13–16] is one in which a two-level atom interacts with a single mode of a cavity as shown in Fig. 1. Although actual atoms have complicated energy-level structures, it is usually only limited pairs of levels (which is on- or near-resonant to the cavity) that contribute to the dynamics of a cavity QED system. In the simplest case, there is only a pair of levels that couples to the cavity. The Hamiltonian for a two-level atom with the ground state |g, the excited state |e, and the energy difference between these levels ωA is given by Hˆ A = ωA |ee| = ωA σˆ + σˆ − ,

(1)

where σˆ + = |eg| and σˆ − = |ge| are the raising and lowering operators, respectively. The operator for the cavity field is given by

Cavity Quantum Electrodynamics with Laser-Cooled Atoms and Optical Nanofibers

ˆ E(r) = E 0 u(r)e(aˆ c + aˆ c† ),

267

(2)

where e is the unit vector in the direction of the field, and u(r) is a dimensionless mode function normalized by the maximum field,  E0 =

ωC 2ε0 V

(3)

|u(r)|2 d r

(4)

is the vacuum field amplitude, and  V =

is the mode volume. Actual cavities have a large number of modes. However, the frequency difference between adjacent modes is very large in a typical cavity QED system (in which cavity length is very short). Therefore, in the simplest case, one considers only one mode whose resonance frequency ωC is closest to the atomic resonance frequency ωA . The Hamiltonian for a single-mode cavity is given by   1 † ˆ , HC = ωC aˆ c aˆ c + 2

(5)

where aˆ c† and aˆ c are the photon creation and annihilation operators, respectively. The most dominant interaction between atom and light is the electric dipole interaction, and its Hamiltonian is given by ˆ A ), ˆ · E(r Hˆ int = µ

(6)

ˆ are the position and the transition dipole moment of the atom, where r A and µ respectively. For an isotropic atom, it holds that µgg = µee = 0,

(7)

ˆ j, and where µi j = i|µ| µ ˆ =



|ii|µ| ˆ j j| = µge σˆ − + µeg σˆ + .

(8)

i, j

For simplicity, we assume that µge is real (ı.e., µge = µeg ), and Hˆ int = g(σˆ − + σˆ + )(aˆ c + aˆ c† ),

(9)

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where g=

E 0 u(r) µge · e 

(10)

is the coupling rate between the atom and the cavity field, which corresponds to the Rabi frequency in a semiclassical theory with the vacuum field amplitude at the position of the atom, E 0 u(r). In the interaction picture, the two terms in Eq. (9) that are proportional to σˆ + aˆ c and σˆ − aˆ c† oscillates slowly with frequency |ωA − ωC |. On the other hand, the other two terms that are proportional to σˆ − aˆ c and σˆ + aˆ c† oscillates rapidly with frequency ωA + ωC . Therefore, these two terms do not affect the dynamics of the system under the condition of g  (ωA , ωC ). By neglecting these terms (rotating-wave approximation), the interaction Hamiltonian is given by  = g(σˆ + aˆ c + σˆ − aˆ c† ). Hˆ int

(11)

Note that the rotating-wave approximation becomes invalid for the ultra- (or deep-) strong coupling regime with g  (ωA , ωC ), that can be realized in the superconducting circuit QED system [17]. The resulting Hamiltonian for the system is the Jaynes–Cummings Hamiltonian, which is given by  Hˆ JC = Hˆ A + Hˆ C + Hˆ int

= ωA σˆ + σˆ − + ωC aˆ c† aˆ c + g(σˆ + aˆ c + σˆ − aˆ c† ),

(12)

where we have neglected the vacuum energy offset, ωC /2. It is straightforwad to diagonalize the Jaynes–Cummings Hamiltonian Hˆ JC to obtain the eigenstates and the corresponding eigenenergies. We denote the state of the whole system by |i, n = |iA ⊗ |nC , where |iA and |nC are the states of the atom and cavity, respectively, and n is the photon number of the cavity. The Jaynes– Cummings Hamiltonian couples only states between pairs of |g, n and |e, n − 1. The ground state and its eigenenergy are given by | (0)  = |g, 0, E

(0)

= 0,

(13) (14)

which are the same as that for the case of no interaction. The excited states are superpositions of |g, n and |e, n − 1. For the case of ωA = ωC , the eigenstates and the corresponding eigenenergies are given by 1 |±(n)  = √ (|g, n ± |e, n − 1) , 2 √ E ±(n) = nωA ±  ng.

(15) (16)

Cavity Quantum Electrodynamics with Laser-Cooled Atoms and Optical Nanofibers

269

Fig. 2 Jaynes–Cummings ladder

√ It can be seen from Eq. (16) that the eigenenergies are shifted by ± ng from the case for no interaction, nωA . The energy level structure of the Jaynes–Cummings Hamiltonian is called the Jaynes–Cummings ladder, and the energy splitting of 2g for n = 1 excited states is called vacuum Rabi splitting (Fig. 2). The dynamics of the system is described by the Schrodinger equation, i

∂ |ψ(t) = Hˆ JC |ψ(t). ∂t

(17)

For example, let us assume that the cavity is resonant to the atom, ωC = ωA , and that at t = 0 the atom in the excited state |e and photon number in the cavity is zero, |ψ(t = 0) = |e, 0. Then the state of the system at later time t > 0 is given by |ψ(t) = cos gt|e, 0 + i sin gt|g, 1.

(18)

The probability that the system stays in the initial state Pe,0 (t) is given by Pe,0 (t) = e, 0|ψ(t)|e, 0 =

1 1 + cos 2gt, 2 2

(19)

which oscillates with a frequency of 2g (see also Fig. 3). This can be interpreted as the Rabi oscillation induced by vacuum.

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Fig. 3 Probability of the system staying in the initial state Pe,0 (t) for the case of |ψ(t = 0) = |e, 0. Blue, red, green lines corresponds to the cases of no dissipation ((g, κ, γ) = 2π × (30, 0, 0) MHz), strong-coupling regime ((g, κ, γ) = 2π × (30, 3, 1) MHz), and Purcell regime ((g, κ, γ) = 2π × (30, 100, 1) MHz), respectively

2.2 Dissipations So far we have considered an ideal closed system. However, in a real world, any system is an open system that is coupled with its environment, and the state of the system in general is no longer a pure state described by a state vector, but a mixed state described by a density matrix. It should be noted that, in optical frequencies, energy of a photon (ω ≈ 4 × 10−19 J for the wavelength of 500 nm) is much larger than the thermal energy of room temperature (kB T ≈ 4 × 10−21 J for T = 300 K), and thermal excitation of the system due to the coupling to the environment can be neglected. Therefore, we only need to consider the decay of the system. Firstly, the atom in the excited state decays to the ground state by irreversibly emitting a photon into the free space (spontaneous emission), as a consequence of coupling to the free-space mode. The energy decay rate by spontaneous emission, or the Einstein A coefficient, is given by  2 1 4ωA3 µge  , = 4πε0 3c3

(20)

where c is the speed of light. The corresponding amplitude decay rate is given by

Cavity Quantum Electrodynamics with Laser-Cooled Atoms and Optical Nanofibers

γ=

1 . 2

271

(21)

Secondly, photons in the cavity decays to the environment as a consequence of finite loss of the cavity. There are two types of cavity losses: external loss and internal loss. The external loss is the escape of photons from the cavity through the finite transmittance of the cavity mirrors. The escaped photons can propagate in a single spatial mode (usually the fundamental mode of Gaussian beam) in free space, and can be coupled to another system (cavity, waveguide, etc.) or detected by a photon detector. On the other hand, the internal loss is the pure loss of photons due to absorption or scattering by the mirrors, or propagation loss inside a cavity. The field decay rate for external and internal losses are given by cTi , 4L cα , κin = 4L

κi,ex =

(22) (23)

where Ti , α, L are the transmittance of the mirror i, internal loss per round trip, and the cavity length, respectively. The total field decay rate is given by κ=



κi,ex + κin .

(24)

i

The total energy decay rate 2κ is equal to the spectral full-width at half-maximum of the resonance FWHM , and to the inverse of the photon lifetime of the cavity, τ = 1/(2κ). The ratio of the external decay rate to the total decay rate, ηi,esc =

κi,ex κ

(25)

is called the cavity escape efficiency (for the mirror i), and it corresponds to the extraction efficiency of a photon to the output mode. Finesse and quality factor are often used to quantify the confinement of photons in the cavity. Finesse F is given by F=

FSR 2π , = FWHM i Ti + α

which depends only on the total loss per round trip, given by Q=

ωC FWHM

=

 i

4ωC L , c(T + α)

(26) Ti + α. Quality factor Q is

(27)

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which depends not only on the total loss per round trip, but also the resonance frequency ωC and the cavity length L.

2.3 Dynamics The dynamics of the system density matrix ρˆ is described by the following master equation [13–16],

 i d ρˆ = − Hˆ JC , ρˆ + γ 2σˆ − ρˆσˆ + − ρˆσˆ + σˆ − − σˆ + σˆ − ρˆ dt  + κ 2aˆ c ρˆaˆ c† − ρˆaˆ c† aˆ c − aˆ c† aˆ c ρˆ .

(28)

As in Sect. 2.1, let us assume that the cavity is resonant to the atom, ωC = ωA , and that at t = 0 the atom in |e and photon number in the cavity is zero, i.e., ρ(t ˆ = 0) = |e, 0e, 0|, the probability that atom stays in |e and the photon number in the cavity is still zero at t > 0 is given by ˆ 0 Pe,0 (t) = e, 0|ρ(t)|e, 2       1 κ − γ − κ+γ −i g˜ t ˜ )t  ) + 1 − κ − γ e−( κ+γ 2 +i g e ( 2 =  +  , 2 i g˜ 2 i g˜

(29)

where  g˜ =

 g2

−

κ−γ 2

2 .

(30)

There are two regimes that show distinct effects of cavity QED. Strong coupling regime (g (κ, γ)) In this regime, Eq. (29) reads as  Pe,0 (t) ≈

 1 1 + cos 2gt e−(κ+γ)t , 2 2

(31)

which decays exponentially with a rate κ + γ but still shows vacuum Rabi oscillation as in the case of a closed system. In this regime, coherent interaction between atom and light dominates the system dynamics. 2 Purcell regime (κ gκ γ ) In this regime, Eq. (29) reads as Pe,0 (t) ≈ e−(1+2C)t , where we have defined the cooperativity parameter,

(32)

Cavity Quantum Electrodynamics with Laser-Cooled Atoms and Optical Nanofibers

g2 C= 2κγ

  g2 = . κ

273

(33)

In this regime, vacuum Rabi oscillation can not be observed, and the system irreversibly decays to the ground state as for the case of an atom in free space. However, the decay rate (1 + 2C) is faster than that for the free-space case . This is because the atom in the excited state decays to the ground state by emitting a photon into the cavity mode with a rate 2C, in addition to the usual spontaneous emission into the free space with a rate . This enhancement of spontaneous emission into the cavity mode is called the Purcell effect. The enhancement factor (Purcell factor) can be also derived from Fermi’s golden rule as [18] 2C =

3 Q 3 λ , 4π 2 V A

(34)

where λA = 2πc/ωA , and it agrees with Eq. (33). The cooperativity parameter C is a figure of merit for various effects in cavity QED, not limited to the Purcell√effect. Let us consider the conditions to achieve high cooperativity. Since g ∝ μge / V , κ ∝ (T + α)/L, γ ∝ μ2ge , and V = AL, where A is the effective mode area at the position of atom, it holds that 1 C∝ (T + α)A



F ∝ A

 .

(35)

Therefore, the cooperativity does not depend on the atomic dipole moment μge or the cavity length L, but only on the total loss of the cavity T + α (or Finesse F) and the mode area A. It is noteworthy that cooperativity is independent of atomic parameters. In order to achieve high cooperativity, one only needs to reduce total internal loss and the mode area of the cavity.

2.4 Deterministic Generation of Single Photons The Purcell effect can be utilized to deterministically generate single photons [19]. When atom is excited to the excited state by e.g., a short π-pulse, a photon is generated selectively in the cavity mode by the Purcell effect. The efficiency of photon generation in the cavity mode (internal efficiency) is calculated as  ηint = 0

∞

Pg,1 (t)dt =

2C κ . κ + γ 1 + 2C

(36)

The overall efficiency of as a photon source is given by the product of this internal efficiency and the escape efficiency of the cavity. That is,

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ηtotal = ηesc ηint =

2C κex . κ + γ 1 + 2C

(37)

For a single photon source based on the Purcell effect, the temporal pulse shape is not controllable. On the other hand, by using cavity-enhanced Raman scattering [20, 21] or vacuum-stimulated Raman adiabatic passage (vSTIRAP) [22–25], it is possible to deterministically generate single photons with controllable pulse shapes. For these methods, the overall photon generation efficiency in the adiabatic (long pulse) limit is given by [26] ηtotal =

κex 2C , κ 1 + 2C

(38)

which is larger than the efficiency for the photon generation based on the Purcell effect. Indeed, Eq. (38) is the fundamental upper limit for the overall efficiency of photon generation using cavity QED systems [26]. In experiments, while the external loss κex is controllable, κin is not. If one increases κex , the escape efficiency ηesc = κex /κ increases, but the cooperativity C decreases (hence the internal efficiency ηint = 2C/(1 + 2C) decreases). Therefore, there is a trade-off between the escape efficiency and the internal efficiency. The overall efficiency takes the maximum value (max) =1− ηtotal

1+

√

2 , 1 + 2Ci

(39)

when the external loss rate takes the optimum value

κ(opt) = κin 1 + 2Ci . ex

(40)

Here, we have defined the internal cooperativity Ci =

g2 2κi γ

 =

1 2α A˜

 ,

(41)

where A˜ = A/σ is the effective mode area at the position of the atom, normalized by the absorption scattering area of the atom, σ=

3λ2A . 2π

(42)

The internal cooperativity Ci is also a figure of merit for quantum gate operations using cavity QED [27].

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275

2.5 Input and Output The coupling between the cavity mode aˆ c and the input and output modes aˆ in , aˆ out is given by the input–output relation [15, 29, 30] aˆ out = aˆ in +

2κex aˆ c .

(43)

We define the operators in the frequency domain by 1 ˆ A(ω) =√ 2π



∞ −∞

eiωt a(t)dt. ˆ

(44)

The amplitude transmission and reflection coefficients of the system for the input from mirror 1 are given by t1 (ω) =

 Aˆ 2,out (ω) ,  Aˆ 1,in (ω)

(45)

r1 (ω) =

 Aˆ 1,out (ω) .  Aˆ 1,in (ω)

(46)

When we input a weak probe beam with the frequency of ωp to the mirror 1, the amplitude transmission and reflection coefficients of the system in the weak-driving limit (i.e., linear regime) are given by [16] (iA + γ) , (iC + κ)(iA + γ) + g 2 (iA + γ) , r1 (ωp ) = 1 − 2κ1,ex (iC + κ)(iA + γ) + g 2

√ t1 (ωp ) = −2 κ1,ex κ2,ex

(47) (48)

where A = ωp − ωA and iC = ωp − ωC are the detuning of the probe beam from the atom and that from the cavity, respectively. Figure 4a, b show the transmission and reflection spectra for the strong-coupling regime with (g, κ1,ex , κ2,ex , κin , γ) = 2π × (30, 1.2, 1.2, 0.6, 1) MHz. Vacuum Rabi splitting with 2g = 2π × 60 MHz can be observed. On the other hand, Fig. 4c, d show the transmission and reflection spectra for the Purcell regime with (g, κ1,ex , κ2,ex , κin , γ) = 2π × (30, 45, 45, 10, 1) MHz. The resonance of the cavityenhanced atom with the width of (1 + 2C) can be observed at the center of a much wider resonance of the cavity with the width of 2κ.

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(a)

(c)

(b)

(d)

Fig. 4 a, c Transmission and b, d reflection spectra in the weak-driving limit. Red and blue lines are for with and without an atom, respectively. The cavity QED parameters are (g, κ1,ex , κ2,ex , κin , γ) = 2π × (30, 1.2, 1.2, 0.6, 1) MHz for a and b (strong-coupling regime), and (g, κ1,ex , κ2,ex , κin , γ) = 2π × (30, 45, 45, 10, 1) MHz for c and d (Purcell regime)

2.6 Transfer Matrix Method The quantum optical model described above assumes a single-mode cavity with high finesse. However, for a nanofiber cQED system that will be described in Sect. 3, strong-coupling regime can be achieved with a cavity length on the order of a meter and finesse on the order of ten. In such conditions, the assumption of the high-finesse single-mode may become invalid. For the calculation of the linear optical response of the system in the weak-driving limit, alternative model using transfer matrices can be used [28]. The transfer matrix for a cavity QED system in Fig. 5 is given by  T (cQED) =

(cQED)

(cQED)

T11 T12 (cQED) (cQED) T21 T22



= T (M1) T (d) T (A) T (l−d) T (M2)      −i kd √ ˜ i 1 − iξ −iξ R1 0 −1 e √ = √ ˜ iξ 1 + iξ T1 − R1 T1 + R1 0 ei kd     √ ˜ i e−i k(l−d) 0 R2 −1 √ × , √ ˜ T2 − R2 T2 + R2 0 ei k(l−d)

(49) (50)

(51)

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Fig. 5 Schematic for modelling a cavity QED system with transfer matrix method. A two-level atom is placed at a distance of d from the mirror 1 in a cavity with a length of l. E i,in(out) are the input (output) field amplitudes for the mirror i. T (M1) , T (M2) , T (d) , T (l−d) , T (A) are the transfer matrices for the mirror 1, 2, free propagation of length l, l − d, and the atom, respectively

and ˜

e−i kl  ˜ (1 − iξ) − iξ R1 e2i kd T11(cQED) = − √ TT

 1 2

˜ ˜ − iξ + (1 + iξ) R1 e2i kd R2 e2i k(l−d) ,

(52)



e ˜ T12(cQED) = − √ −(1 − iξ) + iξ R1 e2i kd R2 T1 T2



˜ ˜ + iξ + (1 + iξ) R1 e2i kd (T2 + R2 )e2i k(l−d) ,

(53)

˜

e−i kl  ˜ T21(cQED) = − √ (1 − iξ) R1 − iξ(T1 + R1 )e2i kd TT

 1 2 ˜ ˜ − iξ R1 + (1 + iξ)(T1 + R1 )e2i kd R2 e2i k(l−d) ,

(54)

˜

e−i kl  ˜ T22(cQED) = − √ −(1 − iξ) R1 + iξ(T1 + R1 )e2i kd R2 TT

 1 2 ˜ ˜ + iξ R1 + (1 + iξ)(T1 + R1 )e2i kd (T2 + R2 )e2i k(l−d) ,

(55)

˜ −i kl

where Ti and Ri is the transmittances and reflectances of the mirror i, k˜ = ω/c − i ln(1 − α)/(4l) is the complex wave number, respectively. ξ is given by [31] ξ=−

1D , i + 2p

where 1D is the energy decay rate into the guided mode and given by

(56)

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T. Aoki

1D =

π 2 g . ωFSR

(57)

For simplicity, we take d = 0 and obtain T

(cQED)

   1 2   (ω) =  (cQED)   T 11  √   (1 − α)T1 T2  =  2πC √   √ √ i  (1 − iξ) − iξ R1 − iξ + (1 + iξ) R1 R2 (1 − α)e ωFSR

(58) 2    ,  

(59) 2   T (cQED)    R (cQED) (ω) =  21(cQED)  (60)  T 11 

 =  (1 − iξ) R1 − iξ(T1 + R1 ) 

2πC 2  i R2 (1 − α)e ωFSR  − iξ R1 + (1 + iξ)(T1 + R1 )  





2πC −2   i ×  (1 − iξ) − iξ R1 − iξ + (1 + iξ) R1 R2 (1 − α)e ωFSR  . (61) If we take the limit of high finesse, (α, 1 − R1 , 1 − R2 )  1, and if we consider only the vicinity of one mode, C  ωFSR , Eqs. (59) and (61) reproduce the quantum results, Eqs. (47) and (48).

3 Nanofiber Cavity QED Remarkable progress has been made by using free-space Fabry–Perot cavities in the study of cavity QED in optical frequencies, and further efforts are being made toward realization of a quantum network by connecting multiple cavity QED systems by optical fibers [11, 12]. Although an elementary quantum network with probabilistic distribution of entanglement has been demonstrated [32], it is technically challenging to connect a large number of these conventional Fabry–Perot cavities with high efficiency. Therefore, novel cavity QED systems based on various nanophotonic cavities are being developed. In the following, we review a nanofiber cavity QED system.

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Fig. 6 Schematic of a nanofiber cQED system

3.1 Strong Coupling with a Nanofiber Cavity and Single Trapped Atoms Figure 6 shows the schematic of a nanofiber cavity QED system. The waist region of the nanofiber has a sub-wavelength diameter [33]. Single mode of light is guided by the large index difference between the fiber and the surrounding vacuum, ı.e., the fiber itself functions as a core and the surrounding vacuum functions as a cladding of a single-mode fiber. This waist region is seamlessly connected to a standard singlemode optical fiber by a tapered region, whose diameter continuously changes from that of the waist region to that of the single-mode fiber. Because of the large index difference between the nanofiber waist and the vacuum, and of the small waist radius, the cross-sectional area A˜ of the nanofiber waist is extremely small, on the order of unity. Therefore, an atom placed in the evanescent field interacts efficiently with the guided light [34]. Atom can be trapped by using the repulsive potential by the evanescent field of the blue-detuned guided light and the attractive potential by the evanescent field of the red-detuned guided light [35, 36]. For Cesium atoms, light shifts of the excited and ground states by the trap light can be cancelled by using the so-called magic wavelength [37–39]. Furthermore, an all-fiber Fabry–Perot cavity can be constructed by placing fiber Bragg gratings (FBGs) on the waist [40] or the outside of the tapered regions [41]. We have realized an all-fiber cavity QED system in the strong coupling regime with a nanofiber cavity and single trapped Cesium atoms, and we have achieved the cavity QED parameters of (g, κ, κin , γ) = 2π × (7.8, 6.4, 3.2, 2.6) MHz with the corresponding internal cooperativity of Ci = 3.7 [42].

3.2 Fabrication of Low-Loss Tapers As described in Sect. 2, the internal cooperativity Ci is a universal figure of merit for a cavity QED, and it is inversely proportional to the product of the round-trip loss ˜ A nanofiber cavity has an extremely of the cavity α and the effective mode area A.

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Fig. 7 Delineation angle as a function of fiber radius, for a step-index single-mode fiber (wavelength 850 nm)

small effective mode area on the order of unity, which is three order of magnitude smaller than the typical value for a conventional free-space Fabry–Perot cavity. On the other hand, it is not trivial to fabricate a nanofiber cavity with small round-trip loss. This is because the tapered regions inside the nanofiber cavity has an inherent loss due to the coupling between the fundamental mode and the higher-order modes, which occurs because of the continuous change of the diameter. This inherent loss can be minimized by designing the shape of the tapered regions to satisfy the adiabatic condition [43]. Specifically, the modal coupling between the mode 1 and 2 is negligible if the local taper angle θ(r ) at the radius r is much smaller than the delineation angle (r ) defined as (r ) =

r (β1 (r ) − β2 (r )), 2π

(62)

where β1 (r ) and β2 (r ) are the propagation constants for the mode 1 and 2, respectively (Fig. 7). Figure 8 shows the schematic of the taper fabrication setup. A piece of optical fiber is held to two motorized stages that translates in opposite directions. A hydrogen torch with small flame scan along the fiber back and forth to heat it. Transmission of a power-stabilized laser through the fiber is continuously monitored during the

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Fig. 8 Schematic of the taper fabrication setup

Fig. 9 Schematic of the FBG fabrication setup

fabrication. We have developed a method to fabricate tapered regions that satisfy the above adiabatic condition and achieved a single-pass transmission through a pair of two tapered regions exceeding 99.7% [44]. Currently we are able to reproducibly fabricate tapers with transmissions >99.9%.

3.3 Fabrication of Low-Loss FBGs In order to reduce the total internal loss of the cavity, It is also important to reduce the internal loss of FBGs. Figure 9 shows the schematic of the setup for fabricating FBGs. A DUV laser beam is incident on a phase mask to be predominantly diffracted into the ±1-st order. A piece of optical fiber is placed in the proximity of the output side of the phase mask, where the ±1-st order diffracted beams overlap and interfere. This interference results in the periodic intensity modulation of the DUV light along the core of the fiber. The core of a standard single-mode optical fiber is doped with germanium, and photo-sensitive to DUV light. That is, its refractive index slightly changes when exposed to DUV light, and the index change depends on the intensity of the DUV light. In this way, a periodic modulation of the refractive index in the optical fiber can be induced. We have developed a method to fabricate FBGs with internal losses as low as 0.3%.

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4 Coupled Cavities QED One of the advantages of nanofiber cavity QED systems is its scalability. Thanks to the all-fiber structure of the cavity, multiple systems can be connected in all-fiber fashion, just by fusion-splicing the fiber. Thus, all-fiber cavity QED network can be constructed. We have constructed a coupled-cavities QED system by connecting two nanofiber cavity QED systems [45, 46]. The Hamiltonian for the coupled-cavities QED system (see Fig. 10) is given by [45] Hˆ CCQED =



Hˆ i + Hˆ m +

i=1,2



Hˆ mi ,

(63)

i=1,2

  Hˆ i = ωCi aˆ i† aˆ i + ωA σˆ i+ σˆ i− + gi aˆ i† σˆ i− + aˆ i σˆ i+ , ˆ Hˆ m = ωm bˆ † b,   Hˆ mi = vi aˆ i† bˆ + aˆ i bˆ † ,

(64) (65) (66)

where Hˆ i are the Jaynes–Cummings Hamiltonian for the cavity QED system i, Hˆ m is the Hamiltonian for the connecting fiber, and Hˆ mi are the Hamiltonian for coupling between the connecting fiber and the cavity i, respectively. The coupling coefficient vi in Hˆ mi is given by  c vi = 2

Ti , Lf Li

Fig. 10 Schematic of an all-fiber coupled-cavities QED system

(67)

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where L i , L f , Ti are the length of the cavity i, the length of the connecting fiber, and the transmittance of the coupling mirror between the connecting fiber and the cavity i, respectively. We define the normal mode operators as 1  v2 aˆ 1 + v1 aˆ 2 , dˆ = √ 2v˜ 1  1 ˆ v1 aˆ 1 + v2 aˆ 2 ± √ b, cˆ± = 2v˜ 2

(68) (69)

where  v12 + v22 . v˜ = 2

(70)

Then the Hamiltonian Hˆ CCQED can be expressed in terms of these normal mode operators as     √  † √  † Hˆ = ωA dˆ † dˆ +  ωC + 2v˜ cˆ+ cˆ+ +  ωC − 2v˜ cˆ− cˆ− + ωA σˆ 1+ σˆ 1− + σˆ 2+ σˆ 2−  

  † † + v1 g1 σˆ1 − + v2 g2 σˆ 2− + H.c. cˆ+ + cˆ− 2v˜ 

 † +√ d v2 g1 σˆ 1− + v1 g2 σˆ 2− + H.c. . 2v˜

(71)

Linear optical response of the coupled-cavities QED system can be also calculated with the transfer matrix method described in Sect. 2.6. Specifically, the transfer matrix for a coupled-cavities QED system (see Fig. 11) is given by

Fig. 11 Schematic of an all-fiber coupled-cavities QED system

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T. Aoki

T (ccQED) = T (CCQED1) T (L f ) T (CCQED2) =

−iπω(1/ωFSR1 +1/ωFSRf +1/ωFSR2 )



√





(72)

R1 −1 1 − iξ1 −iξ1 √ √ iξ1 1 + iξ1 − R T + R η1 ηf η2 T1 T2 T3 T4 1 1 1      √ √ 1 0 1 0 R2 R3 −1 −1 √ √ 2π 2π × C1 Cf i i − R2 T2 + R2 − R3 T3 + R3 0 η1 e ωFSR1 0 ηf e ωFSRf       √ 1 0 R4 −1 1 − iξ2 −iξ2 √ 2πC2 × , (73) i iξ2 1 + iξ2 − R4 T4 + R4 0 η2 e ωFSR2

e

where Ti and Ri are the transmittance and reflectance for the mirror i, ηi = 1 − αi are unity minus the round-trip loss of the cavity i, and η f = 1 − α f is the single-pass transmittance of the connecting fiber, respectively.  2   Figure 12 shows the calculated transmission spectrum, 1/T11(ccQED)  , reflec 2   tion spectrum from mirror 1, T21(ccQED) /T11(ccQED)  , and that from mirror 4,  2   (ccQED) /T11(2cQED)  , for the following set of parameters: R1 = 1 − T1 = −T21 R4 = 1 − T4 = 0.95, R2 = 1 − T2 = R3 = 1 − T3 = 0.9, η1 = η2 = ηf = 0.99, (ωFSR1 , ωFSRf , ωFSR2 ) = (750, 200, 450) × 2π MHz, ωC1 = ωCf = ωC2 = ωC , γ = 2.6 × 2π MHz, gC1 = 9.1 × 2π MHz, and gC2 = 7.1 × 2π MHz. For the case of no atom, three resonances are observed in the transmission and reflection spectra. The center resonance at p = 0 corresponds to the fiber-dark mode √ ˆ while the two side resonances at p = ± 2v˜ to the bright modes cˆ± . d, For the case of one- and two-atom cases, the fiber-dark mode resonance is split into two √ resonances. In order to interpret this, we assume that the normal mode splitting 2v˜ is large compared to all other parameters and derive the reduced Hamiltonian for the vicinity of the resonance p ≈ 0 as Hˆ d = ωC dˆ † dˆ +

 

 ωA σˆ i+ σˆ i− + gdi dˆ † σˆ i− + σˆ i+ dˆ ,

(74)

i=1,2

where v2 v1 gd1 = √ g1 , gd2 = √ g2 . 2v˜ 2v˜

(75)

The Hamiltonian Hˆ d is nothing but the Jaynes–Cummings Hamiltonian for a cavity with a single mode d interacting with atom i with the coupling rate gdi . Therefore, the splitting of the fiber-dark mode resonance can be interpreted as the vacuum Rabi splitting between one (or two) atom(s) and the fiber-dark mode of the coupled cavities. On the other hand, the two resonances of the bright modes show slight shift due to the off-resonant coupling to the atoms. These spectra have been experimentally observed in [45].

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Fig. 12 Transmission and reflection spectra of a coupled-cavities QED system calculated with    (ccQED) 2 a transfer-matrix method. (a) transmission spectrum, 1/T11  , (b) reflection spectrum from      (ccQED) (ccQED) 2  (ccQED) (ccQED) 2 mirror 1, T21 /T11 /T11  , and (c) reflection spectrum from mirror 4, −T21 

The coupled-cavities QED system in the weak-driving limit is analogous to five coupled oscillators in classical mechanics. Indeed, one can diagonalize the full Hamiltonian Hˆ CCQED to find the eigenstates for the first excited states (see Fig. 13) and the corresponding eigenenergies given by:

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T. Aoki

Fig. 13 Five normal modes of the coupled-cavities QED system. Circles and ellipses indicate excitations of atoms and cavities, respectively. Red and blue indicate π out of phase

|BS1 ∝ g|A1  + g|A2  + ζ|C1  + ζ|C2  + 2v|F, |BS2 ∝ g|A1  + g|A2  − ζ|C1  − ζ|C2  + 2v|F,

(ω0 + ζ), (ω0 − ζ),

(76) (77)

|FD1 ∝ |A1  − |A2  + |C1  − |C2 , |FD2 ∝ |A1  − |A2  − |C1  + |C2 ,

(ω0 + g), (ω0 − g),

(78) (79)

ω0 ,

(80)

|CD ∝ v|A1  + v|A2  − g|F,

where |A1  = |e, g, 0, 0, 0, |A2  = |g, e, 0, 0, 0, |C1  = |g, g, 1, 0, 0, |C2  = |g, g, 0, 1, 0, and |Cf  = |g, g, 0, 0, 1 are the first excited states with no interaction (|i 1 , i 2 , n 1 , n 2 , n f  denotes the state of the total system with atom 1 and 2 in the states i 1 and i 2 ; and cavity 1, 2, and the fiber in the Fock states of photon numbers n 1 , n 2 , and n f , respectively). |BS1 and |BS2 are the bright modes that correspond to the two side resonances, while |FD1 and |FD2 correspond to the vacuum-Rabi split peaks of the fiber-dark mode described above. On the other hand, |CD is the cavity-dark mode which can not be observed in the transmission or reflection spectra. This is because the cavity-dark mode has no contribution from the excitation of the either of the two cavities, and this mode can not be either excited or detected by the transmission or reflection measurement. The cavity-dark mode can be observed by inserting a fiber beam splitter in the connecting fiber and directly driving and detecting the connecting fiber mode (see Fig. 14). All five normal modes including the cavity-dark modes have been experimentally observed in [46].

5 Outlook As described above, significant advantage of the nanofiber cavity QED system is its scalability. Observation of dressed states of distant atoms and delocalized photons was made possible because multiple systems can be connected with very low loss, just by fusion-splicing them together. Although the elementary demonstration above

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287

Fig. 14 Setup for observing the cavity-dark mode. Fiber beam splitter is inserted in the connecting fiber

utilized many-atom systems, coupled-cavities QED with a single atom in each cavity will be realized by using recently developed high-finesse nanofiber cavities. With such a system, it will become possible to deterministically generate entanglement between distant atoms, which will be one of the key technologies to realize distributed quantum computing [47].

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31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47.

Robust Quantum Sensing Yuichiro Matsuzaki

Abstract Quantum sensors can be potentially used to outperform the classical sensors. The uncertainty of the estimation of the target field scales as T −0.5 with classical sensors where T denotes the total measurement time. On the other hand, about the uncertainty by using quantum sensors, it is in principle possible to obtain a scaling of 1/T , which is called a quantum scaling. However, since the quantum states are fragile against decoherence, it is not clear whether the quantum sensing is practically useful or not. Here, we propose a scheme to achieve the scaling of 1/T about the uncertainty with quantum sensors under the effect of dephasing. The crucial idea is to use quantum teleportation for the suppression of the dephasing. The quantum teleportation presents a system-environment correlation from growing while the quantum state acquires an information about the target fields. Our results suggest a practical approach to achieve the quantum scaling with the quantum sensing. Keywords Quantum sensing · Entanglement · Quantum teleportation

1 Introduction Magnetic field sensing is one of the important techniques in engineering, medical science, and biology. For example, electron spin resonance and nuclear magnetic resonance are important technique to investigate the materials, and magnetic field sensors can be used to detect these signals. Detection from the tiny magnetic fields from the brain is useful to find the location of tumors. Magnetic resonance imaging (MRI) is also used to provide the imaging of the target materials. Qubit based magnetic field sensor has attracted a lot of attention in recent years. Conventionally, a qubit has been considered as a component for quantum computation. On the other hand, recent studies show a potential of the qubits for the magnetic field sensor. Solid state qubits such as nitrogen vacancy centers [1–3] in diamond and superconducting flux qubit [4–6] interact with magnetic fields. More specifically, the Y. Matsuzaki (B) National Institute of Advanced Industrial Science and Technology, Ibaraki 305-8568, Japan e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 Y. Hirayama et al. (eds.), Hybrid Quantum Systems, Quantum Science and Technology, https://doi.org/10.1007/978-981-16-6679-7_13

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frequency of the qubit can be shifted by applying DC magnetic fields. It is known that Ramsey type measurement provides the information of the qubit frequency. So we can estimate the amplitude of the magnetic field from the Ramsey measurement results with the qubit [7]. It is known that the classical sensor can decrease the uncertainty of the estimation by a square root of the total measurement time T while a quantum sensor can in principle decrease the uncertainty of the estimation with time to 1/T in an ideal condition [8–10]. This shows a potential that the quantum sensor can outperform the classical sensor when the total measurement time is long. However, the quantum states are fragile against dephasing due to an unavoidable coupling with the environment, and so it has not been clear whether we can obtain such a quantum scaling of 1/T under the effect of dephasing. In this article, we introduce a novel scheme to obtain the quantum scaling of 1/T for the uncertainty under the effect of dephasing [11]. The key idea of our scheme is to utilize one way quantum computing based teleportation between qubits [12, 13]. We show that frequent implementations of the quantum teleportation can suppress the dephasing while the globally applied magnetic fields can induce the relative phase with the quantum states of the qubits. Our method has a potential to realize a practical quantum sensor that outperforms the classical sensors.

2 Magnetic Field Sensing with Qubits Without Decoherence Let us describe the details of the conventional magnetic field sensing with qubits without any decoherence [7, 14]. Suppose that L qubits are given, and we will perform the field sensing within the total time of T . In this section, we use separable states for the qubits. The Hamiltonian is given as follows H=

L  ω j=1

2

σˆ z(j)

(1)

where ω denotes the frequency of the qubit. We assume that ω has a linear relationship with the amplitude of the applied DC magnetic field. So the estimation of ω let us know the amplitude of the target magnetic fields. In the magnetic  field sensing with the qubit, we adopt the following steps. Firstly, prepare a state of Lj=1 |+j . Secondly,  let this state evolve for a time t by the Hamiltonian, and obtain Lj=1 |ψ(t)j where ωt ωt |ψ(t)j = √12 (e−i 2 |1j + ei 2 |0j ). Third, we measure each qubit by an observable σˆ y , and obtain L measurement results. Finally, we repeat these three steps with N times. The process explained above is called a Ramsey measurement, and this is the conventional method to measure DC magnetic fields by qubits.

Robust Quantum Sensing

291

We can calculate the uncertainty of the estimation for the magnetic field sensing with separable L qubits as follows. From the process explained above, we obtain NL measurement results {xk }NL k=1 where xk is either +1 or −1, and we can calculate 1 NL the average value xav = NL k=1 xk . By assuming that the target magnetic field is small to satisfy |ω|t  1, the expectation value of the measurement on the j-th qubit is σˆ y  = j ψ(t)|σˆ y |ψ(t)j  ωt, and so we can obtain the estimated value as ωest = xav /t. Obviously, the estimation value should have a finite deviation from the real value ω and the variance can be defined as δ 2 ω (sep) = (ω − ωest )2 where the overline denotes the ensemble average. If the number of the measurement results is sufficiently large, we can calculate the uncertainty of the estimation as 

δω

(sep)

=

δ σˆ y δ σˆ y  d σˆ y  √ | d ω | NL

(2)

where δ σˆ y = σˆ y − σˆ y  [7, 14]. Throughout of this article, we assume that the time for the state preparation and state readout is much shorter than the interaction time t. In this case, we obtain N  T /t where T denotes the total measurement time. Then, we can obtain a simplified form of the uncertainty as follows 1 , δω (sep)  √ LTt

(3)

where we assume that the target magnetic field is small to satisfy sin ωt  ωt. We define the sensitivity as the inverse of δω (sep) in this article. It is worth mentioning that the sensitivity becomes better as we increase t. By setting t = T , we obtain 1 . So we get δω (sep) = (T −1 ), and this is considered as a quantum δω (sep)  T √ L scaling [8–10] while δ (sep) ω = (T −1/2 ) is the typical classical scaling which is derived from the central limit theorem. On the other hand, about the scaling with L, we have δω (sep) = (L−1/2 ), and this is called a standard quantum limit (SQL) [15].

3 Magnetic Field Sensing with Entangled States Without Decoherence It is known that the use of entanglement can improve the sensitivity in the magnetic field sensing [14, 16, 17], and we describe it in this section. Similar to the case of the separable states explained above, we assume that L qubits are available and the field sensing should be performed within the total time of T . We consider the following entanglement with M qubits 1 ) |φ(M GHZ  = √ (|11 · · · 1 + |00 · · · 0) 2

(4)

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Y. Matsuzaki

and this is called a GHZ state. In a conventional scheme for quantum enhanced magnetic field sensing, the GHZ states are used as follows. Firstly, prepare L/M sets of GHZ states with the size of M . Secondly, let this state interact with the magnetic iωMt ) − iωMt √1 2 |11 · · · 1 + e 2 |00 · · · 0). fields with a time t, and we obtain |φ(M GHZ (t) = 2 (e Thirdly, we perform a measurement on each GHZ state by a projection operator ) (M ) (M ) √1 of Pˆ = |φ(M GHZ,⊥ φGHZ,⊥ | where |φGHZ,⊥  = 2 (|11 · · · 1 + i|00 · · · 0) is a state orthogonal with the initial GHZ state, and obtain L/M measurement results. Finally, repeat these three steps N times. We can calculate the uncertainty of the estimation of the magnetic field with such GHZ states as follows. From the process above, we obtain NL/M meaNL/M surement results {yk }k=1 where yk is either +1 or −1, and we can calculate NL/M 1 the average value yav = (NL/M k=1 yk . By assuming that the target magnetic ) field is small to satisfy M |ω|t  1, the expectation value of the measurement is ˆ (M ) (t)j  M ωt, and so we can obtain the estimated value ˆ = j φ(M ) (t)|P|φ P = P GHZ GHZ (en) TL as ωest = yav /Mt. With a large number of measurement results ( NL  tM  1), the M uncertainty of the estimation can be calculated as  δω

(en)

=

) ˆ ˆ (M ) φ(M GHZ (t)|δ Pδ P|φGHZ (t)  | ddPω | N ML

 √

1 MtTL

(5)

where we define δ Pˆ = Pˆ − P as the fluctuation of the measurements [14]. Since the uncertainty becomes smaller monotonically as we increase t, we choose t = T , and we obtain δω (en) = T √1ML . So we obtain δω (en) = (T −1 ), which is the quantum scaling. Moreover, if we have M = (L), we obtain δω (en) = (L−1 ), and this scaling is called the Heisenberg limit [14, 15]. It is worth mentioning that, by substituting M = 1 in the Eq. 5, we can reproduce the uncertainty for the separable states ˆ σˆ 1+

) (M ) y , in the Eq. (3). The projection |φ(M GHZ,⊥ φGHZ,⊥ | with M = 1 corresponds to 2 and this is a measurement along y axis at each qubit, which is effectively the same measurement that we consider in the Sect. 2.

4 Dephasing on the Qubit In the real experiment, we cannot avoid the interaction with the environment to induce the decoherence. One of the typical decoherence on the solid-state qubits is dephasing that decreases the non-diagonal terms of the density matrix [18–21]. So,

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to evaluate the performance of the quantum sensing in the realistic circumstance, it is important to investigate the dynamics of the qubit under the effect of the dephasing. Let us explain how to quantify the effect of the dephasing on the qubit. The Hamiltonian between the qubit and environment is as follows [11, 22, 23]. H = HS + HI + HE L  ω (j) HS = σˆ 2 z j=1 HI =

L 

(6) (7)

λσˆ z(j) ⊗ Bˆ j

(8)

Cˆ j

(9)

j=1

HE =

L  j=1

where Bˆ j and Cˆ j are the operators of the environment at j-th site. In the interac (j) tion picture, we obtain HI (t) = Lj=1 λσˆ z ⊗ Bˆ j (t) where Bˆ j (t) = eiHE t Bˆ j e−iHE t is the environmental operator in the interaction picture. We assume that the initial state  (j) is described as ρ(0) = ρS Lj=1 ρE where ρS denotes the initial state of the system (j)

(j)

ρE is in thermal equilibrium state ([HE , ρE ] = 0). We consider unbiased noise such (j) as (Tr[ρE Bˆ j ] = 0). We solve the Von Neumann equation d ρdtI (t) = −i[HI , ρI (t)] up to the second order as follows. ρI (t) = ρI (0) − i  ρI (0) − i

 t  t

= ρI (0) − iλ

− λ2

0

0

dt [HI (t ), ρI (0)] − dt [HI (t ), ρI (0)] −

L  t  j=1 0

0

0

0

0

0

0

 t  t

(j) dt [σˆ z ⊗ Bˆ j (t ), ρS

L  t  t

 j,j =1

 t  t

dt dt

[HI (t ), [HI (t

), ρI (t

)]] dt dt

[HI (t ), [HI (t

), ρI (0)]]

L  (j

) ρE ]

j

=1



(j) (j ) dt dt

[σˆ z ⊗ Bˆ j (t ), [σˆ z ⊗ Bˆ j

(t

), ρS

By tracing out the environment, we obtain

L  (j

) ρE ]]

j

=1

(10)

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Y. Matsuzaki

ρ(S) I (t) ≡ Tr E [ρI (t)]  ρS − λ2

L  t  j=1

+ λ2

j=1

+ λ2

0

0

j=1

= ρS − λ2

t

t

0

L  t  0

0

0

L  t  j=1

− λ2

0

L  t 

0

t

j=1

(j)

dt dt

Tr[Bˆ j (t )Bˆ j (t

)ρE ]σˆ z(j) σˆ z(j) ρS

(j) dt dt

Tr[Bˆ j (t )ρE Bˆ j (t

)]σˆ z(j) ρS σˆ z(j)

(j) dt dt

Tr[Bˆ j (t

)ρE Bˆ j (t )]σˆ z(j) ρS σˆ z(j)

(j) dt dt

Tr[ρE Bˆ j (t

)Bˆ j (t )]ρS σˆ z(j) σˆ z(j)

L  t  0

t

0

t

(j)

(j)

dt dt

Ct −t

[σˆ z(j) , [σˆ z(j) , ρS ]]

(11)

where ρ(S) I (t) denotes the reduced density matrix of the system in the interaction pic(j) (j) ture and Ct−t = 21 Tr[(Bˆ j (t )Bˆ j (t

) + Bˆ j (t

)Bˆ j (t ))ρE ] denotes a correlation function of the environment. Importantly, in this calculation, we assume that the environment acts on the qubit individually. The correlation function depends on (t − t

) [23]. A typical time scale that the correlation functions decays is called a correlation time [24]. If we are interested in a time scale much shorter than the correlation time of (j) (j) the environment, we can use an approximation of Ct−t  C0 . Also, we consider a (j) homogeneous environment such as C0 = C0 for all j. So we obtain ρ(S) I (t) = (1 − L t )ρS +

L 

(j)

t σˆ z(j) ρS σˆ z(j)

(12)

j=1

where t = λ2 C0 t 2 denotes the error rate. In the Schrodinger picture, we obtain ρS (t) = (1 − L t )e−iHS t ρS eiHS t +

L 

(j)

t σˆ z(j) e−iHS t ρS e−iHS t σˆ z(j)

(13)

j=1

Although this analysis is general for a short time scale, we need a specific noise model that can describe a longer time scale as well. So we adopt a dephasing channel as follows. ρS (t) = Eˆ1 Eˆ2 · · · EˆL (e−iHS t ρS eiHS t ) where Eˆj denotes the individual dephasing channel on the j-th qubit as follows.

(14)

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295

Eˆj (ρS ) = (1 − t )ρS + t σˆ z(j) ρS σˆ z(j) t =

(15)

−γ 2 t 2

1−e 2

(16)

where ρS (t) denotes a density matrix of the system in the Schrodinger picture, γ denotes a decay  rate, and 1/γ denotes a dephasing time. It is worth mentioning that, if we set γ = 2λ2 C0 , we can recover the Eq. (13) in the limit of small t. Actually, a dephasing on some solid state systems such as NV centers [24] and superconducting qubits [18, 19] can be described by such a model in the Eq. (15). On the other hand, if we have a system that has a short correlation time where Ct−t decays much more rapidly than a dephasing time, the dynamic shows an exponential decay, and such a dynamics can be described by a Markovian Lindblad master equation [25]. However, we do not consider such a system described by the Markovian dephasing model in this article, because of the long correlation time of the typical solid state systems.

5 Magnetic Field Sensing with Separable States Under the Effect of Dephasing Let us describe the magnetic field sensing with  separable states under the effect L of dephasing [3, 7]. Firstly, prepare a state of j=1 |+j +|. Secondly, let this  state evolve for a time t according to the Eq. (14), and obtain Lj=1 ρj (t) where ωt ωt ρj (t) = Eˆj (|ψ(t)j ψ(t)|) and |ψ(t)j = √1 (e−i 2 |1j + ei 2 |0j ). Third, we measure 2

each qubit by an observable σˆ y , and obtain L measurement results. Finally, we repeat these three steps with N times. Since we have ρj (t) =

1 1 2 2 |11| + e−iωt−γ t |10| 2 2 1 1 2 2 + eiωt−γ t |01| + |00| 2 2

(17)

we can calculate the uncertainty of the estimation as δω

(sep)

 2 2 δ σˆ y δ σˆ y  eγ t = d σˆ  √ √ TtL | d ωy | NL

(18)

where we use N  Tt . Let us consider a short time scale such as T  1/γ. By choosing t = T , we obtain δω (sep) 

1 + γ2T 2 . √ T L

(19)

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Y. Matsuzaki

On the other hand, if we consider a long time scale such as T  1/γ, we just obtain 1 , we can minimize the uncertainty, and we the classical scaling. By choosing t = 2γ obtain δω

(sep)

1√ e 4 2γ  √ . TL

(20)

It is worth mentioning that the uncertainty scales as δω (sep) = (T −1/2 ) for a time 1 scale T ≥ 2γ , which is the classical scaling from the central limit theorem. This result show that one cannot achieve the quantum scaling of δω (sep) = (T −1 ) for a long time scale in a conventional strategy under the effect of the dephasing [9, 10].

6 Magnetic Field Sensing with Entangled States Under the Effect of Dephasing We explain the magnetic field sensing with entangled states under the effect of dephasing [14, 26–28]. Firstly, prepare L/M sets of GHZ states with the size of M Secondly, let this state evolve for a time t according to the Eq. (14), and obtain .L/M ρ(GHZ) (t) where ρ(GHZ) (t) = Eˆ1 Eˆ2 · · · EˆM (|φ(M ) (t)φ(M ) (t)|) and |φ(M ) (t) = j=1 j iωMt √1 (e− 2 |11 · · · 1 2

GHZ

j

+e

GHZ

GHZ

iωMt 2

|00 · · · 0). Thirdly, we perform a measurement on each ) (M ) (M ) GHZ state by a projection operator of Pˆ = |φ(M GHZ,⊥ φGHZ,⊥ | where |φGHZ,⊥  = 1 √ (|11 · · · 1 + i|00 · · · 0) is a state orthogonal with the initial GHZ state, and obtain 2 L/M measurement results. Finally, repeat these three steps N times. We have 1 1 2 2 |11 · · · 111 · · · 1| + e−iM ωt−M γ t |11 · · · 100 · · · 0| 2 2 1 iM ωt−M γ 2 t 2 1 + e |00 · · · 011 · · · 1| + |00 · · · 000 · · · 0| 2 2

ρj (t) =

ˆ (GHZ) ]  1 + 1 e−M γ 2 t 2 M ωt. The uncertainty of the estiand we obtain P = Tr[Pρ j 2 2 mation can be calculated as follows.  √ 2 2 ˆ Pρ ˆ (GHZ) ] Tr[δ Pδ j P(1 − P) eM γ t δω (en) = =  . (21) √ √ √ | ddPω | NL/M | ddPω | NL/M TLMt Let us consider a short time scale such as T  δω (en) 

√1 γ M

1 + M γ2T 2 √ T LM

. By choosing t = T , we obtain (22)

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for T 

√1 γ M

297

. On the other hand, for a long time scale such as T 

minimize the uncertainty by choosing t = δω

(en)

1 √

2γ M

√1 γ M

, we can

, and obtain

1√ e 4 2γ  √ . TL M

(23)

This means that, under the effect of dephasing, we obtain δω (en) = (T −1/2 ) for a long time scale T  γ √1M , and we cannot obtain the quantum scaling about the

time T . On the other hand, if we choose M = (L), we obtain δω (en) = (L−3/4 ) under the effect of dephasing, which beats the SQL in the scaling of the number of the qubits [26–31]. So the entangled scheme shows a better sensitivity than the separable scheme even under the effect of the dephasing in the conventional strategy.

7 Suppression of the Dephasing by Quantum Teleportation Recently, it was shown that a frequent implementation of the quantum teleportation that was used in a one-way quantum computer can suppress the dephasing [32, 33]. As we explained above, the dephasing shows a quadratic decay of the non-diagonal terms of the quantum states if the correlation time of the environment is longer than the dephasing time. The quadratic decay is slow in a sense that the time derivative for a short time scale is nearly zero. Importantly, a frequent implementation of the quantum teleportation can reset the correlation of the environment, and so we can keep the quantum states in such a slow decay regime. Such a quantum teleportation can be implemented by a combination of a control-phase gate, a single-qubit measurement, and measurement feedforward [12, 13]. There is a concept called quantum Zeno effect (QZE) that can also suppress the dephasing by using a quadratic decay regime [34, 35]. In the QZE, a frequent measurement on a quantum state keeps the dephasing in the slow decay regime. However, when we use the QZE, not only the dephasing but also a time evolution from the target magnetic fields is suppressed. In this sense, it is not clear whether we can use the QZE to improve the sensitivity of the qubit-based magnetic field sensor under the effect of dephasing. On the other hand, by using the quantum teleportation, we can suppress the dephasing while the quantum states can accumulate a phase information from the target magnetic fields, as we will describe in the next section. For this reason, we focus on explaining the quantum teleportation for the suppression of the dephasing in this section. We explain the quantum teleportation that is used for a one-way quantum computer [12, 13]. Firstly, we prepare a quantum state of (α|11 + β|0)|+2 where the first qubit is the target to be teleported and the second qubit is an ancillary system for the teleportation. Secondly, we perform a control-phase gate and obtain α|11 |−2 + β|01 |+2 . Thirdly, we perform a projective measurement about σˆ x(1) , and obtain |±1 ±|(α|11 |−2 + β|01 |+2 ) = |±1 (α|−2 ± β|+2 ). Finally, we

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Y. Matsuzaki

perform a single qubit rotation on the qubits where the rotation and angle depends on the measurement results at the third step, and we obtain |01 (α|12 + β|02 ). This is how we can deterministically teleport an arbitrary quantum state from one place to another. Throughout of this paper, we assume that we can perform a quantum teleportation with a time scale much shorter than other time scale such as a coherence time of the qubit. We describe the theoretical framework how the quantum teleportation can suppress the dephasing [32, 33]. For simplicity, let us consider a two-qubit case, and the Hamiltonian is described as follows. H = HS + HI + HE 2  ω (j) σˆ HS = 2 z j=1 HI =

2 

λσˆ z(j) ⊗ Bˆ j

j=1

HE =

2 

Cˆ j .

(24)

j=1

We consider the dynamics of a state of |+1 when this state is teleported to the site 2. Here, we assume that we can implement perfect quantum teleportation instantaneously. First, we prepare a state of |+1 +| ⊗ |02 0|. Second, let this state evolves by the Hamiltonian for a time τ /2, which decohers the state at the site 1. Third, perform a quantum teleportation, and the state at the site 1 is transferred into the site 2. Fourthly, let this state evolves by the Hamiltonian for a time τ /2, which decohers the state at the site 2. Finally, we study the density matrix of the state at the site 2. From the Eq. (11), in the second step, we obtain the following state at the site 1. ρ(1) I (τ /2)

  |+1 +| − λ

τ /2

2 0



t

0

dt dt

Ct(1) ˆ z(1) , [σˆ z(1) , |+1 +|]]

−t

[σ

(25)

where ρ(1) I (τ /2) denotes a quantum state at the site 1 in the interaction picture. By (1) assuming a long correlation time of the environment, we obtain Ct(1)

−t

 C0 , and we obtain 1 2 2 (1) (1) ˆ z , [σˆ z(1) , |+1 +|]]. ρ(1) I (τ /2)  |+1 +| − λ (τ /2) C0 [σ 2

(26)

At the third step, we teleport this state into the site 2, and so we obtain 1 2 2 (1) (2) ˆ z , [σˆ z(2) , |+2 +|]]. ρ(2) I (τ /2) = |+2 +| − λ (τ /2) C0 [σ 2

(27)

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299

At the fourth step, we let this state evolve by the Hamiltonian for τ , and we obtain 1 2 2 (1) (2) ˆ z , [σˆ z(2) , |+2 +|]] ρ(2) I (τ )  |+2 +| − λ (τ /2) C0 [σ 2  τ  t

dt dt

Ct(2) ˆ z(2) , [σˆ z(2) , |+2 +|]] − λ2

−t

[σ τ /2

τ /2 2

= |+2 +| − λ (τ /2)2 C0 [σˆ z(2) , [σˆ z(2) , |+2 +|]]

(28)

where we assume a homogeneous environment as C0(1) = C0(2) . The fidelity of this 1 2 2 state is calculated as F = 2 +|ρ(2) I (τ )|+2  1 − 2 λ C0 τ We compare this state with another state that interacts with the environment at the site 1 for a time τ where the initial state is also |+1 , and such a state is described as follows. 1 2 2 (1) (1) (1) ˆ z , [σˆ z , |+1 +|]]. ρ˜(1) I (τ )  |+1 +| − λ (τ ) C0 [σ 2

(29)

2 2 (1) The fidelity of this state is F˜ = 1 +|ρ˜(1) I (τ )|+1 = 1 − λ τ C0 . Therefore, the fidelity with the quantum teleportation is higher than that without quantum teleportation, and this shows the suppression of the dephasing. We can easily generalize this result with n-qubit case. First, we prepare a state |+1 . Secondly, let this state evolve for a time τ /n by the Hamiltonian in the Eq. (6). Thirdly, we perform a quantum teleportation to transfer this state into the next site. Fourthly, repeat the second and third process (n − 1) times. Finally, let this state evolve for a time τ /n, and study this quantum state. In this case, the fidelity is given 1 2 2 as F = n +|ρ(n) I (τ )|+n  1 − n λ C0 τ . On the other hand, if we let a state |+ for 2 2 (1) a time τ at the site 1, the fidelity is given as F˜ = 1 +|ρ˜(1) I (τ )|+1 = 1 − λ τ C0 . So, as we increase the number of the quantum teleportation in a time τ , the fidelity increases. It is worth mentioning that, although this analysis is general, we use a perturbation theory, and so this result can be applied only to the short time regime. We now consider the effect of the quantum teleportation for the suppression of the dephasing by using another noise model to consider an arbitrary time scale. We consider a two-qubit case for simplicity. We use an individual dephasing channel described in the Eq. (15). If a quantum state |+1 decohers by this dephasing channel at the site 1 for a time τ /2, we obtain a state of ρ(1) I (τ /2) = −γ 2 (τ /2)2

−γ 2 (τ /2)2

2 2

1−e−γ t . We 2 1 |01| + 2 |00|.

(1 − τ /2 )|+1 +| + τ /2 |−1 −| in the interaction picture where t =

1 e |10| + e 2 can rewrite this as ρ(1) I (τ /2) = 2 |11| + 2 By implementing the quantum teleportation to the site 2, we obtain a state of ρ(2) I (τ /2) = (1 − τ /2 )|+2 +| + τ /2 |−2 −|. Let this state decohers by the individual dephasing channel at the site 2 for a time τ /2, and the state becomes 2 2 ρ(2) I (τ ) = ((1 − τ /2 ) + ( τ /2 ) )|+2 +| + 2 τ /2 (1 − τ /2 )|−2 −|. We can rewrite −2γ 2 (τ /2)2

−2γ 2 (τ /2)2

1 e this as ρ(2) |10| + e 2 |01| + 21 |00|. The fidelity I (τ ) = 2 |11| + 2 1 1 − 21 γ 2 τ 2 is calculated as F = 2 +|ρ(2) . On the other hand, if we let I (τ )|+2 = 2 + 2 e

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Y. Matsuzaki

Fig. 1 Schematic of our system. There is a ring structure composed of 2L qubits. Each qubit is interacted with a local environment to induce dephasing. Half of the qubit is used for the probe with the magnetic fields while the other qubits are ancillary for quantum teleportation. The Hamiltonian is given in the Eq. (6)

the state |+1 decoher by the dephasing channel for a time τ , we obtain ρ˜(1) I (τ ) = (1) ˜ (1 − τ )|+1 +| + τ |−1 −|, and the fidelity is given as F = 1 +|ρ˜I (τ )|+1 = 2 2 1 + 21 e−γ τ . So, from the calculation of the fidelity, we show that a single imple2 mentation of the quantum teleportation reduces the dephasing rate by a factor √ of 2. We can generalize this result with n qubit case. We prepare a state |+1 . We let this state decohers for a time τ /n and perform a quantum teleportation to transfer this state with a next site. By repeating the decoher and teleportation (n − 1) times before we finish by let the state decoher for a time τ /n at the final site, we obtain the state 2 2 2 2 as ρ(n) (τ ) = 21 |11| + 21 e−nγ (τ /n) |10| + 21 e−nγ (τ /n) |01| + 21 |00|. The fidelity 2 2 can be calculated as F = n +|ρ(n) (τ )|+n  21 + 21 e−γ τ /n . So, by increasing the number of the quantum teleportation for a time τ , the fidelity approaches to the unity.

8 Magnetic Field Sensing with Quantum Teleportation Under the Effect of Dephasing Using Separable States Here, we explain our scheme to achieve the quantum scaling δω = (T −1 ) under the effect of dephasing. For simplicity, we now consider a case that a perfect quantum teleportation can be instantaneously implemented, although we will relax this condition in the Sect. 10. We assume the Hamiltonian described in the Eq. 6 where a local environment is independently coupled with the qubit as shown in Fig. 1. The key idea is to use the quantum teleportation to suppress the dephasing during the magnetic field sensing. Suppose that we have a ring structure with 2L qubits as

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Fig. 2 Schematic of the qubit-based sensor to achieve a quantum scaling. There is a ring structure composed of 2L qubits. Half of them are probe qubits to detect the magnetic fields. The other half are ancillary qubits to perform quantum teleportation of the probe qubits. The probe qubits and ancillary qubits are located alternatively where the probe qubit is next to the ancillary qubit. The ancillary qubits are prepared in a ground state that is not affected by neither the magnetic fields nor dephasing. The probe qubits are prepared in |ψj 2j−1 = |+2j−1 (j = 1, 2, . . . , L), and interact with the global magnetic fields that induces a relative phase of the probe qubits. While the probe qubits interact with the magnetic fields, we frequently perform quantum teleportation that transfers the probe qubit to a next site along a clockwise direction. Such a quantum teleportation can suppress the dephasing while the probe qubits can acquire a phase shift

shown in Fig. 2. Half of them are coupled with the magnetic fields, and the other half are ancillary for the quantum teleportation. The probe qubits and ancillary qubits are located alternativelywhere the probe qubit is next to the ancillary qubit. Firstly, we prepare a state of Lj=1 |+2j−1 for the probe qubits, and the ancillary qubits are prepared in a ground state that is not affected by neither the magnetic fields nor dephasing. Secondly, let the state of the probe qubits expose with the magnetic fields for a time τ = t/n under the effect of the dephasing. Thirdly, we perform a quantum teleportation of the qubits to the next site. Fourthly, we repeat the second and third process (n − 1) times. Fifthly, we let the state expose with a time τ = t/n at the last site. Finally, we measure each qubit by an observable σˆ y . We consider a dephasing channel to calculate the uncertainty of the estimation. By assuming that n is even, the state of the probe qubits before the measurement is given as ρ=

L  1 1 2 2 ( |12j−1 1| + e−iωτ −γ τ /n |12j−1 0| 2 2 j=1

1 1 2 2 + eiωt−γ τ /n |02j−1 1| + |02j−1 0|) 2 2

(30)

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Y. Matsuzaki

From the Eq. (18), we obtain the uncertainty of the estimation as δωQT

 2 2 δ σˆ y δ σˆ y  eγ t /n  d σˆ  √  √ . TtL | d ωy | NL

(31)

By choosing t = T and n = γ 2 T 2 , we obtain δωQT 

e √ T L

(32)

Therefore, by using the frequent quantum teleportation during the quantum sensing, we obtain the quantum scaling of δωQT = (T −1 ). In the above calculations, we assume that the dephasing can be suppressed by the quantum teleportation as shown in the Sect. 7. The crucial assumption in the Sect. 7 is that the quantum state interacts with the local environments only one time at each site in the Sect. 7. However, the quantum state could be teleported back to the original site in our scheme, which could provide an opportunity for the quantum state to interact with the same local environment again, as shown in Fig. 3. Fortunately, as long as there are many number of the qubits, we could still use the suppression of the dephasing by the quantum teleportation for the following reason. The key idea is that the environment has a finite correlation time τc , and the environment can be reset after a time much longer than τc has been passed. After such a reset, even when the quantum state is teleported back to the original site, the suppression of the dephasing is still valid. Suppose that n˜ sep = 2L − 1 denotes the maximum number of the quantum teleportation without the quantum state being teleported back to the original site. To make the dephasing suppression valid due to the quantum teleportation, we need to satisfy a condition of n˜ sep τ  τc ⇔ (2L − 1)  γ 2 T τc , which requires a large number of the qubits. We explain the details in the Appendix.

Fig. 3 A schematic when quantum state is teleported back to the original site. The quantum state |ψ1  is repeatedly teleported to the next site after the interaction with an environment at the local site. If we perform the quantum teleportation more than 2L times, the quantum state could interact with the same environment again. More specifically, in this figure, the quantum state |ψ1  interacts with the environment 1 twice, which might degrade the performance of our teleportation-based quantum sensing unless we carefully choose some parameters

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9 Magnetic Field Sensing with Quantum Teleportation Under the Effect of Dephasing Using Entangled States Here, we explain how to achieve the quantum scaling under the effect of dephasing by using entangled states. For simplicity, we assume that we can instantaneously implement a perfect quantum teleportation, although we will relax this condition in the Sect. 11. Again, we consider a ring structure composed of 2L qubits. Half of them are probe qubits that interact with the magnetic fields and the other half of them are ancillary for the quantum teleportation. Firstly, prepare L/M sets of GHZ states of the probe qubits with the size of M , described as  composed M +2kM +2kM L √1 |ψk(GHZ)  = √12 ( M j=1+2kM |1j ) + 2 ( j=1+2kM |0j ) for k = 0, 1, . . . , M − 1. On the other hand, the ancillary qubit is prepared in a separable ground state that is not affected by neither the magnetic fields nor the dephasing. Secondly, let this state interact with the magnetic fields with a time τ = t/n, and we obtain ρk (τ ) =

M +2kM M +2kM 1  1  2 2 ( |1j 1|) + ( |1j 0|e−iM ωτ −M γ τ ) 2 2 j=1+2kM

1 + ( 2

M +2kM j=1+2kM

j=1+2kM

|0j 1|e

iM ωτ −M γ 2 τ 2

M +2kM 1  )+ ( |0j 0|) 2 j=1+2kM

Thirdly, we perform a quantum teleportation of the qubits to the other sites, as shown in Fig. 3. Fourthly, we repeat the second and third process (n − 1) times. Finally, we perform a measurement on each GHZ state by a projection operator ) (M ) (M ) √1 of Pˆ QT = |φ(M GHZ,⊥ φGHZ,⊥ | where |φGHZ,⊥  = 2 (|11 · · · 1 + i|00 · · · 0) is a state orthogonal with the initial GHZ state, and obtain L/M measurement results. The 2 2 probability to project the state into Pˆ QT is PQT  21 + 21 e−M γ t /n M ωt as long as the suppression of the dephasing due to the quantum teleportation is valid. In this case, we can estimate the uncertainty of the estimation as follows. (en) δωQT

 2 2 PQT (1 − PQT ) eM γ t /n = dP √ √ . TLMt | d ωQT | NL/M

(33)

(en) e  √LM . By substituting M = 1 By setting t = T and n = M γ 2 T 2 , we obtain δωQT T with the Eq. (33), we can reproduce the Eq. (31). As we discuss above, the required time for the quantum state being teleported back to the original site should be much longer than the correlation time of the environment in order to suppress the dephasing by the quantum teleportation. The maximum number of the quantum teleportation without the quantum state being − 1. We need to satisfy a teleported back to the original site is given as n˜ en = 2L M condition of n˜ en τ  τc ⇔ 2L  M (1 + M γ 2 T τc ). By assuming that M and T is large such as M γ 2 T τc  1, the condition is simplified as 2L  M 2 γ 2 T τc . If we

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Y. Matsuzaki

choose M = cL 2 −2 where c and denote positive constant numbers, the condition is rewritten as 2  c2 L−4 γ 2 T τc . So, by increase the number of the qubits, we can satisfy the condition for any value of > 0. In this case, the uncertainty is described (en)  √ e3 − . About the scaling of T , we achieve the quantum scaling of as δωQT 1

cL 4

T

(en) (en) = (T −1 ). On the other hand, about the scaling of L, we obtain δωQT = δωQT 3 − 4 − (L ), which beats the SQL.

10 Magnetic Field Sensing with Imperfect Quantum Teleportation Under the Effect of Dephasing Using Separable States In the actual experiments, we cannot implement a perfect quantum teleportation, and so we investigate how an error during the quantum teleportation affects the performance of our scheme in this section. Especially, we discuss the case to use separable states for quantum sensing with imperfect quantum teleportation under the effect of dephasing. Let us investigate the performance of the quantum sensing with imperfect quantum teleportation under the effect of dephasing using separable states. We adopt a model for imperfect quantum teleportation as follows. When we perform the imperfect quantum teleportation on a state of |+j , we obtain (1 − p)|+j+1 +| + p 21 1ˆ j+1 where  p denotes the error rate. The initial state is Lj=1 |+2j−1 . We let the state interact with the magnetic fields under the effect of dephasing and perform the imperfect quantum teleportation. We repeat this n times, and the state just before the readout is described as (j)

ρIQT =

L  1 1 2 2 ((1 − p)n−1 ( |12j−1 1| + e−iωτ −γ τ /n |12j−1 0| 2 2 j=1

1 1 2 2 + eiωt−γ τ /n |02j−1 1| + |02j−1 0|) 2 2 1 + (1 − (1 − p)n ) 1ˆ 2j−1 ) 2 We can calculate the uncertainty of the estimation as (sep) δωIQT

 2 2 δ σˆ y δ σˆ y  eγ t /n = d σˆ  √  √ (1 − p)n−1 TtL | d ωy | NL

where we use N  Tt . We consider a case of a short time scale such as T  and n = γ T , we obtain 2

2

√1 . pγ

(34)

By choosing t = T

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e

(sep)

δωIQT 

(1 −

pγ 2 T 2 )

√ TtL

(35)

and this is almost the same expression as the Eq. (32). This means that, for a short time (sep) scale such as T  √1pγ , we approximately obtain δωIQT = (T −1 ) in the teleportation based scheme. In the conventional scheme with separable states, we approximately obtain δω (sep) = (T −1 ) only for a time scale such as T  γ1 , as shown in the Eq. (19). Therefore, our teleportation-based scheme is advantageous over the conventional scheme in a sense that we can obtain the approximated scaling of δω = (T −1 ) much longer time scale than the conventional scheme. Let us consider a long time scale such as T  √1pγ . About t, by choosing t = √ n , 2γ

(sep)

we can minimize the uncertainty, and obtain δωIQT 

√ 2γe1/4 √√ . n−1 (1−p) nTL

If we

substitute n = 1, we can reproduce the uncertainty δω (sep) when we use the separable state without quantum teleportation, as described in the Eq. (20). We can treat n as a continuous variable, and we can minimize the uncertainty by choosing n = 1 , and we obtain − 41 log(1 − p)  4p (sep) δωIQT

 √ 2 e pγ  √ TL

(36) √

It is worth mentioning that we have a condition of t = 2γn  4γ1√p ≤ T . In order to use the suppression of the dephasing by the quantum teleportation, we need to satisfy √ n˜ sep τ  τc . We can rewrite this condition as L p/γ  τc , and so we can satisfy this condition just by increasing the number of the qubits. With the use of the imperfect (sep) quantum teleportation, we just obtain the classical scaling δωIQT = (T −1/2 ) for a 1 long time scale such as T  √pγ . Let us compare the performance of the magnetic field sensing with imperfect quantum teleportation by using separable states with that of the conventional magnetic field sensing without quantum teleportation for a long time scale. It is worth mentioning that we need to use 2L qubits for the teleportation-based scheme while we just use L qubits for the conventional scheme. So, for the fair comparison, we √ add a factor of 2, which shows a decrease of the uncertainty for the conventional scheme when we use 2L qubits. We obtain  √ √  √ √ 2 e pγ √  √ √ 2γe1/4 √ (sep) 2δωIQT /δω (sep)  2 √ ep /( √ ) = 2 2 e p/( 2e1/4 ) = 2 TL TL

(37)

for 16γ12 T 2  p  1. So, although we cannot obtain the quantum scaling with imperfect quantum teleportation for a long time scale, we can still achieve a constant factor √ ep over the conventional magnetic field sensing for small p. improvement 2

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11 Magnetic Field Sensing with Imperfect Quantum Teleportation Under the Effect of Dephasing Using Entangled States Here, we discuss the case to use the entangled states composed of M qubits with imperfect quantum teleportation under the effect of dephasing. We adopt the following error model to represent the imperfect quantum teleporta(GHZ) ψk(GHZ) | where |ψk(GHZ)  = tion. We consider GHZ states described as ρ(k) 0 = |ψk   M +2kM M +2kM L √1 ( √1 j=1+2kM |1j ) + 2 ( j=1+2kM |0j ) with k = 0, 1, . . . , M − 1. These are tele2 ported to other sites after the interaction with magnetic fields for a time τ , as shown in Fig. 4. Due to possible errors, the quantum state after imperfect quantum teleportation, we could have ρ(k) = (1 − p)M ρ k + (1 − (1 − p)M )ρ

k ρ k = |φ(GHZ) φ(GHZ) | k k M +(2k+1)M M +(2k+1)M   1 1  = |1 ) + |0j ) |φ(GHZ) ( ( √ √ j k 2 j=1+(2k+1)M 2 j=1+(2k+1)M

ρ

k

1 = ( 2

M +(2k+1)M  j=1+(2k+1)M

1 |1j 1|) + ( 2

M +(2k+1)M 

|0j 0|)

(38)

j=1+(2k+1)M

with k = 0, 1, . . . , ML − 1 where p denotes an error probability on a single site, ρ k denotes the state when the quantum teleportation is perfect, and ρ

k denotes a state without any coherence. Let us discuss the performance of our magnetic field sensing with imperfect quantum teleportation using entangled state under the effect of dephasing. The initial state is ρk with k = 0, 1, . . . , ML − 1. Let these states interact with magnetic fields under the effect of dephasing for a time τ = t/n, and these states are teleported to other site as shown in Fig. 4, and obtain ρ˜(k) = (1 − p)M ρ˜ k + (1 − (1 − p)M )ρ

k ρ˜ k =

1 ( 2

M +(2k+1)M  j=1+(2k+1)M

1 + ( 2

1 |1j 1|) + ( 2

M +(2k+1)M  j=1+(2k+1)M

eiM ωτ −M γ

M +(2k+1)M 

e−iM ωτ −M γ

τ

2 2

|1j 0|)

j=1+(2k+1)M

τ

2 2

1 |0j 1|) + ( 2

M +(2k+1)M 

|0j 0|).

(39)

j=1+(2k+1)M

with k = 0, 1, . . . , ML − 1. After repeating these process, we obtain the following state just before the readout.

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Fig. 4 A schematic of our magnetic field sensing with quantum teleportation using entangled states. There is a ring structure composed of L qubits. Half of the qubits are used to probe the magnetic fields, and the other qubits are ancillary for the quantum teleportation. We have L/M sets of GHZ states composed of the probe qubits with the size of M . These entangled states are teleported to the other sites after the interaction with magnetic fields with a time τ . Such a frequency teleportation can suppress the dephasing while the GHZ states can acquire a coherent phase from the target magnetic fields

M (n−1)

ρ˜(k) ρ˜k,fin + (1 − (1 − p)M (n−1) )ρ

k,fin fin = (1 − p)

ρ˜ k,fin

1 = ( 2 +

ρ

k,fin =

1 ( 2 1 ( 2

M +(2k+n)M  j=1+(2k+n)M M +(2k+n)M  j=1+(2k+n)M M +(2k+n)M  j=1+(2k+n)M

1 |1j 1|) + ( 2 eiM ωt−M γ

t /n

2 2

M +(2k+n)M 

e−iM ωt−M γ

|1j 0|)

j=1+(2k+n)M

1 |0j 1|) + ( 2

1 |1j 1|) + ( 2

t /n

2 2

M +(2k+n)M 

M +(2k+n)M 

|0j 0|).

j=1+(2k+n)M

|0j 0|)

(40)

j=1+(2k+n)M

with k = 0, 1, . . . , ML − 1. For the readout, we perform a measurement on each (k) (k) (k) GHZ state by a projection operator of Pˆ IQT = |φ(k) GHZ,⊥ φGHZ,⊥ | where |φ(GHZ),⊥  =   M +2kM M +2kM √1 ( √1 j=1+(2k+n)M |1j ) + i 2 ( j=1+(2k+n)M |0j ) and the probability to project the 2 quantum state into this projection is described as (k) (k) ρ˜fin ] PIQT = Tr[Pˆ IQT

1 1 1 2 2  (1 − p)M (n−1) ( + e−M γ t /n M ωt) + (1 − (1 − p)M (n−1) ) 2 2 2 1 1 2 2 = + (1 − p)M (n−1) e−M γ t /n M ωt. 2 2 So the uncertainty of the estimation is calculated as

(41)

308

Y. Matsuzaki (en) δωIQT

 2 2 PIQT (1 − PIQT ) eM γ t /n =  √ √ dP (1 − p)M (n−1) TLMt | d IQT | L/M ω

(42)

√ Let us consider a short time scale as T  1/( pγM ). In this condition, by choosing t = T and n = M γ 2 T 2 , we obtain (en)  δωIQT

e (1 −

pM 2 γ 2 T 2 )T

√

LM

(43)

and we can obtain almost the same expression as the Eq. (33). This means that, if p is sufficiently small, our scheme with quantum teleportation has an advantage over the conventional scheme described in Sect. 6. Actually, in our teleportation based (en) = (T −1 ) for a time scale scheme, we approximately have a scaling with δωIQT √ with T  1/( pγM ) while the conventional scheme shows δω (en) = (T −1 ) for a √ time scale T  1/(γ M ). So, when we have p  1/(MT γ)2 , our scheme is better than the conventional scheme. √ On the other hand, if we consider a long time scale such as T  1/( pγM ), we cannot obtain the quantum scaling as follows.√ About the interaction time, we can minimize this uncertainty by choosing topt = n/M , and we obtain 2γ (en) δωIQT

√ 1/4  √ 2e γ/ Mn  √ (1 − p)M (n−1) TL

(44)

About M and n, we can also minimize the uncertainty by choosing Mopt = 1 and nopt = 2, and we obtain − 41 log(1 − p)  4p (en) δωIQT

2

3/4

√ e pγ LT

(45)

1 where we assume 4L ≤ p  1. In this case, the number of the quantum teleportation is just twice. This means that the quantum state cannot be teleported back to the original site, and so the suppression of the dephasing due to the quantum teleportation works. Therefore, for a finite value of p, the uncertainty shows a classical scaling (en) (en) = (T −1/2 ) and also shows a SQL such as δωIQT = (L−1/2 ). Compared of δωIQT with our teleportation-based scheme using separable states shown in the Eq. (36), this scheme using entangled states just have a constant improvement for a long time scale, as shown in the Eq. (45). Since it is much easier to use separable states in the current technology, it is natural to adopt the separable states for the teleportation based scheme for a long time scale. Let us discuss whether our teleportation based scheme shows a better performance for a longer time scale than the conventional entangling scheme without teleportation or not. As we mention above, it is more reasonable to use separable states than entangled states when we implement the quantum magnetic field sensing with imperfect

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quantum teleportation for a long time scale, and so we adopt to use the separable states for the teleportation-based scheme. On the other hand, since the use of the entangled states provides us with the sensitivity beyond the SQL for the conventional scheme, we adopt to use the GHZ states for the conventional scheme without teleportation. We assume that the size of the GHZ states is M , which can be interpreted as the maximum size of the GHZ states that experimentalists can create in their technology. While the teleportation based scheme we use 2L qubits, the √ conventional scheme use L qubits. For a fair comparison, we will add a factor of 2 improvement on the conventional scheme. So we obtain √  √ √ √ e pγ TL M

(sep) (GHZ) 2δωIQT /δω  2·2 (46) = 2(eM p)1/4 . √ LT e 14 2γ Therefore, as long as we have 2(eM p)1/4 < 1 is satisfied, our teleportation based scheme has an advantage over the conventional scheme with entangled states.

12 Conclusion In conclusion, we propose quantum magnetic field sensing with quantum teleportation. In the conventional scheme, the uncertainty of the estimation of ω is scaled as δω = (T −1/2 ) under the effect of dephasing in the limit of large T where T denotes a total time for the sensing. This scaling comes from the central limit theorem, and such a scaling is considered as classical. Importantly, we show that, if a perfect quantum teleportation is available, we can achieve a quantum scaling of δω = (T −1 ) even under the effect of dephasing for an arbitrary large T [11]. The key idea is to implement frequent quantum teleportation during the magnetic field sensing where the quantum state can acquire coherent phase shift from the magnetic fields while the dephasing is suppressed. We can use either separable states or entangled states as the initial state for this teleportation bases scheme, and the entanglement sensor has better performance than the separable one as long as the perfect quantum teleportation is available. We show that, even if the quantum teleportation is imperfect, we can approximately obtain the scaling δω = (T −1 ) with separable states (entangled states with a size M ) as long as the total time is rel√ √ atively small such as T  1/ pγ (T  1/ pM γ). For a longer time scale, the scaling becomes classical such as δω = (T −1/2 ) even with our teleportation based scheme. However, the magnetic field sensing with quantum teleportation still has a constant improvement (that becomes better as an error rate of the quantum teleportation decreases) over the conventional scheme without quantum teleportation. Therefore, if high fidelity quantum teleportation becomes available, our proposed scheme could provide an ultra-sensitive magnetic field sensor for practical purposes. We summarize our results in the Table 1.

QT entangle

2 2

√

(1−p)M (n−1) TLMt

2 e(γt) /n

eγ t /n √ (1−p)n−1 TtL

2 2

eM γ t √ TLMt

2 2

eγ t √ TtL

δωIQT 

(en)

δωIQT 

(sep)

δω (en) 

Entangled

QT separable

δω (sep) 

Separable

(en)

for T 

e √ (1−pM 2 γ 2 T 2 )T LM

e √ (1−pγ 2 T 2 )T L

√1 γ M

for T 

for T 

1+γ√2 T 2 T L

2 2 1+M √γ T T LM

δωIQT 

(sep)

δωIQT 

δω (en) 

δω (sep) 

for T 

√1 pγ

1 γ

√1 pM γ

1

(en)

δωIQT 

(sep)



e4 δω (en)  √

δωIQT 

1√ 4 2γ e√ TL

for T 

3 √ 2 2 e pγ LT

1 γ

for T 

√ 2γ for T  √ 1 √ Mγ TL M √√ 2 e pγ √ for T  √1pγ TL

δω (sep) 

√1 pM γ

Table 1 The uncertainty of the estimation for quantum magnetic field sensing with L probe qubits for a given total time T . There are four schemes such as conventional magnetic field sensing with separable states (separable), conventional magnetic field sensing with entangled states (entangled), our scheme with separable states by using imperfect quantum teleportation (QT separable), and our scheme with entangled states by using imperfect quantum teleportation (QT entangled). Here, M denotes the size of the entangled states, γ denotes a dephasing rate, n denotes the number of the quantum teleportation for a single measurement, p denotes an error rate for the quantum teleportation. Except the general form, we substitute n and T to obtain the suitable uncertainty as described in the main text General form of uncertainty Uncertainty for short T Uncertainty for long T

310 Y. Matsuzaki

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Appendix: Suppression of the Dephasing with Quantum Teleportation when a Quantum State Interacts with the Same Environment by Multiple Times Here, we explain the suppression of the dephasing with quantum teleportation when a quantum state interacts with the same environment by multiple times. As shown in Fig. 3, the quantum state could be teleported back to the original site in our scheme, and this could provide an opportunity for the quantum state to interact with the same local environment again. We will show how the suppression of the dephasing works in such a case as long as the environment is reset in a sense that a time much longer than the correlation time has passed. For simplicity, we consider a single qubit interacting with a local environment, and the Hamiltonian is given as follows. H = HS + HI + HE ω HS = σˆ z 2 HI = λf (t)σˆ z ⊗ Bˆ HE = Cˆ

(47)

where Bˆ and Cˆ denote the environmental operators. We define f (t) as follows. ⎧ ⎪ ⎨1 (0 ≤ t ≤ τ /2) f (t) = 1 ((m − 1)τ /2 ≤ t ≤ mτ /2) ⎪ ⎩ 0 (otherwise) This means that the qubits interacts with the environment only when t satisfies either 0 ≤ t ≤ τ /2 or (m − 1)τ /2 ≤ t ≤ mτ /2. We assume that the initial state is described as ρ(0) = ρS ⊗ ρE where ρS denotes the initial state of the system ρE is in thermal equilibrium state ([HE , ρE ] = 0). We consider unbiased noise ˆ = 0). In the interaction picture, the Hamiltonian is described by such as (Tr[ρE B] ˆ ˆ −iCt ˆ ˆ where B(t) ˆ = eiCt . We solve the Von Neumann equaBe HI (t) = λf (t)σˆ z ⊗ B(t) d ρI (t) tion dt = −i[HI (t), ρI (t)] up to the second order as follows. 

t

ρI (t)  ρI (0) − i

dt [HI (t ), ρI (0)] −

0

0



t

= ρI (0) − iλ − λ2

 t 0

0

 t

t

dt dt

[HI (t ), [HI (t

), ρI (0)]]

0

dt f (t )[σˆ z ⊗ Bˆ (t ), ρS ⊗ ρE ]

0 t

dt dt

f (t )f (t

)[σˆ z ⊗ Bˆ (t ), [σˆ z ⊗ Bˆ (t

), ρS ⊗ ρE ]]

(48)

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By tracing out the environment, we obtain ρ(S) I (t) ≡ Tr E [ρI (t)]  t 2  ρS − λ  t 0

+ λ2

t

 t

t

dt dt

f (t )f (t

)Tr[Bˆ (t

)ρE Bˆ (t )]σˆ z ρS σˆ z

0

 t

t

2 0

= ρS − λ

dt dt

f (t )f (t

)Tr[Bˆ (t )ρE Bˆ (t

)]σˆ z ρS σˆ z

0

0

−λ

(j) dt dt

f (t )f (t

)Tr[Bˆ (t )Bˆ (t

)ρE ]σˆ z σˆ z ρS

0

0

+ λ2

t

0  t

dt dt

f (t )f (t

)Tr[ρE Bˆ (t

)Bˆ (t )]ρS σˆ z σˆ z



t

dt dt

f (t )f (t

)Ct −t

[σˆ z , [σˆ z , ρS ]]

2 0

(49)

0

where ρ(S) I (t) denotes the reduced density matrix of the system in the interaction ˆ )B(t ˆ

) + B(t ˆ

)B(t ˆ ))ρE ] denotes a correlation function of picture Ct−t = 21 Tr[(B(t the environment. The correlation function depends on (t − t

). A typical time scale that the correlation functions decays is called a correlation time. By substituting t = mτ /2, we obtain  ρS (mτ /2) = ρS − λ

mτ /2

2



0

  = ρS − λ  −λ

2

0 τ /2

+ 

 2

τ /2



mτ /2 (m−1)τ /2

t

t

)

dt dt

f (t )f (t

)Ct −t

[σˆ z , [σˆ z , ρS ]]

0

dt dt

f (t )f (t

)Ct −t

[σˆ z , [σˆ z , ρS ]]

0



(m−1)τ /2



−λ



τ /2

0 mτ /2

= ρS − λ2

dt dt

f (t )f (t

)Ct −t

[σˆ z , [σˆ z , ρS ]]

0

= ρS − λ2 ( 2

t



0 mτ /2 (m−1)τ /2

t

dt dt

f (t )f (t

)Ct −t

[σˆ z , [σˆ z , ρS ]]

0 t

dt dt

f (t )f (t

)Ct −t

[σˆ z , [σˆ z , ρS ]]

0



τ /2

(

 +

0

t

(m−1)τ /2

)dt dt

f (t )f (t

)Ct −t

[σˆ z , [σˆ z , ρS ]]

 τ /2  t

= ρS − λ2 dt dt

f (t )f (t

)Ct −t

[σˆ z , [σˆ z , ρS ]] 0 0  mτ /2  τ /2 2 −λ dt dt

f (t )f (t

)Ct −t

[σˆ z , [σˆ z , ρS ]]  − λ2

(m−1)τ /2 mτ /2 (m−1)τ /2



0 t

(m−1)τ /2

dt dt

f (t )f (t

)Ct −t

[σˆ z , [σˆ z , ρS ]]

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It is worth mentioning that, if (m − 1)τ /2 is much longer than the correlation time  mτ /2  τ /2 of the environment, the term of λ2 (m−1)τ /2 0 dt dt

f (t )f (t

)Ct −t

[σˆ z , [σˆ z , ρS ]] disappears because we have Ct −t

 0 in this regime. So we have  ρS (mτ /2)  ρS − λ2  − λ2

τ /2

 

(m−1)τ /2 τ /2

= ρS − 2λ2 0

dt dt

f (t )f (t

)Ct −t

[σˆ z , [σˆ z , ρS ]]

0

0 mτ /2



t

t

(m−1)τ /2



t

dt dt

f (t )f (t

)Ct −t

[σˆ z , [σˆ z , ρS ]]

dt dt

Ct −t

[σˆ z , [σˆ z , ρS ]]

(50)

0

If the correlation time of the environment is much longer than τ /2, we can approximate Ct −t

 C0 in this regime. Therefore, we obtain 

τ /2

ρS (mτ /2)  ρS − 2λ2 C0 0



t

dt dt

[σˆ z , [σˆ z , ρS ]]

0

= ρS − λ2 C0 (τ /2)2 [σˆ z , [σˆ z , ρS ]]

(51)

This is the same form as the Eq. (28), and so we can successfully suppress the dephasing via quantum teleportation as long as a condition of mτ /2  τc .

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Transferring Quantum Information in Hybrid Quantum Systems Consisting of a Quantum System with Limited Control and a Quantum Computer Ryosuke Sakai, Akihito Soeda, and Mio Murao

Abstract We consider a hybrid quantum system consisting of a qubit system continuously evolving according to its fixed own Hamiltonian and a quantum computer. The qubit system couples to a quantum computer through a fixed interaction Hamiltonian, which can only be switched on and off. We present quantum algorithms to approximately transfer quantum information between the qubit system with limited control and the quantum computer under this setting. Our algorithms are programmed by the gate sequences in a closed formula for a given interface interaction Hamiltonian. Keywords Quantum algorithm · Quantum information transfer

1 Introduction Quantum information processing requires high controllability to prepare and manipulate quantum systems. However, actual physical systems described by Hamiltonians are not necessarily fully controllable due to the limitations for controllable parameters in the system, even for the cases where dissipation and decoherence are negligible. In spite of the difficulties, constructions of small scale highly-controllable quantum systems have been demonstrated in various quantum systems such as superconducting systems, quantum optical systems, trapped ion systems and NMR systems, where we can apply a wide range of quantum manipulations. However, large-scale realizations of such highly-controllable quantum systems are still under development. Instead of requiring every part of the quantum system to be highly-controllable, one possible way to achieve larger scale highly-controllable quantum information processing is to extend a small scale highly-controllable quantum system by combining with a R. Sakai · A. Soeda · M. Murao (B) Department of Physics Graduate School of Science, the University of Tokyo, Tokyo, Japan e-mail: [email protected] R. Sakai e-mail: [email protected] A. Soeda e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 Y. Hirayama et al. (eds.), Hybrid Quantum Systems, Quantum Science and Technology, https://doi.org/10.1007/978-981-16-6679-7_14

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physical system with limited controllability and introducing algorithms for controlling the physical system. Seeking methods to manipulate a quantum system with limited controllability by combining with a controllable quantum system has been studied as the context of quantum control theory of dynamics [1–13], which aims to implement desired operations by adding extra Hamiltonians with controllable parameters on the original Hamiltonian of the uncontrollable system. In the quantum control theory of dynamics,  the total Hamiltonian is typically given by H (t) = HS + Lk=1 αk (t)Hk where HS is the system Hamiltonian without controllable parameters and {Hk } are the additional control Hamiltonians that their contributions can be independently tuned by adjusting a time dependent parameter αk (t). It is often assumed that HS is the Hamiltonian of a composite system which includes interactions among the subsystems and Hk ’s are local Hamiltonians on the subsystems [5–10]. For the systems such as NMR systems, the time scale of the control Hamiltonian dynamics can be much faster than the uncontrollable system dynamics. That is, the control Hamiltonians can be applied as intensive pulses for such systems and the methods for the time optimal control with intensive pulses have been developed in [5–10]. In many other quantum systems, however, the range of the absolute value of αk (t) is limited and thus it is not possible to straightforwardly apply the methods using intensive pulses. We consider to combine a physical system with limited controllability with a quantum computer instead of Hamiltonian dynamics. Quantum computers are idealized systems, i.e., we can perform arbitrary operations allowed in quantum mechanics, which can be represented by the quantum circuit model. In the preceding works on achieving full control by locally induced relaxation [10–13], the authors also considered a physical system without controllable parameters coupling to a highly controllable system that can be regarded as a quantum computer. They have shown that in principle, any state of a physical system can be transferred to the quantum computer and can be transferred back from the quantum computer to the physical system, if the system Hamiltonian HS and the fixed interaction Hamiltonian Hint describing the coupling satisfy certain conditions that can be interpreted to induce relaxation of the physical system. Since any transformations can be applied in a quantum computer, ability of this type of input-output (IO) operations, namely, the ability to transfer quantum states from the physical system to the quantum computer (an output process) and transferring back quantum states from the quantum computer to the physical system (an input process) achieves the full dynamical control of the physical system via the quantum computer. Algorithms achieving approximate IO operations between a physical system without controllability composed of several subsystems and a quantum computer coupled for a fixed interaction Hamiltonian Hint between the physical system and a part of the quantum computers are also presented in [11–13]. These algorithms assume that the time scale of performing operations on the quantum computer is much faster than the dynamics generated by HS + Hint . However, the time cost for performing operations on the quantum computer is not necessary negligible in general, and the algorithms taking the effect of the time cost are desired for physical implementations.

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In this chapter, we investigate algorithms to achieve the approximate IO operations for a single-qubit physical system even when the time cost on running algorithms in the quantum computer is not negligible. To obtain such algorithms, we relax the requirement for the controllability of the coupling between the physical system and the quantum computer. Namely, we assume that we can switch Hint on and off. That is, the physical system evolves either by HS or HS + Hint whereas we cannot control the choice of Hint . Note that this assumption does not provide much power on the controllability of the physical system comparing to the situations considered in [11– 13], since assuming the infinitesimal time cost for operations in a quantum computer guarantees to be able to neglect the effect of HS and Hint while the operations of quantum computers are applied. We present two concrete algorithms implementing the output process for the approximate IO operations represented by quantum circuits. The first algorithm requires the quantum computer to have a linearly-growing quantum memory, which is referred as the linearly growing size (LS) algorithm. The second algorithm is an improved algorithm that requires only a constant-size quantum memory, which is referred as the constant size (CS) algorithm. The quantum circuits representing both of the two algorithms contain the interface unitary Uint generated by HS + Hint satisfying certain conditions. We derive the conditions for Uint for implementing the approximate IO operations by using each algorithm. We also evaluate the accuracy of the implemented operations in the diamond norm [14], and show that the CS algorithm achieves the IO operation with an arbitrary accuracy by using a finitesize quantum memory. We present the algorithms implementing the input process corresponding to the LS and CS algorithms for the approximate IO operations. This chapter is organized as follows. We present our settings and clarify the assumptions on the operations on a physical system and quantum computers in Sect. 2. We show a lemma which evaluates the accuracy of the desired operation for the physical system by giving trial approximate IO operations in the diamond norm in Sect. 3. In Sects. 4 and 5, we show the LS/CS algorithms programmed by the gate sequences for the trial approximate output operation, and demonstrate the necessary conditions for interface unitary operations. We complete the algorithms for the IO operation by applying a final adjustment for the LS/CS algorithms in Sect. 6. We will show how to prepare the interface unitary operations satisfying such conditions by the given interaction Hamiltonian, and discuss whether two specific systems can be manipulatable in Sect. 7. A summary of our results is presented in Sect. 8.

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2 Settings 2.1 Assumptions for a Physical System Coupling to a Quantum Computer We consider a physical system denoted by S represented by a two-dimensional Hilbert space HS , a qubit system. System S evolves according to a fixed time-independent self-Hamiltonian HS and has limited controllability. Our goal is to apply arbitrary quantum operations on S by coupling to a quantum computer. We assume that the quantum computer in contrast to S is able to perform arbitrary quantum operations represented by quantum circuits. We set the Planck units to be  = 1. For simplicity, we sometimes abbreviate the the joint system of S and I by SI , the joint system of I and R by IR and the total system of S, I , and R by SIR. We refer to a basis {|j SI }j of the joint system SI as a local basis when each element of the basis states can be represented as a product of normalized pure states. We model the coupling between the system S and the quantum computer by introducing two subsystems in the quantum computer, a register system R and an interface system I . Subsystem R is the main processing part of the quantum computer consisting of N qubits represented by a 2N -dimensional Hilbert space HRN . The computational basis of each qubit in R is represented by {|0, |1}. We assume that any quantum operations can be freely performed on R. Subsystem I is a mediating system in the quantum computer which directly couples to S by a fixed interaction between S and I . We set I to be a single qubit system represented by a two-dimensional Hilbert space HI . We assume that we can only control switching on and off of a single fixed interaction Hamiltonian represented by Hint on HS ⊗ HI . The total Hamiltonian of system SI is given by HSI (t) := HS ⊗ II + αon−off (t)Hint ,

(1)

where αon−off (t) is a binary switching parameter for time t that can be designed arbitrarily. We assume that I behaves as a part of the quantum computer, i.e., any quantum operations can be performed on HI ⊗ HRN when αon−off (t) = 0, but it is dominated by the Hamiltonian dynamics according to the Hamiltonian on HS ⊗ HI given by Hon := HS ⊗ II + Hint ,

(2)

when Hint is switched on (αon−off (t) = 1), in which case all additional operations on I are prohibited. The quantum circuit represented by Fig. 1 covers all possible deterministic operations under the assumptions made for systems S, I , and R. We refer to the set of unitary operators on HS ⊗ HI ⊗ HRN of which quantum circuit decompositions are represented by Fig. 1 as LUNHS ,Hint .

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e−iHon t1

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Fig. 1 A quantum circuit representation of a set of most general unitary operators LUN HS ,Hint allowed in our setting. tj and tj (for j = 1, 2, . . .) are duration time of switching on and off, respectively. Vj and Uj are arbitrary unitary operators on HRN and HI ⊗ HRN , respectively S

−iHS t1 −iHS t2 Uint e Uint e

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Fig. 2 A quantum circuit representation of a set of unitary operators LUN HS ,Uint in the simplified setting where the interaction Hamiltonian is switched on for a fixed duration time τ . Vj and Uj are arbitrary unitary operators on HRN and HI ⊗ HRN , respectively

The independent parameters {ti } and {tj } in the quantum circuit representation of LUNHS ,Hint introduce too large degrees of freedom to handle concrete constructions of the approximate IO operations. To handle this difficulty, we introduce a simplified setting by choosing a fixed time duration τ = t1 = t2 = · · · on the switched on time of Hint . This simplification is equivalent to applying a fixed unitary operator Uint = e−iHon τ

(3)

on HS ⊗ HI while Hint is switched on. We refer to Uint as an interface unitary operator. The quantum circuit represented by Fig. 2 covers all possible unitary operators in this simplified setting. We refer to the set of unitary operators on HS ⊗ HI ⊗ HRN represented by Fig. 2 as LUNHS ,Uint which is obviously a subset of LUNHS ,Hint . We construct the gate sequences of approximated IO algorithms in this setting for a given interface unitary operator Uint in LUNHS ,Uint . We also introduce a further simplified setting where we can switch off HS while any Uj is applied on system IR represented by the quantum circuit given by Fig. 3. In this case, the parameters {tj } can be also eliminated. We refer to the set of unitary operators on HS ⊗ HI ⊗ HRN represented by Fig. 3 as LUUint . As the first step, we present algorithms for the approximate IO operations in this settings in Sects. 4 and 5. One of the motivations of our work is to consider the non-negligible time cost for performing operations on the quantum computer, therefore this simplified setting may look similar to the case where the time evolution due to HS during

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S

I R

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.. V1 .

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N Fig. 3 A quantum circuit representation of a set of unitary operators LUU . Vj and Uj are arbitrary int unitary operators on HR and HI ⊗ HR , respectively

the time operating quantum computer is ignored. However, we show that both of the proposed algorithms in the setting for LUUint can be extended to the setting for LUNHS ,Uint , which fully includes the time cost of quantum operation in the quantum computer by introducing a final adjustment utilizing the parameters {tj } in Sect. 6. There are still too many parameters in a whole set of two qubit unitary operators Uint to construct concrete algorithms. We restrict our consideration of Uint to the ones of which matrix representation in a local basis {|iS ⊗ |jI }i,j=0,1 of HS ⊗ HI is given by  i|S j|I Uint |kS |lI i,j,k,l=0,1 ⎤ ⎡ u00,00 u00,01 u00,10 u00,11 ⎢u01,00 u01,01 u01,10 u01,11 ⎥ ⎥ =⎢ ⎣ 0 u10,01 u10,10 u10,11 ⎦ , 0 u11,01 u11,10 u11,11 

(4)

where uij,kl (i, j, k, l = 0, 1) are complex numbers. We denote a set of such unitary operators on HS ⊗ HI by U . We will show that for all Uint ∈ U there exist approximated output processes of the IO operations by presenting the LS and CS algorithm in Sects. 4 and 5. Thus we call the local basis {|iS |jI }i,j=0,1 satisfying Eq. (4) for a given interface unitary operator Uint as the feasible basis of Uint . We assume that the computational bases {|0, |1} of SI are chosen to be the basis of each system for the feasible basis in the following.

3 Approximate IO Operations Evaluated in Terms of the Diamond Norm In our algorithms, we consider a procedure to implement an arbitrary quantum operation denoted by M on S by the input-output (IO) operations consisting of the following three steps. 1. The output process from S to I : Design an operation in LUNHS ,Uint to transfer an arbitrary state from S to I .

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2. Application of M in I : Apply a desired operation M on I , which is always possible while Hint is switched off in our definition. 3. The input process from I to S: Design an operation in LUNHS ,Uint to transfer back the state from I to S. Obviously, if a unitary operator called a half-swap operation Uhswap on HS ⊗ HI satisfying Uhswap |0S |0I = |0S |0I and Uhswap |1S |0I = |0S |1I is in LUNHS ,Uint , the output process is exactly achievable by setting an initial state of I to be |0 and applying Uhswap . We consider the cases where such a convenient unitary is not available in LUNHS ,Uint and investigate a strategy for approximately implementing the output and input processes by applying unitary operations on IR. In the following of this section, we rename a part of R used to implement M by a unitary operation on I and the extra ancilla system as system E, and remove E from R. We set that R consists of M qubits and E consists of N − M qubits. We also set the initial state of IR to be |0I |0⊗M R . For approximation, we evaluate the difference between the desired operation M and the approximated IO operation in terms of the diamond norm to guarantee the worst case to be within a preset bound. Our strategy is to consider a one-parameter family of unitary operations TMout (ξout ) on HS ⊗ HI ⊗ HRM for approximately implementing the output process by transforming an arbitrary state |ψS S := aS |0S + bS |1S of S satisfying |aS |2 + |bS |2 = 1 of IR as for aS , bS ∈ C and |0I |0⊗M R 

2 TMout (ξout )|ψS S |0I |0⊗M = |0S aS |0I + bS 1 − ξout |1I |0⊗M R R + bS ξout |1S |gout IR ,

(5)

where the parameter ξout is set to be 0 ≤ ξout ≤ 1 and |gout IR can be any state. The achieves the exact output process from S to I . action of TMout (0) on |ψS S |0I |0⊗M R Similarly, we consider another one-parameter family of unitary operation TMin (ξin ) on HS HI ⊗ HRM for approximately implementing the input process satisfying, namely, 

†  in a = |0 |0 + b 1 − ξin2 |1I |0⊗M TM (ξin ) |ψS S |0I |0⊗M S S I S R R + bS ξin |1S |gin ,

(6)

where 0 ≤ ξin ≤ 1 and |gin  ∈ S(HI ⊗ HRM ) can be any state. The action of TMin (0) achieves the exact input process from I to S. on |0S |ψS I |0⊗M R We show that we can construct an approximate IO operation by using TMout (ξout ) and TMin (ξin ). In addition to implementing an arbitrary quantum operation M on HS , this construction also guarantees to preserve coherence between S and an external system denoted by E. We introduce a quantum operation on HS ⊗ HE denoted by ξout ,ξin on any M . We define the approximate IO operation as a composite map M  linear operator X on HS and HE for implementing an arbitrary map M by

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Fig. 4 A quantum circuit ξout ,ξin representation of M  defined by Eq. (7) for implementing a map M on HS ⊗ HE . The initial states of systems in HI ⊗ HRM are set to be in |0I |0⊗M R

out ,ξin ΦξM

E

I

S

R

|0 0|I |0 0|1 |0 0|2

.. .

|0 0|M

ξ

,ξ

M'

E

I .. .

out (ξ UTM out )

.. .

in (ξ ) UTM in

 (UTMin (ξin ) ⊗ IE ) ◦ (M ⊗ IS ⊗ IR ) 

 +1 ◦ (UTMout (ξout ) ⊗ IE ) X ⊗ |00|⊗M , IR

.. .

Tr

out in M (X ) := Tr IR 

(7)

where Tr IR represents a partial trace operation on system IR, IA represents an identity map on system A and UV represent a unitary map corresponding to a unitary operator ξout ,ξin is shown in Fig. 4. V on HI ⊗ HRM . The quantum circuit representation of M  ξout ,ξin and The following lemma shows that the difference between the two maps M   M measured in terms of the diamond norm [16] is bounded by the parameters ξout and ξin . The norm • denotes the diamond norm for a map •. Lemma 1 For any M on S and E, and 0 ≤ ξout , ξin ≤ 1, the approximate IO operξout ,ξin satisfies ation M     ξout ,ξin  M − M  ≤

where  := −1 +



√ 2 1 − 2 (  ≥ 0 ) , 2 (