Electrical Control and Quantum Chaos with a High-Spin Nucleus in Silicon (Springer Theses) 3030834727, 9783030834722

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Springer Theses Recognizing Outstanding Ph.D. Research

Serwan Asaad

Electrical Control and Quantum Chaos with a High-Spin Nucleus in Silicon

Springer Theses Recognizing Outstanding Ph.D. Research

Aims and Scope The series “Springer Theses” brings together a selection of the very best Ph.D. theses from around the world and across the physical sciences. Nominated and endorsed by two recognized specialists, each published volume has been selected for its scientific excellence and the high impact of its contents for the pertinent field of research. For greater accessibility to non-specialists, the published versions include an extended introduction, as well as a foreword by the student’s supervisor explaining the special relevance of the work for the field. As a whole, the series will provide a valuable resource both for newcomers to the research fields described, and for other scientists seeking detailed background information on special questions. Finally, it provides an accredited documentation of the valuable contributions made by today’s younger generation of scientists.

Theses may be nominated for publication in this series by heads of department at internationally leading universities or institutes and should fulfill all of the following criteria • They must be written in good English. • The topic should fall within the confines of Chemistry, Physics, Earth Sciences, Engineering and related interdisciplinary fields such as Materials, Nanoscience, Chemical Engineering, Complex Systems and Biophysics. • The work reported in the thesis must represent a significant scientific advance. • If the thesis includes previously published material, permission to reproduce this must be gained from the respective copyright holder (a maximum 30% of the thesis should be a verbatim reproduction from the author’s previous publications). • They must have been examined and passed during the 12 months prior to nomination. • Each thesis should include a foreword by the supervisor outlining the significance of its content. • The theses should have a clearly defined structure including an introduction accessible to new PhD students and scientists not expert in the relevant field. Indexed by zbMATH.

More information about this series at http://www.springer.com/series/8790

Serwan Asaad

Electrical Control and Quantum Chaos with a High-Spin Nucleus in Silicon Doctoral Thesis accepted by UNSW Sydney, Kensington, Australia

Author Dr. Serwan Asaad Niels Bohr Institute University of Copenhagen Copenhagen, Denmark

Supervisor Prof. Andrea Morello School of Electrical Engineering and Telecommunications UNSW Sydney Kensington, Australia

ISSN 2190-5053 ISSN 2190-5061 (electronic) Springer Theses ISBN 978-3-030-83472-2 ISBN 978-3-030-83473-9 (eBook) https://doi.org/10.1007/978-3-030-83473-9 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed 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 Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To Delawer, Yuxian, and Hans To Judith

Supervisor’s Foreword

The recurring theme of Serwan Asaad’s Ph.D. thesis is the exploration of the unknown, the unexpected, and the forgotten. Undertaking a such journey within a hotly-researched, richly-funded and milestone-driven research field adds a mildly transgressive flavor to the enterprise. The result is yet another proof that being daring and transgressive is often the best way to make genuine progress. The broad scientific context of this thesis is spin-based quantum computing. In particular, the development of quantum computers based upon the spin of donor atoms in silicon. This field of research is more than twenty years old, and in the last decade has seen spectacular progress in manufacturing and operating high-quality qubits within silicon nanoelectronic devices. The potential for this research to make an impact on science and technology is enormous. The way forward seems to be clearly mapped out: demonstrate high-fidelity multi-qubit logic operations; construct error-protected logical qubits; integrate the device fabrication with foundry-based manufacturing for production at scale. Are we overlooking something? Is there any fundamental quantum science we can—and should—explore while we’re busy building quantum computers? Fundamental science that may stand in the way of our grand plans, or may help us getting there faster by taking a different route? Answering these questions is the spirit, and the practice, of this Ph.D. thesis. The original ‘plan’ of this research was to explore the emergence of quantum chaos at the single-particle level. Quantum chaos was a hot topic a few decades ago, when electrons in semiconductor quantum dots, or cold atomic clouds in vacuum, were studied as model systems to observe the signatures of chaotic dynamics in quantummechanical objects. The complexity of dealing with such many-body systems had left some questions open around the precise way in which chaos emerges out of the quantum realm. In the meantime, the development of increasingly large quantum processors demands a critical look at how chaotic dynamics may affect the performance of large-scale quantum devices which contain, at their heart, non-linear Hamiltonian terms. This is why we embarked in a project to study the emergence of chaos in a simple, highly coherent, single-atom system: the large nuclear spin (with quantum number I = 7/2) of a 123-Sb atom. The fascinating theory underpinning this project is developed and described in Chap. 8. vii

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Having a clear picture of what to look for—a high-spin nucleus with a large nuclear quadrupole coupling, driven by a strong radiofrequency field—Serwan set off to perform the experiment. What we expected would be the ‘reduction to practice’ of a well-planned project, turned out to reveal something that has an even more remote history than quantum chaos. The device behaved oddly. An expected spin transition seemed impossible to observe. After painstaking trials and test, plenty of head-scratching and heated debates within the group, Serwan realized that he had accidentally (re)discovered Nuclear Electric Resonance (NER). This is the process whereby an oscillating electric field induces transitions in a spin larger than 1/2, placed in a lattice site that lacks point inversion symmetry. It was suggested by Nobel laureate Nico Bloembergen in 1961, investigated with some intensity during the 1960s, then effectively forgotten. It is a very small effect, extremely difficult to observe in an experiment. Fast-forward nearly sixty years, the effect appeared unexpectedly in our lab. This happened because of a combination of experimental accident (a damaged microwave antenna radiating electric fields instead of the intended magnetic fields), and having set up the spin in an extremely quiet environment, where coherence times are exceptionally long (around 0.1 seconds) and even the faintest effects can be observed with utmost clarity. These results are described in Chap. 6, and their microscopic theoretical underpinnings explained in Chap. 7. Adding to the drama of the discovery, these results were published in Nature on the exact day of Nico Bloembergen’s 100th birthday. The remainder of the thesis provides a broad and rigorous introduction to donorbased quantum technologies, including the theory of high-dimensional spin systems (Chap. 2), donors in silicon (Chap. 3), and extensive coverage of the experimental techniques (Chap. 4) and device characterization (Chap. 5). Several appendices provide technical details on aspects of the experiment and the theory of interest to the highly specialized reader. The Ph.D. work of Serwan Asaad suggests an unexpected way forward for spinbased quantum technologies, by providing the option of controlling highly coherent nuclear spins using electric fields. These ideas and discoveries are already being woven into the next generation of spin-based quantum computers in silicon. The fundamental angle of quantum chaos is certain to resurface in the near future, when these devices are put to the test in the lab. Overall, this work is a tribute to curiosity-based research, and the surprising ways in which its finds its way back into transformative technologies. Sydney, Australia April 2021

Scientia Professor Andrea Morello

Abstract

Nuclear spins are highly coherent quantum objects. In large ensembles, their control and detection via magnetic resonance is widely exploited, e.g., in chemistry, medicine, materials science, and mining. Nuclear spins also prominently featured in early ideas and demonstrations of quantum information processing. In silicon, the high-fidelity coherent control of a single phosphorus (31-P) nuclear spin I=1/2 has demonstrated record-breaking coherence times, entanglement, and weak measurements. Replacing 31-P by a high-spin donor leverages all of these features, and unlocks a rich variety of experiments that probe quantum physics at a fundamental level. In this thesis, we demonstrate the coherent quantum control of a single antimony (123-Sb) donor atom, whose higher nuclear spin has eight nuclear spin states. However, rather than conventional nuclear magnetic resonance (NMR), we employ nuclear electric resonance (NER) to drive nuclear spin transitions using localized electric fields produced within a silicon nanoelectronic device. This method exploits an idea first proposed in 1961 but never realized experimentally with a single nucleus, nor in a non-polar crystal such as silicon. Our results are quantitatively supported by a microscopic theoretical model that reveals how the purely electrical modulation of the nuclear electric quadrupole interaction results in coherent nuclear spin transitions. The spin dephasing time, 0.1 seconds, surpasses by orders of magnitude those obtained via methods that require a coupled electron spin for electrical drive. The coherent control of an 123-Sb nucleus opens up the door to experiments on the foundations of quantum mechanics, such as the emergence of chaos in the quantumclassical transition. We present here a realistic proposal to construct a chaotic driven top from the 123-Sb nuclear spin. We show that signatures of chaos are expected to arise for experimentally realizable system parameters, allowing the study of the relation between quantum decoherence and classical chaos, and the observation of dynamical tunneling. These results show that high-spin quadrupolar nuclei could be deployed as chaotic models, strain sensors, and hybrid spin-mechanical quantum systems using allelectrical controls. Integrating electrically-controllable nuclei with quantum dots could pave the way to scalable, nuclear- and electron-spin-based quantum computers in silicon that operate without the need for oscillating magnetic fields. ix

Acknowledgements

I came to Australia seeking adventure, and in this I succeeded. And by this I don’t only mean the deadly spider encounters, but also to have been given the chance of researching truly exciting physics in a group of wonderfully diverse people whom I now call friends. For this, I feel blessed. Although I ventured into the unknown on the other side of the planet, things feel a lot like home. Thank you Andrea, for having given me this opportunity, and for being my mentor. Though I could fill pages listing all the things you have taught me over the past years, I will instead say the two things about you I will remember the most. First of all, it is the raw excitement you have for physics and that you share with us. This excitement can be easily forgotten when trying to figure out why an experiment keeps failing, and I cannot stress enough the importance of being reminded of it. But perhaps more importantly is the trust that you place in each of us. It has given me the courage to explore new territory, make mistakes, and as a result, improve myself. Vincent, without you I never would have embarked on this journey. You have shown me, above all else, the importance of cultivating an independent mind. You have also taught me the fine line between an asset and a liability. I have greatly enjoyed working with you, and I am grateful for your support throughout. Arne, you have been here from the start, and have been integral to my time here. Thank you for your patience in explaining even the most mundane physics, and for the countless discussions that have shaped my understanding of spin physics. Oh, and thank you for the bike. I would like to thank my assessment committee Prof. Martin Brandt and Prof. Peter Zoller for taking the time to read this dissertation and provide insightful comments. These comments have helped improve this thesis and could even spark a next generation of high-spin experiments. Being part of this research group has enriched my experience here in many ways, from the parties and the pub crawls, to the heated and passionate arguments, both physics and otherwise. And we mustn’t forget the frequent birthday cakes, which were always a welcome addition, though by no means forced... Mark, I’m glad to have you as a companion throughout all the experiments. I shall not forget our frequent sparring, which has countless times helped to resolve a problem by looking at it from a fresh new angle. Steffi, even though I claim to be seeking adventure, I xi

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realize through your escapades that that’s not even half true. It’s been great having you along for the ride. Benni, your word in theory and simulations is law. I can count myself lucky to have had you on board this project. Mateusz, thank you for your unparalleled fabrication skills, which have made much of this work possible. And thank you for introducing me to Korean BBQ; may the Soju be ever-flowing. Vivien, your advice has helped me on numerous occasions, and your strong opinions have taught me to be critical. I hope this thesis contains some figures that you at least somewhat approve of. Hannes, I think we’re a really good duo of rubber duckies, you have often helped me get to the crux of any issues I’m facing. Thank you and also Tim for all the memes making fun of my thesis writing; in the words of Abraham Lincoln, “I laugh because I must not cry, that is all, that is all.” Outside our research group, I would like to thank Andrew Baczewski, whose expertise on quantum chemistry has been crucial in understanding our observations. I would also like to thank Philip Blocher, whose visit here in Sydney has proven fruitful and enjoyable. His contributions have paved the way for further exciting experiments that probe quantum chaos. Throughout my life, my parents have always stood by my side. I can always draw inspiration from their lives, from the struggles and the choices that they’ve made. Knowing they’re my parents and I am their son fills me with pride, and I will continue to try and live up to this honor. And lastly Judith, we took a big step into the unknown when we came here, and I must admit I wasn’t sure how this was going to go. But now three and a half years later, I can say without a doubt that this went even better than I possibly could have hoped for. And if we can not only survive, but thrive together on another continent, then I’m certain you and I can make it through anything. you have always been there for me, even during these last months of thesis writing, and this has really helped me to “spin it to win it.”

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Coherent Control of a Quantum Spin . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Manipulating a High-Spin Nucleus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 3 6

2 High-Dimensional Spins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The Discovery of Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Nuclear Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Nuclear Spin States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Spin Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Spin Coherent States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Spin Visualization with the Husimi Q Distribution . . . . . . . . . . . . . . 2.7 Generalized Rotating Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1 Description of the Generalized Rotating Frame . . . . . . . . . . . 2.7.2 Derivation of the Generalized Rotating Frame . . . . . . . . . . . . 2.8 Basis Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Arbitrary State Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 9 11 12 12 14 15 17 17 18 20 23 24

3 Theory of Donors in Silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Solid State Physics of Donors in Silicon . . . . . . . . . . . . . . . . . . . . . . . 3.2 Donor Spin Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Neutral Donor Spin Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Zeeman Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Hyperfine Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Low-Field Versus High-Field Limit . . . . . . . . . . . . . . . . . . . . . 3.2.5 Resonant Driving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.6 Ionized Donor Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25 25 27 27 28 28 29 31 33

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3.3 Nuclear Quadrupole Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Nuclear Quadrupole Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Estimates of Nuclear Quadrupole Interaction . . . . . . . . . . . . . 3.3.3 Nuclear Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Extraction of Quadrupole Parameters . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33 33 36 38 39 41

4 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Architecture of an 123-Sb Sb-Implanted Silicon Nanodevice . . . . . . . . 4.2 The Single Electron Transistor (SET) . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Electrostatics of the SET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 SET Modes of Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Fabrication Protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 123-Sb Sb Implantation Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Device Packaging and Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Instrumentation and Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Phase-Coherent DDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 SilQ Measurement Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43 43 45 45 48 49 51 52 52 54 55 57

5 123-Sb Donor Device Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Charge Sensing with an SET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Calibration of the SET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Charge Stability Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Donor Triangulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Electron Spin Control and Readout . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Donor Electrochemical Potential Regimes . . . . . . . . . . . . . . . 5.3.2 Electron Readout and Initialization Fidelity . . . . . . . . . . . . . . 5.4 ESR Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Flip-Flop Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Flip-Flop Rabi Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Flip-Flop Driven Nuclear-Spin Initialization . . . . . . . . . . . . . 5.6 Continuous Tuning via a Neural-Network . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59 60 60 61 65 67 67 71 73 76 76 77 79 82

6 Nuclear Electric Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Initial Nuclear Resonance Measurements . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Nuclear Spin Initialization, Manipulation, and Readout . . . . 6.1.2 First Nuclear Transition and Rabi Oscillations . . . . . . . . . . . . 6.1.3 Measurements of Subsequent Transitions . . . . . . . . . . . . . . . . 6.2 Nuclear Electric Resonance (NER) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Properties of NER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 m I = ±1 Nuclear Spectrum and Rabi Oscillations . . . . . . 6.2.3 m I = ±2 Nuclear Spectrum and Rabi Oscillations . . . . . . 6.2.4 Power Dependence of Rabi Frequencies . . . . . . . . . . . . . . . . .

83 84 84 87 88 90 90 93 94 97

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6.3 Antenna-Driven NER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Enhanced Electric Fields from a Melted Antenna . . . . . . . . . 6.3.2 Gate-Driven Versus Antenna-Driven NER . . . . . . . . . . . . . . . 6.4 Linear Quadrupole Stark Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Nuclear Coherence Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Possible Quadrupole Orientations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97 97 99 100 102 105 106 107

7 Microscopic Crystalline Origins of the Quadrupole Interaction . . . . . 7.1 Microscopic Origins of the Electric Field Gradient . . . . . . . . . . . . . . 7.2 Finite-Element Model of the Nanostructure Device . . . . . . . . . . . . . . 7.3 Electric-Field-Induced Quadrupole Splitting and NER . . . . . . . . . . . 7.3.1 The Linear Quadrupole Stark Effect (LQSE) . . . . . . . . . . . . . 7.3.2 The Electric-Field Response Tensor . . . . . . . . . . . . . . . . . . . . 7.3.3 Electric-Field Response-Tensor Estimate . . . . . . . . . . . . . . . . 7.3.4 Comparison to LQSE Measurements and Empirical Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Strain-Induced Quadrupole Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 The Gradient-Elastic Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Gradient-Elastic Tensor Calculations . . . . . . . . . . . . . . . . . . . . 7.4.3 Calculation of Quadrupole Splitting Due to Strain . . . . . . . . 7.5 Alternative (Unlikely) Sources of NER . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Direct Effect of Electric Gate Potentials . . . . . . . . . . . . . . . . . 7.5.2 Mechanical Driving Through SiO2 Piezoelectricity . . . . . . . 7.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

109 110 112 113 113 115 118

8 Exploring Quantum Chaos with a Single High-Spin Nucleus . . . . . . . 8.1 Background of Quantum Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Chaos Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.3 Quantum Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.4 Experimental Tests of Quantum Chaos . . . . . . . . . . . . . . . . . . 8.2 The Classical Chaotic Driven Top . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 The Quantum Driven Top . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Experimental Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Comparison Between Classical and Quantum Hamiltonian Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3 Realizing a Quantum Driven Top in the Laboratory Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.4 Eigenbasis Mismatch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.5 Realizing a Quantum Driven Top in the Rotating Frame . . . 8.3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

120 121 122 123 124 125 125 125 127 128 131 132 132 133 135 136 137 140 141 143 144 146 148 150

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8.4 Quantum Dynamics and the Floquet Formalism . . . . . . . . . . . . . . . . . 8.5 Quantum Versus Classical Dynamics: A Comparison . . . . . . . . . . . . 8.5.1 Decoherence as a Precursor of Chaos . . . . . . . . . . . . . . . . . . . 8.5.2 Dynamical Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.3 Dependence of Dynamical Tunneling Rate on System Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

150 152 152 153

9 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Comparison with Other High-Dimensional Quantum Systems . . . . . 9.3 Further Characterization of the Quadrupole Coupling to Strain and Electric Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Strain Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Quantum Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Mutual Coupling of High-Dimensional Spins . . . . . . . . . . . . . . . . . . . 9.7 Quantum Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.8 Quantum Metrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.9 Spin-Mechanical Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

163 163 164

155 157 158

165 166 167 168 169 170 171 173

Appendix A: Finite Element Model Parameters . . . . . . . . . . . . . . . . . . . . . . 177 Appendix B: Hyperfine-Coupled 123 Sb Nucleus . . . . . . . . . . . . . . . . . . . . . . 179 Appendix C: Slope in Δm I = ±2 Rabi Frequencies . . . . . . . . . . . . . . . . . . . 181 Appendix D: Spectral Shift at Charge Transition . . . . . . . . . . . . . . . . . . . . . 183 Appendix E: DFT Simulation Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 Appendix F: Classical Equations of Motion for the Driven Top . . . . . . . . 187 Appendix G: Quantum Driven Top in the Rotating Frame and the Rotating Wave Approximation . . . . . . . . . . . . . . . . . . 189 Appendix H: Simulation Details for Chaotic Dynamics . . . . . . . . . . . . . . . . 193 About the Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

Acronyms

2DEG Al AWG CPW DDS DFT EBL EFG ESR FPGA HF LPCVD LQSE MRI NAR NER NI NMP NMR NQR OTOC PCB PMMA RF SEM SET SRS

Two-dimensional electron gas. Aluminum. Arbitrary waveform generator. Coplanar waveguide. Direct digital synthesis. Density functional theory. Electron beam lithography. Electric field gradient. Electron spin resonance. Field-programmable gate array. Hydrogen fluoride. Low pressure chemical vapor deposition. Linear quadrupole Stark effect. magnetic resonance imaging. Nuclear acoustic resonance. Nuclear electric resonance. National Instruments. N-Methyl-2-pyrrolidone. Nuclear magnetic resonance. Nuclear quadrupole resonance. Out-of-time ordered correlator. Printed circuit board. Poly(methyl methacrylate). Radio-frequency. Scanning electron microscope. Single electron transistor. Stanford Research Systems.

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Chapter 1

Introduction

The advent of quantum physics shook the foundations of our understanding of nature. The prevalent notion of determinism—that the future could be predicted given sufficient knowledge of the present—was found to be incompatible with the observations of quantum experiments. Instead, the inherent uncertainty that is fundamental in quantum physics ushered in a new way of looking at nature, one that needs to be reconciled with our classical picture of nature. The subsequent century of quantum experiments marked itself by increasingly intricate experiments aimed at understanding the effects of quantum mechanics, reconciling quantum mechanics with classical mechanics, and harnessing the quantum for practical applications.

1.1 Coherent Control of a Quantum Spin Particle spin has always stood at the forefront of quantum experiments. One of the most prominent examples is the Stern-Gerlach experiment [1, 2], where silver atoms were shot through an inhomogeneous magnetic field. The interaction of the silver atom’s outer electron spin with the inhomogeneous magnetic field resulted in a two-fold splitting of the final silver atom distribution, as opposed to a continuous distribution. It has thus served as one of the clearest demonstrations of quantization. The nuclear spin of an atom has also been instrumental in the development of quantum physics. One of the primary features of the nuclear spin is that it is extremely coherent, and thus sustains the fragile quantum effects over long periods of time. In large ensembles, their control and detection via magnetic resonance is widely exploited, e.g. in chemistry, medicine, materials science and mining. Nuclear spins also featured in early ideas [3] and demonstrations [4] of quantum information processing. On the road towards miniaturization, systems have now reached the point where individual nuclear spins can be controlled coherently. Individual nuclear spin control is achieved in different systems [5], among which are trapped ions [6, 7], nitrogen-vacancies and © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Asaad, Electrical Control and Quantum Chaos with a High-Spin Nucleus in Silicon, Springer Theses, https://doi.org/10.1007/978-3-030-83473-9_1

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other color centers in diamond [8–10], magnetic molecules [11], and dopant atoms in silicon [12, 13]. In silicon, the idea of manipulating the spins of individual 31 P donor atoms was brought forward in a seminal paper by Kane [3]. A follow-up paper by Morello et al. [14] refined these ideas into the proposal of a device that could read out and control the spins of individual ion-implanted 31 P donor atoms. This proposal was soon turned into a reality by the demonstration of initialization, coherent control, and readout of the 31 P outer valence electron spin [15, 16]. This was followed soon thereafter by coherent control of the 31 P nuclear spin [12]. Both the 31 P electron and nuclear spin were shown to exhibit record-breaking coherence times [17], and highfidelity control [18]. These properties make 31 P donor atoms in silicon an attractive candidate for a quantum computer, and are therefore featured in several proposed architectures for a quantum computer [3, 19, 20]. However, the 31 P nucleus is the smallest of the group-V donor atoms, and has the lowest nuclear spin of I = 1/2. As such, the nucleus only has 2I + 1 = 2 possible spin states, thereby limiting the range of experiments that can be performed. The heavier group-V donors all have a higher nuclear spin, and this opens up the path to new exciting experiments that probe quantum mechanics at a deeper level.

1.2 Manipulating a High-Spin Nucleus In this thesis, we present the coherent control of a single 123 Sb donor atom in silicon, which has a nuclear spin of I = 7/2 and hence 2I + 1 = 8 possible spin states. Whereas the 31 P donor can only be driven directly by magnetic fields, this is not the case for high-spin nuclei. Nuclei with spin I > 1/2 possess an electric quadrupole moment caused by the non-spherical charge distribution [21], which interacts with electric field gradients. We demonstrate that we can modulate the nuclear quadrupole interaction by an electric field, and thus drive nuclear spin transitions electrically via nuclear electric resonance (NER) [22]. This has important advantages, as it is far easier to confine and screen electric fields at the nanoscale level than magnetic fields. This method distinguishes itself from previous methods, where the neutral donor’s outer electron is used to transduce the electric fields via the hyperfine interaction [23], severely limiting the nuclear coherence. In the current method, the covalent bonds between the donor atom and neighboring silicon atoms are responsible for the electric-field-induced quadrupole modulation. This has the advantage that it can be applied to an ionized donor, thereby preserving the exquisite nuclear coherence. This effect has previously only been observed in polar crystals [24–28], and we show here that it also exists in silicon, a non-polar crystal. Using NER, we characterize the full spectrum of the 123 Sb nucleus, and demonstrate coherent driving between the spin states. We measure nuclear coherence times on the order of 0.1 s, opening up the pathway to experiments that probe the subtle effects of quantum physics. The 123 Sb nucleus is perfectly suited to tackle a problem that lies at the heart of the quantum-classical transition: the emergence of chaos (see [29] for an excellent

1.2 Manipulating a High-Spin Nucleus

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guide on quantum chaos). In classical dynamics, chaotic systems are characterized by their exponential sensitivity to perturbations. Chaos is ubiquitous in nature, and is found in numerous fields, such as ecology [30], chemistry [31], physiology [32], and cryptography [33]. In quantum mechanics, however, the inherent quantization leads to the suppression of chaos [34]. Since the correspondence principle states that quantum mechanics should smoothly transform into classical dynamics as the system size increases [35, 36], chaos must be an emergent phenomenon. Understanding this emergence of chaos from a quantum system provides valuable insights into the transition to classical mechanics, as well as neighboring topics such as decoherence [37], and measurements [38, 39]. The nucleus of an 123 Sb donor in silicon can shed light on these issues, as its spin Hamiltonian is the quantum equivalent of a classically-chaotic system: the driven top [40]. We simulate the dynamics of the 123 Sb nuclear spin and compare it to that of the classical driven top. Two experiments are devised, one of which seeks a correspondence between the quantum and classical dynamics by identifying decoherence as a precursor to chaos. The second experiment aims to observe dynamical tunneling, an effect that is classically forbidden and thus highlights the true quantum nature of the system. The execution of these experiments would constitute the first ever experiment on the quantum-chaotic dynamics of a single quantum system. It would provide valuable insights into the emergence of chaos and the potential disappearance of quantum effects. Due to its high dimensionality, high coherence, and high-fidelity control and readout, the 123 Sb donor in silicon is a rich system that enables novel quantum experiments. The topic highlighted in this thesis—quantum chaos—is but one of the many directions that can be pursued. The 123 Sb nucleus’ response to electric fields, combined with the electron-nuclear flip-flop transition [20], opens up the potential for all-electric control of the donor. Furthermore, the nuclear spin can serve as a qudit in a quantum processor, and can be integrated into the existing flipflop qubit proposal [20], leveraging a higher nuclear spin and hence an increased storage of quantum information. Finally, the quadrupole-induced spin coupling to strain opens up avenues to new hybrid systems that utilize spin-mechanical coupling. These examples illustrate the potential of the 123 Sb donor architecture to advance the field of quantum physics.

1.3 Thesis Outline Chapter 2—High-Dimensional Spins This chapter describes the properties of high-dimensional nuclear spins, i.e. nuclei with spin I > 1/2. Examples of high-spin nuclei are those of group-V donors other than 31 P, including 123 Sb with I = 7/2. Special focus is placed on spin coherent states. These are states that have a minimum spin uncertainty, and provide a useful method of

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1 Introduction

visualizing arbitrary high-dimensional spin states. The generalized rotating frame is introduced, which is a convenient frame for the control of high-dimensional quantum states. Using this frame, a protocol is explained that enables the preparation of an arbitrary spin state.

Chapter 3—Theory of Donors in Silicon This chapter discusses the physics of donors in silicon, placing emphasis on those with a high nuclear spin. The different spin interactions are explained, most of which are similar to that of 31 P. The main difference of high-spin nuclei is the addition of the nuclear quadrupole interaction, arising from a non-spherical nuclear charge distribution. The properties of the quadrupole interaction are discussed in detail, such as how it affects the nuclear spectrum. A procedure is described to extract the quadrupole parameters from nuclear spectra for varying static magnetic fields.

Chapter 4—Experimental Setup This chapter is devoted to the various components of the experimental setup. The architecture of the 123 Sb-implanted nanodevice is described. This includes an introduction of the single electron transistor (SET), which is used to detect the donor charge state. A layout is provided of the different instruments used to control the 123 Sb donor. This includes a custom phase-coherent direct digital synthesis (DDS) module that enables high-fidelity control of the 123 Sb nuclear and electron spin. The SilQ measurement framework is also highlighted, whose development has proven crucial for the measurements performed in this thesis.

Chapter 5—123 Sb Device Characterization This chapter goes through the initial measurements performed on an 123 Sb donor device. The first part describes the initial calibration of the—single electron transistor (SET) and detection of donors. It is thus useful for readers interested in how to tune up a device. Next, the spectrum of an 123 Sb donor electron is measured. This is confirmed by the observed eight-fold splitting, corresponding to the eight possible nuclear spin states. The flip-flop transition is also observed, where the nuclear and electron spins simultaneously flip. This transition is exploited for an efficient nuclearspin initialization scheme.

1.3 Thesis Outline

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Chapter 6—Nuclear Electric Resonance This chapter presents the coherent electric control of a single 123 Sb nucleus. The driving mechanism—-the electric modulation of the quadrupole interaction—is surprising, as the corresponding electric field gradient (EFG) is six orders of magnitude higher than expected. The full nuclear spectrum and coherence times of an ionized 123 Sb donor are measured through nuclear electric resonance (NER). These results demonstrate a highly coherent 123 Sb nucleus that can be controlled electrically.

Chapter 7—Microscopic Crystalline Origins of the Quadrupole Interaction This chapter delves into the microscopic origins that result in a sizeable EFG at the nucleus, and hence a quadrupole interaction. In particular, the transducing effect of electric fields into EFGs is attributed to the (LQSE), which has previously been observed in a range of polar crystals. The effect of strain on the quadrupole interaction is also studied using a combination of finite-element modeling and (DFT). The results are found to be in good agreement with observations.

Chapter 8—Exploring Quantum Chaos with a Single 123 Sb Donor The quantum equivalent of the classically-chaotic driven top can be implemented in the nuclear spin of an 123 Sb donor. Two experiments are devised and simulated, one of which identifies decoherence as a precursor of chaos, while the other demonstrates dynamical tunneling, an effect that is classically forbidden. The first experiment highlights the emergence of chaos, while the second experiment highlights the true quantum nature of the nucleus.

Chapter 9—Conclusions and Outlook The primary achievements in this thesis are summarized here. We speculate on several future directions that can be explored with the 123 Sb donor. These highlighted directions are the further characterization of the nuclear quadrupole interaction, quantum computation, quantum chaos, quantum metrology, spin-spin coupling, and spinmechanical coupling.

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References 1. Gerlach W, Stern O (1922) Der experimentelle nachweis des magnetischen mo- ments des silberatoms. Zeitschrift für Physik A Hadrons and Nuclei 8(1):110–111. https://doi.org/10. 1007/BF01329580 2. Friedrich B, Herschbach D (2003) Stern and gerlach: how a bad cigar helped reorient atomic physics. Phys Today 56(12):53–59. https://doi.org/10.1063/1.1650229 3. Kane BE (1998) A silicon-based nuclear spin quantum computer. Nature 393(6681):133–137. https://doi.org/10.1038/30156 4. Vandersypen LM, Chuang IL (2005) NMR techniques for quantum control and computation. Rev Mod Phys 76(4):1037. https://doi.org/10.1103/RevModPhys.76.1037 5. Ladd TD, Jelezko F, Laflamme R, Nakamura Y, Monroe C, O’Brien JL (2010) Quantum computers. Nature 464(7285):45. https://doi.org/10.1038/nature08812 6. Cirac JI, Zoller P (1995) Quantum computations with cold trapped ions. Phys Rev Lett 74(20):4091. https://doi.org/10.1103/PhysRevLett.74.4091 7. Wineland DJ, Monroe C, Itano WM, Leibfried D, King BE, Meekhof DM (1998) Experimental issues in coherent quantum-state manipulation of trapped atomic ions. J Res Nat Inst Stand Technol 103(3):259. https://doi.org/10.6028/jres.103.019 8. Gaebel T, Domhan M, Popa I, Wittmann C, Neumann P, Jelezko F, Rabeau JR, Stavrias N, Greentree AD, Prawer S, Meijer J, Twamley J, Hemmer PR, Wrachtrup J (2006) Roomtemperature coherent coupling of single spins in diamond. Nat Phys 2(6):408. https://doi. org/10.1038/nphys318 9. Hanson R, Mendoza F, Epstein R, Awschalom D (2006) Polarization and read-out of coupled single spins in diamond. Phys Rev Lett 97(8):087 601. https://doi.org/10.1103/PhysRevLett. 97.087601 10. Dutt MG, Childress L, Jiang L, Togan E, Maze J, Jelezko F, Zibrov A, Hemmer P, Lukin M (2007) Quantum register based on individual electronic and nuclear spin qubits in diamond. Science 316(5829):1312–1316. https://doi.org/10.1126/science.1139831 11. Godfrin C, Ferhat A, Ballou R, Klyatskaya S, Ruben M, Wernsdorfer W, Balestro F (2017) Operating quantum states in single magnetic molecules: implementation of grover’s quantum algorithm. Phys Rev Lett 119(18):187 702. https://doi.org/10.1103/PhysRevLett.119.187702 12. Pla JJ, Tan KY, Dehollain JP, Lim WH, Morton JJL, Zwanen-Burg FA, Jamieson DN, Dzurak AS, Morello A (2013) High-fidelity readout and control of a nuclear spin qubit in silicon. Nature 496(7445):334–338. https://doi.org/10.1038/nature12011 13. Pla JJ, Mohiyaddin FA, Tan KY, Dehollain JP, Rahman R, Klimeck G, Jamieson DN, Dzurak AS, Morello A (2014) Coherent control of a single si 29 nuclear spin qubit. Phys Rev Lett 113(24):246 801. https://doi.org/10.1103/PhysRevLett.113.246801 14. Morello A, Escott CC, Huebl H, van Beveren LHW, Hol-lenberg LCL, Jamieson DN, Dzurak AS, Clark RG (2009) Architecture for high-sensitivity single-shot readout and control of the electron spin of individual donors in silicon. Phys Rev B 80:081 307. https://doi.org/10.1103/ PhysRevB.80.081307 15. Morello A, Pla JJ, Zwanenburg FA, Chan KW, Tan KY, Huebl H, Mottonen M, Nugroho CD, Yang C, van Donkelaar JA, Alves ADC, Jamieson DN, Escott CC, Hollenberg LCL, Clark RG, Dzurak AS (2010) Single-shot readout of an electron spin in silicon. Nature 467(7316):687– 691. ISSN: 0028-0836. https://doi.org/10.1038/nature09392 16. Pla JJ, Tan KY, Dehollain JP, Lim WH, Morton JJ, Jamieson DN, Dzurak AS, Morello A (2012) A single-atom electron spin qubit in silicon. Nature 489(7417):541–545. https://doi. org/10.1038/nature11449 17. Muhonen JT, Dehollain JP, Laucht A, Hudson FE, Kalra R, Sekiguchi T, Itoh KM, Jamieson DN, McCallum JC, Dzurak AS, Morello A (2014) Storing quantum information for 30 s in a nanoelectronic device. Nat Nanotechnol 9(12):986–991. https://doi.org/10.1038/nnano.2014. 211

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18. Muhonen JT, Laucht A, Simmons S, Dehollain JP, Kalra R, Hudson FE, Freer S, Itoh KM, Jamieson DN, McCallum JC, Dzurak AS, Morello A (2015) Quantifying the quantum gate fidelity of single-atom spin qubits in silicon by randomized benchmarking. J Phys: Condens Matter 27(15):154 205. https://doi.org/10.1088/0953-8984/27/15/154205 19. Hill CD, Peretz E, Hile SJ, House MG, Fuechsle M, Rogge S, Simmons MY, Hollenberg LCL (2015) A surface code quantum computer in silicon. Sci Adv 1(9). DOI: https://doi.org/10.1126/sciadv.1500707 20. Tosi G, Mohiyaddin FA, Schmitt V, Tenberg S, Rahman R, Klimeck G, Morello A (2017) Silicon quantum processor with robust long-distance qubit couplings. Nat Commun 8:450. https://doi.org/10.1038/s41467-017-00378-x 21. Slichter C (2013) Principles of magnetic resonance. Springer, Berlin, Heidelberg. https://doi. org/10.1007/978-3-662-12784-1 22. Asaad S, Mourik V, Joecker B, Johnson MAI, Baczewski AD, Firgau HR, Madzik MT, Schmitt V, Pla JJ, Hudson FE, Itoh KM, McCallum JC, Dzurak AS, Laucht A, Morello A (2020) Coherent electrical control of a single high-spin nucleus in silicon. Nature 579(7798):205– 209. https://doi.org/10.1038/s41586-020-2057-7 23. Sigillito AJ, Tyryshkin AM, Schenkel T, Houck AA, Lyon SA (2017) All- electric control of donor nuclear spin qubits in silicon. Nat Nanotechnol 12(10):958. https://doi.org/10.1038/ nnano.2017.154 24. Bloembergen N (1961) Linear Stark effect in magnetic resonance spectra. Science 133:1363. https://doi.org/10.1126/science.133.3461.1363 25. Kushida T, Saiki K (1961) Shift of nuclear quadrupole resonance frequency by electric field. Phys Rev Lett 7:9–10.https://doi.org/10.1103/PhysRevLett.7.9 26. Gill D, Bloembergen N (1963) Linear Stark splitting of nuclear spin levels in GaAs. Phys Rev 129:2398-2403. https://doi.org/10.1103/PhysRev.129.2398 27. Brun E, Mahler RJ, Mahon H, Pierce WL (1963) Electrically induced nuclear quadrupole spin transitions in a GaAs single crystal. Phys Rev 129:1965–1970. https://doi.org/10.1103/ PhysRev.129.1965 28. Ono M, Ishihara J, Sato G, Ohno Y, Ohno H (2013) Coherent manipulation of nuclear spins in semiconductors with an electric field. Appl Phys Exp 6(3):033 002. https://doi.org/10.7567/ APEX.6.033002. 29. Haake F (2013) Quantum signatures of chaos, vol 54. Springer Science & Business Media. https://doi.org/10.1007/978-3-642-05428-0 30. May RM (1976) Simple mathematical models with very complicated dynamics. Nature 261(5560):459–467. https://doi.org/10.1038/261459a0 31. Belousov BP (1959) A periodic reaction and its mechanism. Compil Abstr Radiat Med 147(145):1 32. Huberman B (1987) A model for dysfunctions in smooth pursuit eye movementa. Anna New York Acad Sci 504(1):260–273. https://doi.org/10.1111/j.1749-6632.1987.tb48737.x 33. Cuomo KM, Oppenheim AV (1993) Circuit implementation of synchronized chaos with applications to communications. Phys Rev Lett 71(1):65. https://doi.org/10.1103/PhysRevLett.71. 65 34. Berry MV (1987) The bakerian lecture, 1987: quantum chaology. In: Proceedings of the royal society of London a: mathematical, physical and engineering sciences, the royal society, vol 413, pp 183–198. http://rspa.royalsocietypublishing.org/content/413/1844/183.short 35. Born M (1989) Atomic physics, 8th edn. Dover Publications 36. Bokulich A (2014) Bohr’s correspondence principle. In: Zalta EN (ed) The Stanford encyclopedia of philosophy. Spring, Metaphysics Research Lab, Stanford University. https://plato. stanford.edu/archives/spr2014/entries/bohr-correspondence/ 37. Zurek WH, Paz JP (1994) Decoherence, chaos, and the second law. Phys Rev Lett 72(16):2508. https://doi.org/10.1103/PhysRevLett.72.2508 38. Bhattacharya T, Habib S, Jacobs K (2000) Continuous quantum measurement and the emergence of classical chaos. Phys Rev Lett 85(23):4852. https://doi.org/10.1103/PhysRevLett.85. 4852

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39. Eastman JK, Hope JJ, Carvalho ARR (2017) Tuning quantum measurements to control chaos. Sci Rep 7. https://doi.org/10.1038/srep44684 40. Mourik V, Asaad S, Firgau H, Pla JJ, Holmes C, Milburn GJ, McCallum JC, Morello A (2018) Exploring quantum chaos with a single nuclear spin. Phys Rev E 98(4):042 206. https://doi. org/10.1103/PhysRevE.98.042206

Chapter 2

High-Dimensional Spins

The intrinsic angular momentum of a particle is described by its spin. Particle spin has served as the first experimental demonstration of space quantization, and has been instrumental in the development of quantum mechanics. The magnetization of metals is due to spin, and the numerous applications of spin include magnetic resonance imaging (MRI), atomic clocks, and quantum computers. This chapter describes the properties and dynamics of spin systems. The focus is on high-dimensional nuclear spins, such as the 123 Sb nucleus measured throughout this thesis.

2.1 The Discovery of Spin The first experiment that clearly captured the effects of spin was performed by Otto Stern and Walter Gerlach in 1922 [2]. In their experiment, silver atoms were shot through an inhomogeneous magnetic field (Fig. 2.1). Any angular momentum of the silver atoms would result in a force that deflects the atom. If the angular momentum is classical and randomly distributed, the distribution of particles would be continuous, whereas a quantized angular momentum would be reflected in a discretized particle distribution. The resulting particle distribution indeed showed a splitting, which was evidence for angular momentum being quantized. The problem, however, lay with the interpretation of these results. The Bohr atomic model predicted quantized orbital angular momenta of electrons, and so Stern and Gerlach erroneously attributed these results to the orbital angular momentum being quantized [3]. It was unknown at the time that the total angular momentum of a particle can also have a second angular-momentum contribution: spin. However, spin already manifested itself at high magnetic fields as additional splittings in the

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Asaad, Electrical Control and Quantum Chaos with a High-Spin Nucleus in Silicon, Springer Theses, https://doi.org/10.1007/978-3-030-83473-9_2

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Fig. 2.1 Stern-Gerlach experiment. Silver atoms are emitted from the narrow slit of a hot furnace (left), passing through an inhomogeneous magnetic field (middle), and impinge on a glass plate (right), forming a thin layer. The inhomogeneous magnetic field applies a vertical force on the silver atoms whose direction depends on its electron spin. Instead of a continuous distribution of silver atoms, two split lines were observed where the silver atoms reached the collector plate, demonstrating the quantization of spin. Figure adapted from [1]

spectrum of alkali metals [4], which was described as the anomalous Zeeman effect. This mysterious effect led Wolfgang Pauli to postulate a “... classically not describable two-valuedness...” [5] in such systems, though he did not initially attribute this to an intrinsic angular momentum of electrons [6]. This two-valuedness can be described by a fourth quantum number, and from this he formulated his exclusion principle: no two electrons can occupy the same state in an atom, i.e. where all four quantum numbers match.1 Spin owes its name to its initial physical interpretation given by Samuel Goudsmit and George Uhlenbeck, and separately by Ralph Kronig. They proposed that this intrinsic angular momentum represents a rotational degree of freedom, an effective spinning of the electron. Since the electron is a charged particle, this would result in a tiny current flow, which then results in an intrinsic magnetic moment. This classical spin interpretation was initially met with heavy criticism from both Pauli and Hendrik Lorentz [6], who argued that the electron (which was thought to be spherical) would rotate hundreds of times faster than the speed of light. Another serious objection was that the spin g-factor, which couples the angular momentum to the magnetic

1

The exclusion principle was found to be a consequence of the spin-statistics theorem, which followed from the unification of quantum mechanics with special relativity. The spin-statics theorem states that in three spatial dimensions particles can be classified in two categories. Particles whose combined wave function is symmetric under particle exchange have integer spin, and are called bosons. Conversely, particles whose combined wave function is anti-symmetric under particle exchange have half-integer spin, and are called fermions. The exclusion principle is a consequence of a wave function being anti-symmetric, and thus holds for any fermionic particle.

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moment, was a factor two too high.2 Dejected, Uhlenbeck and Goudsmit tried to retract their proposal, but found that it had already been submitted for publication. Luckily, the publication was picked up by Thomas [7], who realized that relativistic corrections were not taken into account [8]. These corrections result in a two-fold increase of the effective g-factor, thus accounting for the factor two discrepancy. This development finally convinced Pauli of the electron spin interpretation. He then successfully applied the interpretation to explain many of the anomalous Zeeman spectra, and developed the framework of spin mechanics.

2.2 Nuclear Spin The angular momentum of an electron is composed of its orbital angular momentum and its spin, both of which are quantized. The electron spin quantum number is S = 1/2, and therefore the electron has 2S + 1 = 2 quantized spin states. These two forms of angular momentum can interact via spin-orbit coupling. For a nucleus, the total angular momentum is the combined angular momenta of its protons and neutrons, each of which has spin 1/2 and can have an orbital angular momentum. Under most circumstances, the nucleus behaves as though it is a single entity having an intrinsic spin with quantum number I that combines all the orbital and spin contributions of its constituent nucleons [9]. The nuclear shell model predicts the filling of protons and neutrons in the nucleus, and is remarkably similar to the atomic shell model. The study of the nuclear structure is a field in its own right, but there are several rules of thumb [10]. First and foremost, protons and neutrons have a strong preference to pair with their own kind to form singlet states with zero angular momentum, in a way similar to electrons in atoms. As a result, atoms with an even number of protons and neutrons always have zero nuclear spin. However, if either the number of protons or neutrons is odd (the other being even), this unpaired particle determines the nuclear spin. Atoms with an uneven mass number always have a half-integer nuclear spin, though it can be greater than 1/2 because the unpaired proton or neutron may also have an orbital angular momentum. Finally, if the number of protons and neutrons are both odd, the nuclear spin is always an integer larger than zero. The vast majority of nuclei with nonzero spin are halfinteger, although the stable atom with highest nuclear spin is 176 Lu with integer spin I = 7.

2

Though not known at the time, an additional argument against this interpretation of spin is that the neutron is also a spin-1/2 particle, even though it has no net charge. One should therefore bear in mind that particle spin lacks a proper classical interpretation.

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2 High-Dimensional Spins

2.3 Nuclear Spin States The nuclear spin has 2I + 1 spin eigenstates, where I is the nuclear spin quantum number, which is either a half-integer or an integer. The lowest nonzero spin is I = 1/2, in which case all nuclear spin states are combinations of spin-up and spindown. For a general spin I , nuclear spin states can be described by the set of basis states |m I  which are simultaneous eigenstates of Iˆ2 and Iˆz   Iˆ2 |m I  = Iˆx2 + Iˆy2 + Iˆz2 |m I  = 2 I (I + 1)|m I ,

(2.1)

Iˆz |m I  = m I |m I ,

(2.2)

where Iˆx , Iˆy , and Iˆz are the spin projection operators along the x-, y-, and z-axes, respectively. The secondary spin quantum number3 m I equals the spin projection along the z-axis, and can take the 2I + 1 values m I ∈ {−I, −I + 1, . . . , I }. A particle’s spin can never be known along all axes simultaneously. This is because Iˆx , Iˆy , and Iˆz are non-commuting operators. Measuring the spin along one axis therefore scrambles the spin along orthogonal axes. This is quantified by the uncertainty principle for spins 2 (2.3) σ 2Iˆ σ 2Iˆ ≥  Iˆz 2 , x y 4  where σ Iˆα =  Iˆα2  −  Iˆα 2 is the spin uncertainty along axis α. This equation holds ˆ2 for any spin state regardless √ of the choice of axes. The eigenvalue of I is the square of the spin magnitude  I (I + 1). The fact that this value is higher than the maximal eigenstate of Iˆz is a reflection of the minimum uncertainty principle. If the maximal eigenstate of Iˆz would coincide with the spin magnitude, this would imply that the spin components along the orthogonal axes would be zero, and thus would not have any uncertainty.

2.4 Spin Operators The minimum uncertainty of spins is directly related to the non-commutativity of the spin operators.4 The spin operators satisfy the following commutation relation

3 The term ‘spin’ is used for both the spin quantum number I and the secondary spin quantum number m I . Generally, ‘spin’ usually refers to I when discussing the nuclear properties of an atom species, while it refers to m I when discussing the spin properties of a single nucleus. 4 For two general operators A ˆ and B, ˆ the uncertainty relation between the operators is σ 2 σ 2 ≥ A B   2   1 ˆ ˆ 4  A, B  [11].

2.4 Spin Operators

13



 Iˆx , Iˆy = i Iˆz ,

(2.4)

and holds for all cyclic permutations of {x, y, z}. The spin raising ( Iˆ+ ) and lowering ( Iˆ− ) operators, which increase and decrease the nuclear spin m I respectively, are given by

Iˆ± |m I  =  I (I + 1) − m I (m I ± 1)|m I ± 1,

(2.5)

and relate to the spin operators via Iˆ± = Iˆx ± i Iˆy .

(2.6)

Combining Eqs. 2.4, 2.5, and 2.6 allows straightforward calculation of the spin operator matrices, which in the case of 123 Sb with I = 7/2 are

Iˆx

=

Iˆy

=

Iˆz

=

√ ⎡ ⎤ 0 7 √ 0 0 0 0 0 0 √ ⎢ 70 12 √ 0 0 0 0 0 ⎥ ⎢ ⎥ ⎢0 √12 0 15 √ 0 0 0 0 ⎥ ⎢ ⎥ √ ⎥ 1⎢ 15 √ 0 16 √ 0 0 0 ⎥ ⎢0 0 ⎢ ⎥, 16 √ 0 15 √ 0 0 ⎥ 0 2 ⎢0 0 ⎢ ⎥ ⎢0 0 12 √ 0 ⎥ 15 √ 0 0 0 ⎢ ⎥ ⎣0 0 0 0 0 12 √ 0 7⎦ 0 0 0 0 0 0 7 0 √ ⎡ 0 − 70√ 0 0 0 0 ⎢√7 0 − 12 0 0 0 0 ⎢ √ ⎢0 √12 0 − 15 0 √ 0 0 ⎢ √ i ⎢ 15 0√ − 16 0 √ 0 ⎢0 0 ⎢ 16 √ 0 − 15 0 √ 0 2 ⎢0 0 ⎢ ⎢0 0 15 0 − 12 0 0 √ ⎢ ⎣0 0 0 0 0 12 0√ 0 0 0 0 0 0 7 ⎡ 7 0 0 0 0 0 0 ⎢ 0 5 0 0 0 0 0 ⎢ ⎢ 0 0 3 0 0 0 0 ⎢ 1⎢ 0 0 0 1 0 0 0 ⎢ ⎢ 0 0 0 0 −1 0 0 2⎢ ⎢ 0 0 0 0 0 −3 0 ⎢ ⎣ 0 0 0 0 0 0 −5 0 0 0 0 0 0 0

(2.7)

⎤ 0 ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥, ⎥ 0 ⎥ 0√ ⎥ ⎥ − 7⎦ 0 ⎤ 0 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥. 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎦ −7

(2.8)

(2.9)

14

2 High-Dimensional Spins

Note that in this definition, we ignore the multiplicative factor  in the spin operators, which allows us to define all our Hamiltonian interactions in units of frequency. The structure of the high-dimensional spin operators shares several properties with their spin-1/2 counterparts, such as Iˆz having nonzero diagonal elements, while Iˆx and Iˆy are nonzero on the first off-diagonal. The off-diagonal elements of Iˆy are all imaginary. However, there are important differences. First, the spin operators of spin-1/2 are equal to the Pauli matrices up to a scaling factor, and therefore these operators all satisfy Iˆx2 = Iˆy2 = Iˆz2 = 41 1. This relation no longer holds for higher-dimensional spin operators, and consequently quadratic interactions such as the quadrupole interaction are possible (Sect. 3.3). Second, the off-diagonal elements of the transverse spin operators Iˆx and Iˆy are not constant, but increase towards the middle.

2.5 Spin Coherent States A nucleus with spin I has a Hilbert space with 2I + 1 dimensions. For 123 Sb with I = 7/2, states in the 8-dimensional Hilbert space are specified by 14 independent real parameters, as each of the dimensions has an amplitude and a phase, minus a global phase and the constraint that the amplitudes are normalized. Exploring this vast multitude of possible states can be daunting, humans were simply not made to navigate through 14 dimensions. Spin coherent states, also known as Bloch states, form a particularly easy-tocomprehend subset of the possible states of a spin-I system. Their defining characteristic is that they are minimum-uncertainty states, meaning that the uncertainty relation σ 2Iˆ  σ 2Iˆ  ≥ 2 /4 Iˆz  2 (Eq. 2.3) becomes an equality for a specific choice of x

y

axes {x  , y  , z  }. The spin coherent states therefore have a spin that is pointing maximally towards an axis (z  ) with minimal spread (σ Iˆx  σ Iˆy ). This is reflected by spin coherent states being the only states whose spin length is maximal |I| =



 Iˆx 2 +  Iˆy 2 +  Iˆz 2 = I.

(2.10)

One straightforward example of a spin coherent state is |I , i.e. a spin is fully pointing up. In fact, the whole set of spin coherent states can be generated by rotating |I  by an angle θ about an axis (x, y, z) = (sin φ, − cos φ, 0). The corresponding rotation operator is given by Rθ,φ = e

  −iθ Iˆx sin φ− Iˆy cos φ

(2.11)

for spherical angles θ (polar) and φ (azimuthal). The spin coherent states |θ, φ are now defined as the state |I  rotated by Rθ,φ [12]. In the basis of Iˆz , a spin coherent state has the form

2.5 Spin Coherent States

15

|θ, φ =Rθ,φ |I   1/2 I    I +m I  1  I −m I 2I = sin 2 θ eiφ(I −m I ) cos 21 θ |m I , I + m I m =−I

(2.12)

I

where |m I  is the eigenstate of Iˆz with spin projection m I along z. A spin coherent state is generally composed of multiple spin states |m I , which are the basis states of Iˆz , depending on the polar angle θ. The two exceptions are the two spin coherent states |7/2 (θ = 0) and | − 7/2 (θ = π). The azimuthal angle φ is only reflected as a phase factor eiφ(I −m I ) that varies linearly with m I (Eq. 2.12), and so it is the phase difference between subsequent levels that determines its azimuthal orientation. Rotated spin operators can be obtained by applying Rθ,φ to the original spin operators −1 , Iˆx  = Rθ,φ Iˆx Rθ,φ −1 Iˆy  = Rθ,φ Iˆy Rθ,φ ,

(2.13)

−1 Iˆz  = Rθ,φ Iˆz Rθ,φ ,

in which case the spin coherent state is an eigenstate of Iˆz  with eigenvalue I . The spin coherent states form an overcomplete normalized basis, having nonzero overlap with each other. Only opposite spin coherent states are orthogonal and have zero overlap, analogous to the orthogonality of | − I  and |I . For I = 1/2, all pure spin states are spin coherent states, leaving the concept unnecessary. However, this concept is very useful for I > 1/2, since now the spin coherent states form a distinct subset of pure states for which the spin is maximally aligned in a certain direction (θ, φ), with a minimum-uncertainty spread around it. As such, these states are the closest quantum analogue to a classical angular momentum, and are therefore used in the quantum driven top experiments as corresponding initial quantum states (Chap. 8).

2.6 Spin Visualization with the Husimi Q Distribution The Husimi Q distribution [13] is a quasi-probability distribution that is used to represent quantum states. It is particularly useful here, as it provides a visualization of high-dimensional quantum states. For a given density matrix ρ, the Husimi Q function is defined as 1 (2.14) Q(θ, φ) = θ, φ|ρ|θ, φ, π where |θ, φ is a spin coherent state. In the case of a pure state (ρ = |ψ ψ| for some state |ψ), the Husimi Q distribution simplifies to Q(θ, φ) = π1 |ψ|θ, φ|2 , i.e. the overlap-squared between |ψ and |θ, φ.

16

2 High-Dimensional Spins

Fig. 2.2 Husimi representation of spin states. The color on the sphere surface at each spherical coordinate (θ, φ) corresponds to the Husimi Q function Q(θ, φ) (Eq. 2.14). Color-scale limits are constant over all panels, varying between Q = 0 (dark blue) and Q = 1/π (bright yellow). (A) Spin coherent state |θ, φ = |π/5, 5π/4 for varying I . As I is increased from 1/2 (left panel, e.g. an 31 P nuclear spin) to 7/2 (middle panel, e.g. an 123 Sb nuclear spin) and beyond (right panel, 31/2, shown for pedagogical reasons only, not corresponding to an actual nuclear spin), the relative uncertainty of the coherent state decreases, effectively localizing the state. (B) Husimi representation of Iˆz eigenstates. Three eigenstates of Iˆz are visualized for I = 7/2. Whereas the state |7/2 is a spin coherent state oriented along +z with minimum uncertainty, as can be seen by its small spread over the spherical surface, the other two eigenstates are uniform bands with a larger spread since these are not minimum-uncertainty states

The Husimi Q distribution satisfies certain properties required for a joint probability distribution, because the distribution is normalized and non-negative with values ranging between 0 ≤ Q(θ, φ) ≤ 1/π. However, since different spin coherent states are non-orthogonal, the values of Q at different coordinates (θ, φ) are not the probabilities of mutually exclusive states, which is a requirement for a joint probability distribution. This reflects the fact that quantum mechanics lacks a clear phase-space description, as opposed to classical mechanics. Although not providing a true phase-space description, quasi-probability distributions such as the Husimi Q distribution provide the closest quantum mechanical proxy to it while still being a complete representation of a quantum state (i.e. invertible to the original density matrix representation). The Husimi distribution allows a spherical representation of spin states (Fig. 2.2), where the value of each point (θ, φ) on the spherical surface corresponds to Q(θ, φ) (Eq. 2.14). This representation extends the notion of the Bloch sphere for

2.6 Spin Visualization with the Husimi Q Distribution

17

two-dimensional spin states: a spin coherent state |θ, φ reaches the maximum possible value 1/π at the spherical coordinates (θ, φ), similar to the Bloch sphere, where an arrow points in the spin direction. However, the Husimi distribution does not immediately drop to zero away from (θ, φ), but instead exhibits a spread, thus visualizing the uncertainty over which a coherent state spreads in phase space. This spread shows that two spin coherent states with different directions still have a nonzero overlap. The spread of coherent states scales linearly with I (σ Iˆx  σ Iˆy = 2 I ). Therefore, in an absolute sense, the minimum spin uncertainty increases with increasing I . However, this scaling should be compared to the total surface of the phase space (4π I 2 ). This leads to a concept of relative uncertainty σ Iˆx  σ Iˆy /4π I 2 ∝ 1/I , which is a measure of how well a quantum state is localized in phase space. The relative uncertainty therefore decreases with increasing I (Fig. 2.2A). Whereas every two-dimensional pure state on the Bloch sphere corresponds to an arrow pointing maximally in a direction, the higher-dimensional spin eigenstates with |m I | < I are not spin coherent states, and correspondingly their Q(θ, φ) peak value is lower than the maximal value 1/π. They are visible as bands in the Husimi representation (Fig. 2.2B), reflecting that these states have a completely indeterminate direction on the plane orthogonal to the z-axis.

2.7 Generalized Rotating Frame The generalized rotating frame is a convenient frame for dealing with the evolution of a high-dimensional quantum system. In this frame, the quantum state appears stationary as it evolves under a static Hamiltonian (see Sect. 2.8 for a comparison of reference frames). This simplifies effects such as the resonant driving of transitions (Sect. 3.2.5), which is of importance for the preparation of arbitrary quantum states (Sect. 2.9). This section begins by describing what happens when a quantum state evolves, and how a rotating frame counteracts this evolution (Sect. 2.7.1). The generalized rotating-frame transformation is then defined (Sect. 2.7.2), based on the derivation by Leuenberger [14], and is shown to transform a time-dependent Hamiltonian into a time-independent Hamiltonian.

2.7.1 Description of the Generalized Rotating Frame When an initial N -dimensional quantum state |ψ0  evolves to a later time t, each of its eigenstate components accumulates a phase dependent on its energy |ψ(t) =

N  k=1

e−i Ek t/ ek |ψ0 |ek ,

(2.15)

18

2 High-Dimensional Spins

where |ek  is the k-th eigenstate with energy E k , and the Hamiltonian is assumed to be time independent. If the system has an energy splitting, these accumulated phases will start to differ, resulting in relative phases between the eigenstate components. For a two-dimensional spin, the accumulating relative phase (E 2 − E 1 )t/ translates to a precession of the spin around the quantization axis, an effect known as Larmor precession. The precession frequency f 1↔2 = (E 2 − E 1 )/ h is referred to as the transition frequency. A mathematical transformation is applied to account for this precession: instead of viewing the system in a static frame (the laboratory frame), the reference frame is rotated at the same frequency as the transition frequency, effectively cancelling the accumulating phase. In this rotating frame, the spin no longer precesses, and this greatly simplifies concepts such as resonant driving of transitions. Higher-dimensional spin systems no longer have a single transition frequency, but instead each pair of eigenstates |ek  and |ek   has its own transition frequency f k↔k  = (E k  − E k )/ h. Fortunately, the concept of the rotating frame can be straightforwardly extended to higher-dimensional Hilbert spaces. The core idea of the generalized rotating frame is that a counter-rotating phase is applied to each eigenstate |ek  that (approximately) matches its phase accumulation E k / h. This negates all the accumulating phases, and additionally converts periodic near-resonant drives into static interactions.

2.7.2 Derivation of the Generalized Rotating Frame In the generalized rotating frame, static interactions are cancelled out, and periodic drive interactions are transformed into static interactions. To illustrate how the dynamics of a system are affected in the generalized rotating frame, we shall apply this transformation to a Hamiltonian of the form ˆ H(t)/ h = Hˆ static / h +

2I  k=1



gk cos(ωk t + φk ) Iˆx , 

ˆ periodic (t)/ h H

(2.16)



where Hˆ static is the static part of the Hamiltonian, assumed to be a diagonal matrix to simplify the derivation.5 Hˆ periodic (t) encapsulates all the oscillating interactions that drive transitions between states. In this example, we have chosen a transverse Zeeman drive6 (Sect. 3.2.5). Each transition k between eigenstates |ek  and |ek+1  has its own drive with strength gk and angular frequency ωk close to the transition If Hˆ static is not diagonal, it can be diagonalized by a change of basis to the eigenbasis of Hˆ static . Note that this change of basis also affects Hˆ periodic (t), though the generalized rotating frame can still straightforwardly be applied. 6 Note that the generalized rotating frame transformation is independent of the drive. The drive interaction is therefore not restricted to periodic Iˆx interactions. 5

2.7 Generalized Rotating Frame

19

frequency (ωk ≈ 2π f k↔k+1 ), plus a phase offset φk . The drive strengths can vary with time, allowing multiple drives to be active simultaneously, or at separate times. We now apply two approximations to simplify Hˆ periodic (t).7 The first approximation is that each periodic interaction only drives one transition, and leaves all other states unaffected, i.e. the amplitude in Rabi’s formula (Eq. 6.1) is close to zero for all other transitions. This is an accurate approximation when the drive strengths gk are significantly weaker than the separations between resonance frequencies, and it imposes an upper limit on the drive strength. We further apply the rotating wave approximation, where we separate the drive interaction into a rotating and a counterrotating term, and ignore the latter: cos(ωk t + φk ) =

 1 1  iωk t+iφk e + e−iωt−iφk ≈ eiωk t+iφk . 2 2

(2.17)

The rotating wave approximation is accurate when the drive strengths gk of Hˆ periodic (t) are weak compared to the resonance frequencies of Hˆ static . This is a valid approximation in our system as the oscillating magnetic field B1 ≈ 1 mT is much lower than the static magnetic field B0 ≈ 1 T. However, the rotating wave approximation breaks down once the drive strength approaches the energy splitting, giving rise to the Bloch-Siegert shift [15, 16]. We illustrate the procedure for brevity using a four-dimensional Hilbert space (I = 3/2), in which case the full Hamiltonian in the laboratory frame (Eq. 2.16) is approximated as ⎞ g˜1 eitω1 +iφ1 0 0 E1/ h ⎟ ⎜ g˜ ∗ e−itω1 −iφ1 E2 / h g˜2 eitω2 +iφ2 0 1 ⎟, Hˆ rwa (t)/ h = ⎜ ∗ −itω −iφ 2 2 ⎝ 0 g˜2 e E3/ h g˜3 eitω3 +iφ3 ⎠ 0 0 g˜3∗ e−itω3 −iφ3 E 4 / h ⎛

(2.18)

where E k = ek |Hˆ static |ek  is the eigenenergy of |ek , and g˜k = 21 gk ek | Iˆx |ek+1  is the interaction strength of transition k. Note the factor 21 , which arises from the rotating wave approximation. We are now in a position to apply the generalized rotating frame transformation, which turns the time-dependent Hamiltonian Hˆ rwa into a time-independent Hamiltonian Hˆ rot upon application of the unitary transformation ∂ Uˆ rot ˆ rot† Hˆ rot = Uˆ rot Hˆ rwa (t)Uˆ rot† + i U . ∂t

(2.19)

We define the unitary operator Uˆ rot as 7

If any of the two approximations do not hold, the resulting generalized rotating frame Hamiltonian

ˆ rot is no longer time-independent. H

20

2 High-Dimensional Spins

⎞ 0 0 0 ei E1 t/ ⎟ ⎜ 0 ei(E2 /−2πδ2 )t 0 0 ⎟, =⎜ i(E 3 /−2πδ3 )t ⎠ ⎝ 0 0 e 0 i(E 4 /−2πδ4 )t 0 0 0 e ⎛

Uˆ rot

(2.20)

where the diagonal components match the phase accumulation of the corresponding eigenstate, plus a potential detuning −δk .8 These detunings are related to the drive frequencies (Eq. 2.16) by ωk = (E k+1 / − 2πδk+1 ) − (E k / − 2πδk ), and their purpose will become clear momentarily. The resulting generalized rotating frame Hamiltonian Hˆ rot calculated from Eq. 2.19 now has the simple form ⎞ 0 g˜1 eiφ1 0 0 ⎜ g˜ ∗ e−iφ1 δ2 g˜2 eiφ2 0 ⎟ 1 ⎟, Hˆ rot / h = ⎜ ⎝ 0 g˜2∗ e−iφ2 δ3 g˜3 eiφ3 ⎠ 0 0 g˜3∗ e−iφ3 δ4 ⎛

(2.21)

which is indeed time independent. By negating all the accumulated phases, the generalized rotating frame greatly simplifies the high-dimensional system. In the rotating frame, a periodic drive with strength gk becomes a static transverse coupling g˜k between two states, inducing a transition between the two in the same way as for a two-dimensional system. The phase offset φk defines the angle of rotation for the transition by cos φk Iˆx + sin φk Iˆy . Finally, the phases of the excited states can be modified by adding a detuning δk to the drive frequency. For a two-dimensional quantum state, this corresponds to a rotation around the z-axis.

2.8 Basis Comparison There are three relevant bases that are important to distinguish between. These depend on two factors: the eigenbasis, and the reference frame. We consider the eigenbasis ˆ These two eigenbases differ of the spin operator Iˆz , and of the static Hamiltonian H. when Hˆ contains interactions that do not commute with Iˆz . The reference frame is either the laboratory frame or the generalized rotating frame (Sect. 2.7). The first basis is the eigenbasis of Iˆz in the laboratory frame. This is the most straightforward basis, as each eigenstate |m I  has a corresponding spin quantum number m I , which is its spin quantization along the z axis. In this basis, the direction of a spin coherent state also physically corresponds to the spin direction, and in the absence of any interactions will forever point in that direction. The eigenbasis of Iˆz is particularly impractical in the presence of interactions that do not commute with Iˆz , as these interactions rapidly change the spin state (Fig. 2.3A– C). This is because the evolution of a spin state is determined by decomposing it 8

The ground-state detuning δ1 has been set to zero, as the global phase is usually irrelevant.

2.8 Basis Comparison

21

Fig. 2.3 Evolution of an initial spin coherent state in different eigenbases. Simulations are performed for 123 Sb with I = 7/2, using a time-independent Hamiltonian Hˆ where the linear interaction strength γn B0 = 5.55 MHz is similar to the quadratic interaction strength Q = 1.5 MHz. (A) Husimi representations of the evolution of an initial spin coherent state at different points in time. The top (bottom) row displays the front (back) of the Husimi sphere, where y > 0 (y < 0). As the spin coherent state evolves, the spin rapidly precesses, and simultaneously experiences a twist due to the quadratic interaction. The decreasing spin length |I| (B) confirms that the evolved state is no longer a spin coherent state. The overlap amplitude |ek |ψ(t)| with each eigenstate |ek  of Iˆz fluctuates (C, varying colors) because the eigenbases of Hˆ and Iˆz differ. In contrast, the eigenstate amplitudes of Hˆ remain constant for both the laboratory frame (D) and the generalized rotating frame (E). The phases of the different eigenstate components accumulate rapidly in the laboratory frame (C, D), while they are negated in the generalized rotating frame (E)

22

2 High-Dimensional Spins

Fig. 2.4 Preparation of a spin coherent state. Final state is the coherent state |ψT  = |θ = 4π/5, φ = π/2, and initial state is the ground state |e1 . (A) A succession of pulses with different frequencies f k↔k  iteratively transfer population from lower-energy eigenstates |ek  to higherenergy eigenstate |ek   (k  > k). To account for the remaining phase accumulation, intermediate fidelities were calculated after leaving the intermediate state idle until the end of the sequence. (B) Fidelity to final spin coherent state |ψT . The final fidelity between evolved state and target spin coherent state is |ψ|θ, φ| = 0.996 for B1 = 1 mT, Q = 1 MHz, B0 = 0.7 T, and can be further increased by reducing the oscillating magnetic-field strength

ˆ As the state evolves, each of the eigenstate into the eigenbasis of the Hamiltonian H. components accumulates a phase dependent on its eigenenergy (Eq. 2.15). Therefore, when the eigenbases of Hˆ and Iˆz differ, the amplitudes of the Iˆz eigenstate components ˆ This occurs will change, since each may be composed of multiple eigenstates of H. on top of a rapid phase accumulation, usually at a frequency of a few MHz (Fig. 2.3C), since the basis is in the laboratory frame. A more practical basis is the eigenbasis of Hˆ in the laboratory frame. In this basis, the amplitudes of a state remain fixed under evolution, though each eigenstate component does accumulate a phase dependent on its eigenenergy (Fig. 2.3D). One point worth emphasizing about this basis is that unless the Hamiltonian only has an Iˆz interaction, the eigenstates will be a superposition of different Iˆz eigenstates. Consequently, a spin coherent state, defined in Eq. 2.12 for the eigenstates {|m I }, will have different coefficients in the Hamiltonian eigenbasis. The final basis discussed here is the eigenbasis of Hˆ in the generalized rotating frame. In this basis, both the amplitudes and phases of a spin state remain fixed (Fig. 2.3E), provided that the Hamiltonian is static. This basis is particularly convenient for manipulation of spin states through resonant driving, as the corresponding Hamiltonian (Eq. 2.21) is time independent. However, even though a spin state is fixed in this basis under a static Hamiltonian, it is important to realize that the actual spin state is changing as it evolves in time. This can be seen in Fig. 2.3A, where the unequal energy level spacings causes an initial spin coherent state to spread out as it evolves.

2.9 Arbitrary State Preparation

23

2.9 Arbitrary State Preparation An interaction that is linear in a spin operator, such as the Zeeman interaction (Sect. 3.2.2), causes an equal splitting of all nuclear spin energy levels. The transition frequencies between neighboring states are therefore equal, and any resonant periodic interaction will therefore drive all transitions simultaneously. The result is a global rotation of the spin state around an axis set by the spin operator of the drive interaction. The ability to create an arbitrary target state |ψT  requires addressability of individual transition frequencies and accurate knowledge of all Hamiltonian parameters. In this case, the procedure described in this section can be used for arbitrary state preparation. Assuming the system to be initialized in the ground state |e1 , the necessary sequence of pulses that results in the target state |ψT  can be found by solving the problem in reverse: go backwards in time and find the pulse sequence to end up in |e1  starting from |ψT . Starting at t = 0 s with initial state |ψT  = k ak |ek , where ak = ek |ψT , pulses are iteratively chosen that transfer population from the populated highest-energy eigenstate to a lower-energy eigenstate. The procedure is as follows: 1. Choose eigenstate |ek  with highest eigenvalue λk and nonzero population |ak |. 2. Find eigenstate |ek   with lower eigenvalue λk  that has the highest coupling     ˆ ek | Ix |ek . 3. Transfer the population from |ek  to |ek   using a pulse with the following properties: • frequency f k↔k  , which is the transition frequency between |ek  and |ek  ; 2 • duration t p =  π  √ |a2k | 2 , where k↔k  is the Rabi frequency; k↔k

|ak | +|ak  |

• phase is the relative phase difference between ak and ak  , minus the phase offset 2π f k↔k  (t − t p ). 4. Update t → t − t p and repeat steps until all population is transferred to |e1 . These pulses can be applied successively to bring any pure state |ψT  back to the ground state |e1 . The following operations must additionally be performed on the pulses to achieve the opposite effect, i.e. reaching |ψT  from the ground state |e1  (Fig. 2.4). First, both the order of the pulses and their phases should be reversed. A phase offset φk = −2π f k↔k  tfinal should then be applied to each pulse, where f k↔k  is the pulse frequency, and tfinal is the time at the end of the last pulse. These phase offsets compensate for the phase accumulation during the pulse sequence (Sects. 2.7 and 2.8), and it is equivalent to a shift in time, such that the end of the last pulse is now at t = 0 s. This method can be followed to create an arbitrary pure state, and in particular the spin coherent states |θ, φ used in the proposed quantum driven-top experiments (Chap. 8). One requirement for high-fidelity state preparation is that each pulse only

24

2 High-Dimensional Spins

drives its target transition, while being off-resonant to all other transitions. This holds when the transition frequencies are separated by significantly more than the Rabi frequencies. Additional effects such as decoherence can further limit the fidelity, although we do not expect this to be an issue for our system (state preparation takes < 1 ms, T2∗ ∼ 0.1 s). Nevertheless, alternative schemes can be employed [17] where pulses at different frequencies are combined to drive multiple transitions simultaneously, thus further reducing the total pulse sequence duration.

References 1. Cresser JD (2009) Chapter 6: particle spin and the stern-gerlach experiment. In: Quantum Physics-MQ course no. PHYS301, MIT OpenCourseWare, Balaclava Rd, Macquarie Park NSW 2109, Australia. https://physics.mq.edu.au/~jcresser/Phys301/Chapters/Chapter6.pdf 2. Gerlach W, Stern O (1922) Der experimentelle nachweis des magnetischen mo-ments des silberatoms. Zeitschrift für Physik A Hadrons and Nuclei 8(1):110–111. https://doi.org/10. 1007/BF01329580 3. Friedrich B, Herschbach D (2003) Stern and gerlach: how a bad cigar helped reorient atomic physics. Phys Today 56(12):53–59. https://doi.org/10.1063/1.1650229 4. Forman P (1970) Alfred landé and the anomalous zeeman effect, 1919-1921. Hist Stud Phys Sci 2:153–261. https://doi.org/10.2307/27757307 5. Pauli W (1925) Über den einfluß der geschwindigkeitsabhängigkeit der elektronen-masse auf den zeemaneffekt, trans by M Jammer, Zeitschrift für Physik A Hadrons and Nuclei, vol 31, no 1, pp 373–385. https://doi.org/10.1007/BF02980592 6. Jammer M (1966) The conceptual development of quantum mechanics. McGraw-Hill, New York. https://doi.org/10.1063/1.3034186 7. Commins ED (2012) Electron spin and its history. Ann Rev Nucl Part Sci 62:133–157. https:// doi.org/10.1146/annurev-nucl-102711-094908 8. Thomas LH (1926) The motion of the spinning electron. Nature 117(2945):514. https://doi. org/10.1038/117514a0 9. Krane KS (1987) Introductory nuclear physics. Third. Wiley. https://doi.org/10.1063/1. 2810884 10. Levitt MH (2001) Spin dynamics: basics of nuclear magnetic resonance. Wiley 11. Sakurai JJ, Commins ED (1995) Modern quantum mechanics, second. AAPT. https://doi.org/ 10.1017/9781108499996 12. Arecchi FT, Courtens E, Gilmore R, Thomas H (1972) Atomic coherent states in quantum optics. Phys Rev A 6:2211–2237. https://doi.org/10.1103/PhysRevA.6.2211. 13. Husimi K (1940) Some formal properties of the density matrix. Proc Phys Math Soc Jpn 22(4):264–314. https://doi.org/10.11429/ppmsj1919.22.4_264 14. Leuenberger MN (2004) The generalized rotating frame. J Magn Magn Mater 272:1974–1975. https://doi.org/10.1016/j.jmmm.2003, 12.440 15. Bloch F, Siegert A (1940) Magnetic resonance for nonrotating fields. Phys Rev 57(6):522. https://doi.org/10.1103/physrev.57.522 16. Laucht A, Simmons S, Kalra R, Tosi G, Dehollain JP, Muhonen JT, Freer S, Hudson FE, Itoh KM, Jamieson DN, McCallum JC, Dzurak AS, Morello A (2016) Breaking the rotating wave approximation for a strongly driven dressed single-electron spin. Phys Rev B 94(16):161 302. https://doi.org/10.1103/physrevb.94.161302 17. Khaneja N, Reiss T, Kehlet C, Schulte-Herbrüggen T, Glaser SJ (2005) Optimal control of coupled spin dynamics: design of NMR pulse sequences by gradient ascent algorithms. J Magn Reson 172(2):296–305. https://doi.org/10.1016/j.jmr.2004.11.004

Chapter 3

Theory of Donors in Silicon

This chapter describes the physics of group-V donors in a silicon lattice. First, the most important relevant solid state concepts of silicon and donor atoms are introduced. Next, the Hamiltonian of a donor in silicon is described, discussing each of the interaction. While most of the chapter is general to any group-V donor in silicon, only donors whose nucleus has a nuclear spin quantum number I > 1/2 possess an electric quadrupole moment, caused by a non-spherical charge distribution. This excludes 31 P, but is present for 123 Sb. The resulting nuclear electric quadrupole interaction couples the nuclear spin to electric field gradients (EFGs), and this plays a crucial role in the remainder of this thesis.

3.1 Solid State Physics of Donors in Silicon Silicon is a group-IV element, and therefore has four electrons in the valence band. The four valence electrons can form covalent bonds with those of other silicon atoms, resulting in a diamond crystalline lattice (Fig. 3.1), which is equivalent to two overlapping face-centered-cubic lattices offset by 1/4 unit cell in all directions. This crystal possesses inversion symmetry in the center of each pair of covalently-bonded lattice sites, and tetrahedral symmetry (Td ) at the lattice sites. Crystalline silicon is a semiconductor material with a bandgap of 1.14 eV [1]. The Fermi level of intrinsic silicon lies in the middle of the band gap, though covalentlybonded electrons can be thermally excited to the conduction band, creating electronhole pairs. At cryogenic temperatures, silicon therefore behaves as an insulator. Doping the silicon with impurities shifts the band structure with respect to the Fermi level, thereby strongly increasing the electrical conductivity. The dopants are classified as either acceptors (p-type doping) or donors (n-type doping), depending on the number of valence electrons the atom has. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Asaad, Electrical Control and Quantum Chaos with a High-Spin Nucleus in Silicon, Springer Theses, https://doi.org/10.1007/978-3-030-83473-9_3

25

26

3 Theory of Donors in Silicon

Fig. 3.1 Unit cell of silicon. The crystalline structure of silicon is the diamond lattice. The unit cell possesses inversion symmetry around the center, while atom sites are tetrahedrally symmetric. The silicon lattice parameter is a = 5.431 Å, and the distance between neighboring silicon atoms is √ 3a/4 = 2.352 Å

Donor atoms are the next group in the periodic table, group-V, and consists of atoms with five valence electrons. For an isolated donor atom, two of the valence electrons are in an s-type atomic orbital, and three in p-type atomic orbitals. The four different stable group-V elements, sorted by increasing atomic mass, are phosphorus (P), arsenic (As), antimony (Sb), and bismuth (Bi), whose properties are summarized in Table 3.1. When a donor atom replaces a Si atom in the lattice, four valence electrons form covalent bonds with the neighboring silicon atoms. In this case, one electron in an s-type orbital and three electrons in p-type orbitals hybridize into four sp3 hybrid orbitals. The fifth electron is in an s-type orbital and remains weakly bound to the donor. These atoms are called donor atoms because they effectively donate an electron at an energy very close to the conduction band. In the neutral charge state (D0 ), the donor atom thus contains one loosely-bound electron, on top of all the strongly-bound electrons that form covalent bonds with the neighboring silicon atoms. This loosely-bound electron lies in an s-like orbital, and hence the donor is referred to as a hydrogen-like atom. Although the remaining electron is bound to the donor, its wave function is spread out over multiple lattice sites. The Bohr radius of a donor electron in silicon is 1–2.5 nm, and varies with donor [2, 3]. The loosely-bound electron can be removed from the donor atom, in which case the donor is in the ionized charge state (D+ ). All the inner and covalently-bonded electrons of a donor atom form singlet pairs, which have zero combined spin. In the absence of a second electron to couple to, the remaining loosely-bound electron has an excess spin S = 1/2. Additionally, the donor nucleus also has a nonzero nuclear spin I that depends on the donor atom species, and generally increases with atomic mass (Table 3.1). In contrast to donor electrons, which usually have a distinct spin and orbital angular

3.1 Solid State Physics of Donors in Silicon

27

Table 3.1 Ground 1s state binding energies taken from Ref. [4]. Hyperfine interaction A taken from Ref. [5]. Nuclear gyromagnetic ratio γn = μμn /I is calculated from the nuclear magnetic moment μ given in units of nuclear magneton μn = 7.62 MHz/T in Ref. [6]. Minimum and maximum values of nuclear quadrupole moment qn are given, based on the range of values reported in Ref. [6] Donor I 1s binding A (MHz) γn (MHz/T) qn (10−28 m2 ) energy (meV) 31 P 75 As 121 Sb 123 Sb 209 Bi

1/2 3/2 5/2 7/2 9/2

45.59 53.76 42.74 42.74 70.98

117.53 198.35 186.80 101.52 1475.4

17.26 7.31 10.26 5.55 6.96

– 0.314 [−0.36 , −0.54] [−0.49 , −0.69] [−0.37 , −0.77]

momentum, the nuclear spin is the combined total angular momentum of all the angular momenta of its constituent nucleons (Sect. 2.2).

3.2 Donor Spin Hamiltonian 3.2.1 Neutral Donor Spin Hamiltonian A donor residing in a silicon lattice experiences different interactions. Some of these interactions, such as the Zeeman interaction, affect the electron and nuclear spins individually, while the hyperfine interaction couples the electron and nuclear spins together. The spin Hamiltonian of a neutral donor atom in silicon is given by Hˆ neutral / h = −γe B0 Sˆ z − γn B0 Iˆz −γe B1 sin (2π f t) Sˆ x − γn B1 sin (2π f t) Iˆx      ˆ Zeeman / h H



+ ASˆ · Iˆ + Q αβ Iˆα Iˆβ .    α,β∈{x,y,z} ˆ hyperfine / h H   

ˆ Zeemandrive / h H

(3.1)

ˆ quadrupole / h H

Each of these interactions is discussed in the following sections. The quadrupole Hamiltonian Hˆ quadrupole is only present for donors with I > 1/2, and is discussed separately in Sect. 3.3. Note that all interactions are given in frequency units, hence a division by h.

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3 Theory of Donors in Silicon

3.2.2 Zeeman Interaction The Zeeman effect, discovered by Dutch physicist Pieter Zeeman in 1896, describes the interaction between a magnetic field and a particle’s magnetic moment. The magnetic moment of an electron is related to the total angular momentum via the gyromagnetic ratio γe = −gμB , where g is the Landé g-factor, and μB ≈ 14.00 GHz T−1 is the Bohr magneton.1 Since the outer bound electron of a neutral donor lies in an s-like orbital, its angular momentum is entirely due to spin. Similarly, the nuclear spin couples to a magnetic moment via the nuclear gyromagnetic ratio γn = gn μn , where gn is a nucleus-dependent g-factor, and μn = 7.623 MHz T−1 is the nuclear magneton. For a magnetic field oriented along the z-axis, the Zeeman contribution Hˆ Zeeman to the Hamiltonian Hˆ is Hˆ Zeeman / h = −γe B0 Sˆ z − γn B0 Iˆz ,

(3.2)

While the electron and nuclear spin both experience a Zeeman interaction, the electron gyromagnetic ratio γe generally dominates by roughly three orders of magnitude. For an isolated electron, the electron gyromagnetic ratio equals γe ≈ −27.97 GHz T−1 [5, 7]. Similarly, γn = gn μ N is the nuclear gyromagnetic ratio, gn is the nuclear g-factor, and μ N is the nuclear magneton. For donor atoms, the nuclear gyromagnetic ratio γn is on the order of ∼ 10 MHz T−1 , and is dependent on the donor atom species (Table 3.1). Note that for donors γn and γe have an opposite sign.

3.2.3 Hyperfine Interaction Whereas the Zeeman interaction is a single-spin effect, the hyperfine interaction is a multi-spin effect that couples the electron and nuclear spins to each other. The hyperfine interaction can be divided into two components. The first is the Fermi contact hyperfine, which is an isotropic interaction (interaction strength is independent of spin orientations). It describes the overlap of the electron wave function with the nucleus, and is therefore nonzero only if the electron occupies an s-orbital, as is the case for neutral donor atoms in silicon. The second contribution to the hyperfine interaction is the electron-nuclear spin dipole coupling. Importantly, this interaction is not isotropic, and therefore the strength depends on the electron and nuclear spin orientations. For a neutral donor atom with a fairly centered outer electron, the Fermi 1

While both the electron spin and orbital angular momentum are related to the magnetic moment via μB , they are related by a different proportionality constant, known as the g-factor. The electron spin g-factor gS ≈ 2.002 is approximately twice as large as the orbital g-factor gL ≈ 1. The two g-factor contributions can be combined into the Landé g-factor g, which for our electron with an s-like orbital is dominated by spin (g ≈ gS ).

3.2 Donor Spin Hamiltonian

29

contact hyperfine dominates over the dipole hyperfine, and so we can approximate the hyperfine interaction as ˆ Hˆ hyperfine / h = ASˆ · I,

(3.3)

where A is the isotropic hyperfine interaction strength that depends on the donor atom.2 Bulk measurements of A for the different donor atoms are summarized in Table 3.1. However, the hyperfine interaction can deviate significantly from its bulk value, as the electron wave function can be affected by external factors such as strain [8] and the external charge distribution. This is especially true for devices such as those measured throughout this thesis, where shallowly-implanted donors experience significant strain from the nearby aluminum gates, as well as significant electric fields from nearby gates, charge traps, and other donor atoms. The hyperfine interaction strength A can also change in response to the electrostatic environment, such as electric noise. This shifts the energy levels of both the electron and nuclear spins, thus adding a source of decoherence. Although the hyperfine interaction is strongest between the electron and the nucleus that it is bound to, there can also be other nearby nuclei that have nonzero spin. These spinful nuclei can be additional donor atoms, but they can also be 29 Si atoms, which have a natural abundance of 4.7% and a nuclear spin I = 1/2. In this case, the electron will also couple to these nuclei, albeit to a lesser extent. These additional hyperfine interaction strengths can range up to a few megahertz in the case of spinful nuclei occupying nearby lattice sites [9]. For a donor, the electron wave function has an exponential envelope, and so the isotropic hyperfine interaction decreases exponentially with distance between electron and nucleus [10], whereas the dipole hyperfine interaction decreases as 1/r 3 . These additional hyperfine couplings can therefore have an anisotropic character. Extra couplings are often undesired, as the state of each coupled nucleus is usually unknown and modifies the energy splitting of the electron spin, and consequently the splitting of the nuclear spin, thereby being a source of decoherence. Recent results have shown [11] that the hyperfine coupling to nearby 29 Si nuclei can severely impact the coherence of the donor nucleus. On the other hand, a low density of coupled nuclei can also be a powerful asset, as a strong coupling to a single nucleus can be harnessed as a quantum memory and a quantum computation resource [11, 12].

3.2.4 Low-Field Versus High-Field Limit The Zeeman interaction is an anisotropic interaction with a quantization axis set by the direction of B0 (defined as the z-axis), while the hyperfine interaction is largely ˆ where A is a second-rank tensor. In the absence of an The full hyperfine interaction is Iˆ · A · S, anisotropic hyperfine interaction, A is diagonal with equal diagonal strengths A. In this case, A may be replaced by a scalar A.

2

30

3 Theory of Donors in Silicon

isotropic. Consequently, the eigenbasis of the combined electron-nucleus system depends on which of the two interactions is strongest. The hyperfine interaction dominates when B0 is sufficiently weak (A  |γe |B0 ), in which case the system approaches the low-field limit. In the low-field limit, the system eigenstates |F, m z  are set by the total spin F of electron and nucleus combined, and the combined projected spin m z along the z-axis, which can be chosen arbitrarily in the absence of B0 . The electron and nuclei can either be in a symmetric superposition (F = I + S) or an anti-symmetric superposition (F = I − S), the latter having a lower energy.3 The allowed values for the secondary quantum number are m z ∈ {−F, −F + 1, ..., F}. In the case of 31 P (I = 1/2), the eigenstates are the singlet state S : |F = 0, m z = 0 and triplet states T + : |F = 1, m z = 1, T 0 : |F = 1, m z = 0, and T − : |F = 1, m z = −1. The high-field limit is reached when the electron Zeeman interaction dominates over the hyperfine interaction (|γe |B0  A). In this case the electron and nuclear spins are separable, set by their projections along the direction of B0 . However, the hyperfine interaction still couples the electron and nuclear spins. As a result, the energy splitting of the electron spin, referred to as the electron spin resonance (ESR) frequency, increases by A for subsequent nuclear spin states. For a nuclear spin m I , the ESR frequency is to first order given by = −γe B0 + m I A. f mESR I

(3.4)

Similarly, the nuclear-spin energy splitting, referred to as the nuclear transition frequency, is higher for a | ↓ electron ( f ↓ = A/2 + γn B0 ) than for a | ↑ electron ( f ↑ = A/2 − γn B0 ).4 The combined spin eigenstates |m S , m I  = |m S |m I  are those of Sˆ z ⊗ Iˆz , where m S (m I ) is the electron (nuclear) spin projection along the z-axis (Fig. 3.2). In the high-field limit, the transverse hyperfine interaction terms can be ignored, which simplifies the hyperfine interaction ASˆ · Iˆ to A Sˆ z ⊗ Iˆz . The electron readout method used in our experiments requires a high spin energy splitting (Sect. 5.3), and therefore operate at magnetic fields approaching the highfield limit (B0 ≈ 1.5T). For this reason, the high-field limit is solely considered for the remainder of this thesis. Note however that the system is not fully in the high-field limit; the hyperfine interaction slightly distorts the eigenbasis from that of Sˆ z ⊗ Iˆz . This second-order effect has two consequences. First, both the electron and nuclear spin transition frequencies experience slight shifts. This has important implications for the nuclear transitions frequencies of a neutral donor, as it separates 3

A derivation for why the anti-symmetric superposition has a lower energy is found by considering ˆ 2 ˆ ˆ 2 ˆ 2 ˆ 2 ˆ ˆ the squared total  spin length ||F|| := ||I + S|| = ||I|| + ||S|| + 2I · S, which can be reorganized 1 2 2 2 ˆ − ||I|| ˆ 2= ˆ − ||S|| ˆ . The individual spin lengths have a definite value ||I|| into Iˆ · Sˆ = 2 ||F||

ˆ 2 = 2 S(S + 1), and so the anti-symmetric superposition (F = I − S) has a 2 I (I + 1) and ||S|| ˆ = −2 (I S + S) than the symmetric superposition (F = I + S) where Iˆ · S ˆ = lower value Iˆ · S 2 I S. 4 Since the electron can only be in one of two spin states, we use the symbols ↑ and ↓ to indicate a spin-up (m S = 1/2) or spin-down (m S = −1/2) electron, respectively.

3.2 Donor Spin Hamiltonian

31

Fig. 3.2 High-field energy level diagram of a neutral 123 Sb donor. In the high-field limit (|γe |B0  A), the nuclear and electron states are separable by their projection along the quantization axis. The Zeeman interaction separates the states into an electron | ↑ (top, blue), and | ↓ (bottom, orange) manifold. These manifolds are then further split by the eight (2I + 1) nuclear spin states |m I , where m I ∈ {−I, −I + 1, . . . I }. The electron spin resonance (ESR) transition frequencies f m I depend on the nuclear spin m I , and subsequent transitions differ by the hyperfine interaction A. The nuclear transition frequencies are higher for the | ↓ manifold, up to first order all equal to f ↓ = A/2 + γn B0 , while being f ↑ = A/2 − γn B0 for the | ↑ manifold. The nuclear transition frequencies are separated by both the transverse hyperfine interaction terms and the quadrupole interaction (Sect. 3.3), allowing individual addressability

2

subsequent transition frequencies to second order by 2γAe B0 . In the case of 123 Sb with A = 101 MHz at B0 = 1.5 T, subsequent nuclear transition frequencies are separated 2 by 2γAe B0 = 121 kHz. This allows for addressing of individual nuclear transitions even in the absence of a quadrupole interaction. Second, this slightly distorted eigenbasis adds a slight probability of flipping the nuclear spin state whenever the electron tunnels on or off the donor [13]. This effect is strongly enhanced by the quadrupole interaction, and is discussed further in Sect. 5.4.

3.2.5 Resonant Driving While static magnetic fields can be used to split the energy levels of spin states, oscillating magnetic fields can be used to drive transitions between them (Sect. 3.2.5). This requires that the oscillating magnetic field B1 sin (2π f t) is along an axis (at

32

3 Theory of Donors in Silicon

least partially) perpendicular to B0 . Ignoring the component of B1 along the z-axis5 results in the driving Hamiltonian Hˆ Zeemandrive / h = −γe B1 sin (2π f t) Sˆ x − γn B1 sin (2π f t) Iˆx ,

(3.5)

where we define B1 to be oriented along the x-axis, though any choice of axis perpendicular to the z-axis is valid. Similar to the static field B0 , the oscillating magnetic field with strength B1 couples to both the electron and nuclear spin, and can therefore be used to drive either depending on the drive frequency f . This effect is known as electron spin resonance (ESR) for the electron spin, and nuclear magnetic resonance (NMR) for the nuclear spin. Because the hyperfine interaction couples the electron and nuclear spins, the transition frequency of each spin is dependent on the spin state of the other (Fig. 3.2). The electron spin can therefore be flipped conditional on the nucleus being in a target nuclear state, or vice versa. Since only the electron spin state can be directly measured, this property is exploited to measure the nuclear state via conditional flipping of the electron (Sect. 6.1.1). A second method to drive the system uses an AC electric field to modify the electron wave-function overlap at the nucleus, thereby changing the hyperfine interaction strength A. The transverse part of the hyperfine interaction A( Sˆ x ⊗ Iˆx + Sˆ y ⊗ Iˆy ) couples states where the total spin is conserved | ↑, m I − 1 ↔ | ↓, m I . These are known as flip-flop transitions, for an increase in spin of either the electron or nuclear spin is matched by an equal decrease in the other. Note that this transition conserves the total spin. For an initially unperturbed electron where A equals the bulk value, a large electric field is required to significantly perturb the wave function. However, if the electron wave function is already perturbed, either through strain or nearby charges, the required electric field is significantly reduced [14, 15]. AC electric fields can also drive nuclear spin transitions without simultaneously driving electron spin transitions. One method relies on the donor-bound electron acting as a transducer [16]. Here, an AC electric field modulates the electron spinorbit coupling, which in turn modulates the quantization axis of the nuclear spin through the hyperfine interaction. This enables the resonant driving of nuclear spin transitions. Chapter 8 describes another method to electrically drive nuclear spin transitions of nuclei with I > 1/2. Instead of using the electron as a transducer, this method does not require an electron, and therefore works on an ionized nucleus. Instead, spin transitions are driven by electric modulation of the nuclear quadrupole interaction, which results in nuclear electric resonance (NER).

5

The component of B1 parallel to B0 modulates the transition frequencies of the system. However, due to the periodic nature of B1 , any phase acquired during each oscillation is cancelled (provided that B1 is small compared to B0 ). We therefore ignore the component of B1 along the z-axis and define B1 to be oriented along the x-axis.

3.2 Donor Spin Hamiltonian

33

3.2.6 Ionized Donor Hamiltonian Since the outer valence electron is weakly bound to the donor, the energy required to remove the outer valence electron from the donor is small. This changes the donor from its neutral state D0 to the ionized state D+ , and significantly simplifies the Hamiltonian to  Hˆ ionized / h = −γn B0 Iˆz + Q αβ Iˆα Iˆβ . (3.6) α,β∈{x,y,z}

One important feature of the ionized donor is the absence of a hyperfine interaction, which therefore no longer presents a channel for decoherence through electric noise. As a result, the nuclear spin is nearly entirely isolated from its surroundings, and this has led to the demonstration of spectacular coherence times in 31 P in excess of a second (T2Echo = 1.75 s) [17], which was further extended to over half a minute using dynamical decoupling sequences (T2CPMG = 35.6 s). However, the lack of a hyperfine interaction also comes at a price, since it separated the nuclear transition frequencies. This enabled addressing of individual transitions, which is needed for full control of the Hilbert space. To separate the transition frequencies, an interaction is needed that is nonlinear in spin operators. The quadrupole interaction is such an interaction, and is discussed in the following section.

3.3 Nuclear Quadrupole Interaction 3.3.1 Nuclear Quadrupole Hamiltonian Assuming perfect spherical symmetry of the nuclear charge distribution, its electrical response to its environment is described by the Coulomb potential of a point charge. However, deviations from spherical symmetry require correction terms to this simple model. The lowest-order, nonzero correction term to the point charge model is the nuclear quadrupole moment qn , which captures oblate or prolate deviations from spherical symmetry (Fig. 3.3). Values for qn for different donors are given in Table 3.1. A nucleus with a non-spherical charge distribution will have a preferred alignment in response to an external electric field gradient (EFG) (Fig. 3.4), resulting in the quadrupole interaction. The quadrupole interaction scales linearly with the EFG of the local electric field [19], and is quadratic in the nuclear spin operators. This interaction can only be nonzero for nuclei having I ≥ 1, and so is non-existent for 31 P donor with I = 1/2. A general Hamiltonian Hˆ quadrupole of the nuclear quadrupole interaction may be written as  Hˆ quadrupole / h = Q αβ Iˆα Iˆβ , (3.7) α,β∈{x,y,z}

34

3 Theory of Donors in Silicon

Fig. 3.3 Prolate or oblate nucleus resulting in a quadrupole moment. When the nuclear charge distribution deviates from spherical symmetry, this gives rise to the electric quadrupole interaction. This interaction couples the nuclear spin to external EFGs. A positive nuclear quadrupole moment qn > 0 corresponds to a prolate nucleus (A), while a negative nuclear quadrupole moment qn < 0 corresponds to an oblate nucleus (B) [18]

Fig. 3.4 Orientation of a non-spherical nucleus in response to an EFG. (A) Excess charge of a prolate nucleus (qn > 0). The deviation of a prolate nucleus (gray) from spherical symmetry (dotted line) results in an excess positive charge (blue) at the ends of the semi-major axis, and consequently an effective build-up of negative charge (red) at the ends of the semi-minor axes. (B, C) Orientations of a prolate nucleus in the presence of an external EFG. Here the EFG is caused by two surrounding positive (blue) and negative (red) charges in a quadrupolar arrangement. The energy of the prolate nucleus is minimized when its semi-major axis is oriented towards the negative charges (B), and is increased when oriented towards the positive charges (C)

where Q αβ determines the strength of the different terms and is given by Q αβ =

eqn Vαβ , 2I (2I − 1)h

(3.8)

where e is the elementary charge, and Planck’s constant h arises from our convention of specifying all interactions in frequency units. The EFG tensor components Vαβ = ∂ 2 V (x, y, z)/∂α∂β are defined as the second partial derivatives of the electric potential V (x, y, z) experienced by the nucleus. Furthermore, the Sternheimer anti-shielding factor, which captures the enhancement of the EFG at the nucleus due

3.3 Nuclear Quadrupole Interaction

35

to the rearrangement of the core electrons [20] (Sect. 3.3.2), is incorporated into Vαβ . Finally, the EFG tensor is traceless and symmetric, as V experienced at the nucleus obeys the Laplace equation ∇ 2 V = 0 and its partial derivatives commute. Therefore, Q αβ = Q βα , allowing Eq. 3.7 to be expanded as Hˆ quadrupole = Q x x Iˆx2 + Q yy Iˆy2 + Q zz Iˆz2 + Q yz ( Iˆy Iˆz + Iˆz Iˆy ) + Q x z ( Iˆx Iˆz + Iˆz Iˆx ) + Q x y ( Iˆx Iˆy + Iˆy Iˆx ).

(3.9)

Multipole effects such as the electric quadrupole interaction are typically weak effects. To quantify this using Eq. 3.8 and the parameters of 123 Sb (Table 3.1), an interaction strength Q αβ = 1 kHz would require 2I (2I − 1)h Q αβ eqn ≈ −3 × 1018 V m−2 ,

Vαβ =

i.e. a change in electric field of 3GV m−1 over a single nanometer. This is over six orders of magnitude higher than the expected EFG in the 123 Sb-implanted nanodevice measured throughout this thesis (Sect. 7.5.1). Typically, only a microscopic mechanism in a crystal lattice or molecule, such as the distortion of covalent bonds in the vicinity of the nucleus, creates a significant EFG. Furthermore, the presence of the outer valence electron also influences the local EFG at the nucleus, and so Hˆ quadrupole may differ for an ionized versus neutral donor. An in-depth discussion of the microscopic origin of the EFG at the position of the nucleus in our experiment, and its dependence on electric fields, may be found in Chap. 7. For the following, it suffices to know that such an EFG is present, and that it can be changed by varying an external electric field. Thus far, the quadrupole interaction Hamiltonian has been considered for an arbitrary frame. However, since the EFG is a real, traceless, and symmetric tensor, a set of principal axes {x  , y  , z  } exists for which the tensor is diagonal [19]. The  is given by corresponding diagonalized Hamiltonian Hˆ quadrupole  /h = Hˆ quadrupole



Vx  x  − V y  y   ˆ2 eqn Vz  z  2 2 2 ˆ ˆ 3 Iz  − I + Ix  − I y , 4I (2I − 1)h Vz  z 

(3.10)

with the principal axes chosen such that |Vx  x  | ≤ |V y  y  | ≤ |Vz  z  |. The z  axis is then the quadrupole’s primary axis, and we often refer to this simply as the quadrupole’s axis. We define the quadrupole interaction strength Q as the factor that precedes the quadratic Iˆz2 term6 in Eq. 3.10,

6

In literature, Q is sometimes used to denote the nuclear quadrupole moment qn , as is Q n . It should not be confused with our definition of Q as being the quadrupole interaction strength.

36

3 Theory of Donors in Silicon

Q=

3eqn Vz  z  . 4I (2I − 1) h

(3.11) V

  −V  

It is standard practice to define an asymmetry parameter η = x xV   y y to characterize z z the deviation from cylindrical symmetry (0 ≤ η ≤ 1, η = 0 corresponding to perfect cylindrical symmetry). In summary, the five independent degrees of freedom of the quadrupole Hamiltonian may now be understood as an overall interaction strength Q, an asymmetry η, and three angles defining the orientation of the principal axes. In the absence of any other interaction, the quadrupole interaction results in a double degeneracy, since spins with equal magnitude and opposite sign have the same energy. This is a consequence of Kramers degeneracy theorem, which states such a double degeneracy must exist for systems having half-integer spin and time reversal symmetry. Since electric fields obey time reversal symmetry, the quadrupole interaction will always have such a degeneracy. An additional effect such as the Zeeman effect, which depends on magnetic fields and thus does not possess time reversal symmetry, is required to lift this degeneracy.

3.3.2 Estimates of Nuclear Quadrupole Interaction Accurate calculation of Q for donors is complicated by the Sternheimer anti-shielding effect, a phenomenon that describes the re-arrangement of the inner electron shells in response to an external EFG, effectively enhancing the EFG experienced by the nucleus [20, 21]. This effect is captured phenomenologically in the Sternheimer anti-shielding factor γs , a multiplicative term in the quadrupole interaction strength. Its effect can be considerable, theoretical calculations [22] for isolated As and Bi ions show an enhancement of about one order of magnitude for As and up to three orders of magnitude for Bi. To the best of our knowledge, no such calculations have been completed for Sb. Furthermore, it is unknown how the covalent bonding of the donor to the silicon lattice affects γs . As a result of the uncertainty in γs , it is difficult to make purely theoretical predictions of Q for donors in silicon. Numerical simulations using density functional theory (DFT) have been employed to calculate the quadrupole interaction strength in response to strain and an electric field, and this is discussed extensively in Chap. 7. Recent experiments [23–26] on quadrupole effects in silicon devices have produced some quantitative results that can be used to estimate Q for As, Sb and Bi donors in Si. We will present an analysis for each of the donors in sections below, predicting the quadrupole coupling in the ionized charge state D+ , where the EFG is produced by the crystal lattice alone.

3.3 Nuclear Quadrupole Interaction

3.3.2.1

37

Arsenic

While arsenic has the lowest nuclear spin of all quadrupolar donor nuclei, it is a relatively well-studied donor for its quadrupole properties. In Refs. [23, 25], spectroscopy of As donors in a strained silicon sample has been performed (uniaxial strain ⊥ ≈ 3 × 10−4 ). Strain couples linearly to the EFG via V = S · , where V is the EFG tensor, S is the gradient-elastic tensor, and is the strain tensor (Sect. 7.4). Measurement of the quadrupole shifts in two samples of different surface planes [(100) and (111)] enabled the extraction of the non-trivial gradient-elastic tensor components, S11 = 1.5 × 1022 V m−2 and S44 = 6.8 × 1022 V m−2 , where the tensor indices are given in Voigt notation (Sect. 7.1). These components (which include the Sternheimer anti-shielding factor γs ≈ −7 for As [22]) can be used to provide a rough estimation of the D+ quadrupole coupling for Bi and Sb (see below). In a nanodevice, strains of order 10−3 are expected directly underneath the metallic surface electrodes (Sect. 7.4) due to the mismatch in thermal expansion coefficients [27], similar in magnitude to those observed in the pre-strained devices in Refs. [23, 25]. This allows to estimate the quadrupole interaction strength in a device: Q ≈ 210 kHz for a (111) surface and Q = 60 kHz for a (100) surface.

3.3.2.2

Bismuth

In order to estimate Q in the D+ state for Bi, we require the anti-shielding factor γs . We take an order-of-magnitude estimate, only serving as a rough guide, of γs ≈ 100 for the anti-shielding factor of Bi. This value is based on the simulations of measured data reported in Ref. [24]. Using the estimated magnitude of γs for Bi, the measured gradient elastic tensor components S11 and S44 of As (converted to strain using the theoretical magnitude of |γs | ≈ 7 for As), we estimate the quadrupole interaction strength achievable in a nanodevice for the D+ state to be: Q ≈ 240 kHz for a (111) silicon surface and Q = 45 kHz for a (100) surface.

3.3.2.3

Antimony

Antimony is the least-understood of the group-V donors for its quadrupole properties. There are no theoretical calculations for the Sternheimer anti-shielding factor, and no experimental data on the interplay between strain and quadrupole interaction. In the EDMR experiments of Ref. [24], Q was found for the neutral 121 Sb donor to be approximately half that of the measured value for 209 Bi, with the caveat that the implantation conditions were different for the 121 Sb and 209 Bi samples, and no estimate was given for the likely separation between the donors and the readout centers (which strongly influences the EFG) in the 121 Sb sample.

38

3 Theory of Donors in Silicon

3.3.3 Nuclear Spectrum In the absence of any quadrupole interaction, all nuclear transition frequencies of the ionized donor are equal. Any resonant driving will therefore drive all nuclear transitions simultaneously. The quadrupole interaction splits these transition frequencies, enabling the addressing of individual transitions. The nuclear spectrum can therefore be used to extract the quadrupole interaction parameters. For the neutral donor, the nuclear transition frequencies are already separated by the transverse hyperfine interaction terms. We shall from hereon focus on the nuclear spectrum of the ionized donor, though similar techniques can be applied to the nuclear spectrum of the neutral donor. As the experiments reported in this thesis are performed near B0 = 1.5 T, the level splitting due to the Zeeman interaction is of order 8 MHz, far higher than the expected range of quadrupole strengths. This justifies treating the quadrupole interaction as a perturbation to the Zeeman interaction. First-order perturbation theory can be applied to find the quadrupole interaction correction to the energy E m I of state |m I  E m I = −hγn B0 m I + m I |Hˆ quadrupole |m I ,

(3.12)

which results in a transition frequency f m I −1↔m I between neighboring nuclear states |m I − 1 and |m I  given by   f m I −1↔m I = −γn B0 + m I − 21 Q x x + Q yy − 2Q zz ,

(3.13)

The quadrupole splitting f Q is defined as the first-order splitting between two subsequent spectral lines f Q = f m I ↔m I +1 − f m I −1↔m I = Q x x + Q yy − 2Q zz .

(3.14)

Although the splitting between subsequent spectral lines is dominated by the firstorder quadrupole splitting f Q , variations between subsequent spectral line splittings can provide additional information about the quadrupole interaction. For instance, the second-order splitting is usually positive when the primary quadrupole axis is aligned with B0 , and negative when perpendicular to B0 . The full quadrupole interaction tensor with components Q αβ cannot be determined from a single nuclear spectrum. This is due to a strong dependence of the spectral line splitting on the value of the asymmetry parameter as well as an inherent rotational symmetry around the axis of B0 . In the absence of any a-priori knowledge of the orientation and strength of the EFG, f Q may take on any value in the range −2Q ≤ f Q ≤ 2Q.

3.3 Nuclear Quadrupole Interaction

39

3.3.4 Extraction of Quadrupole Parameters Combining nuclear spectroscopy with full control of both the direction and strength of B0 allows extraction of all five quadrupole parameters. This relies on the underlying assumption that each of the nuclear spin transitions is individually addressable, a condition that can be satisfied by assuming a Q larger than the nuclear transition linewidth (Fig. 3.5B). The first step is to find the quadrupole’s primary axis z  through successive spectroscopy measurements while rotating B0 . Without prior knowledge of the quadrupole’s coordinate system, B0 is initially rotated around an arbitrary axis. Figure 3.6B shows the nuclear spectrum if B0 passes through the z  -axis. Note that the spectrum is π-periodic because the quadrupole interaction has a double degeneracy. In this case, the quadrupole splitting f Q varies between −2Q (B0 along z  ) and +Q (B0 perpendicular to z  ), though the latter increases if η > 0 (Fig. 3.6C). Since the rotation axis is chosen randomly, B0 is not expected cross the z  -axis, and there-

Fig. 3.5 Energy levels and transition frequencies of an ionized 123 Sb nuclear spin. (A) Energy level diagram. A positive static magnetic field B0 along z separates the nuclear spin states (green), resulting in 8 equidistantly-spaced energy levels. The quadrupole interaction, oriented along x, further separates these energy levels, resulting in distinct nuclear transition frequencies f m I −1↔m I between spin states |m I − 1 and |m I . (B) Transition frequencies for varying quadrupole strength Q. A nonzero quadrupole interaction strength separates the nuclear transition frequencies by a constant amount f Q . The relation between Q and f Q is dependent on the quadrupole orientation (Eq. 3.14). Here, Q is oriented perpendicular to B0 with η = 0, in which case f Q = Q (C)

40

3 Theory of Donors in Silicon

Fig. 3.6 Quadrupole splitting f Q for varying magnetic field orientation B0 . The nuclear spectra for varying B0 allows extraction of all the five quadrupole parameters. The quadrupole principal axis coordinate system {x  , y  , z  } is used. (A) B0 is rotated along the y  -axis by angle θ in B and C. (B) Nuclear spectrum of 123 Sb for varying B0 orientation (A) in the high-field limit. While rotating B0 along y  from being parallel to the quadrupole axis (B0  z  , θ = 0) to being perpendicular (B0  x  , θ = π/2), the order of the transition frequencies switch. This is reflected in the quadrupole splitting f Q (C), which changes sign. The value of f Q when B0  x  (θ = π/2) depends on the asymmetry η, varying between f Q = Q when η = 0 (red solid), to f Q = 2Q when η = 1 (blue dashed). (D) Same as A, but the quadrupole frame is also rotated around x  by angle φ, used in E and F. The rotation axis of Q is perpendicular to B0 , and when φ (mod π) = 0, B0 is never aligned with z  regardless of the angle θ. (E) Quadrupole splitting f Q while varying θ and φ (D). Although f Q does not always change sign while varying θ (e.g. when φ = π/2), two symmetry axes are always visible. Rotating B0 again along one of these symmetry axes will align B0 with Q at some angle, thereby revealing both the strength and primary axis of Q. (F) Quadrupole splitting f Q for varying θ and η, while φ = π/2. The variations in f Q while rotating B0 perpendicular to the quadrupole axis reveals the asymmetry η and secondary axis of the quadrupole

3.3 Nuclear Quadrupole Interaction

41

fore the minimal quadrupole splitting7 min( f Q ) ≥ −2Q will depend on the minimal angle between B0 and the z  -axis (Fig. 3.6E). However, it is guaranteed that such a spectroscopy will reveal two symmetry axes, one of which is perpendicular to z  . A second rotation of B0 around this particular symmetry axis, perpendicular to the former, will align B0 with z  at some specific angle. This angle is characterized by a minimal quadrupole splitting ( f Q = −2Q), and a minimal variation in splittings (B0 and Q z  z  share the same quantization axis), thereby revealing the quadrupole’s primary axis z  and interaction strength Q. The two remaining unknown parameters of the quadrupole interaction, the asymmetry η and its orientation, can be found through a final rotation of B0 perpendicular to the z  -axis (Fig. 3.6F). If η = 0, spectral lines will be independent of this rotation, while for η > 0 again two symmetry axes will be revealed. The symmetry axes where f Q is largest corresponds to B0 being parallel to the secondary y  -axis, and the strength of spectral line variation is determined by the size of η. The sequence sketched here allows variation of a single experimental handle (NMR frequency, B0 direction) to isolate the effect of each parameter. This allows an accurate determination of all the relevant Hamiltonian parameters and a detailed understanding of the system.

References 1. Streetman BG, Banerjee S (1995) Solid state electronic devices, vol 4. Prentice Hall Englewood Cliffs, NJ 2. Koiller B, Hu X, Sarma SD (2001) Exchange in silicon-based quantum computer architecture. Phys Rev Lett 88(2):027 903. https://doi.org/10.1103/PhysRevLett.88.027903 3. Saraiva A, Baena A, Calderón M, Koiller B (2015) Theory of one and two donors in silicon. J Phys: Condens Matter 27(15):154 208. https://doi.org/10.1088/0953-8984/27/15/154208 4. Ramdas AK, Rodriguez S (1981) Spectroscopy of the solid-state analogues of the hydrogen atom: donors and acceptors in semiconductors. Rep Prog Phys 44(12):1297. https://doi.org/ 10.1088/0034-4885/44/12/002 5. Feher G (1959) Electron spin resonance experiments on donors in silicon. I. electronic structure of donors by the electron nuclear double resonance technique. Phys Rev 114(5):1219. https:// doi.org/10.1103/PhysRev.114.1219 6. Stone NJ (2014) Table of nuclear magnetic dipole and electric quadrupole moments. Int Nucl Data Comm Doc INDC (NDS)-0658. https://doi.org/10.1016/j.adt.2005.04.001 7. Rahman R, Park SH, Boykin TB, Klimeck G, Rogge S, Hol-lenberg LC (2009) Gate-induced g-factor control and dimensional transition for donors in multivalley semiconductors. Phys Rev B 80(15):155 301. https://doi.org/10.1103/PhysRevB.80.155301 8. Mansir J, Conti P, Zeng Z, Pla JJ, Bertet P, Swift MW, Van de Walle CG, Thewalt MLW, Sklenard B, Niquet YM, Morton JJL (2018) Linear hyperfine tuning of donor spins in silicon using hydrostatic strain. Phys Rev Lett 120:167 701. https://doi.org/10.1103/PhysRevLett.120. 167701 7

The quadrupole splitting f Q can be positive or negative, depending on whether subsequent nuclear transitions have increasing ( f Q > 0 Hz) or decreasing frequencies ( f Q < 0 Hz). Differentiating between the two cases requires matching of the transition frequencies to their respective transitions. This identification is straightforward in our system because the nuclear spin can be measured via the electron; the highest ESR frequency always corresponds to the nuclear spin m I = I (Sect. 6.1.1).

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3 Theory of Donors in Silicon

9. Ivey JL, Mieher RL (1975) Ground-state wave function for shallow-donor electrons in silicon. I. isotropic electron-nuclear-double-resonance hyperfine interactions. Phys Rev B 11(2):822. https://doi.org/10.1103/PhysRevB.11.822.63 10. Calderón MJ, Koiller B, Hu X, Das Sarma S (2006) Quantum control of donor electrons at the Si-SiO2 interface. Phys Rev Lett 96(9):096 802. https://doi.org/10.1103/PhysRevLett.96. 096802. 11. Hensen B, Huang WW, Yang C-H, Chan KW, Yoneda J, Tanttu T, Hudson FE, Laucht A, Itoh KM, Ladd TD, Morello A, Dzurak AS (2020) A silicon quantum-dot-coupled nuclear spin qubit. Nat Nanotechnol 15(1):13–17. https://doi.org/10.1038/s41565-019-0587-7 12. Pla JJ, Mohiyaddin FA, Tan KY, Dehollain JP, Rahman R, Klimeck G, Jamieson DN, Dzurak AS, Morello A (2014) Coherent control of a single si 29 nuclear spin qubit. Phys Rev Lett 113(24):246 801. https://doi.org/10.1103/PhysRevLett.113.246801 13. Pla JJ, Tan KY, Dehollain JP, Lim WH, Morton JJL, Zwanen-burg FA, Jamieson DN, Dzurak AS, Morello A (2013) High-fidelity readout and control of a nuclear spin qubit in silicon. Nature 496(7445):334–338. https://doi.org/10.1038/nature12011 14. Tosi G, Mohiyaddin FA, Schmitt V, Tenberg S, Rahman R, Klimeck G, Morello A (2017) Silicon quantum processor with robust long-distance qubit couplings. Nat Commun 8:450. https://doi.org/10.1038/s41467-017-00378-x 15. Laucht A, Muhonen JT, Mohiyaddin FA, Kalra R, Dehollain JP, Freer S, Hudson FE, Veldhorst M, Rahman R, Klimeck G, Itoh KM, Jamieson DN, McCallum JC, Dzurak AS, Morello A (2015) Electrically controlling single-spin qubits in a continuous microwave field. Sci Adv 1(3):e1500022. https://doi.org/10.1126/sciadv.1500022 16. Sigillito AJ, Tyryshkin AM, Schenkel T, Houck AA, Lyon SA (2017) All-electric control of donor nuclear spin qubits in silicon. Nat Nanotechnol 12(10):958. https://doi.org/10.1038/ nnano.2017.154 17. Muhonen JT, Dehollain JP, Laucht A, Hudson FE, Kalra R, Sekiguchi T, Itoh KM, Jamieson DN, McCallum JC, Dzurak AS, Morello A (2014) Storing quantum information for 30 s in a nanoelectronic device. Nat Nanotechnol 9(12):986–991. https://doi.org/10.1038/nnano.2014. 211 18. Vij D (2007) Handbook of applied solid state spectroscopy. Springer Sci Bus Media. https:// doi.org/10.1007/0-387-37590-2.64 19. Slichter C (2013) Principles of magnetic resonance. Springer Berlin Heidelberg. https://doi. org/10.1007/978-3-662-12784-1 20. Sternheimer RM (1967) Quadrupole antishielding factors of ions. Phys Rev 159:266–272. https://doi.org/10.1103/PhysRev.159.266 21. Kaufmann EN, Vianden RJ (1979) The electric field gradient in noncubic metals. Rev Mod Phys 51(1):161. https://doi.org/10.1103/RevModPhys.51.161 22. Feiock F, Johnson W (1969) Atomic susceptibilities and shielding factors. Phys Rev 187(1):39. https://doi.org/10.1103/PhysRev.187.39 23. Franke DP, Hrubesch FM, Künzl M, Becker H-W, Itoh KM, Stutz-mann M, Hoehne F, Dreher L, Brandt MS (2015) Interaction of strain and nuclear spins in silicon: quadrupolar effects on ionized donors. Phys Rev Lett 115(5):057 601. https://doi.org/10.1103/PhysRevLett.115. 057601 24. Mortemousque PA, Rosenius S, Pica G, Franke DP, Sekiguchi T, Truong A, Vlasenko MP, Vlasenko LS, Brandt MS, Elliman RG, Itoh KM (2016) Quadrupole shift of nuclear magnetic resonance of donors in silicon at low mag-etic field. Nanotechnology 27(49):494 001. https:// doi.org/10.1088/0957-4484/27/49/494001 25. Franke DP, Pflüger MP, Mortemousque P-A, Itoh KM, Brandt MS (2016) Quadrupolar effects on nuclear spins of neutral arsenic donors in silicon. Phys Rev B 93(16), 161 303. https://doi. org/10.1103/PhysRevB.93.161303 26. Pla JJ, Bienfait A, Pica G, Mansir J, Mohiyaddin FA, Zeng Z, Niquet YM, Morello A, Schenkel T, Morton JJL, Bertet P (2018) Strain-induced spin-resonance shifts in silicon devices. Phys Rev Appl 9(4):044 014. https://doi.org/10.1103/PhysRevApplied.9.044014 27. Thorbeck T, Zimmerman NM (2015) Formation of strain-induced quantum dots in gated semiconductor nanostructures. AIP Adv 5(8):087 107. https://doi.org/10.1063/1.4928320.

Chapter 4

Experimental Setup

This chapter is devoted to the experimental setup used to manipulate a single 123 Sb donor atom in silicon. This includes a description of the device architecture and fabrication, as well as the measurement apparatus that has been used and developed. The device architecture of the 123 Sb-implanted silicon chip is described in Sect. 4.1. Charge detection in the device is performed with a single electron transistor (SET), the workings of which are described in Sect. 4.2. The fabrication protocol used to develop the device is summarized in Sect. 4.3. A key component of the fabrication is the implantation of 123 Sb donors, the full details of which are given in Sect. 4.4. Once the device is fabricated, it is packaged and then cryogenically cooled in a dilution refrigerator (Sect. 4.5). A range of instruments are used to control the 123 Sb donor (Sect. 4.6), including a custom phase-coherent direct digital synthesis (DDS) module that drives the electron and nuclear spins (Sect. 4.7). To control and measure the 123 Sb donor, fairly intricate measurement sequences needed to be programmed on these instruments. To this end, the SilQ measurement software was developed, which significantly simplified the execution of measurement sequences (Sect. 4.8). Note that the measurement sequences themselves are described in Chaps. 5 and 6.

4.1 Architecture of an 123-Sb Sb-Implanted Silicon Nanodevice The 123 Sb-implanted silicon nanoscale device (Fig. 4.1) consists of several components that together enable the coherent control and measurement of a single 123 Sb donor. The device design is similar to those implanted with 31 P [1, 2]. A 500 µm © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Asaad, Electrical Control and Quantum Chaos with a High-Spin Nucleus in Silicon, Springer Theses, https://doi.org/10.1007/978-3-030-83473-9_4

43

44

4 Experimental Setup

Fig. 4.1 False-colored scanning electron micrograph of the silicon metaloxide- semiconductor device used in the experiment. The top gate and barrier gates (green) define the SET, which is used as a charge detector, underneath at the 28 Si/SiO2 interface (Sect. 5.1). Four donor gates (blue), placed with the ends surrounding the donor-implanted region, are used to tune the donor electrochemical potential (Sect. 5.3) and electrically drive nuclear transitions (Chap. 6). The microwave antenna (red) is used to drive NMR and ESR transitions. Note the melted gaps in the nominally short-circuited antenna terminations, the effect of which is discussed in Sect. 6.3.1

thick silicon wafer is used as the device substrate, with a 900 nm top layer of isotopically-purified 28 Si. This isotopic purification largely removes the spinful 29 Si nuclei (I = 1/2) in the surrounding of the 123 Sb donors, which are implanted in the top few nanometers. These 29 Si nuclei would otherwise act as a spin bath that couples to the donor atoms, creating a source of decoherence [3]. Adding this purified 28 Si layer has been shown to drastically increase the coherence times of 31 P in similar devices by orders of magnitude [4]. The 123 Sb donor atoms are implanted in a 90 nm × 90 nm window, at a peak implantation depth of 2 nm in the purified 28 Si substrate (Fig. 4.5). The 123 Sb dose is sufficiently low such that there are usually no significant couplings between donor atoms. Full details of the implantation parameters are given in Sect. 4.4. Aluminum nanostructures are fabricated in the surrounding of the implantation window to interact with the 123 Sb donors. The aluminum is separated from the silicon substrate by a thin layer of high-quality SiO2 , which acts as an electrically-insulating layer. This prevents electrons from tunneling between donors and the aluminum. A positive potential applied to the aluminum gates also attracts electrons at the 28 Si/SiO2 interface from nearby n+ -doped regions (Sect. 4.3). The aluminum gates can

4.1 Architecture of an 123-Sb Sb-Implanted Silicon Nanodevice

45

therefore define a two-dimensional electron gas (2DEG) underneath. In particular, this 2DEG can form a (SET), which is a very sensitive charge detector (Sect. 4.2 and Sect. 5.1). Importantly, this SET is used to read out the electron and nuclear spin of the 123 Sb donor by combining spin-to-charge conversion and charge detection (Sect. 5.3). The electrochemical potentials of the donors are controlled by four gates, referred to as donor gates, that surround the donor-implanted region. This allows control over the donor’s charge state, for an electron will tunnel from the SET 2DEG to the donor if it has a lower electrochemical potential, and vice versa (Sect. 5.3). The capacitive coupling between a donor and a gate depends on their distance. Each donor will have a different capacitive coupling to each of the gates, and so varying gate potentials separately allows some control over which donors are addressed. The oscillating magnetic fields used to drive ESR and NMR transitions are generated by an aluminum microwave antenna next to the donor gates. The antenna is a coplanar waveguide with shorted ends placed as close as possible to the donors, in an effort to maximize the magnetic fields and minimize electric fields [5].

4.2 The Single Electron Transistor (SET) 4.2.1 Electrostatics of the SET The 123 Sb-implanted device contains the necessary gates to form a single electron transistor (SET). The primary functionality of the SET is that of a very sensitive charge sensor. In particular, it can be used to detect the 123 Sb donor charge state. This can be combined with spin-to-charge conversion to measure the donor electron spin (Sect. 5.3), as well as the nuclear spin (Sect. 6.1.1). The SET additionally acts as an electron reservoir, enabling electron spin initialization. Three aluminum gates are used to form an SET at the 28 Si/SiO2 interface (Figs. 4.2A, B). When a sufficiently positive voltage is applied to the top gate (TG), electrons are sourced from nearby n+ -doped reservoirs to create a 2DEG under the top gate. This creates a conductive path between the source and drain contacts (Fig. 4.6). A left barrier gate (LB) and right barrier gate (RB) cross underneath a segment of the top gate, and decreasing their voltages can restrict the 2DEG from forming underneath them. This divides the 2DEG into an SET island containing ∼100 electrons, and the source and drain leads (Fig. 4.2B). Due to the finite barrier potentials, electrons can still tunnel between the island and the leads depending on the electrostatic configuration. To understand the operating principle of the SET, it is instructive to consider the SET circuit representation [6, 7] (Fig. 4.3A). Each nearby gate electrode k has a potential Vk and is capacitively coupled to the SET island with a capacitance Ck . This includes the source and drain 2DEG, with the addition that they are connected to the SET island by a tunnel barrier, allowing electrons to tunnel back and forth.

46

4 Experimental Setup

Fig. 4.2 Location of the single electron transistor (SET). (A) SEM top view of the SET region (green). The SET is defined by the top gate (TG), left barrier (LB) and right barrier (RB) gates, and can additionally be tuned with the plunger gate (PL). (B) Side-view schematic of SET and defining gates (pink line-cut in A). A positive potential applied to the top gate can form a 2DEG under the 28 Si/SiO interface that is connected to the source and drain leads. The barrier gates, which are 2 separated from the top gate by a layer of Al2 O3 (brown), and create tunnel barriers that divides an SET island from the source and drain 2DEG (pink). A positive bias potential VSD applied between source and drain can result in electron tunneling from drain to source via the SET island. This results in a current ISD between source and drain

Fig. 4.3 Electrostatics of an SET. (A) Circuit representation of an SET. The SET island with N excess electrons has a tunnel coupling to a source and drain lead, allowing electrons to tunnel back and forth. A source-drain bias potential VSD may result in a current flow ISD through the SET island depending on the electrochemical potential of the island. This can be tuned by capacitively-coupled gates, which in this circuit is shown as a single gate at potential Vk with capacitive coupling Ck . (B) SET electrostatic energy. For a fixed N , the electrostatic energy has a quadratic dependence on the offset charge q induced by capacitively-coupled gates. Tunnel coupling to the leads allows the number of electrons to relax to the minimum-energy configuration (red) for which |q/e − N | ≤ 1/2. The minimum-energy configuration is degenerate when q/e = N ± 1/2

The difference between each gate potential Vk and the island potential Vi induces a charge qk = Ck (Vk − Vi ) at the respective capacitor on the side of the gate electrode. The total electrostatic energy of the system is E el =

 1 Ck (Vi − Vk )2 − qk Vk , 2 k k

(4.1)

where k runs over all the gate electrodes, including the source and drain gates. The first component is the energy stored on each capacitor Ck and the second is the work

4.2 The Single Electron Transistor (SET)

47

done by the voltage source Vk to transfer the charge qk to the gate electrode. The relation qk = Ck (Vk − Vi ) allows us to rewrite Eq. 4.1 as E el =

1 1 Ck Vi2 − Ck Vk2 . 2 k 2 k

(4.2)

The second term does not depend on the SET island’s potential but only on the gate potentials. It is irrelevant for the remainder of this derivation and is therefore disregarded. When the tunnel barriers are sufficiently strong, such that the conductance is much 2 smaller than the conductance quantum G 0 = 2eh , the total charge on the island is quantized and satisfies the relation eN = −



qk =



k

Ck (Vi − Vk ),

(4.3)

k

where N is the electron occupancy number, i.e. the number of electrons on the island. To simplify calculations, we introduce the offset charge q :=



Ck Vk ,

(4.4)

k

which equals the total charge on the gate capacitors when the island is grounded (Vi = 0 V). These can be combined to arrive at the expression for the island potential eN − q =



Ck Vi ⇒ Vi =

k

eN − q , C

(4.5)

 where the shorthand notation C = k Ck is used for the total capacitance of the island. The electrostatic energy (Eq. 4.2) can now be written as a function of the island charge occupancy E el (N ) =

e2 1 (N − q/e)2 = E c (N − q/e)2 . 2C 2

(4.6)

2

Here we introduce the charging energy E c = Ce .1 The electrostatic energy has a quadratic dependence on the offset charge q for a fixed N (Fig. 4.3B), and is minimal when N = q/e. Each charge offset q therefore has a corresponding N for which the total energy is minimized, and the SET island will relax to this configuration due to its tunnel coupling with the leads. Note that q/e = N + 1/2 results in a degeneracy, where both N and N + 1 electrons are at the minimum energy.

1

In literature, the charging energy is sometimes defined to be a factor two smaller [6], i.e. E c =

e2 2C

.

48

4 Experimental Setup

The electrochemical potential associated with adding the N -th electron to the SET island is given by2  μ(N ) = E el (N ) − E el (N − 1) = E c

 1 N − q/e − . 2

(4.7)

Consequently, subsequent electrochemical potentials differ by the charging energy E c = μ(N + 1) − μ(N ). Gates can be used to tune the electrochemical potential of the SET island since μ(N ) depends on q. varying the potential Vk of gate k by Vk shifts the island electrochemical potential by μ(N ) = −|e|αk Vk ,

(4.8)

where αk = Ck /C < 1 is the gate lever arm. Gates with a stronger capacitive coupling to the SET island have a higher lever arm and consequently their gate potential has a stronger effect on the electrochemical potential. Of all the gates in our device, the top gate and barrier gates have the highest lever arms.

4.2.2 SET Modes of Operation The finite potential barrier between the SET island and the source and drain leads allows electrons to tunnel between them. A small bias potential VSD applied between the source and drain leads can therefore result in a current ISD between the leads through the SET island. However, an electron will only tunnel to an energetically favorable location, and this depends on the island’s electrochemical potential μ(N ) with respect to the source potential μS and drain potential μD . Assuming μD − μS = |e|VSD > 0, a current ISD will only flow when μD > μ(N ) > μS for some N . Figure 4.4 depicts this situation, where the energy cost to add the N -th electron to the SET island is less than the energy gained by the electron leaving the drain lead. Once the N -th electron is on the island, it can further reduce its energy by tunneling on to the source (μ(N ) > μS ). This frees up the SET island to accept a new N -th electron from the drain, and the repetition of this process results in a current ISD > 0 from source to drain. Conversely, when μ(N ) does not lie between μS and μD , electrons cannot continuously flow to a lower potential (Fig. 4.4B). The resulting lack of current (ISD = 0) is known as Coulomb blockade. The current flow ISD , caused by a single electron tunneling through the SET island at a time, can therefore be switched on or off by changing μ(N ), hence the name single electron transistor. 2

Since the SET island has a finite size, the electrochemical potential can also contain an orbital energy contribution. However, this contribution is small compared to the electrostatic charging energy due the relatively high number of electrons in the SET island.

4.2 The Single Electron Transistor (SET)

49

Fig. 4.4 Operation of the SET. (A) Electrons can tunnel from drain to source via the SET island when the N -electron SET island electrochemical potential μ(N ) lies between the source and drain electrochemical potentials μS and μD , respectively. This configuration (μD > μ(N ) > μS ) requires a bias potential VSD between source and drain, and results in a current flow ISD > 0 from source to drain. (B) When the SET island electrochemical potentials lie outside the source and drain potentials, electrons require energy to cross through the SET island, restricting current flow (ISD = 0). Here the system is in Coulomb blockade. (C) A source-drain bias potential VSD exceeding the charging energy E c allows electrons to flow regardless of the island potential, since there is always at least one accessible charge state within the bias window

Coulomb blockade requires a bias potential below the charging energy (|eVSD | < E c ). If this is not the case, there will always be an N for which μ(N ) lies between μS and μD (Fig. 4.4C), thereby suppressing Coulomb blockade. A nearby gate k has a lever arm αk to the SET island (Eq. 4.8), and can thus be used to tune μ(N ). Varying the gate potential Vk therefore causes the SET to alternate between being in and out of Coulomb blockade, resulting in Coulomb oscillations of ISD . The regions with zero current are in Coulomb blockade, and have a well-defined number of electrons on the island. The areas between these Coulomb blockade regions have a peak in current, known as a Coulomb peak. The height and width of a Coulomb peak depends on VSD , the electron temperature, and the height of the potential barriers. Since μ(N ) depends on all capacitively-coupled charges, it will also shift when a nearby single charge is displaced. This can result in a measurable difference in ISD , provided that the charge has a strong enough capacitive coupling to the island. The SET can thus act as a charge detector, and this is exploited to measure the charge state of a donor (Sect. 5.3).

4.3 Fabrication Protocol Devices are fabricated using standard microelectronic fabrication techniques. The substrate is a p-type 100 silicon (Si) wafer (resistivity 10-20  cm), with a (LPCVD) grown epitaxial layer of isotopically enriched 28 Si (concentration of residual 29 Si = 730 ppm) of 900 nm thickness on top.

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4 Experimental Setup

The main fabrication steps are: 1. Definition of negative optical alignment markers via tetramethylammonium hydroxide (TMAH) etc.h of the silicon, using optical lithography and a wet thermally grown masking oxide. The masking oxide is subsequently removed using buffered hydrogen fluoride (HF) etc.hing. 2. Creation of p+ -doped regions in between the n+ -doped metallic leads of step 4, defined using optical lithography and a wet thermally grown masking oxide. This step is designed to suppress spurious leakage currents flowing at the silicon/silicon dioxide (28 Si/SiO2 ) interface between the n+ -doped leads. Doping is achieved via thermal diffusion of boron (B). The masking oxide is subsequently removed using buffered HF etc.hing. 3. Two-step thermal oxidation process to repair defects and drive B into the silicon, resulting in a final wet thermal oxide of 200 nm thickness. 4. Creation of n+ -doped metallic leads, defined by optical lithography using the thermally grown masking oxide of the previous step. Doping is achieved via thermal diffusion of phosphorus (P). Masking oxide is subsequently removed via buffered HF etc.hing. 5. Single-step thermal oxidation to drive P into the silicon, resulting in a wet thermal oxide of 200 nm thickness which serves as thick field oxide. 6. Etching of a central window of 20 µm × 40 µm in the field oxide using buffered HF etc.hing, and subsequent growth of an 8 nm thick, high-quality dry thermal oxide. The p+ -doped regions and n+ -doped leads of steps 2 and 4 extend for ∼ 2 µm underneath the high-quality gate oxide. 7. Definition of positive alignment markers via optical lithography. Markers are created by electron-beam evaporation of 15/75 nm titanium/platinum (Ti/Pt) and lift-off using warm N-Methyl-2-pyrrolidone (NMP). 8. Definition of positive electron-beam alignment markers by patterning a poly (methyl methacrylate) (PMMA) mask aligned to the optical alignment markers using electron beam lithography (EBL). Markers are created by electron-beam evaporation of 15/75 nm titanium/platinum (Ti/Pt) and lift-off using warm NMP. 9. Definition of a 90 nm×90 nm implantation window, patterned in a PMMA mask using EBL. 10. Low-dose implantation of 123 Sb (see Sect. 4.4 for details), followed by a rapid thermal anneal for 5 s at 1000 ◦C to achieve donor activation and repair implantation damage to the silicon lattice. 11. Definition of ohmic contacts to the n+ -doped metallic leads via optical lithography; ohmic contact is achieved by etc.hing through the field oxide using buffered HF, evaporation of 200 nm aluminum (Al), and a 15 min forming gas (5% hydrogen in nitrogen) anneal at 400 ◦C. 12. Definition of all gates necessary to control and read out the 123 Sb donor, including gates to tune the electrochemical potential of the donor, an SET for read-out, and a microwave antenna. This is achieved via two steps of standard EBL using PMMA masks, thermal evaporation of Al and lift-off using warm NMP. The first layer consists of 20 nm of Al, and contains the SET barrier gates and two of the

4.3 Fabrication Protocol

51

donor tuning gates. The second layer consists of 40 nm of Al, and contains the top-gate of the SET, the two remaining donor tuning gates, and the microwave antenna. The two layers are electrically isolated by the native oxide formed on the first Al layer upon exposure to air. 13. Forming gas anneal for 15 min at 400 ◦C, to passivate traps in the gate oxide. 14. Dicing and packaging, including bonding (see also Sect. 4.5).

4.4

123-Sb Sb

Implantation Parameters

Standard modeling using SRIM/TRIM software [8] was used to predict the 123 Sb dopant profile. The implantation energy was chosen such that the peak of the 123 Sb dopant profile is located 2 nm below the 28 Si/SiO2 interface. Such a shallow implantation was chosen to maximize the donor’s exposure to static strain upon cool-down of the sample (see Sect. 7.4 for details on strain calculations). Given a gate oxide thickness of 8 nm, the optimal implantation energy was found to be 8 keV; the simulated vertical doping profile is shown in Fig. 4.5. The implantation profile can be approximated by a modified Gaussian distribution homogeneous in the lateral dimensions, with a mean depth at −2.5 nm, standard deviation of 2.6 nm and slight skewness (0.54) and kurtosis (3.59). A low dose of 2 × 1011 cm−2 123 Sb donor atoms was implanted. This corresponds to an average of 14 donors per implantation window, i.e. each device is expected to have a few donors that are tunnel coupled to the SET.

Fig. 4.5 Expected 123 Sb concentration versus depth. Ion implantation parameters are an acceleration voltage of 8 keV at a fluence of 2 × 1011 cm−2 . The distribution is simulated with SRIM/TRIM and scaled by the 90 × 90 nm area of the implantation window. The dashed vertical line at 0 nm denotes the 28 Si/SiO2 interface. The peak of the implantation profile is at 2 nm below this interface; integrating the distribution over the Si volume underneath the implantation window results in an expected number of 14 123 Sb donors per implantation window

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4 Experimental Setup

4.5 Device Packaging and Cooling The experiments on single 123 Sb donors require cooling of the sample to cryogenic temperatures. This is primarily because the electron-spin-readout method requires an energy splitting between the electron spin states that is higher than the thermal broadening of the SET (Sect. 5.3). Higher temperatures also introduce unwanted interactions, such as lattice vibrations that decrease the coherence and relaxation times of the donors. Device packaging consists of a custom-made printed circuit board (PCB) with microwave and low-frequency lines, positioned around a central cutout accommodating the fabricated chip. The PCB is mounted to a copper enclosure with K connectors (2.92 mm, rated to 40 GHz), and MMCX connectors (rated to 6 GHz). The device is glued to the enclosure and Al wire bonded to the PCB. The copper enclosure is mounted on a gold-plated copper cold-finger bolted to the mixing chamber of the dilution refrigerator. A Bluefors BF-LD400 cryogen-free dilution refrigerator is used to cool the device to a base temperature of ∼12 mK3 [9]. The necessary electron spin energy splitting requires a static magnetic field of B0  1 T. The static magnetic field in the experiment is produced by a superconducting magnet placed inside the dilution refrigerator at a temperature of 4 K. The magnet is equipped with a low-drift persistent mode switch that results in a typical magnetic field drift of less than 50 ppb/hour.

4.6 Instrumentation and Connectivity Three different types of control lines are present in the experimental apparatus. A single high-frequency microwave line is used for ESR in the ∼ 40 GHz regime, and has an inner/outer DC block at room temperature and a 10 dB attenuator at the 4 K stage of the dilution refrigerator. The coaxial microwave line has a silverplated copper-nickel inner conductor, a copper-nickel outer conductor and a PTFE dielectric. Six radio-frequency coaxial lines are used for static and dynamic tuning of the donor electrochemical potential, driving via nuclear electric resonance (NER) (Chap. 6), and for the readout signal from the SET. These lines have a graphite coating on the dielectric to reduce triboelectric noise effects [10], and are low-pass filtered with a 145 MHz cut-off. Three more lines of a Constantan loom, low-pass filtered to a 20 Hz cut-off, are used for the static electrical tuning of the SET. 3

While the device lattice temperature approximately equals the base temperature, the electron temperature of the SET is significantly higher, estimated to be at least a hundred millikelvin. The high electron temperature is due to the thermal load provided by the radiation coming through the high-frequency cables, combined with the weak electron-phonon coupling at low temperatures, which scales as T 5 [9]. The electron temperature can be reduced by line filtering. The SET electron temperature affects the electron spin readout fidelity (Sect. 5.3).

4.6 Instrumentation and Connectivity

53

Fig. 4.6 Schematic of the experimental measurement setup. A DC source (PXIe-4322) is used to define gate voltages and tune the SET via the TG, LB, RB and PL gates. A total of eight AWG channels (Keysight M3300A and M3201A) are used to apply dynamical electrical stimuli to the donor gates (DFL, DFR, DBL, and DBR) of the device, and IQ modulation inputs to the microwave source. The DC and AC signals are combined in a resistive voltage adder, reducing each input by a factor of 13. A vector microwave source (Keysight E8267D) is used to perform ESR via the microwave antenna. A small voltage bias (SIM928) is applied to the ohmic source contact, and the resulting SET current is subsequently measured from the drain contact. The SET current passes through a transimpedance amplifier (FEMTO DLPCA-200) with a gain of 1 × 107 V A−1 and bandwidth of 50 kHz. The current is then further amplified (SIM911) by a factor of 102 and filtered to a 50 kHz cut-off (SIM965), after which it is recorded with a digitizer (Keysight M3300A)

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4 Experimental Setup

The electronic setup, as depicted in Fig. 4.6, includes two instruments as DC sources, a Stanford Research Systems (SRS) SIM928 and (NI) PXIe-4322. We use a total of eight arbitrary waveform generator (AWG) channels between the Keysight M3201A and M3300A modules, which are bandwidth limited to 200 MHz, and a 100 megasamples-per-second digitizer channel of the M3300A is used to record SET current traces. These modules have on-board field-programmable gate array (FPGAs) where a variety of custom code is implemented, including an in-house direct digital synthesis (DDS) system allowing on-the-fly generation and sequencing of sinusoidal and linear chirp pulses (Sect. 4.7). These are used as the IQ modulation inputs for the Keysight E8267D PSG vector microwave source that is used to perform ESR. Using the DDS for IQ modulation has the advantage that it allows rapid switching of ESR frequencies. Single-sideband modulation, either upper or lower, is used to address the relevant ESR transitions. To avoid simultaneously driving multiple transitions with leakage of the alternate sideband, the microwave carrier frequency is offset from the center of any two ESR frequencies. The SET current passes through the following chain of amplifiers and filters: FEMTO DLPCA-200 transimpedance amplifier, SRS SIM911 BJT amplifier and an SRS SIM965 low-pass filter.

4.7 Phase-Coherent DDS Full coherent control of the 123 Sb nuclear and electron spins requires applying multiple radio-frequency (RF) pulses at different frequencies within a single pulse sequence. The phase of each RF pulse is set with respect to a fixed point in time, see the generalized rotating frame for details (Sect. 2.7). This means that switching between frequencies needs to be phase coherent.4 One approach is to pre-compute each of the RF-pulse waveforms necessary for a pulse sequence, and upload them to the AWG’s. These directly drive nuclear spin transitions, and IQ-modulate the microwave source to drive different ESR transitions. However, since the duration of a single pulse ranges between microseconds to a few milliseconds, uploading these waveforms at the required sampling rate (500 mega samples per second) can take several seconds. As a result, performing a simple measurement such as varying a single pulse frequency can have a significant instrument configuration overhead that becomes a bottleneck for experiments. Long pulse sequences can additionally run into memory issues. An alternative approach that circumvents these issues is to use phase-coherent (DDS) [11]. Instead of pre-computing an entire waveform, a DDS module combines a counter with a look-up table to calculate the output signal at each clock cycle. The counter is usually incremented by an amount based on the accumulated phase during 4

An alternative to phase-coherent switching is phase-continuous switching, where each pulse starts with the phase of the previous pulse. Phase-continuous switching does not properly take the accumulated phase between quantum states into account, and is therefore not suited for coherent spin control.

4.7 Phase-Coherent DDS

55

Fig. 4.7 Phase continuous versus phase coherent switching between pulses. Three sinusoidal pulses are applied, where the first and third pulses (I) share the same frequency, while the second pulse (II) has a higher frequency. (A) Phase-continuous switching between pulses maintains the phase of the previous pulse. In this case, switching the frequency back to a previously-used frequency generally does not maintain the same phase relation (red dashed line). (B) Phase-coherent switching between pulses calculates the phase based on a fixed point in time. Here switching back to a previously-used frequency maintains the same phase relation (red dashed line)

a clock cycle, which depends on the pulse frequency. This results in phase-continuous switching (Fig. 4.7A), which is not practical for higher-dimensional spin systems that have multiple transition frequencies (Sect. 2.7). Our modified phase-coherent DDS module increments the counter by one every clock cycle, regardless of the frequency. The pulse frequency, phase offset, and amplitude are then used to convert the counter into a value used by the look-up table. The resulting output signal is the same as for the standard phase-continuous DDS methods, but there is one important difference. By separating the counter increment from the pulse frequency, switching between frequencies is ensured to be phase coherent (Fig. 4.7B). Another feature added to the phase-coherent DDS module is frequency ramping, resulting in chirp pulses that adiabatically invert the spin population [12]. An example pulse sequence output of the DDS module is shown in Fig. 4.8. As waveforms do not need to be loaded, the phase-coherent DDS module provides extremely lowlatency programming of pulses, and switching between them. It can also be combined with other FPGA components, ensuring easy integration with the next generation of experiments that incorporate feedback routines.

4.8 SilQ Measurement Software The control of an 123 Sb donor requires reasonably complex pulse sequences involving several instruments that need to output their respective pulses at specific times. This also includes ancillary pulses, such as triggering pulses to signal other instruments to output their pulses. The programming of these instruments can become rather intricate, and is dependent on their connectivity. On top of this, the measured 123 Sb device

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4 Experimental Setup

Fig. 4.8 Pulse sequence using DDS. The DDS module has been programmed with a sequence of pulses. Every time the DDS module receives a trigger (spikes in A), it outputs the next pulse in the sequence (B) with a latency of a few hundred nanoseconds. In this example, the pulse sequence consists of the following four pulses: (I) a sinusoidal pulse, (II) a DC pulse, (III) a chirp pulse, and (IV) a sinusoidal pulse. Instead of storing a pre-calculated waveform, the next output voltage is calculated every cycle. Switching between pulses maintains phase coherence, which makes this DDS method suitable for the control of high-dimensional quantum systems (Sect. 2.7)

was switched between two dilution refrigerators which had different equipment and connectivity, significantly complicating the instrument programming. We developed the SilQ measurement software [13] to manage the growing complexity of such experiments. The software uses the Python-based QCoDeS data acquisition framework [14], which facilitates tasks such as running a measurement loop, data storage, and plotting results. The key principle behind SilQ is modularity: one should be able to use the exact same measurement code to run the same experiments on completely different experimental setups having different instruments and connectivity between them (provided that the instruments are able to output the requested pulse sequence). The measurement code in SilQ is based on setup-independent pulse sequences, such as the following pulse sequence to perform ESR: PulseSequence([ DCPulse("load_electron_spin_down", amplitude=0 V, t_start=0 s, duration=1 ms),

DCPulse("neutralize", amplitude=20 mV, t_start=1 ms, duration=5 ms), SinePulse("ESR", frequency=42 GHz, amplitude=50 mV, t_start=2 ms, duration=10 us),

DCPulse("read_electron", amplitude=0 V, t_start=6 ms, duration=10 ms)])

An example of a measurement would then consist of varying the of the ESR pulse, thereby measuring an ESR spectrum.

frequency

property

4.8 SilQ Measurement Software

57

The instruments and their connectivity are stored in SilQ for a given experimental setup. This is used to direct pulse instructions to the appropriate instruments. Every instrument has its own interface that can translate the pulse instructions it needs to output into instrument-specific instructions. It can also request additional input pulses such as triggering pulses at specific times, which are then redirected to the triggering instrument using the connectivity layout. This alleviates the end-user from having to deal with low-level setup-specific issues such as setting up instruments or programming trigger pulses, and enables the user to write high-level setup-independent measurements that can be shared between different setups and experiments.

References 1. Morello A, Escott CC, Huebl H, Van Beveren LW, Hollenberg LC, Jamieson DN, Dzurak AS, Clark RG (2009) Architecture for high-sensitivity single-shot readout and control of the electron spin of individ- ual donors in silicon. Phys Rev B 80(8):081 307. https://doi.org/10. 1103/PhysRevB.80.081307 2. Muhonen JT, Laucht A, Simmons S, Dehollain JP, Kalra R, Hudson FE, Freer S, Itoh KM, Jamieson DN, McCallum JC, Dzurak AS. Quantifying the quantum gate fidelity of single-atom spin qubits in silicon by randomized benchmarking. J Phys: Condens Matter 27(15):154 205. https://doi.org/10.1088/0953-8984/27/15/154205 3. Witzel WM, Carroll MS, Morello A, Cywi´nski Ł, Sarma SD (2010) Electron spin decoherence in isotope-enriched silicon. Phys Rev Lett 105(18):187 602. https://doi.org/10.1103/ PhysRevLett.105.187602 4. Muhonen JT, Dehollain JP, Laucht A, Hudson FE, Kalra R, Sekiguchi T, Itoh KM, Jamieson DN, McCallum JC, Dzurak AS, Morello A (2014) Storing quantum information for 30 seconds in a nanoelectronic device. Nat Nanotechnol 9(12):986–991. https://doi.org/10.1038/nnano.2014. 211 5. Dehollain JP, Pla JJ, Siew E, Tan KY, Dzurak AS, Morello A (2012) Nanoscale broadband transmission lines for spin qubit control. Nanotechnology 24(1):015 202. https://doi.org/10. 1088/0957-4484/24/1/015202 6. Nazarov YV, Nazarov Y, Blanter YM (2009) Quantum transport: introduction to nanoscience. Cambridge University Press. https://doi.org/10.1017/CBO9780511626906 7. Ihn T (2010) Semiconductor nanostructures: quantum states and electronic transport. Oxford University Press 8. Ziegler JF, Ziegler MD, Biersack JP (2010) Srim—the stopping and range of ions in matter. Nucl Instrum Methods Phys Res B 268:1818–1823. https://doi.org/10.1016/j.nimb.2010.02. 091 9. Pobell F (2007) Matter and methods at low temperatures, vol 2. Springer. https://doi.org/10. 1007/978-3-540-46360-3 10. Kalra R, Laucht A, Dehollain JP, Bar D, Freer S, Simmons S, Muhonen JT, Morello A (2016) Vibration-induced electrical noise in a cryogen-free dilution refrigerator: characterization, mitigation, and impact on qubit coherence. Rev Sci Instrum 87(7):073 905. https://doi.org/10.1063/ 1.4959153 11. Firgau HR (2018) Development of FPGA hardware and software for spin qubit control. Master’s thesis, The University of New South Wales 12. Laucht A, Kalra R, Muhonen JT, Dehollain JP, Mohiyaddin FA, Hudson F, McCallum JC, Jamieson DN, Dzurak AS, Morello A (2014) High- fidelity adiabatic inversion of a 31p electron spin qubit in natural silicon. Appl Phys Lett 104(9):092 115. https://doi.org/10.1063/1.4867905 13. Asaad S, Johnson M, Firgau H (2017) Silq measurement software. In: MathWorld- a wolfram web resource. https://nulinspiratie.github.io/SilQ/

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14. Nielsen JH, Nielsen WH, Astafev M, Johnson AC, Vogel D, Chatoor S, Ungaretti G, Smiles AM, Pauka S, Eendebak P, Saevar Q, Een- debak P, van Gulik R, Pearson N, damazter, Corna A, Droege S, damazter2, Larsen T, Geller A, euchas, Hartong V, Asaad S, Granade C, Drmi´c L, Borghardt S, mltls, Qcodes/qcodes: Qcodes 0.2.1 (2019). https://doi.org/10.5281/zenodo. 2649884.86

Chapter 5

123-Sb Donor Device Characterization

This chapter presents the first measurements performed on a single 123 Sb donor in silicon. The start of the chapter discusses the tuning of the SET and subsequent detection of an 123 Sb donor. An eight-fold splitting of the ESR spectrum is observed, matching the eight nuclear spin states of the 123 Sb donor. Flip-flop transitions are also observed, where both the nuclear and electron spins flip simultaneously, and this is exploited for fast high-fidelity nuclear spin initialization. This chapter includes results from the following publication: S. Asaad∗ , V. Mourik∗ , B. Joecker, M. A. I. Johnson, A. D. Baczewski, H. R. Firgau, M. T. Ma˛dzik, V. Schmitt, J. J. Pla, F. E. Hudson, K. M. Itoh, J. C. McCallum, A. S. Dzurak, A. Laucht, A. Morello. “Coherent electrical control of a single high-spin nucleus in silicon”. Nature 579.7798, pp. 205–209 (2020). The author acknowledges (i) M. T. Ma˛dzik for fabrication and initial characterization of the 123 Sb device, (ii) B. Joecker for finite-element modeling of the electrostatics in the device for donor triangulation, and (iii) A. Laucht for support in analyzing measurement results. After initial calibration of the single electron transistor (SET), it is used to detect the charge state of an electron on an 123 Sb donor, i.e. if the donor is ionized or neutral (Sect. 5.1). The position of this donor is triangulated by comparing measurements to an electrostatic model of the device (Sect. 5.2). Spin-dependent tunneling is then used to read out and initialize the donor electron spin (Sect. 5.3). Measurements of the ESR spectrum reveal the resonance frequency jumping between eight equallyspaced frequencies, confirming that this electron indeed belongs to a single 123 Sb donor (Sect. 5.4). © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Asaad, Electrical Control and Quantum Chaos with a High-Spin Nucleus in Silicon, Springer Theses, https://doi.org/10.1007/978-3-030-83473-9_5

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5 123-Sb Donor Device Characterization

Flip-flop transitions are also observed, where both electron and nuclear spins are flipped simultaneously, while preserving the total spin (Sect. 5.5). This proved fortuitous, as it provides a method to initialize the nuclear spin with high fidelity and without measurements (Sect. 5.5.2). Nuclear spin resonance is explored in the next chapter.

5.1 Charge Sensing with an SET 5.1.1 Calibration of the SET A single electron transistor (SET) is used to detect the donor charge state, which can be either ionized (D+ ) or neutral (D0 ). The SET is formed by a two-dimensional electron gas (2DEG) underneath the top gate that acts as a conductive path between source and drain (Sect. 4.2). This 2DEG forms when the top gate and barrier gate potentials are simultaneously increased sufficiently. The gate potential at which a 2DEG is formed is referred to as the turn-on voltage. Its value lies somewhere between 1 and 2 V, and depends on many factors, such as fixed charges in the SiO2 layer, charge traps at the 28 Si/SiO2 interface, the SiO2 thickness, and the workfunction of the metal used as gate electrode. The turn-on voltage can be measured by increasing the gate potentials while applying a small source-drain bias voltage VSD , and measuring the onset of a current ISD between source and drain that flows through the 2DEG (Fig. 5.1A). Once the 2DEG has been formed, each of the two barrier gates can pinch off the 2DEG underneath the gate by lowering its potential well below the turn-on voltage. This can be measured by starting with all three gates above the turn-on voltage, and lowering one of the barrier gate potentials until ISD is suppressed. This indicates that the 2DEG is depleted underneath the barrier gate. The highest barrier gate potential at which this happens is referred to as the pinch-off voltage, and for the measured device it is found to be nearly identical for both gates (Fig. 5.1B). This suggests that the fabricated barrier gates have a fairly symmetric shape and no significant disorder. When both barrier gate potentials are close to their pinch-off voltage, a 2DEG region between the barriers is separated from that of the source and drain leads. This separated region serves as the SET island. Varying either of the two barrier gate potentials around this tuning point results in oscillations of ISD (Fig. 5.1C). These are Coulomb oscillations, which arise from the SET alternating between being in and out of Coulomb blockade (Sect. 4.2.2). Observation of these oscillations indicates the successful formation of the SET.

5.1 Charge Sensing with an SET

61

Fig. 5.1 Formation of the SET. A source-drain bias VSD = 150 µV is applied during all measurements. (A) Forming a 2DEG between the n+ -doped source and drain leads. Increasing the top gate (TG) and barrier gate potentials (LB, RB) simultaneously above the turn-on voltage forms a 2DEG underneath. This creates a conductive path between source and drain, resulting in a current ISD . The turn-on voltage is slightly above 1 V. (B) Pinching off the 2DEG with the barrier gates. Decreasing either of the barrier gate potentials while keeping the top gate and other barrier gate above turn-on (1.1 V) pinches off the 2DEG, restricting current flow. Both barrier gates have a similar pinch-off voltage of ∼0.55 V . (C) ISD for varying barrier gate potentials while keeping the top gate above turn-on (1.1 V). A line-cut of ISD at equal barrier gate potentials (green) reveals Coulomb oscillations (D). Here an SET island is separated from the source and drain leads. These Coulomb oscillations indicate that the SET is tuned up and can function as a charge detector. Note the varying slopes of the Coulomb peaks in panel C, which could indicate that the SET island shape varies with the barrier gate potentials, or possibly multi-dot physics

5.1.2 Charge Stability Diagram Once the SET has been tuned up, it can be used as a charge detector. Potential donors can be detected by measuring ISD while varying one gate that is strongly coupled to the SET, such as the top gate (TG), versus one or more gates that are close to the donors, resulting in a charge stability diagram (Fig. 5.2). Each Coulomb peak shifts diagonally and corresponds to an equipotential of the SET island. Here, the increase in island electrochemical potential due to one gate is counteracted by a decrease due to the other gate. The slope thus corresponds to the ratio of gate lever arms to the SET island (Sect. 4.2.1). Charge stability diagrams can show sudden upward Coulomb peak shifts as the donor gate potentials are increased (Fig. 5.2). These charge transitions indicate that an electron has been attracted to somewhere close to the SET island, possibly a

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5 123-Sb Donor Device Characterization

Fig. 5.2 Charge stability diagram. The Coulomb peaks shift diagonally, at a slope equal to the ratio of gate lever arms (Sect. 4.2.1). As the donor gate voltages are increased, charge transitions occur that shift the Coulomb peak (first transitions indicated by blue arrows). Each transition corresponds to a movement of a single charge near the SET island, and could be due to a donor transitioning from the ionized (D+ ) to neutral (D0 ) charge state upon increase of the donor gate voltage. Note that these charge transitions have a stronger coupling to the donor gates than to the top gate, as is observed by the nearly vertical slopes of the charge transitions. This indicates that the charges lie closer to the donor gates, as is expected for the implanted donors

5.1 Charge Sensing with an SET

63

hitherto ionized donor atom (or alternatively an empty charge trap). This system can be described by considering the donor as a second quantum dot that is capacitively coupled to the SET island (see [1] for a detailed analysis of the relevant electrostatics). An electron tunneling onto a donor (or nearby charge trap) increases the SET island’s electrochemical potential μ(N ) by μ(N ) = E c

Cm , Cd

(5.1)

where E c is the charging energy, Cm is the mutual capacitance between the donor and SET island, and Cd is the total capacitance of the donor, which includes Cm . A higher gate potential is needed to compensate this increase in μ(N ), resulting in an upward Coulomb peak shift in the charge stability diagram. The ratio Cm /Cd ≤ 1 can be determined from the charge stability diagram, as it matches the ratio Vt /VC , where Vt is the Coulomb peak shift at the charge transition, and VC is the potential difference between successive Coulomb peaks1 (Fig. 5.3A). This ratio provides information about how close the electron has moved to the SET island, since the maximum Cm /Cd = 1 corresponds to an electron moving from infinitely far to being on the SET island. The SET functions as a charge detector by utilizing this Coulomb peak shift at a charge transition. By tuning to the Coulomb peak on one side of the charge transition2 a change in the donor charge state shifts the Coulomb peak, and hence reduces ISD (Fig. 5.3C). This is easiest with a clearly measurable Coulomb peak shift, and hence a significant fraction Cm /Cd .Most devices have a few donors that satisfy these requirements. The donor and SET island usually have a nonzero tunnel coupling due to their close proximity (Fig. 5.3B). Therefore, aligning the electrochemical potential of the donor to that of the SET island results in electron tunneling events between the two, which can be subsequently detected by the SET (Figs. 5.3C, D). This does require that the tunnel coupling between donor and SET island is significantly lower than that between the SET island and leads, and also lower than the measurement rate of the electronics. In our experiment, the measurement rate is limited by a 50 kHz low-pass filter, which we use to improve the signal-to-noise ratio of the measured SET current ISD . (Sect. 4.6). Conversely, the tunnel coupling should also not be too low, as this slows down the measurements. Donors that have a measurable Coulomb peak shift also often have a tunnel coupling that satisfies both these criteria.

The ratio Vt /VC is independent of the varied gate, since the gate lever arm cancels out. For a ) Vt = αk αμ(N = CCmd (Eq. (5.1)). gate k with lever arm ak , the ratio equals V C k Ec

1

2

The system is usually tuned to the Coulomb peak when the donor is ionized, while being in Coulomb blockade when the donor is neutral. This configuration is chosen because the neutral donor is far more sensitive to charge noise, which might be enhanced by a nonzero SET current ISD .

64

5 123-Sb Donor Device Characterization

Fig. 5.3 Donor charge transition. (A) The SET island is tunnel coupled to nearby 123 Sb donors. This allows an electron to tunnel between a donor and SET island, resulting in a neutral (D0 ) or ionized (D+ ) donor. (B) A charge stability diagram exhibits a Coulomb-peak shift at a donor charge transition (dashed line). At a charge transition, the ratio between the Coulomb peak shift Vt and the spacing between Coulomb peaks VC equals the ratio between the donor-SET mutual capacitance Cm and donor capacitance Cd (Eq. (5.1)). (C) Coulomb oscillations for the ionized (D+ ) and neutral (D0 ) donor. The Coulomb peak shift due to a neutral donor enables detection of the donor charge state. (D) Charge discrimination in an ISD current trace. When the electrochemical potentials of the donor and SET island are aligned, random electron tunneling events occur between the two. By aligning the potential to lie on a Coulomb peak when the donor is ionized (blue dashed line in C), a high (low) SET current ISD corresponds to an ionized (neutral) donor

5.2 Donor Triangulation

65

Fig. 5.4 Donor gate lever arms. In all panels, the plunger gate (PL) voltage is varied on the vertical axis, while the donor gate voltage of DFL (A), DFR (B), DBL (C), or DBR (D) is varied on the horizontal axis (see Fig. 4.6 for gate labeling). The charge stability diagrams all show the charge transition of the measured 123 Sb donor (white dashed line). Each transition slope corresponds to the ratio of the lever arms: one lever arm is between the SET plunger gate (PL) and donor, and the other between one of the donor gates and the donor. Note that additional lines of discontinuous Coulomb peaks are visible in (A) and (B). These are caused by additional donors that are sufficiently far in gate space from the donor of interest, and do not bear relevance to the work presented here

5.2 Donor Triangulation Similar to the SET island (Eq. (4.8)), the donor’s electrochemical potential has a different capacitive coupling to each of the gates. This allows triangulation of the donor location by comparing the experimental ratios of different gate capacitances with electrostatic COMSOL simulations (Sect. 7.2). Figure 5.4 shows the charge stability diagram around the donor charge transition, varying a different donor gate in each panel. The electrostatic potential V ( r0 ; VPL , VDFL , . . .) at the donor position r is constant along the donor charge transition. Consequently, the gate potentials in Fig. 5.4A along the charge transitions must satisfy

66

5 123-Sb Donor Device Characterization

d V ( r0 ; VPL , VDFL , . . .) d V ( r0 ; VPL , VDFL , . . .) VPL + VDFL = 0, d VPL d VDFL

(5.2)

where VPL and VDFL are the shifts along VPL and VDFL , respectively. This can be rearranged as VPL d V ( r0 ; VPL , VDFL , . . .) − = VDFL d VDFL



d V ( r0 ; VPL , VDFL , . . .) , d VPL

(5.3)

where the left-hand side is the experimentally measured slope sDBL = −VPL /VDBL = tan(θ ),

(5.4)

which is set by the ratio in gate capacitances to the donor charge. The same approach can be applied to the other donor gates, resulting in four slopes. The measured slopes are assumed to have an equal angle uncertainty σθ . Gaussian error propagation then leads to an estimated standard deviation of the slope σDFL given by 2 + 1)σθ . σDFL = (sDFL

(5.5)

The donor is triangulated by finding the position r at which an electrostatic simulation of the right-hand side in Eq. (5.3) best matches the measured slopes. The electrostatic potential landscape across the model has been calculated in COMSOL using the experimental gate voltages (Sect. 7.2 and Appendix A). Subsequently, the relevant gate voltages are varied by 10 mV, and the electrostatic potential landscape across the model is computed again for variation of each gate voltage. This results in a spatially-varying simulated slope, which, at a given position r, is defined as sim ( r) = sDFL

r ; VPL , VDFL , . . .) V ( r ; VPL , VDFL + 10mV, . . .) − V ( . V ( r ; VPL + 10mV, VDFL , . . .) − V ( r ; VPL , VDFL , . . .)

(5.6)

Here again the DFL gate is used as an example to illustrate the procedure, and the plunger gate is used as the common gate. To compare the simulated slopes to the measured ones at each point within the model space, a least-squares estimate [2] is used, defined by ⎡

1   Ptriangulation ( r ) = N · exp ⎣− g∈ DFL,DFR, 2 DBL,DBR



r ) − sg sgsim ( σg

2 ⎤ ⎦,

(5.7)

with Ptriangulation ( r ) the probability density, which is normalized over the given volume via N . Two approaches have been compared to incorporate the 2DEG of the SET island and leads into the finite-element model. The first method approximates the 2DEG as a metallic layer that is fixed at ground potential. The thickness and exact geometry

5.2 Donor Triangulation

67

of the 2DEG is found to have a negligible influence on Ptriangulation ( r ). The second approach uses the Thomas-Fermi approximation to calculate the charge density of the two-dimensional electron gas. In this approximation, the temperature is assumed to be at 0 K, and the electrons are non-interacting. Both methods result in a nearly r ). For this reason, the less computationally-expensive, metallic identical Ptriangulation ( approximation is used to triangulate the donor. The resulting triangulation probability density function is shown in Figs. 5.5A, C. The planar gate layout leads to a low sensitivity of the capacitance triangulation method in the out-of-plane y direction (i.e. the donor depth), as reflected by a large uncertainty along this axis (Fig. 5.5C). However, a significant section of this region constitutes highly unlikely positions of the donor, as revealed by the donor implantation depth profile (Sect. 4.4). A more accurate probability distribution of the donor’s position is therefore given by r )Pimplantation ( r ), P( r ) = N Ptriangulation (

(5.8)

r ) the estimated dopant probability distribution (Sect. 4.4) and N a with Pimplantation ( normalization factor. This distribution is cut off at the 28 Si/SiO2 interface, and donor diffusion due to the rapid thermal anneal of 5 s at 1000◦ has a negligible effect. Further methods to narrow P( r ), e.g. by modeling the donor-SET tunnel coupling [3], were not adopted here. The combined probability density function provides a best estimate for the donor position [maximum of P( r )] at a lateral position between the DFL and DFR gates (Fig. 5.5B), and a depth of −5 nm below the 28 Si/SiO2 interface (Fig. 5.5D).

5.3 Electron Spin Control and Readout 5.3.1 Donor Electrochemical Potential Regimes A static magnetic field B0 causes an energy splitting between the donor electron spin states due to the Zeeman interaction (Sect. 3.2.2). This section describes how this energy splitting can be utilized to read out the donor electron spin, and also to deterministically initialize a |↓ electron onto the donor. Due to the capacitive coupling, both the donor and SET electrochemical potentials depend on the charge occupancy of the other. We define μ↑ (μ↓ ) as the electrochemical potential of a |↑ (|↓ ) electron on a neutral donor (D0 ) when the SET island contains N − 1 electrons. Similarly, μ(N ) is the N -electron SET island electrochemical potential for an ionized donor (D+ ). We assume a positive B0 aligned along the zˆ -axis, in which case μ↓ < μ↑ . We further assume that μD > μ(N ) > μS , where μD and μS are the drain and source potentials, respectively. Three regimes can be distinguished based on how μ(N ) is positioned with respect to μ↑ and μ↓ . When μ(N ) < μ↓ < μ↑ , the lowest-energy configuration is an ionized

68

5 123-Sb Donor Device Characterization

Fig. 5.5 Position triangulation of the 123 Sb donor. (A) (SEM) image near donor-implanted region. The lateral top view region (B, C) is indicated in red, and the transverse cross-section (D, E) in blue. (B, D) Probability density of donor location, estimated by comparing simulated gate-to-donor coupling strengths (Sect. 7.2) with the experimentally observed strengths (Fig. 5.4). A lateral topview (B) and a transverse cross-section (D) are shown with a cross indicating the most likely position. (C, E) Probability density including the implantation profile. To improve on the low resolution of the triangulation method in the y direction, the triangulation probability density function is multiplied with the donor implantation probability density function (Sect. 4.4). Incorporating the donor implantation parameters significantly confines the likely depth range of the donor, as shown in the transverse cross-section (E). This depth confinement also leads to a lateral confinement of the donor, as can be seen from the lateral topview (C). The most likely donor position (indicated by a cross) is at a lateral location (x, z) = (13 nm, 8 nm) at a depth of y = −5 nm. Probability density functions are normalized over the model volume and are integrated over the out-of-plane axis in both panels, specifically P(x, z) = P( r )dy and P(y, z) = P( r )d x. The contour lines mark the 68% and 95% confidence regions

5.3 Electron Spin Control and Readout

69

donor (D+ ). The loosely-bound electron of a neutral donor will therefore tunnel to the SET island regardless of the electron spin (Fig. 5.6A). This is the operation regime when manipulating the nuclear spin of an ionized donor (Chap. 6). Aligning μ(N ) between the two donor potentials (μ↓ < μ(N ) < μ↑ ) results in spin-dependent tunneling [5] (Fig. 5.6B). The most energetically-favorable configuration is when a |↓ electron resides on the donor, and a |↓ electron will therefore likely remain on the donor. On the other hand, it is energetically favorable for a |↑ electron to tunnel onto the SET island, ionizing the donor. The energy can then further be reduced when a |↓ electron tunnels back onto the donor. This brief ionization can be detected by the SET as a brief increase in current, henceforth referred to as a current blip. The observation of a current blip therefore indicates a |↑ electron on the donor, while the lack thereof indicates a |↓ electron. This spin-to-charge conversion is used to read out the donor electron spin. Additionally, in both cases the final configuration is a neutral donor with a |↓ electron. Aligning μ(N ) between μ↓ and μ↑ therefore also serves as a method to initialize a |↓ electron. The final regime is μ↓ < μ↑ < μ(N ), i.e. when the donor electrochemical potential is below that of the SET island (Fig. 5.6). In this case, the loosely-bound electron cannot escape the neutral donor (D0 ), regardless of its spin state. The donor electrochemical potential is therefore plunged into this regime when performing ESR (Sect. 5.4). If the donor is ionized prior to plunging, an electron is loaded with a random spin.3 The donor electrochemical potential can be modified without affecting μ(N ) by also changing the potential of a second gate. The plunger gate (PL) is usually used to compensate any shift in μ(N ) caused by a varying donor gate potential, as the plunger gate is more strongly coupled to the SET island than to the donor atoms. To ensure that a high (low) SET current ISD corresponds to an ionized (neutral) donor, a compensated plunger gate potential is applied to remain on the Coulomb peak when the donor is ionized (Fig. 5.6D). Spin-dependent tunneling can be observed by cycling through these three regimes (Fig. 5.6E). Starting from an ionized donor, the donor electrochemical potential is plunged below that of the SET island (μ↓ < μ↑ < μ(N )), thus loading an electron with a random spin. Next, the system is tuned into the readout position (μ↓ < μ(N ) < μ↑ ). A current blip is observed if the electron spin is |↑ , and is thus observed roughly half the time. Importantly, these current blips have the highest probability of occurring early on. Consequently, averaging over many traces reveals a higher average SET current ISD at the start, due to the congregation of current blips. Finally, the donor is ionized by increasing its electrochemical potential (μ(N ) < μ↓ < μ↑ ), preparing the donor for the next iteration of the pulse sequence.

3 When μ < μ < μ(N ), the probability of loading a |↑ electron versus a |↓ electron is roughly ↓ ↑ equal, though the precise ratio depends on the details of the spin filling of the SET island at the μ↑ and μ↓ electrochemical potentials.

70

5 123-Sb Donor Device Characterization

Fig. 5.6 Spin-dependent tunneling. The N -electron SET electrochemical potential μ(N ) is tuned between the source and drain potential when the donor is ionized (D+ ). A static magnetic field B0 separates the donor electrochemical potentials μ↑ and μ↓ by the Zeeman splitting. (A) A donor electron will prefer to tunnel onto the SET when μ(N ) < μ↓ < μ↑ , thereby ionizing the donor. (B) Spin-dependent tunneling is observed when μ↓ < μ(N ) < μ↑ . A |↓ donor electron is in the lowest-energy configuration, while a |↑ donor electron reduces its energy by tunneling onto the SET, followed by a |↓ electron tunneling back onto the donor. This brief ionization is measured as an SET current blip, enabling electron spin readout. (C) The donor neutralizes when μ↓ < μ↑ < μ(N ), loading an electron with random spin if the donor is initially ionized. (D) SET charge stability diagram near the donor charge transition (white dashed line). The three tuning regimes are defined for compensated gate voltages (red dashed line) that follow the Coulomb peak of an ionized donor : neutralize (purple), read (green) varied around the charge transition by Vread , and ionize (blue). (E) Averaged current trace while cycling through neutralize → read → ionize for varying Vread . When Vread is tuned to the readout regime, current blips are observed whenever the electron is initially |↑ . This is visible in the averaged traces as an increased current near the start of the read stage (indicated by B). This spin-dependent tunneling potential window equals the Zeeman splitting scaled by the donor gate lever arm [4]. The width of the transition between A and B is set by the electron temperature

5.3 Electron Spin Control and Readout

71

5.3.2 Electron Readout and Initialization Fidelity Electron tunneling between the donor and SET island is a stochastic process with rates governed by Fermi’s golden rule. At the readout tuning (μ↓ < μ(N ) < μ↑ ), the tunneling time τ|↑ →SET of a |↑ electron from the donor to the SET island depends on the SET island’s density of unoccupied |↑ states at potential μ↑ . The probability of a |↑ electron tunneling off the donor within a given readout time tread is

P↑ (tread ) = 1 − exp −

tread τ|↑ →SET

 .

(5.9)

The donor electron spin readout fidelity is adversely affected by thermal broadening of the SET island’s Fermi distribution. Since the density of unoccupied |↓ states at μ↓ is nonzero, the tunneling time τ|↓ →SET of a |↓ donor electron to the SET island is finite, though generally much longer than τ|↑ →SET . Such a tunneling event of a |↓ electron is measurable as a current blip which would be wrongly attributed to tunneling of a |↑ electron, resulting in a readout error. Readout errors also occur when the donor electrochemical potential is kept at the read position, and the tunneling time τSET→|↓ of a |↓ electron from the SET island to the donor is so low that it approaches the limit set by the low-pass filters in the measurement setup (Sect. 4.6). In this case, a current blip that indicates a tunneling event can be so brief that it is not detected. The readout fidelity is dependent on the readout time tread , which is the interval during which the source-drain current ISD is measured to detect current blips. The optimal tread lies between τ|↑ →SET and τ|↓ →SET . If tread is not significantly longer than τ|↑ →SET , a |↑ electron on the donor does not always tunnel out. Conversely, if tread is not much lower than τ|↓ →SET , a |↓ electron has a significant probability of tunneling out. Both cases result in a readout error. The ability to discriminate between a |↑ and |↓ donor electron is quantified by the readout visibility. Neglecting additional readout errors due to current blips that are too brief, the readout visibility is the probability P↑ (tread ) of a |↑ electron tunneling out of the donor, minus the probability P↓ (tread ) of a |↓ electron tunneling out during a readout time tread , and is given by visibility(tread ) = P↑ (tread ) − P↓ (tread ), 



tread tread + exp − , = − exp − τ↑ τ↓

(5.10)

where we have used the shorthand notation τ↑ := τ|↑ →SET and τ↓ := τ|↓ →SET . Note that visibility(tread ) is zero at tread = 0 and at tread → ∞, and the shape of the curve depends on the ratio between τ↑ and τ↓ (Fig. 5.7A). The optimal readout duration that maximizes the visibility is

72

5 123-Sb Donor Device Characterization

Fig. 5.7 Readout visibility. (A) Readout visibility versus readout time tread . The visibility (Eq. (5.10)) is set by the ratio between τ↓ and τ↑ , which are the tunneling times of a |↓ and a |↑ electron, respectively, from the donor to the SET island. Each ratio has an optimal readout time max (dots) that maximizes the visibility. (B) Maximum readout visibility for varying tunneling tread times. The maximum readout visibility is shown (Eq. (5.12), blue), along with an approximated curve (Eq. (5.13), red) that is valid when τ↑ τ↓ . Additional readout errors caused by current blips that are too brief are not considered here

max tread =

 τ↑ τ↓ τ↑ , log τ↑ − τ↓ τ↓

(5.11)

with a corresponding maximal readout visibility

max visibility(tread )

=

≈

τ↑ τ↓ τ↑ τ↓

− τ τ−τ↑ ↑

 ττ↑ ↓

↓

−

− τ↑ , τ↓

τ↑ τ↓

− τ τ−τ↓ ↑

↓

,

(5.12) (5.13)

where the approximation only deviates appreciably at low τ↓ /τ↑  1. The maximum readout is shown for varying τ↓ /τ↑ (Fig. 5.7B). A readout visibility of 0.9 requires a tunneling time ratio of τ↓ /τ↑ ≈ 50. The SET island’s electron temperature also causes a nonzero density of occupied |↑ states at electrochemical potential μ↑ . A |↑ electron can thus tunnel from the SET island to an ionized donor with tunneling time τSET→|↑ . While this does not affect the readout fidelity, it can wrongly initialize a |↑ electron, causing an initialization error. Both the readout and initialization fidelities worsen with increasing electron temperature, and improve with higher Zeeman splitting. Consequently, the errors can be reduced by increasing the magnetic field B0 .

5.4 ESR Spectrum

73

5.4 ESR Spectrum The 123 Sb donor has a nuclear spin I = 7/2 and thus has eight nuclear spin states (Sect. 2.3). The neutral donor ESR frequency depends on the nuclear spin due to the hyperfine coupling (Sect. 3.2.3), and therefore has eight possible ESR frequencies that are separated by the hyperfine interaction strength. The eight ESR frequencies can be detected by trying to excite an initialized |↓ with an ESR pulse and measuring the resulting electron spin. Instead of using a singlefrequency ESR pulse, a linear chirp pulse is applied to adiabatically flip the electron spin via ESR [6, 7]. This has the advantage that the electron spin can be flipped whenever the ESR frequency lies within the swept frequency range. A wide range of frequencies can therefore be covered in a single measurement, significantly speeding up ESR spectrum scans at the cost of less precision in measured ESR frequency. An additional advantage of adiabatic pulses is that they give a high inversion fidelity without the need for a well-calibrated pulse amplitude, provided that the chirp-pulse frequency ramp rate (chirpyness) is chosen to be below the adiabatic limit set by the Landau-Zener equation [7], and above the limit set by the homogeneous linewidth. The fidelity of a chirp pulse is also insensitive to slow resonance frequency drifts. The pulse sequence to measure the ESR frequencies consists of four steps: 1. initialize a |↓ electron by remaining at the readout level for a duration exceeding τ|↑ →SET and τSET→|↓ ; 2. plunge the donor electrochemical potential below the Fermi level of the SET island to ensure the donor electron cannot tunnel away from the donor; 3. apply an adiabatic ESR pulse with a specific frequency range, which can flip the electron to |↑ when crossing the ESR frequency; 4. return to the readout level to measure the resulting electron state. Without nuclear spin initialization (Sect. 5.5.2), the ESR frequency can have any of its eight possible values. However, due to occasional nuclear flips, the eight ESR frequencies are found by repeated measurements of the ESR spectrum (Fig. 5.8). The chirp pulses initially have a wide frequency window to cover the large range of ESR frequencies. Overlaying multiple spectra reveals eight ESR frequencies, corresponding to the eight nuclear spin states. The eight-fold splitting of the ESR spectrum indicates that the measured electron indeed belongs to an 123 Sb donor atom. Once all eight ESR frequencies have been found, chirp pulses with a narrow window can be used to accurately measure each of the ESR frequencies (Fig. 5.9). The hyperfine interaction strength, equal to the average spacing between successive ESR frequencies, is measured at A = 96.5 MHz, significantly lower than the bulk value of 101.52 MHz. One possible cause for this deviation is strain, which is known to modify the hyperfine interaction [8, 9]. Replacing the ESR chirp pulse by a single-tone ESR pulse on resonance results in Rabi oscillations at a Rabi frequency of 94(1) kHz at an input power of 15 dBm (Fig. 5.10). However, the ESR frequency exhibited frequent jumps of a few hundred kilohertz, possibly attributable to a second coupled ionized 123 Sb donor

74

5 123-Sb Donor Device Characterization

Fig. 5.8 123 Sb ESR spectra. At each frequency, an initialized |↓ electron is adiabatically inverted to |↑ by a chirp pulse if its ESR frequency is in the 5 MHz chirp frequency window. Superimposing multiple individual ESR spectra (B) reveals eight distinct ESR peaks (A), corresponding to the eight nuclear spin states of 123 Sb. The frequency is swept from high to low, causing the flip-flop transitions to decrease the nuclear spin (Sect. 5.5.2), resulting in the frequent observation of multiple ESR peaks

(Appendix B). This severely limited the ability to perform non-adiabatic ESR pulse sequences. The nucleus regularly jumps between its nuclear spin states, at a rate much higher than observed in 31 P [6]. This could be caused by the quadrupole interaction (Sect. 3.3), whose transverse components can cause a nuclear flip during every donor ionization/neutralization event. The repeated ionization/neutralization that occurs during readout therefore yields a nonzero probability for the nuclear spin to transition to a different spin state. Calculations of the ionized and neutral Hamiltonians reveal that a quadrupole interaction of 50 kHz oriented perpendicular to a static magnetic field B0 = 1.4T increases the nuclear flip probability by roughly an order of magnitude compared to solely isotropic-hyperfine-induced flipping. These chosen parameters likely match the quadrupole parameters of the measured 123 Sb donor (see Sect. 3.3.4 for details). This partly explains why a single scan over the ESR spectrum often exhibits multiple resonance peaks. The nuclear spin is observed to have a strong preference to increase its spin projection along the z-axis, and is therefore most often found in the |7/2 state. It is unlikely that this is caused by a nuclear T1 relaxation process: the true nuclear spin-lattice relaxation time is unmeasurably long, and the device temperature is rather high compared to the nuclear spin energy splitting. Instead, Tx cross-relaxation is the most likely cause for the preference in nuclear spin orientation. Here a |↑

5.4 ESR Spectrum

75

Fig. 5.9 123 Sb ESR spectra for different nuclear spin states. Measurements were performed at a magnetic field B0 = 1.496 T. (A) Schematic of ESR frequencies for eight nuclear spin states. Successive ESR transition frequencies are separated by the hyperfine interaction A. (B) Measured ESR frequencies for eight nuclear states. The hyperfine interaction strength A = 96.5 MHz is estimated as the average difference between successive ESR transition frequencies, as shown by the fitted black line. (C) ESR spectral lines. For each nuclear state, the nucleus is initialized at the start of each microwave sweep, and adiabatic ESR pulses with 1 MHz frequency deviation are applied to excite the electron

76

5 123-Sb Donor Device Characterization

Fig. 5.10 ESR Rabi oscillations. While the nuclear spin state is in |7/2 , a resonant ESR pulse is applied for varying duration. A sinusoidal fit (black) through the measured Rabi oscillations gives Rabi,ESR an estimated Rabi frequency of f 7/2 = 94(1) kHz at 15 dBm input power

electron can transfer its spin to the nucleus via the flip-flop transition |↑, m I → |↓, m I + 1 , reducing the combined energy by emission of a phonon into the silicon lattice. Cross-relaxation requires an initial |↑ electron, and is therefore only relevant if Tx approaches the electron T1 relaxation time; otherwise a |↑ electron is much likelier to relax to a |↓ electron via T1 electron-spin relaxation than via Tx flip-flop relaxation. For 31 P, Tx cross-relaxation is much longer than electron T1 relaxation [10] and is therefore irrelevant. However, the effect is much stronger for donors with I > 1/2 [11] and is even reported to dominate over electron T1 relaxation [10, 12].

5.5 Flip-Flop Transition 5.5.1 Flip-Flop Rabi Oscillations We have observed that the microwave antenna can drive electron-nuclear flip-flop transitions, which correspond to a simultaneous flipping of both the electron and nuclear spin |↑, m I − 1 ↔ |↓, m I , thereby preserving the total spin. Flip-flop states are coupled by the transverse hyperfine interaction terms A Sˆ x ⊗ Iˆx + A Sˆ y ⊗ Iˆy , and can therefore be induced by electrically modulating the hyperfine strength A at the FF = f mESR − A/2 + γn B0 . flip-flop transition frequency f ↑m I I −1↔↓m I The likely explanation for the strong flip-flop drive is that the microwave antenna is damaged (Fig. 4.1), resulting in much stronger electric fields that modulate the hyperfine interaction (Sect. 6.3.1). Additionally, the measured hyperfine interaction strength A = 96.5 MHz deviates significantly from the bulk value A = 101.52 MHz. This indicates that the electron wave function is already distorted, and is therefore more sensitive to electric fields. The observation of coherent flip-flop oscillations is an important step for future quantum-computing systems such as the flip-flop qubit [13], which rely on the flip-flop transition. Rabi oscillations have been measured for the flip-flop transition |↑, 5/2 ↔ |↓, 7/2 (Fig. 5.11) By starting with a |↓ electron, and adiabatically inverting the electron if the nucleus is in state |5/2 , the initial state is one of the two flip-flop

5.5 Flip-Flop Transition

77

Fig. 5.11 Rabi oscillations for the flip-flop transition |↑, 5/2 ↔ |↓, 7/2 . The flip-flop transition is driven by electric modulation of the hyperfine interaction. Readout and extraction of the nuclear flip probability is described in Sect. 6.1.1. A decaying sinusoidal fit (black) shows that the Rabi frequency 9.2(2) kHz is very close to the fitted decay rate 8.3(17) kHz. Error bars show the 95% confidence interval

states. The flip-flop pulse is then applied for varying duration. The readout sequence consists of measuring the nuclear state, and then repeating the pulse sequence to extract a flipping probability (Sect. 6.1.1). Several Rabi oscillations are observed at a Rabi frequency 9.2(2) kHz, close to the decay rate 8.3(17) kHz. Increasing the microwave power further increases the decay rate, while only marginally increasing the Rabi oscillation frequency. This suggests that the strong microwave signals add a source of decoherence for the electron. This could either be due to the strong electric fields, or due to local heating of the device. Both the Rabi frequency and the decay rate can likely be improved by using a dedicated electric antenna. When an ESR spectrum is measured from low to high frequency, the nuclear spin is most often found in the |7/2 state, in line with a flip-flop relaxation process. However, when the spectrum starts at the highest frequency and is then decreased, successive peaks are often observed in a single scan, and the nuclear spin no longer has a strong preference for the |7/2 state (Fig. 5.8B). The cause for this discrepancy and f mESR , the microwave is that while sweeping between two ESR frequencies f mESR I I −1 FF frequency crosses the flip-flop transition frequency f ↑m I −1↔↓m I . Since the electron is initialized to |↓ , this transition is a one-way process that can reduce the nuclear spin |↓, m I → |↑, m I − 1 . This explains why a single ESR spectrum taken with decreasing microwave frequency often shows multiple successive resonance peaks: crossing the flip-flop transition has a high probability of reducing the nuclear spin, which is then visible as the next ESR resonance peak.

5.5.2 Flip-Flop Driven Nuclear-Spin Initialization The flip-flop transitions have proven to be highly useful for our experiments, as they can be used for high-fidelity fast nuclear spin initialization (Fig. 5.12). Using high-power microwave chirp pulses (henceforth referred to as a flip-flop pulse), a

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5 123-Sb Donor Device Characterization

Fig. 5.12 Nuclear-spin initialization scheme using flip-flop transitions. (A, B) Pulse sequence to decrease nuclear spin from m I to m I − 1, while leaving all other nuclear spin states unaffected. The evolutions of an initial |m I − 1 (blue) and |m I (red) nuclear spin state are shown with a corresponding energy level diagram (B). Starting with an initialized |↓ electron, the donor gate voltage is increased to keep the donor neutral (D0 ). A flip-flop chirp pulse is applied with center FF frequency f ↑m (dashed). This pulse changes the state |↓, m I → |↑, m I − 1 , effectively I −1↔↓m I decreasing the nuclear spin. If the nucleus starts with spin m I − 1, the flip-flop pulse is off resonant, and a final electron-spin initialization (wavy arrow) also brings the final state to |↓, m I − 1 . C, D Pulse sequence to increase nuclear spin from m I to m I + 1. The pulse sequence is similar to A, with the addition of two ESR pulses before the flip-flop pulse. These ESR pulses invert the electron to |↑ spin is m I or m I + 1. (E) Nuclear-spin initialization scheme, illustrated for target state if the nuclear   target = |−1/2 . When the pulse sequence does not modify the nuclear spin, the corresponding m I arrows are left out. Separate pulse sequences are used to increase (left, A, B) or decrease (right, C, target D) the nuclear spin m I to m I = −1/2. each of the pulse sequencesover the  By iterating   nuclear  target  target , the nuclear spin is pumped to m I without spins below (A, B) and above (C, D) m I needing readout nor feedback

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transition between two flip-flop states could be addressed with a fidelity of ∼ 0.5, being limited by the relatively poor ratio of flip-flop Rabi frequency and coherence time. For an initial |↓ electron, application of a flip-flop pulse followed by a donor ionization results in a 50% chance of decreasing the nuclear spin (m I = −1). A few iterations of this sequence increases this probability to above 90%. The nuclear target by performing this sequence while spin can be pumped down to a target state m I FF for decreasing |m I , cycling through all flip-flop transition frequencies f ↑m I −1↔↓m I target where m I > m I (Fig. 5.12A, B). One cycle through the flip-flop frequencies can already initialize the nuclear spin with high-fidelity, but the fidelity can be improved even further by performing multiple cycles. A similar sequence is used for all nuclear target spin states |m I below the target state m I < m I , with the modification of starting and with an initial |↑ electron by applying two ESR pulses with frequencies f mESR I . In this case, the flip-flop transition will increase the nuclear spin (m = +1), f mESR I I +1 and cycling through the flip-flop transition frequencies effectively pumps all lower   target (Fig. 5.12C, D). nuclear states up to m I One of the main advantages of this initialization scheme is that it does not require knowledge of the initial nuclear state, and instead cycles through the different flip-flop frequencies to pump the nuclear spin into a target state. This is an entirely open-loop control sequence to initialize the nuclear spin. The use of chirp pulses removes the requirement of an accurately-tuned flip-flop frequency, thereby negating the need to periodically recalibrate the flip-flop pulses. An additional benefit is that the flip-flop transition frequencies can be accurately estimated from the ESR frequencies alone, and thus do not require knowledge of nuclear transition frequencies.

5.6 Continuous Tuning via a Neural-Network The measured device regularly suffered from both slow and sudden voltage drifts that cause the system to go out of tune. These drifts need not be large; a shift of 2 mV on one of the donor gates is sufficient to all but diminish the readout visibility. They are especially detrimental for long measurements, such as those involving the nuclear spin. To still perform these long measurements, routines are needed to retune the system when needed. The two key requirements of such a retuning sequence are efficiency and reliability; an inefficient routine may significantly slow down the measurement, while a routine that does not consistently retune the system correctly may cause more harm than good. The first step is diagnosing when the system is out of tune by periodically measuring the readout visibility, and triggering the retuning sequence when the visibility is too low. The first and most straightforward retuning sequence consisted of measuring the readout visibility (Sect. 5.3.2) for a range of gate voltages close to the present gate voltages, and then tuning to the found optimum voltages. While this method is successful at dealing with small drifts, it is incapable of correcting for larger drifts. The sequence is also rather inefficient, as the readout visibility needs to be measured

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for a two-dimensional range of gate voltages, at a measurement rate of roughly 1 s per pair of voltages. The retuning sequence was significantly improved, both in terms of speed and reliability, by using a two-stage process that utilizes a neural network. The first stage calibrates the Coulomb peak by measuring ISD for varying plunger (PL) gate voltages, and tuning to the maximum. The second stage consists of measuring current traces for compensated gate voltages along the Coulomb peak, crossing the charge transition. Three quantities related to current blips are extracted from each current trace: • the current blips per second, which should be zero well below/above the charge transition; • the average duration per current blip, which equals the total duration of the current trace at voltages well below the charge transition as the donor is always ionized; • the average duration between current blips, which equals the total duration of the current trace at voltages well above the charge transition as the donor is always neutral. These three quantities can be measured quite accurately in ∼100 ms, and provide excellent information for the neural network to predict the optimal tuning voltage. The neural network receives the above three quantities for varying gate voltages as input values, and then predicts the gate voltage with the best readout visibility (Sect. 5.3.2). A double-layered neural network is used with a hyperbolic tangent activation function for both layers. The first layer combines the three quantities to an overall score, and all scores are weighted in the second layer to predict the optimal tuning voltage. To train the neural network, repeated scans have been measured, along with the readout visibility to serve as the verification data. The performance of the neural-network-based retuning sequence is measured by repeatedly correcting for perturbations applied to the donor gate DBR (Fig. 5.13A) and the plunger gate PL (Fig. 5.13B). In all cases, the retuning sequence is able to correctly predict the optimal tuning position to an average accuracy of 0.2 mV (DBR) and 0.9 mV (PL), well within the required tolerance. By including the retuning sequence into existing measurements, the donor gate tuning can be tracked over time, providing insight into the charge stability of a device. Whereas sometimes the donor electrochemical potential remains stable for several days (Fig. 5.14B), other times the donor electrochemical potential experiences sudden shifts of several millivolts (Figs. 5.14A, C). The neural-network retuning sequence is a fast and accurate approach to continuously keep the system tuned, enabling continuous measurements that run over the course of multiple days without the system going out of tune. One limitation of the current approach is that it is targeted to a charge transition with specific tunneling times. It would therefore be less effective when applied to another charge transition, and the neural network could need to be retrained. The scheme could therefore be improved by generalizing the neural network to work independent of the charge transition. This could be achieved by collecting training data for different charge transitions, and train the neural network on all the data sets simultaneously.

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Fig. 5.13 Testing the performance of the retuning sequence. An automated retuning process calibrates to the Coulomb peak, and then measures current blips while crossing the donor charge transition. These current-blip statistics are then processed by a pre-trained neural network to predict the optimal tuning position that maximizes the readout visibility. To test the performance of the retuning sequence, the system is first manually tuned to the optimal readout position. Next, every measurement iteration, random perturbations VDBR , VPL are applied (red filled) to the donor gate DBR (A) and plunger gate PL (B), ranging between −10 and 10 mV. The retuning sequence then predicts the perturbations and corrects for them (blue unfilled), with a near-unity success rate, and an average accuracy of 0.2 mV (DBR) and 0.9 mV (PL).

Fig. 5.14 Continuous tuning of donor charge transition. Periodically tuning the device during long measurements using an automated retuning sequence allows tracking of any voltage drifts occurring in the device. The measurement progression is visualized with a color gradient between blue (measurement start) and green (measurement end). Whereas the device sometimes remains stable for several days (B measured for 3 d), other measurements exhibit jumps and drifts on the order of hours (A 3 h, C 18 h)

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References 1. Van der Wiel WG, De Franceschi S, Elzerman JM, Fujisawa T, Tarucha S, Kouwenhoven LP (2002) Electron transport through double quantum dots. Rev Mod Phys 75(1):1. https://doi. org/10.1103/RevModPhys.75.1 2. Sivia D, Skilling J (2006) Data analysis: a Bayesian tutorial. OUP Oxford 3. Mohiyaddin FA, Rahman R, Kalra R, Klimeck G, Hollenberg LC, Pla JJ, Dzurak AS, Morello A (2013) Noninvasive spatial metrology of single-atom devices. Nano Lett 13(5):1903–1909. https://doi.org/10.1021/nl303863s 4. Morello A, Pla JJ, Zwanenburg FA, Chan KW, Tan KY, Huebl H, Mottonen M, Nugroho CD, Yang C, Van Donkelaar JA, Alves ADC, Jamieson DN, Escott CC, Hollenberg LCL, Clark RG, Dzurak AS (2010) Single-shot readout of an electron spin in silicon. Nature 467(7316):687– 691. issn: 0028-0836. https://doi.org/10.1038/nature09392 5. Elzerman JM, Hanson R, Willems van Beveren LH, Witkamp B, Vandersypen LMK, Kouwenhoven LP (2004) Single-shot read-out of an individual electron spin in a quantum dot. Nature 430(6998):431. https://doi.org/10.1038/nature02693 6. Pla JJ, Tan KY, Dehollain JP, Lim WH, Morton JJ, Zwanenburg FA, Jamieson DN, Dzurak AS, Morello A (2013) High-fidelity readout and control of a nuclear spin qubit in silicon. Nature 496(445):334–338. https://doi.org/10.1038/nature12011 7. Laucht A, Kalra R, Muhonen JT, Dehollain JP, Mohiyaddin FA, Hudson F, McCallum JC, Jamieson DN, Dzurak AS, Morello A (2014) High-fidelity adiabatic inversion of a 31p electron spin qubit in natural silicon. Appl Phys Lett 104(9):092 115. https://doi.org/10.1063/1.4867905 8. Wilson D, Feher G (1961) Electron spin resonance experiments on donors in silicon. III: investigation of excited states by the application of uniaxial stress and their importance in relaxation processes. Phys Rev 124(4):1068. https://doi.org/10.1103/PhysRev 9. Mansir J, Conti P, Zeng Z, Pla JJ, Bertet P, Swift MW, Van de Walle CG, Thewalt ML, Sklenard B, Niquet YM, Morton JJL (2018) Linear hyperfine tuning of donor spins in silicon using hydrostatic strain. Phys Rev Lett 120(16):167 701. https://doi.org/10.1103/PhysRevLett.120. 167701 10. Feher G (1959) Electron spin resonance experiments on donors in silicon. i. electronic structure of donors by the electron nuclear double resonance technique. Phys Rev 114(5):1219. https:// doi.org/10.1103/PhysRev.114.1219 11. Pines D, Bardeen J, Slichter CP (1957) Nuclear polarization and impurity-state spin relaxation processes in silicon. Phys Rev 106(3):489. https://doi.org/10.1103/PhysRev.106.489 12. Culvahouse J, Pipkin F (1958) Experimental study of spin-lattice relaxation times in arsenicdoped silicon. Phys Rev 109(2):319. https://doi.org/10.1103/PhysRev.109.319 13. Tosi G, Mohiyaddin FA, Schmitt V, Tenberg S, Rahman R, Klimeck G, Morello A (2017) Silicon quantum processor with robust long-distance qubit couplings. Nat Commun 8:450. https://doi.org/10.1038/s41467-017-00378-x

Chapter 6

Nuclear Electric Resonance

After having found an 123 Sb donor and measuring all its eight ESR peaks (Sect. 5.4), the system stood poised for measurements on the nuclear spin. However, we were quickly thrown a curve ball, for the observations were in clear disagreement with NMR. A melted microwave antenna was found to be the cause, blocking any magnetic fields that would result in NMR. Instead, a different driving mechanism altogether is responsible for the observed nuclear transitions: the electric modulation of the quadrupole interaction, resulting in nuclear electric resonance (NER). We measure the full nuclear spectrum of an ionized 123 Sb donor using NER, including m I = ±2 transitions. The coherence times of all transitions are measured, and are found to be possibly affected by magnetic noise, as well as electric noise that couples through the quadrupole interaction. Spectral lineshifts are also observed in response to a DC bias potential, stemming from the same electric modulation of the quadrupole interaction. These results demonstrate that NER is a successful approach for coherent electrical control of a high-dimensional nuclear spin. This chapter includes results from the following publication: S. Asaad∗ , V. Mourik∗ , B. Joecker, M. A. I. Johnson, A. D. Baczewski, H. R. Firgau, M. T. Ma˛dzik, V. Schmitt, J. J. Pla, F. E. Hudson, K. M. Itoh, J. C. McCallum, A. S. Dzurak, A. Laucht, A. Morello. “Coherent electrical control of a single high-spin nucleus in silicon”. Nature 579.7798, pp. 205–209 (2020). The author acknowledges (i) M. T. Ma˛dzik for fabrication of the 123 Sb device, (ii) M. A. I. Johnson for aiding in the measurements, and (iii) V. Mourik and A. Laucht for support in analyzing measurement data.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Asaad, Electrical Control and Quantum Chaos with a High-Spin Nucleus in Silicon, Springer Theses, https://doi.org/10.1007/978-3-030-83473-9_6

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Fig. 6.1 Nuclear resonance pulse sequence. Donor gate voltages control the donor electrochemical potential, and thereby whether the donor is ionized (D+ ), neutral (D0 ), or at the electron readout and initialization tuning. An initial RF pulse, in this example with nuclear transition frequency f 5/2↔7/2 , is applied to drive the |5/2 ↔ |7/2 transition of the D+ donor. The two nuclear states are then measured sequentially by initializing a |↓ electron, and then applying an ESR chirp pulse ESR ( f ESR ), which adiabatically flips the electron to |↑ if the nucleus is in with center frequency f 5/2 7/2 state |5/2 (|7/2). The nuclear spin state can then be inferred by measuring the electron spin for each of the two nuclear states. The readout sequence is repeated Nrep times to suppress measurement errors and ensure a correct nuclear spin readout

6.1 Initial Nuclear Resonance Measurements 6.1.1 Nuclear Spin Initialization, Manipulation, and Readout The nuclear resonance measurements in this thesis always focus on two chosen states out of the eight nuclear spin eigenstates of 123 Sb. We assume the nucleus to be initialized in one of these two relevant spin states, using the nuclear spin initialization protocol described in Sect. 5.5.2. Next, we perform nuclear spin manipulation, always restricting ourselves to the ionized donor (D+ ).1 In the example in Fig. 6.1, RF pulses are applied to drive Rabi oscillations between two nuclear spin states.

6.1.1.1

Single-Shot Nuclear Spin Readout

After completing the nuclear spin manipulation, its final state is measured by using the electron as an ancilla (Fig. 6.1). Although the nuclear spin has eight possible states, the coherent nuclear spin manipulation induces transitions between only two states at a time. Therefore, we are interested in measuring the probability of the nuclear spin to occupy one or the other of those two states connected by the nuclear

1

In principle, nuclear spin manipulation can also be performed on the neutral donor (D0 ). However, the nuclear coherence times of a neutral donor are much lower, and consequently fairly high Rabi frequencies are needed for coherent control.

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spin transition. For illustration purposes, we describe here the protocol to measure whether the spin is in the |7/2 or |5/2 state. The measurement sequence begins by first loading a |↓ electron onto the donor. Then, the electrochemical potential μ↓ (μ↑ ) of the neutral donor D0 with a |↓ (|↑) electron is tuned below the SET electrochemical potential μ(N ), i.e. μ↑ , μ↓  μ(N ), to ensure that the electron remains bound to the donor (Sect. 5.3). Next, the electron spin is selectively inverted from |↓ to |↑ by applying a chirped ESR pulse (Fig. 6.1) that sweeps the excitation frequency around the ESR resonance corresponding to a specific nuclear spin states. The chirp pulse frequency range is chosen such that the targeted resonance is close to the center frequency of the pulse. Its frequency ramp rate (chirpyness) is chosen to be below the adiabatic limit set by the Landau-Zener equation [1], and above the limit set by the homogeneous linewidth. The main advantage of adiabatic pulses is that they give a high inversion fidelity without the need for a well-calibrated pulse amplitude, and are insensitive to slow resonance frequency drifts. The electron spin state is then read out by spin-dependent tunneling, by tuning the device such that μ↓ < μ(N ) < μ↑ . Measuring an electron tunneling event indicates that the electron was successfully flipped, and hence that the nucleus is in the specific nuclear spin state targeted by the applied ESR chirp pulse. The single-shot nature of the electron spin readout therefore translates into the nuclear spin readout being single-shot as well. Two additional procedures are implemented to improve the single-shot readout fidelity of the nuclear spin. First, to ensure the nuclear spin did not escape the twodimensional subspace of the nuclear spin transition, we perform the same process of electron spin initialization, nuclear-spin dependent flipping, and electron spin readout for the ESR frequency corresponding to the other nuclear state (in this example, the |7/2 state). The combined sequence should result in one positive (i.e. electron spin |↑) and one negative (i.e. electron spin |↓) outcome. Second, since the single-shot electron readout method only has ∼ 90% fidelity, the two ESR adiabatic inversions and electron spin reads are repeated Nrep ≈ 10 times. The number of |↑ counts at each ESR frequency is counted, and the nuclear spin state is assigned with a majority vote. In this way, the resulting nuclear readout fidelity can easily exceed 99.9%. Upon completing this repeated readout for both ESR frequencies, we occasionally find the outcomes of both nuclear readouts to be positive or negative. This could be caused by the nuclear state randomly flipping to one of the other six nuclear states, or simply due to the readout fidelity being less than 100%. In such a case, we discard the data point and perform an initialization sequence to return the nuclear spin to one of the two targeted nuclear spin states (Sect. 5.5.2). We stress that, although multiple electron spin readouts are performed for each measurement, the process as a whole remains a single-shot readout of the nuclear state, since the nuclear spin preparation is unique, and the first single-shot readout already projects the nuclear spin into one of the eigenstates.

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6 Nuclear Electric Resonance

Coherent Control: Measuring Nuclear Spin Transition Probabilities

When performing coherent nuclear spin control experiments, the relevant quantity to be measured is the probability Pflip that the nuclear spin has changed state (“flipped”) as a result of an applied RFpulse. For this purpose, the sequence of initialization, nuclear spin manipulation and single-shot nuclear readout is repeated Niterations times at fixed nuclear transition frequency and pulse duration. The number of nuclear flips Nflips is then counted from the combined outcomes of the individual experiments. Nflips This provides an estimate of the nuclear transition probability as Pflip = Niterations − 1 (Fig. 6.2). We perform repetitive signal averaging to account for effects such as slow drifts.

Fig. 6.2 Extraction of nuclear flip probability Pflip . (A) Electron |↑ fraction of two ESR frequencies for varying electric drive frequency, using the pulse sequence described in Fig. 6.1 and repeated Niterations = 10 times. Each double column corresponds to the indicated electric drive frequency in B (gray dashed line). In each double the left (right) column is the measured  column,  ESR f ESR . Each pixel pair in a row of a double column electron |↑ fraction for ESR frequency f 5/2 7/2 corresponds to an individual iteration of the whole pulse sequence for both nuclear spin states. The color of a pixel is the electron |↑ fraction estimated from Nrep ≈ 10 electron spin readout repetitions. The nuclear spin state is assigned to the state whose pixel has an electron |↑ fraction above (blue) 0.5 (black). If both electron |↑ fractions are low (red), this indicates that the nucleus is in one of the other six nuclear spin states (e.g. at electric drive frequency 8.4457 MHz, from the second iteration onward). In this case, an initialization routine is performed to return the nuclear spin to either |5/2 or |7/2 (Sect. 5.5.2). (B) Nuclear transition probability Pflip . Each data point is extracted from the corresponding double column in A. A nuclear flip is counted whenever the nuclear state switches between the two states in subsequent pulse sequence iterations, visible as a change from red to blue (or vice versa) in successive rows of a double column. The nuclear transition probability Pflip is then estimated by dividing the number of nuclear flips Nflips by the total number of possible flips Niterations − 1. If the nuclear state cannot be determined for each iteration, the data point is discarded (e.g. at electric drive frequency 8.4457 MHz). The data shown here has not been subjected to repetitive signal averaging, which is performed additionally to account for slow drifts. The data points have been fitted (blue line) with Rabi’s formula (Eq. (6.1))

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Fig. 6.3 Nuclear spectrum of the |5/2 ↔ |7/2 transition. The nuclear spin is driven by the antenna using an RF pulse. For this spectrum, the pulse duration (3.378 ms) has already been calibrated from a previous measurement to correspond to a π -pulse when on-resonance. Errorbars are within the data points, and are fitted (blue line) with Rabi’s formula (Eq. (6.1)). The slight discrepancy between the data points and the fit is due to the repetitive signal averaging combined with slow frequency drifts. The measured transition frequency is f 5 /2 ↔ 7/2 = 8.479360(2) MHz at B0 = 1.5 T

6.1.2 First Nuclear Transition and Rabi Oscillations The most straightforward nuclear spin transition to measure is |5/2 ↔ |7/2, as the nuclear spin naturally relaxes to higher spin states (Sect. 5.4). All nuclear frequencies are expected in the vicinity of the Zeeman splitting γn B0 , and are separated by the nuclear quadrupole splitting f Q (Sect. 3.3). The nuclear transition frequency can lie within a wide range of RF frequencies, as both the quadrupole interaction strength and orientation are initially unknown. The |7/2 ↔ |5/2 ionized nuclear transition frequency is found at 8.479360(2) MHz at a static magnetic field B0 = 1.5 T (Fig. 6.3). This lies 152 kHz above the Zeeman splitting γn B0 = 8.327 MHz, which suggests a quadrupole splitting f Q ≈ 50 kHz. However, B0 has a few mT uncertainty, and so additional resonance peaks need to be measured to provide an accurate estimate of f Q . The nuclear Rabi frequency is measured at f 5/2↔7/2 = 146(1) Hz (Fig. 6.4A). Measurement of Rabi oscillations at varying drive frequency f drive shows a clear chevron Rabi pattern (Fig. 6.4B), given by Pflip =

γ2 sin2 t, 2

(6.1)

  2 where γ is the drive strength,  = γ 2 + 2π( f drive − f 5/2↔7/2 ) /4, and t is the drive duration. The measured Rabi frequency is over an order of magnitude lower than expected, as measurements on 31 P at similar drive strengths usually have Rabi frequencies of at least several kilohertz, and usually tens of kilohertz.

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Fig. 6.4 Rabi oscillations of the |5/2 ↔ |7/2 transition. (A) Antenna-driven Rabi oscillations on-resonance. The data points are fitted to a decaying sinusoid (blue line), yielding a Rabi frequency f Rabi = 146(1) Hz. (B) Rabi oscillation for varying drive frequency around f 5/2↔7/2 , resulting in a chevron Rabi pattern

6.1.3 Measurements of Subsequent Transitions The first nuclear transition provides a way to initialize the nuclear spin in the |5/2 state whenever it is in the |7/2 state. This allows measuring the |3/2 ↔ |5/2 transition, which in turn can be used to initialize the nuclear spin in the |3/2 state, et cetera.2 Using this method, the first three nuclear spin transitions of the ionized donor have been found, down to the |1/2 ↔ |3/2 transition (Fig. 6.5B). The quadrupole splitting separating these frequencies is f Q = 65.5 kHz. The Zeeman splitting is to first order equal to the resonance frequency of the middle transition, which lies f Q below the |1/2 ↔ |3/2 resonance frequency. We extract a Zeeman splitting 8.410 MHz, which corresponds to a magnetic field B0 = 1.514 T using the bulk gyromagnetic ratio of γn = 5.55 MHz (Table 3.1). The extracted B0 is slightly above the set value B0 = 1.5 T. Many attempts have been made to measure the |−1/2 ↔ |1/2 middle transition, but all were unsuccessful. Even though the resonance frequency is already known very accurately from the measured spectrum,3 transitions between the two nuclear spin states could not be induced, even at maximum drive power and drive durations far exceeding the expected Rabi frequency. Since the middle transition was needed to prepare the nuclear spin in the |−1/2 state, the remaining nuclear transitions could not be measured either.

2

At the time of these measurements, the flip-flop transitions were not yet discovered, and therefore could not be used to initialize the nuclear spin state (Sect. 5.5.2). 3 The |−1/2 ↔ |1/2 transition frequency is to first order f less than the |1/2 ↔ |3/2 transition Q frequency.

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Fig. 6.5 Spectrum and Rabi frequencies of the first three ionized nuclear spin transitions. (A) Energy level diagram of ionized 123 Sb nuclear spin states, with colored arrows indicating driven transitions. The |−1/2 ↔ |1/2 transition (red with a cross) could not be driven. (B) The spectrum of the first three measured transitions, each with a drive duration matching a resonant π -pulse. The spectra appear as vertical lines since the power-broadened spectral linewidth is orders of magnitude less than the quadrupole splitting f Q = 65.5 kHz. RF pulses with durations up to 15 ms were applied in a broad range around the expected transition frequency of the |−1/2 ↔ |1/2 transition (red), but no excitation was observed. (C) Rabi frequencies of nuclear transitions at constant drive power. Outer transitions should have a lower f Rabi than inner transitions when driven via NMR (triangles). Instead, the opposite trend is observed, with the outer transition having the highest f Rabi , while the central transition could not even be driven. The red filled dot is inferred by the absence of the |−1/2 ↔ |1/2 transition

The measured Rabi frequencies are also not in line with expectations. Assuming a magnetic drive, e.g. −γn B1 sin (2π f t) Iˆx , the Rabi frequency of a transition |m I − 1 ↔ |m I  is given by NMR f mRabi, =| m I − 1|γn B1 Iˆx |m I |, I −1↔m I

=γn B1 αmNMR , I −1↔m I

(6.2)

given by with the NMR coupling coefficients αmNMR I −1↔m I = αmNMR I −1↔m I

1 I (I + 1) + m I (m I − 1). 2

(6.3)

In case an additional driving component along Iˆy is present, B1 is replaced by the remains unchanged. The NMR coupling modulus of the total field, but αmNMR I −1↔m I NMR coefficients αm I −1↔m I increase closer to the central transitions (Table 6.1).

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Table 6.1 Rabi-frequency coefficients for NER and NMR transitions. Transitions are labeled NER by their secondary spin quantum number m I . The coefficient αm governs the NER transition I −1↔m I NER between neighboring levels with m I = ±1, βm I −2↔m I governs the NER transition between nextNMR nearest neighboring levels with m I = ±2 (hence the absence of the first element), and αm I −1↔m I governs the NER transition between neighboring levels with m I = ±1. mI −5/2 −3/2 −1/2 1/2 3/2 5/2 7/2 √ √ √ √ √ √ 63 48 15 0 15 48 63 NER αm I −1↔m I √ √ √ √ √ √ 21/2 45/2 60/2 60/2 45/2 21/2 βmNER −2↔m I I √ √ √ √ √ √ √ 7/2 12/2 15/2 16/2 15/2 12/2 7/2 NMR αm I −1↔m I

While the absolute Rabi frequencies depend on the drive strength, the relation , provided between Rabi frequencies of different transitions is fixed by αmNMR I −1↔m I that the drive strength is kept constant. NMR thus predicts that the Rabi frequency should be higher for the central transitions than for the outer transitions at fixed B1 , and should notably be maximal for |−1/2 ↔ |1/2. the measured Rabi frequencies are in strong disagreement with the trend of NMR (Fig. 6.5C), as the Rabi frequency decreases towards the central transition, and the |−1/2 ↔ |1/2 transition could not even be driven. To rule out any strong frequency-dependence in the microwave line or at the antenna, the same measurements have been performed at lower magnetic fields (1.364 T and 1.4 T), and indeed show the same trend in Rabi frequencies. As a further verification of our observations, a second measured 123 Sb donor exhibited very similar properties: the |−1/2 ↔ |1/2 transition could not be driven, the Rabi frequencies followed the same trend, and the quadrupole splitting had the same order of magnitude ( f Q = 28.2 kHz). These results suggest that a driving mechanism alternative to NMR is responsible for the observed nuclear spin transitions.

6.2 Nuclear Electric Resonance (NER) 6.2.1 Properties of NER Could the nuclear quadrupole interaction (Sect. 3.3) be the interaction that drives the observed nuclear spin transitions? In principle, the gates produce inhomogeneous electric fields, and the corresponding electric field gradient (EFG) could be used to induce transitions between nuclear spin states via the quadrupole interaction. However, the direct EFG produced from a 1 V electric AC stimulus applied to a donor gate results in a Rabi frequency below one millihertz, as determined from electrostatic device simulations (Sect. 7.5.1). This is over 6 orders of magnitude

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lower than observed. On top of this, these transitions are driven via the microwave antenna, which has shorted ends and should thus predominantly emit magnetic fields, not electric fields. Nevertheless, let us postulate the existence of a modulation of the EFG components at the nucleus. This modulation has a peak amplitude δVαβ and is of the form cos (2π f t)δVαβ , where f is the drive frequency. This EFG modulation causes a modulation of the quadrupole interaction components (Eq. 3.8), which we shall ˆ to the static Hamiltonian Hˆ refer to as δ Q αβ . The time-dependent addition δ H(t) (given by Eq. 3.9) is 

ˆ δ H(t) = cos (2π f t)δ Hˆ Q = cos (2π f t)

α,β∈{x,y,z}

δ Q αβ Iˆα Iˆβ .

(6.4)

This quadrupole-induced driving mechanism has actually been observed experimentally in a range of polar crystals [2–4], where the permanent dipole moment transduces electric fields into significant EFGs at the nucleus. However, silicon is a non-polar crystal, and thus does not possess a permanent dipole. This driving mechanism is referred to as nuclear electric resonance (NER) since the drive is electric in origin, distinguishing itself from the conventional nuclear magnetic resonance (NMR). Note also that NER is radically different from the well-known technique called nuclear quadrupole resonance (NQR), which is another magnetic resonance technique used for spectroscopy. In In nuclear quadrupole resonance (NQR), the spin splitting is dominated by the quadrupole interaction; a static magnetic field is either entirely absent (known as pure NQR) or otherwise sufficiently low such that the Zeeman interaction acts as a perturbation [5, 6]. However, the spin transitions are still driven magnetically via NMR. Transitions between neighboring states |m I − 1 and |m I  require the first offˆ diagonal matrix elements to be nonzero. Since δ H(t) contains only products of two spin operators, this requires one of the spin operators to be Iˆz , which has nonzero diagonal matrix elements, and the other to be Iˆx or Iˆy , which have nonzero first offdiagonal matrix elements (see Sect. 2.4 for spin operator matrices). This is satisfied by four spin-operator combinations: Iˆx Iˆz , Iˆz Iˆx , Iˆy Iˆz , and Iˆz Iˆy , which we can combine into Iˆx Iˆz + Iˆz Iˆx and Iˆy Iˆz + Iˆz Iˆy since the EFG is traceless and symmetric (Sect. 3.3). Therefore, when f equals the transition frequency for the transition between |m I − 1 NER is given by and |m I  (m I = ±1), the resonant Rabi frequency f mRabi, I −1↔m I NER f mRabi, =| m I − 1|δ Hˆ Q |m I |, I −1↔m I

=|δ Q x z m I − 1| Iˆx Iˆz + Iˆz Iˆx |m I +δ Q yz m I − 1| Iˆy Iˆz + Iˆz Iˆy |m I |. (6.5) The operator Iˆx Iˆz + Iˆz Iˆx is given by

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Iˆx Iˆz + Iˆz Iˆx

=

⎡ 0 ⎢√63 ⎢ ⎢ 0 ⎢ ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎣ 0 0

√

63 0 √ 48 0 0 0 0 0

√0 48 0 √ 15 0 0 0 0

⎤ 0 0 0 0 0 0 0 0 0 ⎥ ⎥ √0 15 0 0 0 0 ⎥ ⎥ ⎥ 0 0 0 0 0 ⎥ √ ⎥, ⎥ 15 0 0 0 0 − √ √ ⎥ 0 − 48 √ 0 ⎥ 0 − 15 √ ⎥ 0 0 − 48 √ 0 − 63⎦ 0 0 0 − 63 0

(6.6)

and the purely imaginary operator Iˆy Iˆz + Iˆz Iˆy has matrix elements with equal modulus. The relation between the m I = ±1 Rabi frequencies is determined by the relative magnitudes of the first off-diagonal matrix elements     ˆ ˆ ˆ ˆ = − 1| I + I |m  I I αmNER  m , I x z z x I I −1↔m I  1 = |2m I − 1| I (I + 1) − m I (m I − 1). 2

(6.7)

Values of αmNER for the different transitions are given in Table 6.1. This yields I −1↔m I the Rabi frequencies  NER NER 2 2 = α f mRabi, m I −1↔m I δ Q x z + δ Q yz . I −1↔m I

(6.8)

The relation between the NER coupling coefficients αmNER presents a trend I −1↔m I unique to electric driving via the nuclear quadrupole interaction. This relation exposes a startling feature: outer transitions have a stronger coupling than inner transitions. In fact, the central transition |−1/2 ↔ |1/2 has no coupling whatsoever, and therefore cannot be driven via NER. These features are in line with our observed Rabi frequencies, including the lack of a central transition (Fig. 6.5). This indicates that the transitions are not driven via NMR, but are instead driven via NER through electric modulation of the quadrupole interaction. The quadrupole interaction terms contain products of two spin operators, and this enables first-order transitions between next-nearest-neighboring levels |m I − 2 ↔ |m I  (m I = ±2). Again, inspecting the spin matrices reveals that matrix elements coupling such states must stem from operator products Iˆx2 , Iˆy2 , Iˆx Iˆy , and Iˆy Iˆx , as only such products result in nonzero matrix elements on the second off-diagonal. Explicitly, the Rabi frequencies for such transitions are given by NER =|δ Q x x m I − 2| Iˆx2 |m I  + δ Q yy m I − 2| Iˆy2 |m I  f mRabi, I −2↔m I

+ δ Q x y m I − 2| Iˆx Iˆy + Iˆy Iˆx |m I |. The operator Iˆx2 is given by

(6.9)

6.2 Nuclear Electric Resonance (NER)

√ 7 0 84 √ 0 0 0 0 ⎢ 0 19 0 180 0 0 0 ⎢√ √ ⎢ 84 0 27 0 240 0 0 ⎢ √ √ 1⎢ 240 √ 0 180 √ 0 31 0 ⎢ 0 ⎢ 0 240 √ 0 31 0 180 4⎢ 0 ⎢ ⎢ 0 0 0 240 0 27 0 √ ⎢ ⎣ 0 0 0 0 180 √0 19 0 0 0 0 0 84 0 ⎡

Iˆx2

=

93

⎤ 0 0 ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥, √0 ⎥ ⎥ 84⎥ ⎥ 0 ⎦ 7

(6.10)

showing the coupling matrix elements on the second off-diagonal. The operator Iˆy2 has elements on the second off-diagonal with the same magnitude but opposite sign, whereas the operator Iˆx Iˆy + Iˆy Iˆx is purely imaginary and its second off-diagonal elements have twice the strength of those of the other operators. This leads to a Rabi frequency given by  NER NER = β (δ Q x x − δ Q yy )2 + 4δ Q 2x y , f mRabi, −2↔m m −2↔m I I I I

(6.11)

with βmNER given by I −2↔m I = βmNER I −2↔m I

1 (I − m I − 7)(I − m I − 6)(I − m I + 1)(I − m I + 2). (6.12) 4

Values of βmNER for the different transitions are given in Table 6.1. Importantly, I −2↔m I m I = ±2 transitions are to first order forbidden in NMR, and so observation of these transitions would serve as an additional verification of NER.

6.2.2 m I = ±1 Nuclear Spectrum and Rabi Oscillations If the driving mechanism is indeed electric in origin, the transitions would not need to be driven via the antenna. In fact, the antenna is designed to minimize electric fields, and therefore driving via a gate instead should produce significantly stronger electric fields, which would amplify the drive strengths. This hypothesis has been tested by applying an RF pulse to the DFR donor gate (see Sect. 4.6 for gate labeling). It is found that the nuclear transitions can indeed be driven by a donor gate, and as expected, the drive strengths are several times higher than with the antenna. This demonstrates that the drive cannot be magnetic, since the donor gate is open-ended, and therefore does not generate any significant magnetic fields. The insight that the transitions are induced by the modulation of the quadrupole interaction allows us to bypass the “forbidden” |−1/2 ↔ |1/2 transition using m I = ±2 transitions (see Sect. 6.2.3). By preparing the state in |3/2 and driving the transition |−1/2 ↔ |3/2, the nuclear spin can be transferred to |−1/2.

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From hereon, the remaining transitions involving negative nuclear spin states can be measured. This enabled us to continue measuring the remaining m I = ±1 transitions using the DFR gate (Fig. 6.6B–H). Successive transitions are separated by a quadrupole splitting f Q = 65.5 kHz, with a second-order splitting (difference between successive splittings) of -30 Hz. These splittings provide information about the quadrupole orientation (Sects. 3.3.4 and 6.6). All non-center transitions could be driven coherently (Figs. 6.6I–O), and most importantly, their Rabi frequencies accurately follow the trend predicted by NER, (Table 6.1). The Rabi frequency (Fig. 6.6I), given by the NER coefficients αmNER I −1↔m I is maximal for the outer transitions |±5/2 ↔ |±7/2, and decrease towards the center transition. Again, the center transition |−1/2 ↔ |1/2 could not be driven, as expected from NER. If the driving of transitions is due to NMR, the Rabi frequency should be lowest for the outer transitions and highest for the center transition (Fig. 6.6I). This is clearly not observed, and so these results rule out NMR. The alternative explanation, NER, matches the observed trend well, and is thus likely to be the underlying driving mechanism.

6.2.3 m I = ±2 Nuclear Spectrum and Rabi Oscillations Modulation of the Q x x , Q yy or Q x y quadrupole terms allows driving of m I = ±2 transitions (Fig. 6.7A). The resonance frequency of a transition |m I − 2 ↔ |m I  is equal to the sum of the two corresponding m I = ±1 transition frequencies f m I −2↔m I = f m I −2↔m I −1 + f m I −1↔m I .

(6.13)

All six m I = ±2 transitions were indeed observed at the expected transition frequencies (Figs. 6.7B–H). Their Rabi frequencies again accurately match the NER (Table 6.1). For the trend (Figs. 6.7I–O), given by the NER coefficients βmNER I −2↔m I m I = ±2 transitions, the Rabi frequency is maximal for the two center transitions. Importantly, m I = ±2 transitions are forbidden to first order in NMR. Although two-photon NMR could drive m I = ±2 transitions, such a second-order process would need to be driven at half the expected transition frequency, instead of at the expected transition frequency as is the case here. A minor decrease can be observed for the Rabi frequencies of successive transitions compared to the NER trend. Additional measurements were performed to reveal that this slope is dependent on the NER drive frequency, and is likely due to a non-uniform transmission profile of the gate line (Appendix C)

6.2 Nuclear Electric Resonance (NER)

95

Fig. 6.6 NER spectrum and Rabi frequencies of m I = ±1 transitions. All transitions are driven with the DFR donor gate at a magnetic field B0 = 1.496 T on an ionized 123 Sb donor. (A) Energy level diagram with colored arrows indicating m I = ±1 transitions. The |−1/2 ↔ |1/2 transition (red with a cross) cannot be driven via NER. (B) Spectrum of m I = ±1 transitions. Each individual transition (C–H) is measured with the NER pulse duration corresponding to a resonant π -pulse, and is fitted with Rabi’s formula (Eq. 6.1). The central transition |−1/2 ↔ |1/2 could not be driven, in agreement with NER. Successive peaks are split by the quadrupole splitting f Q = 65.5 kHz. (I) Measured Rabi frequencies (dots), along with individual Rabi oscillations (J– gate O), fitted with sinusoids. All transitions were driven at a peak drive amplitude VRF = 20 mV. The Rabi frequencies can only be scaled by a global drive strength, while the Rabi-frequency ratio of different transitions is fixed by the driving mechanism (Table 6.1). Importantly, the measured Rabi frequencies accurately follow the trend predicted by NER (stars), and not the trend of NMR (triangles).

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Fig. 6.7 NER spectrum and Rabi frequencies of m I = ±2 transitions. All transitions are driven with the DFR donor gate at a magnetic field B0 = 1.496 T on an ionized 123 Sb donor. Transitions involving m I = ±2 are forbidden to first order in NMR. (A) Energy level diagram with colored arrows indicating m I = ±2 transitions. (B) Spectrum of m I = ±2 transitions. Each individual transition (C–H) is measured with the NER pulse duration corresponding to a resonant π -pulse, and is fitted with Rabi’s formula (Eq. 6.1). Successive peaks are split by double the quadrupole splitting 2 f Q = 131 kHz. (I) Measured Rabi frequencies (dots), along with individual Rabi oscillations (J–O), fitted with sinusoids. All transitions were driven at a peak drive amplitude gate VRF = 40 mV. The measured Rabi frequencies accurately follow the trend predicted by NER (stars), which can only be scaled by a global drive strength

6.2 Nuclear Electric Resonance (NER)

97

gate

Fig. 6.8 Rabi frequencies for varying drive amplitude VRF . A linear relation between f Rabi and gate VRF is found for both the m I = ±1 transition |5/2 ↔ |7/2 (A) and the m I = ±2 transition gate |3/2 ↔ |7/2 (B). The fitted ∂ f Rabi /∂ VRF slopes (blue lines) are 34.21(3) Hz mV−1 (A) and −1 1.995(4) Hz mV (B). These results lead to the conclusion that the transitions are driven by a single-photon process such as NER, for a two-photon process would have a quadratic relation

6.2.4 Power Dependence of Rabi Frequencies The relation between the Rabi frequency f Rabi and drive strength provides information on the transition type. For a first-order process, f Rabi scales linearly with the gate drive amplitude VRF . A second-order process, such as a two-photon process, will gate instead have a quadratic dependence of f Rabi on VRF [7]. gate The Rabi frequency f Rabi was measured for varying drive amplitudes VRF , both for the |5/2 ↔ |7/2 and |3/2 ↔ |7/2 transitions (Fig. 6.8). In both cases, the gate observed relation between f Rabi and VRF is linear. This confirms that the driving mechanism for both m I = ±1 and m I = ±2 transitions is a first-order process, in agreement with NER. The linear dependence of m I = ±2 transitions excludes a two-photon m I = ±2 driving process, which exhibits a quadratic dependence [8, 9]. This is further confirmed by the transitions being driven at their expected transition frequencies, and not at half the expected transition frequencies as would be the case for a two-photon process.

6.3 Antenna-Driven NER 6.3.1 Enhanced Electric Fields from a Melted Antenna One remaining question is why NER could be induced by the microwave antenna, which was designed to minimize electric fields. The antenna that was originally designed to perform NMR and ESR is an on-chip (CPW) terminated by a short

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6 Nuclear Electric Resonance

circuit. Due to the short circuit termination, the electric field at the tip of the antenna will be minimized, whilst a time-varying magnetic field is generated by the current flowing through the short. In an effort to maximize the magnetic field B1 at the donor site, multiple steps were taken to ensure the donor is located as close as possible to the antenna. This includes placing the implantation window no further than 150 nm away from the antenna and making the shorting wire as narrow as possible. The microwave antenna is made of aluminum (see step 12 of the fabrication procedure outlined in Sect. 4.3) and was designed to have a length of 500 nm on each side of the CPW, a thickness of 40 nm and a width of 50 nm. However, we have noticed across multiple devices that a short of these dimensions is not robust enough to withstand our standard experimental procedures such as connecting and disconnecting the microwave line, or applying high-power RF pulses (of the order of −10 to 0 dBm at the antenna). Upon imaging this device with an SEM, the short is indeed found to be ‘molten’ (Fig. 4.1), which suggests that the cross-sectional area of the short was too small to sustain high-power microwaves or current spikes. One can approximate the behaviour of a broken antenna, to first order, by modeling it as a series RC circuit. The produced electric field is proportional to the voltage at the capacitor, and the magnetic field is proportional to the current flowing through the circuit. Thus, the electric field will be at a relatively constant value for all frequencies until it starts rolling off at higher frequencies. The capacitor strongly attenuates any low-frequency current and thus causes the magnetic field to increase with frequency. At high frequencies, the impedance of the broken part of the antenna decreases to the point where a reasonable amount of current can flow. This all points to the antenna acting as an electric antenna at low frequencies (∼ 8 MHz needed for NER) and a (albeit poor) magnetic antenna at higher frequencies (∼ 40 GHz needed for ESR). To study the electromagnetic response in some depth, the width and location of the antenna breaks were extracted from the device SEM image (Fig. 4.1). We simulated the broken antenna using CST Studio Suite, and extracted the electric and magnetic fields as a function of frequency at the expected donor site (Sect. 5.2). The original intact antenna was also simulated, allowing the definition of a relative antenna performance, where the electric and magnetic fields generated by the broken antenna at the donor site are divided by those of the working antenna. The simulation results show a general trend that the magnetic field is reduced for a broken antenna, while the electric field is enhanced (Fig. 6.9), which matches the experimental observations. When analyzing the simulation results, we find that the simulations for a broken antenna suggest that the magnetic field is on the order of 60 dB lower than that of a working antenna at ESR frequencies. A significantly reduced magnetic field at ESR frequencies has indeed been observed, but comparison to previous functional 31 P-donor devices in our group suggests that the magnetic field is ∼30 dB weaker. The difference between experimentally observed values and simulation is likely due to the simplifications (exact shape of the discontinuity, grain structure, etc.) that were made in the CST model for ease of simulation.

6.3 Antenna-Driven NER

99

Fig. 6.9 Simulated response of damaged and intact antenna. The AC electric and magnetic fields produced by either an intact or a damaged antenna at the triangulated donor location (Sect. 5.2) are simulated using finite-element methods. The ratios E damaged /E intact 0 dB and Bdamaged /Bintact  0 dB indicate that the damaged antenna produces chiefly an electric field at its tip, rather than a magnetic field. This is due to the nanoscale open-circuit termination at its tip, which lies in proximity to the donor. These effects are more pronounced at the lower frequencies, explaining the driving of nuclear transitions via NER. With increasing frequency, the capacitive impedance created by the gap in the damaged antenna becomes small enough to allow some AC current to flow. This produces an oscillating magnetic field at microwave frequencies that is sufficient to weakly drive ESR transitions

The antenna simulations in the range of the nuclear resonance frequencies (∼8 MHz) show a highly attenuated magnetic field (∼100 dB attenuation), which would inhibit any coherent driving via NMR. Instead, in this frequency range, the structure acts as a decent electric antenna with ∼ 80 dB higher electric fields, enabling driving via NER. This explains how we were able to drive transitions via NER instead of NMR with a structure that was designed to be a broad-band magnetic antenna (Fig. 6.5). The validity of this interpretation is confirmed by our observation of NER using a nearby electrostatic gate as an electrical antenna.

6.3.2 Gate-Driven Versus Antenna-Driven NER The full NER spectrum has also been measured while driving with the damaged microwave antenna, from which the Rabi frequencies have been extracted for all m I = ±1 (Fig. 6.10A) and m I = ±2 (Fig. 6.10B) transitions. In both cases, the relation between the Rabi frequencies matches well with that expected for NER (Table 6.1). Additionally, the transition |−1/2 ↔ |1/2 could not be driven, further confirming that the driving mechanism from the antenna in the ∼MHz frequency range is electric instead of magnetic.

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Fig. 6.10 Antenna-driven NER Rabi frequencies. The damaged short of the microwave antenna suppresses magnetic fields and enhances electric fields, resulting in NER instead of NMR. The Rabi frequencies are extracted by fitting Rabi’s formula (Eq. 6.1) to the antenna-driven NER spectrum. For both the m I = ±1 (circle) and m I = ±2 (square) transitions, the relation between the Rabi frequencies match well with those expected for NER (stars). Error bars show the 95% confidence interval

When driven with the antenna, the Rabi frequencies of the m I = ±2 transitions are significantly higher than those of the m I = ±1 transitions. This is opposite to the results obtained while driving with an electric gate, where the m I = ±1 Rabi frequencies are much higher (Fig. 6.8). This could be caused by the differing AC electric field orientation at the donor site when driving with an antenna versus with a donor gate, which will affect the ratio of m I = ±1 to m I = ±2 Rabi frequencies.

6.4 Linear Quadrupole Stark Effect The quadrupole splitting f Q is to first order set by the diagonal quadrupole components Q x x , Q yy , and Q zz (Eq. 3.14). Since the observation of NER implies that the quadrupole interaction couples to electric fields, modifying the diagonal quadrupole components shifts f Q , and consequently Stark shifts each transition frequency f m I −m I ↔m I by an amount  f m I −m I ↔m I = m I (m I − m I /2) f Q ,

(6.14)

where  f Q is the shift of f Q , and m I is positive. gate The nuclear spectrum has been measured for varying negative bias voltage VDC applied to the DFL donor gate (Fig. 6.11). All m I = ±1 transitions (Fig. 6.11A) and m I = ±2 transitions (Fig. 6.11B) indeed shift in accordance with Eq. 6.14.

6.4 Linear Quadrupole Stark Effect

101

Fig. 6.11 NER spectral line shifts for varying DC gate voltage. The spectral lines of all m I = ±1 transitions (A) and m I = ±2 transitions (B) are measured for varying NER DC gate bias voltage VDC (columns), each of which is fitted with Rabi’s formula (Eq. 6.1). Varygate ing VDC causes a quadrupole Stark shift of each transition frequency (Eq. 6.14), due to the dependence of the quadrupole interaction on electric fields. Note that the bias voltage gate gate VDC is applied on top of a significant gate voltage (VDC ≈ 0.5 V), necessary to electrostatically tune the device to enable its operation. Ordered from top to bottom transition, the drive gate strengths VRF are [20 mV, 20 mV, 25 mV, 25 mV, 20 mV, 25 mV] for m I = ±1 transitions, and [30 mV, 30 mV, 40 mV, 40 mV, 40 mV, 40 mV] for m I = ±2 transitions. Error bars show the 95% confidence interval

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Fig. 6.12 Quadrupole shifts for varying DC gate voltage. The measured gate-induced spectral ∂ fQ gate line shifts (Fig. 6.11) are due to a shift of the quadrupole splitting  f Q = gate VDC (Eq. (6.14)). ∂ VDC

The gate-dependent quadrupole shift  f Q is determined from a single fit (solid lines) to the resonance frequency shifts of all m I = ±1 (A) and m I = ±2 (B) transitions. The estimated coef∂ fQ ficient of  f Q is gate = 9.9(3) kHz V−1 . ∂ VDC

gate

A second charge transition is crossed at a higher negative bias potential VDC = −45 mV (Appendix D). Crossing this transition results in a discrete shift of the spectral lines, measured at 1.43 kHz for the |5/2 ↔ |7/2 transition. A single least-squares fit of Eq. 6.14 to all the spectral lines gives a best estimate of ∂ fQ = 9.9(3) kHz V−1 . The γn B0 = 8.246301(6) MHz, f Q = 65.503(8) kHz, and ∂ V gate DC resulting shifts in quadrupole splitting  f Q have been calculated for all spectral lines using Eq. 6.14 with the fitted values (Fig. 6.12). This effect, known as the linear quadrupole Stark effect (LQSE), has been previously observed in a range of polar crystals [10–13], and is discussed further in Chap. 7.

6.5 Nuclear Coherence Times Magnetic and electric noise have a different effect on the coherence between spin states of high-dimensional nuclei. Magnetic noise couples to the nuclear spin via the Zeeman interaction. Since the m I = ±1 transitions are split by γn B0 , whereas the m I = ±2 are split by 2γn B0 , the m I = ±2 transitions are twice as sensitive to magnetic noise. The observation of the linear quadrupole Stark effect (LQSE) indicates that electric noise couples to the nuclear spin via the quadrupole interaction, and thus shifts the quadrupole splitting f Q . Outer transitions are more sensitive to shifts in f Q since their resonance frequencies are shifted from the Zeeman splitting by higher multiples of f Q (Eq. 6.14). In fact, the |−1/2 ↔ |1/2 transition is to first order unaffected by

6.5 Nuclear Coherence Times

103

electric noise, i.e. its resonance frequency equals the Zeeman splitting. On the other hand, the |±3/2 ↔ |±7/2 transition frequencies are shifted by ±5 f Q from the Zeeman splitting. Any shift in f Q thus has a five-fold impact on the outer m I = ±2 transition frequencies, resulting in an enhanced sensitivity to electric noise. The coherence times of ionized 75 As donors in silicon have been studied by Franke et al. [14]. In their system, the readout mechanism relies on charge defects close to the donor atoms. Their initialization routine excites these charge defects, and the resulting relaxation of charge defects couples to the nuclear spin via the quadrupole interaction. This is found to be the dominant source of decoherence for 75 As. However, if the excited defects were first given time to relax, the dominant decoherence mechanism is consistent with magnetic-field noise. This could be due to fluctuations in the external magnetic field, or due to nearby 29 Si atoms, which are more abundant in their sample containing natural silicon. The coherence of all m I = ±1 and m I = ±2 transitions has been measured using two metrics. The first one is the pure dephasing time T2∗ , which has been measured by a Ramsey experiment, where two π/2 pulses are applied, separated by a wait time τ . Both pulses have been calibrated to be on resonance, and in the absence of any dephasing, will result in a full spin inversion (Pflip = 1). Dephasing reduces Pflip , until all phase information is lost, resulting in a fully mixed state (Pflip = 0.5) The rate at which the system dephases is characterized by the dephasing time T2∗ . The Ramsey sequence does not have refocusing pulses that filter low-frequency noise, and since Each Ramsey measurement is taken over the course of several hours, T2∗ is sensitive to extremely slow frequency drifts that are well below 1 MHz. The second metric is the echo time T2Echo . The corresponding spin-echo measurement is very similar to the Ramsey measurement, with the addition of a π -pulse in the middle of the wait time (at τ/2). This pulse inverts the spin population, thereby negating any accumulated phase caused by noise whose frequency is below 1/2π τ . The corresponding coherence time is therefore generally higher than the pure dephasing time T2∗ . The measured coherence times all show a higher T2Echo than T2∗ (Fig. 6.13). This indicates that low-frequency noise significantly impacts the donor coherence, which is in agreement with our observations of slow drifts of the resonance frequencies. The m I = ±1 coherence times are all around T2∗ ≈ 0.1 s and T2Echo ≈ 0.25 s. Both coherence times are approximately six times lower than those measured on a 31 P donor in a similar device [15] (T2∗ = 0.6 s and T2Echo = 1.75 s). This reduction is even more significant when considering that 123 Sb has a nuclear gyromagnetic ratio γn = 5.55 MHz that is roughly a factor three lower than 31 P with γn = 17.26 MHz (Table 3.1), and is therefore three times less affected by magnetic noise. This means that the measured 123 Sb donor experiences roughly an 18-fold increase in dephasing noise. It is unclear why the coherence times of 123 Sb are adversely affected. The m I = ±2 transitions have notably lower coherence times than the m I = ±1 counterparts. This is quite possibly due to the m I = ±2 transitions being twice as sensitive to magnetic-field noise, since they are split by twice the Zeeman energy. Furthermore, the T2Echo coherence times of the m I = ±2 transitions seem to increase towards the inner transitions. This would match with electric noise

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Fig. 6.13 Ionized nuclear T2∗ and T2Echo coherence times. (A, B) Ramsey measurement of the transition |5/2 ↔ |7/2 (A) and |3/2 ↔ |7/2 (B), consisting of two π/2 pulses separated by a wait time τ . The system dephases after a characteristic dephasing time T2∗ . (C, D) Spin-echo measurements of the same transitions, which have an added π refocusing pulse at τ/2 that negates low-frequency noise, thus resulting in an increased coherence time T2Echo . To improve the fitting accuracy, oscillations are induced by increasing the phase of the final pulse in both sequences linearly with τ . (E, F) Coherence times for m I = ±1 transitions (E) and m I = ±2 transitions (F). The higher coherence times of the m I = ±1 transitions can be attributed to the resonance frequencies being half that of the m I = ±2 counterparts, and could indicate magnetic noise being a significant dephasing channel. The coherence times of the |−1/2 ↔ |1/2 transition were not measured because the transition could not be driven via NER

coupling to the nuclear spin via the quadrupole interaction, which affects the outermost m I = ±2 transitions five times more than the innermost transitions.4 In this regard, it would be interesting to measure the |−1/2 ↔ |1/2 coherence times, as this transition is to first order unaffected by quadrupole-induced noise (Fig. 3.5). Driving this transition would either require NMR, or alternatively a combination of a m I = ±1 and m I = ±2 NER pulse, and is a topic for future research.

The outermost m I = ±2 transitions are shifted by ±5 f Q from 2γn B0 , while the innermost m I = ±2 transitions are only shifted by ± f Q (Eq. 6.14). Any fluctuations in f Q would therefore affect the outermost transitions five times more than the innermost transitions.

4

6.6 Possible Quadrupole Orientations

105

6.6 Possible Quadrupole Orientations The full quadrupole tensor cannot be determined from a single spectrum, as the direction of the external magnetic field B0 constitutes a rotational symmetry axis (Sect. 3.3.4). Nevertheless, the measured first-order quadrupole splitting f Q = 65.5 kHz provides valuable information about the quadrupole orientation and strength, as it directly relates to the diagonal quadrupole components (Eq. 3.13) The second-order quadrupole splitting f Q(2) = −30 Hz, which is the difference between successive splittings, provides additional information. The fact that f Q(2) is negative considerably narrows down the possible quadrupole orientations to two cases 1. a positive quadrupole interaction strength Q > 0, oriented nearly perpendicular to B0 ; 2. a negative quadrupole interaction strength Q < 0, oriented nearly parallel to B0 , with a high asymmetry η. To find the possible quadrupole parameters that match the measured spectrum, a nonlinear constrained optimization algorithm in MATLAB has been employed that calculates the nuclear spectrum for varying quadrupole parameters, aiming to minimize the difference with the measured spectrum. A total of 10,000 separate instances were run, each with randomized initial quadrupole parameters. By starting with different initial parameters every iteration, the optimizer is able to find many different configurations that match the measured frequencies. Figure 6.14 shows the top 100 instances, which matched all transition frequencies to within 100 Hz. These results may serve as crude probabilities for the donor quadrupole parameters, though it should be stressed that these probabilities may be skewed by the optimization algorithm. The primary quadrupole axis is nearly always found to be oriented perpendicular to B0 , with a positive quadrupole strength Q, at an average Q = 52(2) kHz. However, a few instances have successfully found a configuration with Q oriented parallel to B0 , with Q = −33.4(2) kHz. In all cases, the asymmetry η is considerable, nearly always exceeding 0.5. Contrary to the primary axis, the secondary quadrupole axis is found to be fairly aligned with B0 in most instances. Since the asymmetry is likely high, the orientation of the secondary quadrupole axis has a significant effect on f Q . The third quadrupole rotation angle cannot be estimated from a single spectrum due to the inherent rotational symmetry. It is also not possible with certainty to say if Q is positive or negative, and if it is oriented parallel or perpendicular to B0 , though a positive Q oriented perpendicular to B0 is far more likely. Additional measurements of the nuclear spectrum are therefore needed to fully determine all quadrupole parameters (Sect. 3.3.4). Nevertheless, this method shows that a single spectrum provides a lot of information about the quadrupole parameters. Additionally, rotating B0 by π/2 for each of the top hundred quadrupole configurations results in wildly varying quadrupole splittings f Q . This suggests that only two to three spectra, measured for B0 in perpendicular orientations, should suffice to characterize the full quadrupole interaction, as the spectral resolution is on the order of a few Hertz.

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Fig. 6.14 Potential quadrupole parameters. Starting from randomized initial parameters, a nonlinear optimization algorithm is used to repeatedly find the quadrupole parameters that matches the measured spectrum. From a total 10,000 randomized runs, the top 100 solutions that most accurately match the measured spectrum are shown here, each of which matches all nuclear resonance frequencies to within 100 Hz. The quadrupole primary axis (B) is nearly always oriented perpendicular to B0 , with an average quadrupole strength Q = 52(2) kHz (A). However, a small fraction of the solutions have a negative Q oriented parallel to B0 . The secondary axis (C) is most likely predominantly aligned with B0 . All solutions have a fairly high asymmetry η (D)

6.7 Conclusion These measurements on the ionized 123 Sb nucleus have resulted in several key observations. First, the Rabi frequencies of the transitions between neighboring states accurately follow the trend predicted for NER in Eq. 6.7 (including the absence of the |−1/2 ↔ |1/2 transition), and do not follow the prediction of Eq. 6.3 for NMR. Second, transitions between next-nearest-neighboring states, which are first-order forbidden for NMR, accurately follow the prediction of Eq. 6.12 for NER. Finally, the transitions can be driven via oscillating biases applied to the donor gates, and the resulting drive strengths scale linearly with drive amplitude, confirming an electric first-order driving mechanism. These combined observations enable the main conclusion of this chapter: the demonstration of coherent electrical control of the 123 Sb nuclear spin via NER, without transducing the RF electric field into a magnetic field. The NER drive strengths can be increased through several approaches. Importantly, the NER drive strengths depend on the direction of the oscillating electric fields (see Sect. 7.3.2 for details). This is evident from the differing ratios between

6.7 Conclusion

107

the m I = ±1 and m I = ±2 NER drive strengths when driven by the donor gate versus the damaged antenna (Sect. 6.3.2). Compensated voltages on the different donor gates can therefore increase the modulation of the NER quadrupole terms (Eqs. 6.8 and 6.11). The modulation of the NER quadrupole terms can be further enhanced by a well-chosen orientation of the external magnetic field B0 , e.g. using a 3D vector magnet. A dedicated electric antenna, as well as higher gate voltages, can further increase the NER drive strengths. The first- and higher-order quadrupole splittings in the nuclear spectrum suggest that the quadrupole interaction is oriented perpendicular to B0 , with a strength Q = 52(2) kHz. Determining all five quadrupole parameters will require spectra at different magnetic-field orientations, though two to three well-chosen orientations should be sufficient. The measured quadrupole strength is promising for future experiments involving quantum chaos (Chap. 8), as it is exactly in the range to implement the driven top Hamiltonian in the rotating frame. However, this Hamiltonian does require a magnetic drive, and so a future device containing an 123 Sb donor with a similar quadrupole strength but with a working microwave antenna would allow the implementation of the quantum driven top. The ionized 123 Sb nucleus is measured to have long spin coherence times, with m I = ±1 transitions having T2Echo coherence times of a few hundred milliseconds. Still, the coherence times are significantly lower than those usually observed in ionized 31 P donors. The results suggest that, depending on the transition, the coherence is limited by either electric or magnetic noise. Future measurements of the noise spectroscopy can likely shed further light on the matter. Finally, the observation of the flip-flop transition (Sect. 5.5) demonstrates that the electron spin can also be manipulated electrically. Combining the flip-flop transition with NER would enable the all-electrical control of the 123 Sb donor, alleviating the need for a dedicated microwave antenna. While the results clearly demonstrate that the driving mechanism stems from modulation of the quadrupole interaction, this effect can only be understood by studying its microscopic origins. This is the topic of the next chapter.

References 1. Laucht A, Kalra R, Muhonen JT, Dehollain JP, Mohiyaddin FA, Hudson F, McCallum JC, Jamieson DN, Dzurak AS, Morello A (2014) High- fidelity adiabatic inversion of a 31p electron spin qubit in natural silicon. Appl Phys Lett 104(9):092 115. https://doi.org/10.1063/1.4867905 2. Brun E, Hann R, Pierce W, Tanttila W (1962) Spin transitions induced by external RF electric field in GaAs. Phys Rev Lett 8(9):365. https://doi.org/10.1103/PhysRevLett.8.365 3. Kushida T, Silver A (1963) Electrically induced nuclear resonance in Al2 O3 (ruby). Phys Rev 130(5):1692. https://doi.org/10.1103/PhysRev.130.1692 4. Meyer WJ, Lang D, Slichter C (1973) Electric field nuclear double resonance in Ag+-doped NaCl. Phys Rev B 8(5):1924. https://doi.org/10.1103/PhysRevB.8.1924 5. Zax D, Bielecki A, Zilm K, Pines A, Weitekamp D (1985) Zero field NMR and NQR. J Chem Phys 83(10):4877–4905. https://doi.org/10.1063/1.449748

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6. Vij D (2007) Handbook of applied solid state spectroscopy. Springer Science & Business Media. https://doi.org/10.1007/0-387-37590-2 7. Gentile TR, Hughey BJ, Kleppner D, Ducas TW (1989) Experimental study of one-and twophoton Rabi oscillations. Phys Rev A 40(9):5103. https://doi.org/10.1103/PhysRevA.40.5103 8. Bodenhausen G (1980) Multiple-quantum NMR. Prog Nucl Magn Reson Spectrosc 14(3):137– 173. https://doi.org/10.1016/0079-6565(80)80007-0 9. Munowitz M (2007) Double quantum coherence. eMagRes. https://doi.org/10.1002/ 9780470034590.emrstm0131 10. Bloembergen N (1961) Linear Stark effect in magnetic resonance spectra. Science 133:1363. https://doi.org/10.1126/science.133.3461.1363 11. Dixon R, Bloembergen N (1964) Electrically induced perturbations of halogen nuclear quadrupole interactions in polycrystalline compounds. II. microscopic theory. J Chem Phys 41(6):1739–1747. https://doi.org/10.1063/1.1726153 12. Dixon RW, Bloembergen N (1964) Linear electric shifts in the nuclear quadrupole interaction in Al2 O3 . Phys Rev 135(6A):A1669–A1675. https://doi.org/10.1103/PhysRev.135.A1669 13. Gill D, Bloembergen N (1963) Linear Stark splitting of nuclear spin levels in GaAs. Phys Rev 129(6):2398–2403. https://doi.org/10.1103/PhysRev.129.2398 14. Franke DP, Hrubesch FM, KKünzlnzl M, Becker HW, Itoh KM, Stutzmann M, Hoehne F, Dreher L, Brandt MS (2015) Interaction of strain and nuclear spins in silicon: Quadrupolar effects on ionized donors. Phys Rev Lett 115(5):057 601. https://doi.org/10.1103/PhysRevLett. 115.057601 15. Muhonen JT, Dehollain JP, Laucht A, Hudson FE, Kalra R, Sekiguchi T, Itoh KM, Jamieson DN, McCallum JC, Dzurak AS, Morello A (2014) Storing quantum information for 30 seconds in a nanoelectronic device. Nat Nanotechnol 9(12):986–991. https://doi.org/10.1038/nnano.2014. 211

Chapter 7

Microscopic Crystalline Origins of the Quadrupole Interaction

This chapter describes how an electric field gradient (EFG) can be produced at a lattice site in the silicon crystal. The EFG results in a nuclear quadrupole interaction when applied to a high-spin nucleus. A finite-element model is used to estimate the strain and electric fields that are present in our device. We then develop a microscopic model that relates the strain and electric fields to the EFG at the site of the donor, which directly relates to the quadrupole interaction strength. The observed NER and quadrupolar Stark shift are attributed to the LQSE, which has previously only been observed in polar crystals. This chapter includes results from the following publication: S. Asaad* , V. Mourik* , B. Joecker, M. A. I. Johnson, A. D. Baczewski, H. R. Firgau, M. T. Ma˛dzik, V. Schmitt, J. J. Pla, F. E. Hudson, K. M. Itoh, J. C. McCallum, A. S. Dzurak, A. Laucht, A. Morello. “Coherent electrical control of a single high-spin nucleus in silicon”. Nature 579.7798, pp. 205–209 (2020). The author acknowledges (i) V. Mourik, B. Joecker, and A. D. Baczewski for their contributions in developing a microscopic model of the quadrupolar interaction, (ii) B. Joecker for finite-element modeling of the device strain and electric fields, and (iii) A. D. Baczewski for DFT simulations. The experimental demonstration of coherent nuclear spin control via NER has been described in Chap. 6. In particular, we provided conclusive evidence that an oscillating electric field causes nuclear spin transitions by modulating the electric quadrupole interaction. Here we investigate the microscopic mechanism that makes NER possible, i.e. the relation between electric field and quadrupole coupling. The direct EFG from the inhomogeneous electric fields are insignificant (Sect. 7.5.1), which means that the electric fields must somehow be transduced into significant EFGs (Sect. 7.1).

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Asaad, Electrical Control and Quantum Chaos with a High-Spin Nucleus in Silicon, Springer Theses, https://doi.org/10.1007/978-3-030-83473-9_7

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7 Microscopic Crystalline Origins of the Quadrupole Interaction

We postulate that both NER and the quadrupole Stark shift are manifestations of an effect known as the LQSE (Sect. 7.3.1). This is substantiated by comparing the inferred coupling between electric field and EFG (Sect. 7.3) to historical measurements of the LQSE in bulk materials (Sect. 7.3.4). The portion of the quadrupole interaction that is due to external applied electric fields is estimated by comparing the electric-field profile from our finite-element model (Sect. 7.2) with the measured NER shifts and Rabi frequencies. By combining the strain profile from our finite element model with DFT calculations for the strain-EFG coupling, we estimate the portion of the quadrupole interaction that is due to strain arising from the thermal contraction of metallic gates in the device (Sect. 7.4). Concluding this section, we use our finite element model to quantitatively rule out other conceivable physical mechanisms that could lead to NER (Sect. 7.5).

7.1 Microscopic Origins of the Electric Field Gradient The quadrupole interaction is generated by an EFG at the site of the 123 Sb nucleus, as discussed in Sect. 3.3. This EFG is due to inhomogeneities in both the external applied electric field and the internal electric field arising from the charged electrons and nuclei composing the host crystal. The external-field inhomogeneity can be ignored due to the relatively large physical length scales of the gates that supply the external field, as quantitatively supported by a finite element model (Sect. 7.5.1). Here, we focus entirely on a microscopic model for the distortion of the electronic charge distribution around the 123 Sb nucleus. We propose that both strain and externally applied electric fields distort the chemical environment coordinating the 123 Sb nucleus (Fig. 7.1), and thus give rise to a nonzero EFG. The coupling of the externally applied electric field to the EFG provides the physical mechanism that enables the driving of 123 Sb transitions via NER, using an AC electric field (Chap. 6). It also leads to a manifestation of the LQSE (Sect. 7.3.1), in which the quadrupole splitting can be tuned with a DC electric field (Sect. 6.4). We first consider a generic expression that relates the total charge density ρ( r ) of the electrons and nuclei to the EFG tensor V. Taking the 123 Sb nucleus as the origin of coordinates, the EFG tensor components are given by [1] Vαβ =

1 4πε0

 

3rα rβ − δαβ | r |2 5 | r|

 ρ( r )d 3r,

(7.1)

where r = (r x , r y , r z ) is the position vector, δαβ is the Kronecker delta, and α, β ∈ {x, y, z}. Strain and electric fields modify the charge density ρ( r ). Their effects can

7.1 Microscopic Origins of the Electric Field Gradient

111

Fig. 7.1 123 Sb donor in a silicon lattice. The valence charge density (isosurface shown in red) near the ionized 123 Sb atom (gold) and its 16 closest Si atoms (black) is calculated using DFT (Sect. 7.4.2). The donor’s positive charge causes an asymmetric charge density along the 123 Sb-Si bond, but vanishes by symmetry at the 123 Sb site in the absence of strain or external electric fields. The larger core of 123 Sb also slightly displaces the neighboring Si atoms (Sect. 7.4.2)

be included by expanding ρ( r ) as a first-order functional Taylor series about its form at zero electric field and zero strain Vαβ

1 = 4πε0



⎛ ⎞         3rα rβ − δαβ | r |2 ∂ρ( r ) ∂ρ( r )   ⎝ρ( ⎠ 3 + Eγ εγδ r )  ε=0 +  ε=0 d r. ε=0 ∂ E ∂ε | r |5 γ γδ γ  E=0

 E=0

γδ

 E=0

(7.2) Here E γ are external electric field components, and εγδ are strain tensor components, where γ, δ ∈ {x, y, z} index their respective components. This first-order expansion permits a simplified functional form 0 + Vαβ = Vαβ

 γ

Rαβγ E γ +



Sαβγδ εγδ ,

(7.3)

γδ

0 where Vαβ is the EFG arising due to the unperturbed charge density. The third-rank electric-field response tensor Rαβγ captures the relationship between the electric field and the EFG at the 123 Sb nucleus. The fourth-rank gradient-elastic tensor Sαβγδ describes the effect of strain on the EFG at the 123 Sb nucleus. The symmetry properties of the tensors that appear in Eq. (7.3) suggest a more compact representation. The symmetric tensor Vαβ has 6 independent components that we represent using Voigt notation with integer indices, i.e. V1 = Vx x , V2 = V yy , V3 = Vzz , V4 = V yz = Vzy , V5 = Vx z = Vzx , and V6 = Vx y = V yx . Similarly, because R and S obey the same symmetries as Vαβ , they can be represented by 18 and 36 independent components, respectively, rather than 27 and 81. The tetrahedral symmetry Td of the 123 Sb donor site permits an even more dramatic reduction in the number of independent components of R and S that will be discussed in Sect. 7.3.2 and in Sect. 7.4.1.

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We first consider whether the strain- and electric-field-independent contribution, 0 , vanishes by symmetry. A first remark is that any axis of symmetry in the charge Vαβ distribution is also a principal axis of the EFG. If this were not the case, the symmetry operation would generate a new set of principal axes, which would violate the uniqueness property of the principal axes [2]. Secondly, a rotation symmetry larger than two-fold further implies that the principal axes perpendicular to the symmetry axis are degenerate. Since the tetrahedral symmetry (Td ) has four symmetry axes, 0 indeed vanishes each of which is three-fold degenerate, it can be concluded that Vαβ by symmetry. Our measurements indicate a nonzero EFG, so the symmetry of the 123 Sb site must be effectively lowered by strain and/or an external applied electric field. The following sections discuss both the leading-order electric contribution (Sect. 7.3) and strain contribution (Sect. 7.4). It is worth noting that the higher-order mixed terms may be of interest in future experiments. Specifically, shear strains may break symmetries that lead to a permanent dipole moment that can enhance the electric field’s coupling to the EFG, relative to uniaxial strains. However, this would require measurements of samples under different or variable strains to be substantiated.

7.2 Finite-Element Model of the Nanostructure Device The finite-element solver COMSOL is used to simulate the electrostatics and strain in the device due to the different thermal contraction of Si and Al upon cooling down of the device. The specific packages used are the electrostatics package (part of the AC/DC package), and the solid mechanics package, respectively. The model consists of a 2 µm × 2 µm × 2 µm silicon substrate that is mechanically fixed and electrically grounded at the bottom. An 8 nm thick SiO2 layer is defined on top of the silicon, and the aluminum gates are defined on top of this layer. Following the fabrication procedure described in Sect. 4.3, the aluminum consists of a 20 nm bottom layer and 40 nm top layer, and is covered by a 2 nm aluminum oxide layer. The gate layout is matched to that of the actual device upon its imaging (Fig. 4.1), including slightly misaligned donor gates (Fig. 7.2). In the electrostatic simulations, the gate-induced 2DEG forming the SET leads and island is modeled by a 1 nm thick metallic layer at the 28 Si/SiO2 interface, with the lateral dimensions of the top gate (TG), and gaps of the width of the barrier gates (LB and RB). The two SET leads and island are kept at a potential of 0 V. A tetrahedral mesh was used, with a variable resolution that is maximally 2 nm in the donor implantation region. The structural deformation during the cool-down is modeled in two simulation steps that mimics the device cool-down. The device is assumed to be strain free at 850 ◦C, i.e. the growth temperature of the silicon oxide. At this stage, no aluminum gates are present. In the first step, the 28 Si/SiO2 chip is cooled down from 850 ◦C to 400 ◦C. The aluminum gates are then added to the structure, followed by a forming

7.2 Finite-Element Model of the Nanostructure Device

113

Fig. 7.2 Device geometry in COMSOL model. The gate layout is adapted to match the imaged device in Fig. 4.6. All gates are labeled for identification. The coordinate system on the bottom left indicates the laboratory axes used in this work as well as the silicon crystal orientation. The [001] crystal direction is along the y-axis normal to the device surface and the static magnetic field B0 is applied along the [110] z-axis. The origin of this model is located at the 28 Si/SiO2 interface (not visible here) in the center of the implantation region between the DFL and DFR gates. The dashed line marks the y − z cut-plane at x = 13 nm that cuts through the most probable donor position (Sect. 5.2). This plane is chosen to show simulations results throughout this chapter

gas anneal of the Al/Alx O y gates. We assume that the 400 ◦C forming gas anneal strain-relaxes the aluminum structures at this temperature. In the second step, the deformation is modeled during the cool-down from 400 ◦C to 200 mK, including the initial strain of the 28 Si/SiO2 . Note that below ∼40 K, the thermal deformation has saturated, and further cool-down has a negligible effect (see references in Table A.1 for details). The aluminum oxide is assumed to be isotropic. The thermal expansion coefficients used in the strain simulations can be found in Appendix A. All other material properties are taken from the COMSOL material library.

7.3 Electric-Field-Induced Quadrupole Splitting and NER 7.3.1 The Linear Quadrupole Stark Effect (LQSE) The observed modulation of the quadrupole interaction by an external electric field means that the electric fields must somehow be transduced into significant EFGs. A theoretical idea in this direction was proposed by Bloembergen as early as 1961 [3]:

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7 Microscopic Crystalline Origins of the Quadrupole Interaction

If a nucleus or paramagnetic ion with spin greater than one-half occupies a site in a crystal lattice without inversion symmetry, an external applied electric field will cause, in general, a change in the quadrupole coupling constant or crystal-line field splitting proportional to [the electric field] E.

The effect, later termed the LQSE [4], was first invoked to explain contemporary observations of electric-field-induced shifts in the zero-field quadrupolar spectra of ionic crystals containing halogen nuclei (81 Br [5] and 35 Cl [6, 7]). Subsequently, it was measured for 27 Al in Al2 O3 [8], and 69 Ga, 71 Ga, and 75 As in GaAs [9]. The same quadrupole modulation can induce transitions between nuclear states, resulting in NER [3, 6]. This resonant version of the LQSE has been shown in GaAs [10, 11], Al2 O3 [12], and Na in NaClO3 [13] and in NACl [14]. The coherent driving via NER was only demonstrated more recently in a bulk GaAs crystal [15]. The physical mechanism by which the external applied electric field couples to the nuclear quadrupolar interaction is through a distortion of the chemical environment of a given nucleus, which gives rise to a change in the EFG at the nucleus. Typical measured proportionalities between the external applied electric field and the resulting EFG are on the order of 1012 m−1 [6, 9, 16]. The LQSE requires the nucleus to lie on a site that lacks inversion symmetry [3, 9]. Because the tetrahedral symmetry group (Td ) of the 123 Sb donor site is not inversion symmetric, it is plausible in principle that such an electric-field-induced shift can explain our observations. It may nevertheless sound surprising that this effect would be observed in silicon, because even though the donor site lacks point inversion symmetry, silicon is a non-polar crystal. However, as discussed by Gill et al. [9], there are two sources contributing to the LQSE. The first contribution is due to a polar crystal comprising different ionic nuclear species. These nuclear species undergo a relative displacement in response to an external electric field, giving rise to a permanent dipole interaction, i.e. piezoelectricity. While this effect is not present in the silicon crystal, an external electric field also distorts the covalent bonds, mixing the electronic ground state of the ionized 123 Sb donor with excited states of opposite parity. These two states differ in their charge distribution, which gives rise to a transition dipole moment between them (Fig. 7.3). This polarization leads to a symmetry breaking in ρ( r ) that gives rise to a linear coupling between the electric field and the EFG through the R tensor. Additionally, the tetrahedral symmetry can be broken by lattice strain, and can cause a permanent dipole moment. This direct coupling between the electric field and the EFG is the physical mechanism that allows us to drive NER with an external applied AC electric field. That this coupling is present and of the correct order of magnitude to rationalize our observations is further substantiated by the observed shift of the nuclear spectral lines in response to a DC electric field. This shift is caused by the same coupling mechanism, and in the NER literature, the resulting shift in transition frequencies is known as the LQSE. We provide an estimate of this coupling from our measurements of the DC electric field induced shifts in Sect. 7.3.3 and a comparison to measurements of the LQSE in bulk crystals in Sect. 7.3.4.

7.3 Electric-Field-Induced Quadrupole Splitting and NER

115

Fig. 7.3 Electric-field-induced distortion of covalent bonds. The electric field (green) affects the covalent bonds (red) between the ionized 123 Sb donor (gold) and neighboring Si atoms (black). This creates a transition dipole moment, which results in an EFG at the 123 Sb nucleus, and consequently a modulation of the quadrupole interaction. This transducing mechanism is characterized by the electric-field response tensor R, and is the likely cause of the observed LQSE and NER

7.3.2 The Electric-Field Response Tensor The effect of an applied electric field E on the EFG can be represented by the thirdrank R-tensor as  V = R E, (7.4) where V is the EFG tensor induced by the electric field. The structure of the R-tensor is dependent on the crystal symmetry. For a lattice site with Td symmetry, the R-tensor is fully described by one unique nonzero element R14 = R25 = R36 [11].1 In the crystal frame given by the axes x  along [010], y  along [001], and z  along [100],2 the R-tensor equals ⎛ ⎞ ⎛ Vx  x  0 ⎜V y  y  ⎟ ⎜ 0 ⎜ ⎟ ⎜ ⎜ Vz  z  ⎟ ⎜ 0 ⎜ ⎟ ⎜ ⎜ V y  z  ⎟ = ⎜ R14 ⎜ ⎟ ⎜ ⎝ Vx  z  ⎠ ⎝ 0 Vx  y  0

  V 

1

0 0 0 0 R14 0  R 

⎞ 0 ⎛ ⎞ 0 ⎟ ⎟ E x  ⎟ 0 ⎟⎝ ⎠ E y  , 0 ⎟ ⎟ E z  0 ⎠   R14 E  

(7.5)

The first index refers to the electric field component at the donor site in the crystallographic frame (i.e. x  = 1, y  = 2, z  = 3) and the second index refers to the resulting EFG component in Voigt notation. 2 The axes of this crystal frame are a permutation of the standard crystal axes. These permuted axes have a simplified transformation to the laboratory frame by a rotation of π/4 along y  (Eq. (7.6)).

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7 Microscopic Crystalline Origins of the Quadrupole Interaction

where the EFG tensor V  , the R-tensor R  , and the electric-field vector E  are all in the crystal frame. Note that in the crystal frame, the R-tensor only give rise to off-diagonal contributions to the EFG tensor. The structure of the R-tensor is different in the laboratory frame, which has axes ¯ x along [110], y along [010], and z along [110]. The laboratory frame is transformed into the crystal frame by a −π/4 rotation along the y axis, as represented by the unitary transformation U(1) as ⎞⎛ ⎞ ⎛  ⎞ ⎛ √1 √1 0 x x 2 2 ⎟ ⎝ y  ⎠ = ⎜ ⎝ 0 1 0 ⎠ ⎝y⎠ . z z  − √12 0 √12  

(7.6)

U(1)

Similarly, the laboratory-frame EFG-tensor V transforms into the crystal-frame V  by the transformation matrix U(2) as ⎞ ⎛ 1 Vx  x  2 ⎜V y  y  ⎟ ⎜ 0 ⎟ ⎜ ⎜ 1 ⎜ Vz  z  ⎟ ⎜ 2 ⎟=⎜ ⎜ ⎜ ⎜ V y  z  ⎟ ⎜ 0 ⎟ ⎜ ⎝ Vx  z  ⎠ ⎜ ⎝ − 21 Vx  y  0

   ⎛

V

⎞⎛ ⎞ 0 21 0 1 0 Vx x ⎜ 10 0 0 0 ⎟ ⎟ ⎜V yy ⎟ ⎟ 0 21 0 −1 0 ⎟ Vzz ⎟ ⎟⎜ ⎜ ⎟. 0 0 √12 0 − √12 ⎟ V yz ⎟ ⎟⎜ ⎜ ⎟ ⎟ 0 21 0 0 0 ⎠ ⎝ Vx z ⎠ Vx y 0 0 √12 0 √12     U(2)

(7.7)

V

The relations V  = U(2) V and E  = U(1) E can be used to obtain the laboratory-frame R-tensor as follows V  = R  E  ,  U(2) V = R  U(1) E, −1   V = U(2) R U(1) E,

  R

⎛ ⎞ ⎛ Vx x 0 −R14 ⎜V yy ⎟ ⎜ 0 0 ⎜ ⎟ ⎜ ⎜ Vzz ⎟ ⎜ 0 R 14 ⎜ ⎟=⎜ ⎜ V yz ⎟ ⎜ 0 0 ⎜ ⎟ ⎜ ⎝ Vx z ⎠ ⎝ 0 0 Vx y −R14 0

   V

⎞ 0 ⎛ ⎞ 0 ⎟ ⎟ Ex ⎟ 0 ⎟⎝ ⎠ Ey . R14 ⎟ ⎟ Ez 0 ⎠   0 E 

(7.8)

R

We note that the above assumption of Td symmetry is not entirely obvious, as strain breaks this symmetry. However, to first order, the effects of electric field and strain

7.3 Electric-Field-Induced Quadrupole Splitting and NER

117

are independent; the development of a more elaborate theory that takes mixing terms into account is a topic of future theoretical research. The R-tensor can be used to derive analytic expressions for the electric-field dependence of the m I = ±1 (Sect. 6.2.2) and m I = ±2 transition Rabi frequencies (Sect. 6.2.3) under NER. Applying a sinusoidal pulse to an electric gate gate with peak amplitude VRF will cause an electric-field modulation with amplitude δ E = {δ E x , δ E y , δ E z } at the donor position. This in turn gives rise to the EFG modulation δVαβ , postulated in Sect. 6.2. In the laboratory frame, the quadrupole terms responsible for the m I = ±1 transition Rabi frequencies (Eq. (6.8)) can be written in terms of the EFG modulations as      eqn NER 2 + δV 2  , αm −1↔m (7.9) = δV f mRabi, I xz yz   I I −1↔m I 2I (2I − 1)h which in turn relate via the R-tensor (Eq. (7.8)) to electric-field modulations δ E as NER f mRabi, I −1↔m I

    eqn  R14 δ E z  . = αm I −1↔m I 2I (2I − 1)h

(7.10)

Note that only the electric field component δ E z along the applied magnetic field contributes. Repeating the procedure for the m I = ±2 transition Rabi frequencies (Eq. (6.11)), we find      eqn NER 2 + 4δV 2  , βm −2↔m = (δV − δV ) f mRabi, xx yy I xy  I I −2↔m I 2I (2I − 1)h      eqn = βm I −2↔m I R14 (δ E y )2 + 4(δ E x )2  . 2I (2I − 1)h

(7.11)

The same approach can be applied to derive an analytic expression for the shift of the quadrupole splitting with electric field (Eq. 3.14) eqn (Vx x + V yy − 2Vzz ), 2I (2I − 1)h eqn (−3)R14 E y . = 2I (2I − 1)h

 fQ =

(7.12)

Note that when the external magnetic field B0 is oriented along one of the crystallographic axes, the electric-field shift of the quadrupole splitting  f Q is always zero. This is because the R-tensor in the crystal frame (Eq. (7.5)) has no elements coupling electric fields to Vx x , V yy , or Vzz (Eq. (7.12)). This may prove useful for extracting the quadrupole splitting f Q that is solely caused by strain.

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7.3.3 Electric-Field Response-Tensor Estimate The measured Rabi frequencies and quadrupole shifts can be used to estimate R14 by comparing them to the electric fields at the triangulated donor position (Sect. 5.2). These electric fields are extracted from finite-element modeling in COMSOL (Sect. 7.2). To compare the measured Rabi frequencies, the electric response is gate calculated when a peak voltage VRF = 20 mV of an RF pulse is applied to the DFR gate, matching the experimental conditions. The relevant electric-field components determining the m I = ±1 (Eq. (7.10)) and m I = ±2 (Eq. (7.11)) Rabi frequencies are shown across the device in Fig. 7.4A and Fig. 7.4C. Similarly, the relevant LQSE electric-field component (Eq. (7.12)) is the out-of-plane DC electric-field shift gate E y , which is modeled across the device for increased gate voltage VDC = 20 mV applied to the DFL gate (Fig. 7.4E). The unknown constant R14 can be estimated by matching the simulated electric fields to the experimental values. The measured NER Rabi frequencies are gate Rabi,NER Rabi,NER = 684 Hz and f 3/2↔7/2 = 38.5 Hz for VRF = 20 mV drive amplitude f 5/2↔7/2 applied to the DFR gate (Fig. 6.8). Similarly, a spectral shift  f Q = 200 Hz was gate observed in response to a DC bias voltage VDC = 20 mV applied to the DFL gate (Fig. 6.11). The simulated electric field at the triangulated donor position matches the measured m I = ±1 Rabi frequencies when R14 = 1.7 × 1012 m−1 (Fig. 6.11A, B). Similarly, agreement with the measured m I = ±2 Rabi frequencies is found when R14 = 0.7 × 1012 m−1 (Fig. 6.11C, D), and the quadrupole shift matches when R14 = 10 × 1012 m−1 (Fig. 6.11E, F). Although there is a significant spread in the R14 values, the results can be considered respectable given the complexity of the models and the absence of any free parameters in them. We combine these results and determine a unique value of R14 at the triangulated donor position by minimizing the sum of the square of the normalized residuals χ2 ( r) =

   Mi − R14 ( r ) · Si ( r) 2 , σ Mi i

(7.13)

where Mi is the observed measurement value, i.e. the left-hand side in Eqs. (7.10), r ) is the corresponding right-hand side using the electric(7.11), or Eq. (7.12), and Si ( r ) is the targeted minimization field components found at the donor location. R14 ( value. σ Mi is the uncertainty of the measurement value Mi . Assigning an equal measurement error to each observation, we find R14 = 1.7 × 1012 m−1 at the most likely donor position. This estimate is in good agreement with the observed m I ± 1 and m I = ±2 Rabi frequencies (Fig. 7.5). However, the resulting quadrupole shift  f Q = 34 Hz is significantly lower than the observed  f Q = 200 Hz, reflecting the spread in R-tensor estimates. The observed quadrupole splitting f Q in the spectrum is largely caused by static strain (Sect. 7.4). However, a small part of this should be due to the LQSE, as static electric fields are present, caused by the gate voltages required for device operation. Using the static electric field from our COMSOL model and the value of R14 we

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119

Fig. 7.4 Electric fields at the ionized 123 Sb donor. Electric fields are calculated using COMSOL (Sect. 7.2). The y − z cut-plane is shown along x = 13 nm as indicated in Fig. 7.2; the cross indicates the most likely donor position and contour lines correspond to the 68% and 95% confidence intervals for the triangulated donor  position (Sect. 5.2) (A, C) Spatial dependence of the electric-field components δ E z (A) and

δ E y2 + 4δ E x2 (C) upon varying the DFR gate (right) by

gate VRF

= 20 mV. This is the gate varied in the experiment to achieve NER, and these electric field components are responsible for the m I = ±1 and m I = ±2 transitions, respectively. (E) Spatial gate dependence of the out-of-plane electric field difference E y upon applying VDC = 20 mV to the DFL gate (left), which is responsible for the measured quadrupole shift  f Q = 200 Hz (Sect. 6.4). (B, C, D) Spatial dependence of R14 . Each value is found by solving Eq. (7.12) (B), Eq. (7.10) (D) or Eq. (7.11) (F) for R14 using the measured Rabi frequencies and spectral line shifts. At the most likely donor position we find R14 = 1.7 × 1012 m−1 (B), R14 = 0.7 × 1012 m−1 (D), and R14 = 10 × 1012 m−1 (F)

found, we estimate an electric contribution to f Q of 1.7 kHz, i.e. ∼ 2.5% of the experimentally observed f Q = 65.5 kHz (Sect. 6.2.2). This electric-field contribution to f Q increases to 10 kHz, or 15%, when using the higher R14 estimate from the quadrupole shifts only. Although the electric-field contribution might be significant, this suggests that the largest fraction of f Q is still caused by strain. In what follows we aim to justify the value of R14 on the basis of a microscopic theory.

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Fig. 7.5 Comparison between measured Rabi frequencies and R14 estimate.Calculation of the NER Rabi frequencies caused by electrical EFG modulation (green lines), compared to experimental results (Fig. 6.8) of the m I = ±1 transition |5/2 ↔ |7/2 (dots), and the m I = ±2 transition |5/2 ↔ |7/2 (squares). The calculated f Rabi are obtained by combining the simulated electric fields at the triangulated donor position with the estimated R14 = 1.7 × 1012 m−1 , which is calculated via finite-element modeling and electronic structure theory. No free fitting parameters were used

7.3.4 Comparison to LQSE Measurements and Empirical Theory The manifestation of LQSE in our experiment is distinguished from previous measurements in three important aspects. First, both the LQSE and NER have not previously been observed in a single nucleus. Second, the nuclear quadrupole moment is between 2 (75 As) and 8 (35 Cl) times larger than any previous measurements. Finally, the effect is observed in a host material that is non-polar in bulk. We stress that the necessary condition for LQSE of broken point inversion symmetry is naturally fulfilled by silicon; the Td symmetry of Si lacks point inversion symmetry at the lattice sites. Comparison to the GaAs measurements is the most relevant because the individual nuclei experience the same tetrahedral coordination as our 123 Sb donor. In the preceding section (Sect. 7.3.3) we estimated a value of R14 = 1.7 × 1012 m−1 for the 123 Sb substitutional donor in silicon under study here. This is comparable to the values reported for 75 As in bulk GaAs: 1.55 × 1012 m−1 [9], 2.0 × 1012 m−1 [11], and 3.16 × 1012 m−1 [16]. The microscopic theory used to rationalize the bulk GaAs measurements of Gill and Bloembergen [9] decomposes the electric-field response tensor into ionic and ion cov + R14 . The ionic contribution accounts for the covalent contributions, R14 = R14 EFG due to an electric-field-induced distortion of the crystal lattice (i.e. piezoelectricity), whereas the covalent contribution accounts for the EFG due to an electric-fieldinduced distortion of the valence orbitals. In the case of 75 As, their theory predicts that roughly 2/3 of the measured value of R14 is covalent, while the remaining 1/3 cov ion /R14 ≈ 2/3 and R14 /R14 ≈ 1/3. Because the silicon host crystal is ionic, i.e. R14 is non-polar, we anticipate that our inferred value of R14 is almost entirely covalent,

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121

cov R14 ≈ R14 . Thus, in comparing the strength of the LQSE in our experiment to bulk GaAs, we compare to Gill and Bloembergen’s covalent value of 1.1 × 1012 m−1 [9]. To account for the difference in nuclear quadrupole moment qn , nuclear spin I , we introduce the nucleus-specific constant of proportionality

K cov =

cov qn R14 , 2I (2I − 1)

(7.14)

which directly relates the electric field to the quadrupole interaction strength. We find K cov values of 5.8 × 10−18 m for the covalent contribution to 75 As in GaAs and 2.8 × 10−18 m for 123 Sb in silicon. That K cov is higher for 75 As than for 123 Sb supports the claim that the DC-bias-induced shift in the NER spectral lines (Fig. 6.11) is due to the LQSE. This also rationalizes the observed m I = ±1 and m I = ±2 NER Rabi frequencies, as these also depend on electric fields, albeit different components (Sect. 7.3.3). The transition dipole moment of 123 Sb is estimated to be on the order of 0.1 eV Å, as obtained from preliminary quantum chemistry calculations on cationic hydrogenterminated Sb-doped crystalline Si clusters using the NWChem software package [17]. This is an order of magnitude lower than the transition dipole moment 1.5 eV Å of GaAs used by Gill and Bloembergen [9], and would suggest that R14 should decrease accordingly. However, since the 123 Sb electrons lie in a 5p orbital, as opposed to the 4p orbital of 75 As in GaAs, it is expected that the EFG response at the nucleus increases. The Sternheimer anti-shielding effect can also be invoked for even further increases, particularly due to the positive ionization state of the 123 Sb and the observation that the effect generally grows with atomic number [18]. These combined effects allow some compensation for the lower transition dipole coupling of 123 Sb It is worth noting again that the sensitivity of our inference of R14 to details in the COMSOL model suggests that we should interpret it as an order of magnitude estimate. Further experiments and the development of a first-principles theory will provide more clarity. Such a theory is a topic of ongoing theoretical work. While there are reasonably mature first-principles methods for computing the electric field gradient with and without strain (see Sect. 7.4.2), the direct evaluation of the LQSE parameters has not yet been demonstrated. Such a theory would circumvent the need for a description in terms of empirical parameters like bond ionicities, atomic EFGs, and Sternheimer factors. Another item of interest for a more detailed theory is the possibility of a second-order coupling of the EFG to both the strain and electric field, due to the development of a permanent electric dipole moment under shear strains as alluded to in Sect. 7.1.

7.4 Strain-Induced Quadrupole Splitting Both uniaxial and shear strains will change the positions of the silicon atoms that coordinate the 123 Sb donor such that the Td symmetry of the substitutional site is broken. This in turn gives rise to significant EFGs, and consequently a quadrupole splitting.

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Fig. 7.6 Shear-strain-induced distortion of covalent bonds. Shear strain displaces the Si atoms (black) and covalent bonds (red) neighboring the 123 Sb nucleus (gold), creating an EFG which results in a quadrupole shift. The quadrupole shift is also affected by uniaxial strain, though to a lesser extent

Uniaxial strain will reduce the site’s symmetry to D2d , giving rise to a linear coupling between the strain and the diagonal components of the EFG tensor. Shear strain will reduce the site’s symmetry to C2v (Fig. 7.6), giving rise to a linear coupling between the strain and the off-diagonal components of the EFG tensor. These couplings correspond to the two non-trivial components S11 and S44 of the gradientelastic strain tensor S for Td symmetry, as discussed in Sect. 7.4.1. DFT calculations of these couplings are given in Sect. 7.4.2. These calculations include the direct evaluation of the EFG as a function of strain and confirm a linear relationship that extrapolates to zero in the absence of strain. These couplings are then incorporated in a detailed model of the device strain, and the resulting quadrupole splitting is compared to the observed splitting in Sect. 7.4.3.

7.4.1 The Gradient-Elastic Tensor For tetrahedral symmetry (Td ), the gradient-elastic tensor S has two different nonzero components S11 and S44 , corresponding to uniaxial and shear strain respectively. Here, Voigt notation in the conventional crystallographic, non-rotated basis is used. We adopt engineering strain measures, i.e. strain is defined as the ratio between the deformation and the original dimensions. In this frame (given by x  , y  , z  , see Sect. 7.1), the EFG tensor is given by ⎛ ⎞ ⎛ S11 −S11 /2 −S11 /2 0 0 Vx  x  ⎜V y  y  ⎟ ⎜−S11 /2 S11 −S11 /2 0 0 ⎜ ⎟ ⎜ ⎜ Vz  z  ⎟ ⎜−S11 /2 −S11 /2 S11 0 0 ⎜ ⎟ ⎜ ⎜ V y  z  ⎟ = ⎜ 0 0 0 S 44 0 ⎜ ⎟ ⎜ ⎝ Vx  z  ⎠ ⎝ 0 0 0 0 S44 Vx  y  0 0 0 0 0

⎞⎛ ⎞ εx  x  0 ⎜ ⎟ 0⎟ ⎟ ⎜ε y  y  ⎟ ⎜ εz  z  ⎟ 0⎟ ⎟⎜ ⎟. ⎜ ⎟ 0⎟ ⎟ ⎜ γ y  z  ⎟ 0 ⎠ ⎝ γx  z  ⎠ γx  y  S44

(7.15)

Note that the relationship between shear components of the engineering and infinitesimal strain is γ y  z  = 2ε y  z  . In the experiment, the magnetic field is applied along the

7.4 Strain-Induced Quadrupole Splitting

123

[110]√crystal direction, which in the crystal frame corresponds to Hˆ = γn B0 ( Iˆx  + Iˆz  )/ 2. Computing the first-order perturbative correction, as in Sect. 3.3.3, yields the quadrupole splitting fQ =

 3 eqn S11 (2ε y  y  − εx  x  − εz  z  ) − 4S44 γx  z  , 2I (2I − 1)h 4

(7.16)

which transforms into the laboratory frame (Sect. 7.1) as fQ =

 3 eqn S11 (2ε yy − εx x − εzz ) − 4S44 (εx x − εzz ) . 2I (2I − 1)h 4

(7.17)

7.4.2 Gradient-Elastic Tensor Calculations We use Kohn-Sham density functional theory (DFT) to develop an atomistic understanding of the impact that strain has on the EFG at the ionized 123 Sb donor. We perform a series of supercell calculations to determine the manner in which the silicon atoms coordinating the 123 Sb site relax as a function of strain. These calculations also provide us with first-principles values for the S tensor that avoid the need for empirically derived Sternheimer factors. Additional simulation details are found in Appendix E. We self-consistently relax the Si crystal geometry around the ionized 123 Sb donor for both uniaxial strain on [100] and shear strain at 5 different values (−10−2 , 3×10−3 , −10−3 , −3 × 10−4 , and −10−4 ). Our starting point for each calculation is a pristine Si crystal with the strain applied relative to the experimental lattice constant (5.431 Å) and a single substitutional 123 Sb donor. To account for the +1 ionization state of the Sb donor, we include 4 electrons per atom in our supercells, and a homogeneous background charge is assumed to maintain the overall neutrality necessary to obtain a finite total energy. We relax the atomic positions at a fixed −1 supercell volume until the inter-atomic forces are all less than 1 meV Å , and find 123 that in all cases the primary impact of the ionized Sb donor is to displace the 4 silicon atoms that coordinate it by ∼ 0.2 Å. The next-nearest neighbors are displaced by ∼0.05 Å, and beyond that all displacements are below ∼ 0.025 Å. This is consistent with a simple picture in which the ionized Sb occupies a larger volume than neutral Si by merit of its filled M and N shells. For each strain type and amplitude we compute the EFG and perform a linear regression on the computed values to extract estimates for S11 and S44 . For the case of uniaxial strain, we evaluate S11 for two Poisson ratio’s, i.e. the ratio of transverse extension strain to longitudinal contraction strain in the direction of the contraction force. For a Poisson ratio of 0 (i.e. unphysical “cork-like” Si), the resulting value of S11 is 1.9 ×1022 Vm−2 , while for the experimental bulk Poisson ratio of 0.28, S11 = 2.4×1022 Vm−2 . For shear strain, we compute S44 to be 6.1×1022 Vm−2 . These values are comparable to those reported for As in Si by Franke et al., which are

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7 Microscopic Crystalline Origins of the Quadrupole Interaction

respectively 1.5×1022 Vm−2 and 6.0×1022 Vm−2 [19]. That the ratio of the shear component to the uniaxial component is larger for As than Sb is to be expected because Sb occupies a larger volume than As and will thus have an EFG that couples more strongly to strains that change volume. The relaxed supercell geometries expose microscopic details of the symmetry breaking at the 123 Sb site. In the absence of strain, this site has the symmetry of the point group Td , which is apparent from the tetrahedral coordination of the four nearest silicon atoms. Uniaxial strain reduces the symmetry of this site to D2d , whereas shear strain reduces it to C2v . These strain-induced symmetry reductions may invalidate the first-order approximation of Eq. (7.3) that strain and electric fields separately affect the EFG. Instead, higher-order contributions may become significant where strain and electric fields are coupled.

7.4.3 Calculation of Quadrupole Splitting Due to Strain Because the S-tensor component of shear strain is stronger than that of uniaxial strain (S44 /S11 ≈ 2.5), shear strain along the magnetic field aligned with the [110] axis has the strongest influence on the quadrupole splitting f Q (Eq (7.17)). The thermal deformation is computed using COMSOL, and the resulting shear strain εx x − εzz is shown in Fig. 7.7A. The computed strains are related to the EFG via the S-tensor (Eq. (7.15)), which allows calculation of the eigenenergies (Eq. 3.6) and consequently the quadrupole splitting f Q (Fig. 7.7B). The variation in quadrupole splitting shows a good qualitative correspondence with the variation of the shear strain, reflecting the larger contribution of S44 to the quadrupole splitting. Values found for the strain-induced quadrupole splitting range between 90 and 160 kHz

Fig. 7.7 Simulated device strain and resulting quadrupole splitting f Q . (A) Simulated shear strain in the vicinity of the donor. The strain, calculated using COMSOL, is predominantly due to the much higher thermal expansion coefficient of aluminum. (B) Quadrupole splitting f Q of an ionized 123 Sb donor due to simulated uniaxial and shear strain. Both shear strain and uniaxial strain result in a quadrupole splitting f Q , though shear strain has a stronger effect due to its S-tensor component (S44 = 6.1 × 1022 V m−2 ) being higher than that of uniaxial strain (S11 = 2.4 × 1022 V m−2 ). The y − z cut-plane is shown along x = 13 nm as indicated in Fig. 7.2; the cross indicates the most likely donor position and contour lines correspond to the 68% and 95% confidence intervals for the triangulated donor position (Sect. 5.2)

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125

throughout the area where the donor is estimated to be. This result is in good agreement with the experimentally-observed value of f Q = 66 kHz, especially in light of the complex combination of strain modeling, DFT calculations and donor triangulation required in our modeling effort.

7.5 Alternative (Unlikely) Sources of NER Two possible alternative explanations for the origin of a dynamical EFG are considered here, both of which are found to be in disagreement with the observed NER rate by orders of magnitude. This further corroborates the claim that our data is most convincingly explained by an electric-field-induced modulation of the nuclear quadrupole tensor, caused by a local distortion of the atomic bonds.

7.5.1 Direct Effect of Electric Gate Potentials The electric gate potentials directly lead to an EFG at the donor site, since the produced electric field is spatially inhomogeneous. To estimate the scale of this effect, the direct EFG has been calculated using electrostatic COMSOL simulations in the absence of any other EFG-generating effects such as strain or the LQSE. The EFG due to the gate potentials at the tuning point has been converted to a quadrupole splitting f Q (Fig. 7.8A). Additionally, the spectral line shift  f Q has gate been calculated for a DC bias potential VDC = 1 V (Fig. 7.8B), as well as the NER gate Rabi,NER Rabi frequency f 5/2↔7/2 for a peak drive amplitude VRF = 20 mV (Fig. 7.8C). Their strengths are in the 100 µHz, 1 mHz V−1 and 1 µHz range, respectively, as opposed to the experimentally observed values of the order of 100 kHz, 10 kHz V−1 and 1 kHz. Although some enhancement of this EFG is predicted at the 123 Sb nucleus due to the Sternheimer anti-shielding effect, this is not expected to surpass two orders of magnitude. As the direct effect of the electric gates results in an underestimation of the experimental observations by at least six orders of magnitude, we conclude this effect is insignificant. This illustrates that in our experiment a significant quadrupole interaction and NER strength requires a microscopic mechanism to be at play in direct vicinity of the donor site.

7.5.2 Mechanical Driving Through SiO2 Piezoelectricity Recent results by Lazovski et al. [20] suggest that thin-film SiO2 thermally grown on Si can exhibit piezoelectric properties in the first few monolayers of the SiO2 film. This piezoelectric layer would cause a local deformation of the sample due to varying

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7 Microscopic Crystalline Origins of the Quadrupole Interaction

Fig. 7.8 Effect of the direct EFG from electric gate potentials. The EFG considered here is solely due to the inhomogeneous electric field from the gate potentials, ignoring strain and LQSE contributions. The y − z cut-plane is shown along x = 13 nm (Fig. 7.2). The contour lines correspond to the 68% and 95% confidence intervals for the triangulated donor position (Sect. 5.2). The quadrupole splitting f Q (A) is calculated for the gate voltages used in the experiment (Table A.1). Small bias voltages are applied to determine the line shift per applied volt on gate DFL,  f Q /VDFL (B), and Rabi, NER the NER driving strength for 20 mV driving amplitude on gate DFR, f 5/2↔7/2 (C). All values are more than 6 orders of magnitude smaller than the experimentally-observed values, confirming that this effect is entirely negligible

electric gate potentials. The resulting strain modulation generates a time-dependent EFG (Eq. (7.15)), which in turn modulates the quadrupole interaction (Eq. (6.4)). This would enable electrical driving of the nuclear spin. The measured mechanical displacement was found to coincide with an initial 3.5 nm thick quartz layer [20]. We therefore incorporate this quartz layer at the Si/SiO2 interface in our COMSOL model. This requires adjusting the orientation of the piezoelectric and stiffness matrices to coincide with the silicon crystal axes. The crystal structure of α-quartz is trigonal trapezohedral. Note that the quartz crystal directions [100]quartz , [010]quartz , and [001]quartz do not correspond to the chosen silicon crystal axes x  along [010], y  along [001], and z  along [100] (Sect. 7.2). In the x  − y  plane, the quartz unit cell has the shape of an equilateral parallelogram (a = b = 4.91 Å), with an angle γ = 120◦ between the crystal directions [100]quartz and [010]quartz . The silicon x  and y  axes therefore correspond to the [010]quartz and [210]quartz crystal directions, respectively. Perpendicular to these (α = β = 90◦ ) extends the [001]quartz direction of the unit cell with a length of c = 5.40 Å. This is the exceptional three-fold rotation axis of α-quartz. Ref. [21] found that the growth direction of α-quartz on a (100) silicon substrate, using our crystal axis, is [210]quartz  [001]silicon . Hence in our model the exceptional [001]quartz axis will be in plane and

7.5 Alternative (Unlikely) Sources of NER

127

Fig. 7.9 Effect of a piezoelectric quartz layer at the 28 Si/SiO2 interface. A possible piezoelectric drive is modeled by assuming a 3.5 nm thick, piezo-electric, quartz layer (blue) at the 28 Si/SiO interface, instead of a fully amorphous SiO layer. The experimental time-varying electric 2 2 potential δVDFR = 20 mV is applied to the DFR gate (right). The resulting strain is calculated using the COMSOL model and subsequently converted into quadrupole interaction strength using S11 and S44 as found in DFT calculations (Sect. 7.4.2.). Shown is the y − z cut-plane along x = 13 nm, as indicated in Fig. 7.2. The contour lines correspond to the 68% and 95% confidence intervals for the Rabi,NER triangulated donor position (Sect. 5.2). The piezoelectrically-driven Rabi frequency f 5/2↔7/2 = 5 − 10 Hz is two orders of magnitude smaller than the experimental value of 684 Hz. This indicates that dynamical strain is an unlikely cause of NER in our device

aligned with the [100]silicon or [010]silicon directions. The results for different quartz orientations were all within an order of magnitude. The first orientation is considered in what follows. A time-varying potential δVDFR = 20 mV applied to the DFR gate results in the Rabi frequencies shown in Fig. 7.9. The estimated Rabi frequency on resonance is 5 to 10 Hz at the expected donor position, two orders of magnitude smaller than Rabi,NER = 684 Hz. We therefore conclude that the experimentally observed value f 5/2↔7/2 piezoelectricity is an unlikely cause of NER.

7.6 Conclusion In Chap. 6, we showed that the experimentally observed nuclear electric resonance is a manifestation of the LQSE. However, until now, the LQSE had only been observed in polar crystal, which silicon is not. To provide a microscopic validation of the presence of the LQSE in our system, a combination of finite-element modeling and donor triangulation has been employed to determine the electric fields at the donor location. The resulting electric-field response-tensor component R14 is found to be in good agreement with existing measurements of LQSE in GaAs. This corroborates that the electric fields are transduced into EFGs through a distortion of the 123 Sb covalent bonds. This mechanism is found to result in both the LQSE and NER depending on the electric-field component. Two main alternative effects, the direct EFG produced

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7 Microscopic Crystalline Origins of the Quadrupole Interaction

by inhomogeneous electric fields, and a thin quartz-like piezoelectricity at the SiO2 , have also been considered. However, the strength of both effects are found to be orders of magnitude too low, leaving the LQSE as the most likely explanation for the observed NER and quadrupole shift. The R14 estimate can be further improved in several ways. First, the NER Rabi frequencies and spectral shifts can be measured for each of the different donor gates. This provides many more data points, and therefore smaller R14 uncertainty margins. Since each of these quantities has a straightforward relation to specific electric-field components, this also allows the extraction of additional information about the device geometry and donor position. Furthermore, a vector magnet would enable varying the orientation of the static magnetic field B0 . This modifies the electric-field components causing NER and LQSE, thus allowing an even more accurate characterization of the electric fields and the R14 component, as well as the static quadrupole parameters (Sect. 3.3.4). Although the LQSE does influence the quadrupole interaction, strain is still the primary contribution to the static quadrupole splitting f Q . The gradient-elastic tensor components S11 and S44 were found to be similar to those reported for 75 As [19], supporting the validity of the DFT simulations. These extracted components were then used in a strain map, computed from a finite-element model of the device geometry incorporating the fabrication process, and the resulting quadrupole splitting matched the observed splitting to within a factor two. The agreement between the simulated and measured quadrupole splitting is rather remarkable considering the intricacies of the simulations. Adding a piezoelectric layer above the donor would open up new avenues for future experiments. For instance, the assumption made in Sect. 7.1 that the electric field and strain independently affect the quadrupole interaction is only valid up to first order. Second-order couplings that depend on the product of the two effects could be detected by measuring the NER Rabi frequencies for varying piezoelectric strain. Our device would be perfectly suited to measure such subtle effects due to its exquisite coherence times and lack of ensemble averaging. Additionally, the strong dependence of the quadrupole interaction on strain enables piezoelectric driving of nuclear transitions.

References 1. Slichter C (2013) Principles of Magnetic Resonance. Springer, Berlin, Heidelberg. https://doi. org/10.1007/978-3-662-12784-1 2. Hunt M (1969) The symmetry of the electric field gradient in tetrahedral environments and its application to 14N pure quadrupole resonance in amino acids. J Mag Resonance 15(1):113– 121, 1974. https://doi.org/10.1016/0022-2364(74)90180-2 3. Bloembergen N (1961) Linear Stark effect in magnetic resonance spectra. Science 133:1363. https://doi.org/10.1126/science.133.3461.1363

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4. Kempf JG, Weitekamp DP (2000) Method for atomic-layer-resolved measurement of polarization fields by nuclear magnetic resonance. J Vac Sci Technol B: Microelectron Nanometer Struct Process Meas Phenom 18(4):2255–2262. https://doi.org/10.1116/1.1305287 5. Kushida T, Saiki K (1961) Shift of nuclear quadrupole resonance frequency by electric field. Phys Rev Lett 7:9–10. https://doi.org/10.1103/PhysRevLett.7.9 6. Armstrong J, Bloembergen N, Gill D (1961) Linear effect of applied electric field on nuclear quadrupole resonance. Phys Rev Lett 7:11–14. https://doi.org/10.1103/PhysRevLett.7.11 7. Armstrong J, Bloembergen N, Gill D (1961) Linear effect of electric field on the 35Cl quadrupole interaction in paradichlorobenzene. J Chem Phys 35(3):1132–1133. https://doi. org/10.1063/1.1701194 8. Dixon RW, Bloembergen N (1964) Linear electric shifts in the nuclear quadrupole interaction in Al2O3. Phys Rev 135:A1669-A1675, 6A. https://doi.org/10.1103/PhysRev.135.A1669 9. Gill D, Bloembergen N (1933) Linear Stark splitting of nuclear spin levels in GaAs. Phys Rev 129:2398–2403. https://doi.org/10.1103/PhysRev.129.2398 10. Brun E, Hann R, Pierce W, Tanttila W (1962) Spin transitions induced by external rf electric field in GaAs. Phys Rev Lett 8(9):365. https://doi.org/10.1103/PhysRevLett.8.365 11. Brun E, Mahler RJ, Mahon H, Pierce WL (1963) Electrically induced nu- clear quadrupole spin transitions in a GaAs single crystal. Phys Rev 129:1965–1970. https://doi.org/10.1103/ PhysRev.129.1965 12. Kushida T, Silver A (1963) Electrically induced nuclear resonance in Al2O3 (ruby). Phys Rev 130(5):1692. https://doi.org/10.1103/PhysRev.130.1692 13. Luukkala M (1964) Electrically induced population inversion of the nuclear magnetic resonance of 23Na in NaClO3. Phys Lett 10:20–21. https://doi.org/10.1016/0031-9163(64)90550-5 14. Meyer WJ, Lang D, Slichter C (1973) Electric field nuclear double resonance in Ag+ -doped NaCl. Phys Rev B 8(5):1924. https://doi.org/10.1103/PhysRevB.8.1924 15. Ono M, Ishihara J, Sato G, Ohno Y, Ohno H (2013) Coherent manipulation of nuclear spins in semiconductors with an electric field. Appl Phys Express 6(3):033 002. https://doi.org/10. 7567/APEX.6.033002 16. Dumas K, Soest J, Sher A, Swiggard E (1979) Electrically induced shifts of the GaAs nuclear spin levels. Phys Rev B 20(11):4406. https://doi.org/10.1103/PhysRevB.20.4406 17. Valiev M, Bylaska E, Govind N, Kowalski K, Straatsma T, Dam HV, Wang D, Nieplocha J, Apra E, Windus T, de Jong W (2010) Nwchem: A compre- hensive and scalable open-source solution for large scale molecular simulations. Comput Phys Commun 181(9):1477–1489. https://doi. org/10.1016/j.cpc.2010.04.018 18. Feiock F, Johnson W (1969) Atomic susceptibilities and shielding factors. Phys Rev 187(1):39. https://doi.org/10.1103/PhysRev.187.39 19. Franke DP, Hrubesch FM, Künzl M, Becker H-W, Itoh KM, Stutz-n mann M, Hoehne F, Dreher L, Brandt MS (2015) Interaction of strain and nuclear spins in silicon: Quadrupolar effects on ionized donors. Phys Rev Lett 115(5):057 601. https://doi.org/10.1103/PhysRevLett. 115.057601 20. Lazovski G, Wachtel E, Lubomirsky I (2012) Detection of a piezoelectric e?ect in thin ?lms of thermally grown SiO2 via lock-in ellipsometry. Appl Phys Lett 100(26): 262 905 (2012). https://doi.org/10.1063/1.4731287 21. Carretero-Genevrier A, Gich M, Picas L, Gazquez J, Drisko G, Boissiere C, Grosso D, Rodriguez-Carvajal J, Sanchez C (2013) Soft-chemistry-based routes to epitaxial α-quartz thin films with tunable textures. Science 340(6134):827–831. https://doi.org/10.1126/science. 1232968

Chapter 8

Exploring Quantum Chaos with a Single High-Spin Nucleus

The emergence of chaos in a quantum system is of integral importance to the understanding of the quantum-classical transition, and yet many aspects of it are unexplored. This is in part due to the scarcity of experiments that can probe the dynamics of quantum-chaotic systems in a time-resolved manner, because the subtle chaotic dynamics are often masked by other prominent effects such as decoherence. High-spin nuclei are prime candidates to shed light on quantum chaos, due to their exquisite coherence and control. In this chapter, we propose the implementation of the quantum driven top—whose classical equivalent is chaotic— in a single high-spin nucleus of a group-V donor atom, such as 123 Sb, in silicon. Two experiments are devised, the first of which seeks a correspondence between classical and quantum dynamics, while the second demonstrates dynamical tunneling, which is classically forbidden and thus exposes the true quantum nature of the system. These would constitute the first ever experiments that probe the dynamics of quantum chaos in a single quantum system, and over unprecedented timescales. This chapter includes results from the following publication: V. Mourik* , S. Asaad* , H. Firgau, J. J. Pla, C. Holmes, G. J. Milburn, J. C. McCallum, A. Morello. “Exploring quantum chaos with a single nuclear spin.” Physical Review E 98.4 (2018): 042206. Copyright 2018 by the American Physical Society. The author acknowledges (i) V. Mourik for studying the driven top with the author, (ii) H. Firgau for optimization of classical simulation code, and (iii) R. B. Liu for suggesting the rotating-frame driven-top Hamiltonian. This chapter starts by providing a background of chaos and quantum chaos (Sect. 8.1). Next, the classical driven top is introduced, and its regions of regular and chaotic dynamics are studied (Sect. 8.2). Using spin as the quantum equivalent of classical angular momentum, the quantum driven top is discussed for high-spin nuclei of © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Asaad, Electrical Control and Quantum Chaos with a High-Spin Nucleus in Silicon, Springer Theses, https://doi.org/10.1007/978-3-030-83473-9_8

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group-V donors in silicon (Sect. 8.3). The quantum driven top has a periodic Hamiltonian; its evolution can be described using the Floquet formalism (Sect. 8.4). Two experiments are proposed to compare the quantum dynamics to the classical equivalent. The first experiment studies the purity of an initial spin coherent state while evolving under a perturbed Hamiltonian, and finds a correspondence with the classical dynamics of the driven top (Sect. 8.5.1). In the second experiment, the evolution of spin coherent states is tracked, and is in some cases found to result in dynamical tunneling (Sect. 8.5.2). This is classically forbidden and is only possible due to the inherent uncertainty of quantum states.

8.1 Background of Quantum Chaos 8.1.1 Introduction Determinism was one of the prevalent ideas in science until the twentieth century. Given the premise that the laws of nature are such that the future is a direct consequence of the present, sufficient knowledge about the present should allow the future to be determined. At the turn of the twentieth century, two discoveries were made that challenged this notion. The first of the two discoveries was that of chaotic dynamics. It was found that for certain systems, any sort of perturbation would have an exponentially amplifying effect on its dynamics as the system evolves. These systems are chaotic, and this observation implied that unless everything about the system is known up to infinite precision, its future cannot be determined accurately for sufficient timescales. Chaos does not fundamentally oppose determinism, but it does place a greater restriction on determinism, namely that “sufficient knowledge about the present” means “infinitely exact knowledge about the present.” The second discovery that challenged determinism was the quantization at the microscopic level, which spawned the field of quantum mechanics. In contrast to classical mechanics, systems at the microscopic level can exist in a superposition of mutually-exclusive states. Measuring such a quantum system can collapse this superposition, forcing the system into a discrete state. Quantum systems are described by their wave functions, which contain a probability distribution of all the possible measurement outcomes. Even if the wave function of a quantum system is known exactly, and hence the probability distribution for obtaining each of the possible final states, the individual occurrence of a specific final state is truly probabilistic and therefore unpredictable.1 The probabilistic nature of quantum physics is fundamentally incompatible with determinism. Much work has been done in explaining why we do not notice these quantum effects in our daily life. The correspondence principle states that classical mechan1

This statement does not hold if quantum physics is underpinned by a hidden-variable theory [1].

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ics is obtained as the limiting case of quantum mechanics for sufficiently large systems [2, 3]. However, it is not clearly understood how the correspondence principle can be applied to non-integrable, i.e. chaotic systems. This topic lies at the heart of the field known as quantum chaos. It is one of the major questions that needs to be answered to truly harmonize quantum and classical mechanics.

8.1.2 Chaos Theory When Poincaré was studying the classical three-body problem at the turn of the 20th century, he showed that analytically-conserved quantities of momentum and position could not exist [4, 5]. His results led him to the conclusion that small perturbations can have an amplifying effect as the system evolves. Although theoretical work on nonlinear systems evolved considerably since the work of Poincaré [6], it took several decades and the advent of the computer before these discoveries culminated in a distinct field known as chaos theory. In 1963, Edward Lorenz was performing computer simulations on a weather model, when he discovered that interrupting and subsequently continuing a simulation resulted in entirely different results compared to the same simulations without any interruption. The cause for the discrepancy was found to be due to the parameters being rounded off when pausing the simulations. Although rounding the parameters resulted in small differences, these differences would propagate and rapidly amplify. Edward Lorenz published his computer simulation results of a weather model [7]. Herein, he related the hypersensitivity of initial conditions to the weather, and discussed if these results meant that the weather is inherently unpredictable over larger timescales. This marked the quantitative discovery of chaotic dynamics, and has since spawned similar discoveries in many different fields, including ecology [8], chemistry [9], physiology [10], and cryptography [11]. It also plays a fundamental role in establishing the validity of classical statistical mechanics and thermodynamics. Chaos arises from nonlinear terms in the equations of motion, and from there being fewer constants of motion compared to the degrees of freedom of the system [5]. There is no unique definition of chaos, but in one definition that is easily explained, a system is chaotic when it satisfies four properties [13]: 1. 2. 3. 4.

hypersensitivity, determinism, boundedness, aperiodicity.

Chaotic dynamics are marked by their hypersensitivity to initial conditions; any perturbations will be amplified at an exponential rate (Fig. 8.1). This exponential divergence is characterized by the Lyapunov exponent λ, which can be defined by considering two initial points in phase space x(0) and x  (0) separated by a small distance x  (0) − x(0) = δ x(0). The distance at a later time t is then given by

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A

B

Fig. 8.1 Chaotic evolution in a logistic map. In the logistic map, the population of successive generations are calculated as x(t + 1) = r x(t)[1 − x(t)], where x(t) is the population at generation t, and r is the growth rate. (A) Diverging evolution of two nearly identical initial states x1 (0) = 0.5 (blue) and x2 (0) = 0.5001 (orange). At the growth rate r = 3.8 used here, the resulting dynamics is chaotic. This is reflected by their exponentially increasing divergence (B), characterized by the positive Lyapunov exponent. Figure based on Ref. [12]

| x  (t) − x(t)| ≈ eλt |δx(0)|.

(8.1)

Note that the Lyapunov exponent does not have a precise definition. This is because the rate of divergence depends on both the position x in phase space, as well as the direction δx of the displacement [14]. The Lyapunov exponent described in Eq. (8.1) is the average Lyapunov exponent over the course of a trajectory. A positive Lyapunov exponent therefore characterizes chaotic dynamics. Although this hypersensitivity results in systems exhibiting chaos to be unpredictable in the long term, the equations governing the evolution are deterministic, and so its dynamics are not random. A dynamical system can be characterized by its phase space, where each dimension corresponds to a degree of freedom, and each point in phase space uniquely represents a state of the system. The region in phase space for which the system displays chaotic dynamics must be bounded, meaning that it cannot extend to infinity. These chaotic bounded regions also have a sharp boundary with nonchaotic (regular) regions, if any such regular regions are present. Finally, a chaotic system is aperiodic, and so a chaotic trajectory never returns exactly to a previous point in phase space. That being said, the trajectory can come arbitrarily close to said point.

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8.1.3 Quantum Chaos The correspondence principle, as formulated by the Copenhagen school of quantum mechanics, states that the dynamics of quantum systems should converge towards classical dynamics, in the limit where the system becomes large [2, 3]. Appealing (and, for simple cases, often correct) as it may sound, this point of view is afflicted by a plethora of complications and controversies around the precise nature of the quantum-classical transition [15], such as decoherence [16] and the quantum measurement problem [17]. Another key aspect of the quantum-classical transition concerns reconciling the chaotic dynamics of certain classical systems with the unitary evolution of their quantum-mechanical counterparts. The usual description of quantum systems, in terms of states vectors that evolve according to the Schrödinger equation, can appear puzzling when examined in the context of the chaotic behaviour of the equivalent classical Hamiltonian. Consider for example two slightly different quantum states |ψ1 (0) and |ψ2 (0) at time t = 0, having an initial overlap |ψ1 (0)|ψ2 (0)|2 = 1 − δ 2 , with δ  1. As time progresses, these states evolve according to the time evolution operation U (t). The overlap at later times is thus ψ(t)|ψ(t) = |ψ1 (0)|U † (t)U (t)|ψ2 (0)|2 . Since the time evolution is unitary, U † (t) = U −1 (t), we find that ψ(t)|ψ(t) = ψ(0)|ψ(0) = 1 − δ 2 , i.e. the overlap remains constant at all times. Since the exponential divergence of trajectories typical of classical systems appears ruled out, does this mean that there cannot be chaos in quantum dynamics? A more appropriate and illuminating comparison between classical and quantum dynamics is obtained by describing the classical system in terms of a density f in phase space, and calculating its time evolution using the Liouville equation i ∂ f∂t(t) = L f (t), where L is the Liouville operator [18]. One then finds that, given two initially overlapping densities f 1 (0) and f 2 (0), the Liouville equation for a conservative Hamiltonian system ensures that their overlap remains constant at all times [19]. This property mirrors the quantum behaviour described earlier, so now the question may be reversed: in what way, if at all, does classical chaos differ from the dynamics of quantum systems? The answer to this question can be rather subtle. At its heart, quantum mechanics requires that the classical phase space is coarse-grained into volumes of size  N (with N the number of degrees of freedom), and forbids specifying the state of the system to a precision finer than that. An illuminating example of how this affects the dynamics of chaotic systems was provided by Korsch and Berry [20], who analyzed a classically chaotic iterative map while varying the value of . In the classical limit ( → 0), the map ‘shreds’ the initially smooth distribution into thin chaoticlooking ‘tendrils’. Conversely, when  becomes sizeable on the scale of the map’s effect, one finds that the distribution remains smooth and seems to lose its chaotic features, displaying instead a slow spread from the initial shape. More generally, since bounded quantum systems have a discrete spectrum, their dynamics exhibits quasiperiodic features that are at odds with the ‘true chaos’ seen in classical systems [21].

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The discrepancy between quantum dynamics and classical chaos was noted by Einstein in 1917, in an article where he expanded on the work of Sommerfield and Epstein regarding quantization [22]. Here, he discussed issues on the quantization of systems that are classically non-integrable, i.e. chaotic [23]. In the decades that followed, the interest in the topic was modest. In 1977, the Como conference was held on “Stochastic behaviour in classical and quantum Hamiltonian systems” [24]. This conference brought together researchers in several disciplines that were all studying problems related to the quantization of classical non-integrable systems. The conference helped demonstrate that many of the observed chaotic features in these disciplines were universal, distinguishing the field of quantum chaos [25]. The issue of how to observe and interpret signatures of chaos in quantum mechanics has profound repercussions on many important topics in physics. For example, classical chaos underpins the ergodic hypothesis in statistical mechanics, and it is expected that its quantum equivalent plays a fundamental role in the thermalization of isolated quantum systems [26–29]. Chaos is also thought to be related to the issue of decoherence [30], which is crucial in the modern topic of quantum information science. There, one must answer the delicate question of whether an onset of chaos may harm the operation of a large-scale quantum computer [31–33]. On the other hand, it has been suggested that the inherent ability of chaotic systems to quickly explore a vast configuration space can be used for the purpose of demonstrating ‘quantum supremacy’ in multi-qubit devices without error correction [34, 35].

8.1.4 Experimental Tests of Quantum Chaos Highly excited hydrogen-like atoms were among the first physical systems found to display chaotic behaviour in their energy spectra. [36–38]. A simplified classical version of this system is that of a charged particle in a two-dimensional potential well in the presence of a magnetic field, which displays chaotic behaviour for sufficiently strong fields. At low magnetic fields, the spectrum is regular and can be determined using perturbation theory. At sufficiently strong fields, however, perturbation theory fails, and the energy spectrum becomes highly irregular. The magnetic field needed to reach this transition regime has a 1/n 4 dependence on the principal quantum number n, and reaches attainable magnetic fields for sufficiently high n. By comparing the spectra of many such atoms, they were all found to possess eigenvalue spacings equal to those of random matrices having the same symmetry class [39]. These results lead to the Bohigas-Giannoni-Schmit conjecture, which asserts that random matrix theory can accurately describe the spectra of quantum systems whose classical equivalent is chaotic [40]. The same effect was found soon after in quantum billiards [41–44]. Here, the system boundaries confine a region – often shaped like a stadium – wherein classical trajectories can exhibit chaotic dynamics. Experiments probing the time evolution of quantum-chaotic systems are rare. Crudely speaking, this is because most quantum systems decohere and randomize for unrelated reasons (noise, uncontrolled environments, etc.) over time scales that are

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too short for signatures of chaotic behaviour to reveal themselves. Systems with long coherence times, such as ensembles of nuclear spins in liquids, can show signatures of chaos in the dynamics of their macroscopic magnetization [45]. The experimental state of the art for truly quantum chaotic dynamics is found in ensembles of cold gases [46–49], whereas only very recently an experiment on three superconducting qubits has provided experimental insight into the link between chaos and thermalization in a small-scale quantum system [50]. What is still missing is an experimental study of the quantum signatures of chaos in an individual quantum system. Such a study will be an important complement and extension to experiments conducted on ensembles of particles, since an individual quantum system allows a much broader choice of measurement strategies. Although the chaotic dynamics we will describe here are the result of the Hamiltonian evolution alone, the use of an individual quantum system will allow us in the future to explore the interplay between the emergence of chaos and the measurements performed on the system. Theoretical studies [51, 52] predict that the measurement strength can be used as an additional experimental knob to tune the chaoticity of the system’s dynamics. The measurement strength on a single object can be tuned continuously [53, 54] from projective single-shot readout [55–57] to arbitrarily weak measurements, partial wave function collapse [58] and even measurement reversal [59]. Strings of individual measurement outcomes could be analyzed with sophisticated statistical techniques to extract the most accurate information on the trajectory of the quantum object [60], providing unprecedented insights into the chaotic dynamics of a monitored quantum system. Variable-strength measurements have already been experimentally demonstrated in the 31 P donor system [61]. Moreover, in the realm of classical computation, it has been recently shown that a network of individually chaotic electronic components can solve computationally hard problems faster than one using nonchaotic elements [62]. The construction of a single quantum chaotic system amenable to networking and controlled interactions could provide insights into how the equivalent quantum circuit would perform in complex computational problems.

8.2 The Classical Chaotic Driven Top Here we present a detailed and quantitative proposal to experimentally realize a single-atom version of the classical chaotic ‘kicked top’, which is one of the best studied systems for quantum chaos [48, 63, 64]. For experimental convenience, we will focus on the case where the top is periodically driven, instead of kicked with δ functions. The classical driven top becomes chaotic in the presence of a term in the Hamiltonian that is quadratic in the angular momentum, and has a classical Hamiltonian of the form Hˆ classical = αL z + β L 2x + γ cos (2π f t) L y ,

(8.2)

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A

B

C

D

E

F

G

Fig. 8.2 (A) Stroboscopic map of two trajectories of L corresponding to initial conditions in a regular (blue) and chaotic (orange) region, with parameters shown to the left. (B)–(G) Same as (A), but where a single parameter is modified. (B), (C) Modifying the quadratic interaction strength β shrinks the chaotic region and displaces the enclosed regular regions. (D), (E) Increasing (decreasing) the periodic drive strength γ leads to an enlarged (reduced) chaotic region. (F), (G) Shifting the drive frequency f away from precession frequencies at the boundaries of regular regions results in a reduced chaotic region

8.2 The Classical Chaotic Driven Top

A

139

B

Fig. 8.3 Chaotic percentage of phase space for the driven top. Chaotic percentage of total phase space as a function of quadratic interaction strength β and periodic drive frequency f for weak drive strength γ = 0.02α (A) and moderate drive strength γ = 0.05α (B). Simulation details are given in Appendix H

T   = constant), α, β, and γ proportionwith angular momentum L = L x L y L z (| L| ality constants, and f the frequency of the drive. The equations of motion governing the evolution of a trajectory are discussed in Appendix F When α, β γ > 0, the classical phase space of the driven top contains three regular regions (Fig. 8.2A). One of these regions lies in the upper hemisphere, where the dominant energy contribution is the linear term αL z , and consequently trajectories therein precess around the z-axis. The other two regular regions lie towards the ±xaxes, and have a higher quadratic energy contribution β L 2x . Note that the quadratic interaction produces two regular regions, since the quadratic energy contribution is independent of the sign of L x . A weak periodic drive γ cos (2π f t) resonates with a trajectory when the drive frequency f roughly equals the trajectory’s precession frequency. Such resonances mix the regular regions, resulting in the formation of a chaotic region at the boundaries between these regular regions. The size and shape of the regular and chaotic regions in classical phase space are determined by the Hamiltonian parameters α, β, γ , and f (Fig. 8.2). Varying β changes the quadratic interaction strength, and therefore changes the size of the two corresponding regular regions (Fig. 8.2B, C). The periodic drive strength γ directly affects the size of the chaotic region (Fig. 8.2D, E). However, this does require the drive frequency f to be comparable to the energies of the system, i.e. the precession frequencies of the trajectories. Otherwise, the periodic drive is off-resonant and thus does not lead to any significant mixing of regular regions (Fig. 8.2F, G). Since the phase space of the driven top is bounded, we can define the percentage of phase space that is chaotic. This percentage can be estimated by initializing trajectories that are equidistantly spaced over the phase space, and measuring which are chaotic. The percentage of chaotic phase for different parameters is quantitatively summarized in Fig. 8.3. The region of chaos is largest when the linear and quadratic

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8 Exploring Quantum Chaos with a Single High-Spin Nucleus

A

B

C

Fig. 8.4 Phase space of the driven top for high drive strength gamma symbol. A strong periodic drive γ = 0.24α results in a sizeable chaotic region even when the linear and quadratic interac results in a highly tion strengths strongly differ. (A) A dominant linear interaction α = 2.5β| L| asymmetric chaotic region, indicating that the periodic drive no longer acts as a weak perturbation.  results in a chaotic region surrounding the yz(B) A dominant quadratic interaction β = 10α/| L|  plane. (C) Chaotic percentage of phase space for varying β| L|/α and drive frequency f . Significant  = 10α, but are chaotic regions can still exist for a strongly dominating quadratic interaction of β| L|  = 0.3α. The presence of chaos nearly fully suppressed for a dominating linear interaction of β| L| is therefore more amenable to a dominant quadratic interaction than a dominant linear interaction

 2 The size of the chaotic region interactions are of similar strength (β ≈ α/| L|). strongly depends on γ , and at a moderate strength γ = 0.05α the chaotic region can exceed 20% of the total phase space (Fig. 8.3B). The drive frequency resulting in the largest chaotic region is found to increase with increasing β, reflecting that the average energy of the system is also increasing. At a high drive strength γ , significant chaotic regions are still present even when the quadratic interaction strength dominates the linear strength by over an order of magnitude (Fig. 8.4). Note that the converse is not true; chaos is still nearly fully  When α > β| L|,  the chaotic region is mostly confined suppressed when α  3β| L|. to the lower hemisphere, where the linear energy contribution is weakest (Fig. 8.4A).  the chaotic region forms a band around the yz-plane Similarly, when β α/| L|, (Fig. 8.4B). Note that in this situation, the regular region in the northern hemisphere disappears. These results show that even when the linear and quadratic interaction strengths differ significantly, a sizeable chaotic region can still be attained by increasing the drive strength γ .

8.3 The Quantum Driven Top The obvious quantum equivalent of a classical spinning top is a spin. The challenge here is to find a spin system whose Hamiltonian maps onto that of the chaotic driven top. This requires in particular a quadratic term in the Hamiltonian, which is only 2

 accounts for the difference in dimensionality between α and β (Sect. 8.3.2). The factor 1/| L|

8.3 The Quantum Driven Top

A

141

B

C

Fig. 8.5 Localization of a spin-coherent state within regions of classical phase space. Colors of the sphere surface correspond to the Husimi Q distribution (Eq. (2.6)) of the coherent state |θ = 2π/3, φ = 0 evaluated at those spherical coordinates. Color-scale limits are constant over all panels. Stroboscopic maps of two trajectories of the classical driven top are superimposed on the same spherical surface, with the same parameters as chosen in Fig. 8.2A. The chaotic trajectory (orange) encloses an island of stability in which the regular trajectory (blue) resides. As I is increased from 3/2 (A, e.g. an 75 As nuclear spin) to 7/2 (B, e.g. an 123 Sb nuclear spin) and beyond (C, 31/2, shown for pedagogical reasons only, not corresponding to an actual nuclear spin), the relative uncertainty of the coherent state decreases (Sect. 2.6). This effectively localizes the state within the different regions of the corresponding classical phase space. This underlines the importance of choosing a donor with a large nuclear spin for a meaningful comparison between the dynamics of quantum states and the classically regular or chaotic counterpart

possible for a spin quantum number I > 1/2. Moreover, a larger spin is crucial in comparing its dynamics to the structure of the classical phase space, as its smaller relative uncertainty spread allows for better localization of the quantum state in a certain area of interest in classical phase space (Fig. 8.5). This is because the minimum spin uncertainty (σ Iˆx  σ Iˆy = 2 I ) decreases relative to the surface of the phase space (4π I 2 ) for increasing I (Sect. 2.6). Using a single spin with access to high-fidelity single-shot state readout and/or variable-strength weak measurements opens up the largely unexplored area of the interplay between chaos and quantum measurements. Lastly, it is of paramount importance that this system does not lose coherence for trivial reasons, unrelated to chaos, on time scales short compared to the chaotic dynamics. This requires a long intrinsic quantum coherence time of the system.

8.3.1 Experimental Platform Our proposed system meeting these requirements is the nuclear spin of a heavy group-V substitutional donor in isotopically enriched 28 Si [65]. In Chap. 6, we have demonstrated the full coherent control of the 123 Sb donor atom, with exquisite coherence times in the range of 100 ms. This suggests that donor spin systems would be ideal platforms to study the subtle effects of dynamical chaos, if it were possible to engineer a suitable spin Hamiltonian. This is not the case with 31 P, since its nuclear

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8 Exploring Quantum Chaos with a Single High-Spin Nucleus

spin I = 1/2 forbids the presence of quadratic terms in the Hamiltonian. Heavier donors, such as 75 As, 121 Sb, 123 Sb and 209 Bi, all have nuclear spins I > 1/2, and thus have a quadrupole interaction that contains quadratic spin interaction terms (Sect. 3.3). Of the heavy donors, the most suitable are 123 Sb and 209 Bi, as their nuclear spins are the highest, being respectively I = 7/2 and I = 9/2 (Table 3.1). Below we show that, under realistic conditions of quadrupole interaction and periodic drive, a heavy group-V donor can become a single-atom solid-state implementation of a chaotic driven top. To implement the quantum version of the driven top, a suitable equivalent interaction needs to be found for each of the terms in the classical driven top. The Zeeman interaction γn B0 Iˆz is linear in spin operators (Sect. 3.2.2), and therefore serves as the quantum equivalent of the classical linear interaction αL z . The quantum linear interaction can be enhanced by the hyperfine interaction of the neutral donor to (−γn B0 + m S A) Iˆz (Sect. 3.2.4). However, since the nuclear coherence times of a neutral donor are orders of magnitude lower than for an ionized donor [66], the ionized donor is considered from hereon. A magnetic antenna used for NMR provides an oscillating magnetic field −γn B1 sin(2π f t) which matches the classical drive interaction γ cos (2π f t) L y . To simplify the comparison with classical dynamics, we shall from hereon remove the minus sign preceding the Zeeman interaction, which is valid if we assume B0 is aligned along −ˆz and B1 along − yˆ . All terms of the nuclear quadrupole interaction are products of two spin operators (Sect. 3.3), which makes it a candidate to replace the classical quadratic interaction β L 2x . One caveat is that the nuclear quadrupole interaction can have multiple terms, which complicates the Hamiltonian. However, it is often the case that the electric field gradient tensor has approximately axial symmetry (η = 0) [67], in which case the quadrupole interaction has only one significant term in its principal-axes frame3 (Sect. 3.3.1). If, in addition, the static magnetic field B0 can be oriented in an arbitrary direction (for example using a 3-axis vector magnet), the linear and quadratic terms in the spin Hamiltonian can be made orthogonal. Ignoring the static energy offset Q I 2 /3, and assuming for simplicity that the symmetry axis of the electric field gradient is orthogonal to the periodic driving field B1 , the Hamiltonian of a donor with nuclear spin I > 1/2 takes the form Hˆ quantum = γn B0 Iˆz + Q Iˆx2 + γn B0 cos (2π f t) Iˆy .

(8.3)

Therefore, Hˆ quantum represents the quantum equivalent of the Hamiltonian of a classical periodically driven top (Eq. (8.2)).

While a nonzero asymmetry η does add additional quadratic terms to the Hamiltonian, the classical dynamics of the corresponding Hamiltonian can still exhibit chaos. The condition that η = 0 is therefore not a strict requirement, though it does simplify the Hamiltonian. Without exact knowledge of the full Hquadrupole of the particular donor under study, there is little point in investigating these details. This should instead be addressed once the system parameters are accurately known, in particular those related to Hquadrupole . 3

8.3 The Quantum Driven Top

143

8.3.2 Comparison Between Classical and Quantum Hamiltonian Parameters The different parameters of the classical and quantum driven top can be meaningfully compared by converting them to their dimensionless equivalent. For the classical  t) and the paramdriven top, dimensionless variants are needed for the variables ( L, ˆ  eters (α, β, γ , f ). Dividing Hclassical by | L| (including a factor 2π to convert from rads−1 to Hz) we obtain Ly Hˆ classical α Lz β L 2x γ = + + , cos (2π f t)     2π 2π 2π 2π | L| | L| | L| | L| α  β  2 γ = L + | L|L x + cos (2π f t) L y , 2π z 2π 2π

(8.4)

 L|  is introduced. Next where the normalized angular momentum variable L  = L/|  we divide by α/2π and time t = (α/2π )t is introduced:    2 γ Hˆ classical 2π f  β| L|   ˆ = Lz + L x + cos 2π t L y , Hclassical =  α α α α| L|   2 = L z + β  L x + γ  cos 2π f  t  L y .

(8.5)

This makes the parameters β  , γ  and f  dimensionless and relative to α. α itself has units of rad s−1 and time variable t  has units of 2π/α. These dimensionless  parameters show that α should not be directly compared to β, but instead to β| L|. The above implies that the same convention of a normalized angular momentum has to be followed quantum mechanically. Hence the quantum Hamiltonian Hˆ quantum is divided by h and transformed with I = I/I , thus converting to frequency units and normalizing the spin operators: Hˆ quantum = γn B0 Iˆz + Q I Iˆx2 + γn B1 cos (2π f t) Iˆy . hI

(8.6)

This assumes units of Hz T−1 for γn , units of Hz for Q, and dimensionless spin operators I. Note that after introducing I an additional factor I also appears in the second term due to its quadratic nature. Next, as in the classical case, dividing by γn B0 and introducing time variable t  = γn B0 t results in the dimensionless Hamiltonian   Hˆ quantum Q I ˆ2 B1 2π f  ˆ   ˆ ˆ Hquantum = = Iz + cos t Iy , I + h I γn B0 γn B0 x B0 γn B0   = Iˆz + Q  Iˆx2 + B1 cos 2π f  t  Iˆy .

(8.7)

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8 Exploring Quantum Chaos with a Single High-Spin Nucleus

Table 8.1 Each parameter of the classical driven top (left columns, Eq. 8.2) has a corresponding parameter in the quantum driven top (right columns, Eq. 8.3). In both cases, the parameters can be meaningfully compared by considering their equivalent dimensionless counterparts, given by Eq. 8.5 (classical) and Eq. 8.7 (quantum). Note in particular that the dimensionless quadratic term is multiplied by the magnitude of the angular momentum. As a result, the linear classical parameter  and similarly the linear quantum parameter γn B0 α should be compared to the quadratic term β| L|, should be compared to Q I . Classical Quantum Original Dimensionless Original Dimensionless L    L L = I I = I α 2π β 2π γ 2π

f t

 | L|

1  β| L| α γ γ = α f  = 2πα f α t  = 2π t

β =

I

γn B0

1

Q

Q =

γn B1

B1 =

QI γn B0 B1 B0 f γn B0

f

f =

t

t  = γn B0 t

Again, the linear interaction strength γn B0 should not be directly compared to the quadratic interaction strength Q, but to Q I . Table 8.1 gives an overview of the different parameters used throughout the classical and quantum simulations.

8.3.3 Realizing a Quantum Driven Top in the Laboratory Frame We now estimate the parameters of the spin Hamiltonian of the quantum driven top, in order to compare them to the parameters that are known to lead to chaotic dynamics in the equivalent classical case. The key parameter values of the group-V donors are summarized in Table 3.1. The value of I increases with atomic mass, while the hyperfine interaction A has a non-monotonic behaviour, with a significant jump for the heavy 209 Bi donor. A large nuclear spin I is desirable to reduce the relative quantum uncertainty of the spin state (Fig. 8.5), whereas from the analysis of the classical driven top we know that interesting chaotic dynamics arises when linear and quadratic terms in the spin Hamiltonian are of comparable strength. Even in the ionized charge state, the linear interaction strength γn B0 is expected to be higher than the quadratic interaction strength Q. The goal therefore is to minimize the linear strength and maximize the quadratic strength. The nuclear Zeeman term is minimized by operating at low B0 , with the caveat that reducing B0 affects the electron readout and initialization fidelity (Sect. 5.3.2). Using a minimum value of

8.3 The Quantum Driven Top

145

B0 = 0.5 T that still allows high-fidelity readout, and an ionized donor, the linear term in the spin Hamiltonian thus takes values of order 3 MHz. Next, we wish to obtain a comparable value for the quadratic term, which is achieved by maximizing the strength of the quadrupole interaction, and hence the electric field gradient (Eq. (3.8)). Finite-element modeling of the measured 123 Sbimplanted device (Sect. 7.4) has found shear-strain values up to 0.2% in the donorimplanted region, with corresponding quadrupole strengths in excess of 100 kHz (Fig. 7.6). The resulting quadrupole strength is dependent on the donor atom, as each has its own Sternheimer anti-shielding factor [68, 69]. As the Sternheimer anti-shielding factor generally grows with atomic number, the quadrupole strength is expected to be higher for 209 Bi. Since the classical linear interaction α needs to be compared to the quadratic term  (Sect. 8.3.2), the Zeeman interaction must hence also be compared to Q I . This β| L| makes the expected Q exceeding a hundred kHz rather promising, since for large I the corresponding Q I ∼ 0.5 MHz is within an order of magnitude of the linear interaction strength (γn B0 ∼ 2.8 MHz). If this proves to be insufficient, strain engineering could be deployed to further increase the electric field gradient to reach a target value of Q I ∼ 1 MHz, at which point the classical phase space exhibits a significant chaotic region (Fig. 8.2). This can be achieved either in MOSFET structures [70] or in Si/SiGe devices, where the ability to electrically detect a single dopant atom coupled to a quantum dot has recently also been demonstrated [71]. All the above options will deliver a fixed value of strain, set by the thermal expansion of the metallic electrodes and/or the built-in strain in the substrate. As a next step in experimental sophistication and control, one could consider fabricating on-chip piezoactuators to dynamically control the strain [72], allowing the study of the chaotic dynamics of a nuclear spin as a function of its Hamiltonian parameters. Finally, another option to tune in situ the quadrupole splitting could be to distort the electron wave function using strong voltages on gates placed above the donor, to the point where the electron wave function is significantly displaced from the nuclear site, potentially generating substantial electric fields. These are then transduced into strong electric field gradients and hence quadrupole interactions via the LQSE. This type of electron wave function distortion has been discussed in numerous papers [73–75], but no calculation of the resulting nuclear quadrupole splitting in the case of a I > 1/2 nucleus has been performed to date. The broadband antenna near the donor can be used to apply an RF periodic drive. Previous work has been conducted with drive strengths up to B1 ∼ 2 mT [76], which correspond to radio-frequency powers of order 0.5 mW (at the chip). Those values were sufficient to achieve high-fidelity coherent control of the 31 P nuclear spin qubit. Here, we wish to compare γn B1 to the classical parameter γ . Chaos arises when γ  0.02α in the classical model. For B0 ∼ 0.5 T and thus α ∼ 3 MHz, this implies B1 ∼ 10 mT. The observation of a damaged antenna tip in the measured device (Sect. 4.1) might suggest that this value is beyond reach. However, the diameter of the damaged antenna tip had been reduced compared to previous devices, and measurements across multiple similar devices have shown that this severely reduces the critical microwave power. Increasing the diameter of the antenna tip can therefore allow for much higher

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8 Exploring Quantum Chaos with a Single High-Spin Nucleus

microwave powers. If this still proves insufficient, high B1 values with low incident power could be obtained by using LC resonators and optimized NMR coils.

8.3.4 Eigenbasis Mismatch Nuclear spin initialization and readout can be adversely affected by a strong quadrupole interaction. This is because the presence of a strong quadrupole interaction with axis perpendicular to that of the linear interaction inhibits defining a clear quantization axis (Fig. 8.6A). Upon adding an electron to the ionized donor, the accompanying hyperfine interaction enhances the linear interaction by about an order of magnitude for 123 Sb (two orders for 209 Bi), in which case the eigenstates Since the eigenstates of the ionized are nearly fully quantized along Iˆz (Sect.  3.2.4).  donor {|ek+ } and of the neutral donor |ek0  differ significantly, there is no one-to2  one correspondence between eigenstates of the two, i.e. ek+ |ek0  is significantly lower than unity regardless of the chosen pair of indices k and k  (Fig. 8.6). A nuclear spin eigenstate of an ionized donor will therefore be a superposition of eigenstates when the donor is neutralized, and vice versa. The overlap between eigenbases is quantified by the participation ratio [77] PR =

D

−1 |φk |k  |4

,

(8.8)

k,k  =1

where {|φk } and {|k } are two different eigenbases. When {|φk } and {|k } are identical, the participation ratio attains its minimum PR = 1/D, where D is the system dimensionality. A maximum participation ratio PR = 1 indicates that the eigenbases differ maximally, i.e. |φk |k  |2 = 1/D ∀ k, k  . An increasing quadrupole strength Q corresponds to a higher participation ratio (Fig. 8.6C), meaning that the ionized and neutral eigenbases differ more strongly. The existing techniques for initialization and readout of the nucleus rely on electron tunneling events between donor and SET island (Sect. 5.3). Each such tunneling event is accompanied by a change of eigenbasis. These changes in eigenbasis can in principle be accounted for if all Hamiltonian parameters are known accurately and the moment of electron tunneling can be measured at a resolution that significantly exceeds the nuclear transition frequencies. However, in our setup the measurement resolution is limited by a 50 kHz low-pass filter (Sect. 4.6). As a result, each tunneling event adds a degree of scrambling to the nuclear spin state, effectively limiting the initialization and readout fidelity. Several methods can be employed to counteract the nuclear spin scrambling. One approach is to study the driven-top dynamics on a second ionized donor that is weakly coupled to the primary donor by the hyperfine interaction. The primary donor can then be used to read out the second donor. This has the advantage that for the second donor

8.3 The Quantum Driven Top

147

Fig. 8.6 Eigenstates of an ionized 123 Sb nucleus for varying quadrupole strength Q. A magnetic field B0 = 0.5 T is chosen along the z-axis, oriented perpendicular to the quadrupole axis. (A) When Q = 0 Hz, the nuclear spin eigenstates {|ek+ } of the ionized 123 Sb donor (colored lines) match the eigenstates {|m I } of Iˆz . In this case, each eigenstate |ek+  satisfies ek+ | Iˆz |ek+  = m I for some m I . However, as Q increases, the eigenstates {|ek+ } start to deviate from {|m I }. This is reflected by their shifting spin projections ek+ | Iˆz |ek+ . (B) Overlap of ionized-donor eigenstates {|ek+ } and neutral-donor eigenstates {|ek0 } of the electron | ↓ manifold. Both sets of eigenstates are ordered from lowest (blue) to highest (gray) eigenenergy. The eigenstates {|ek0 } closely match those of Iˆz , whereas the eigenstates {|ek+ } deviate as Q is increased. The overlap-squared |ek+ |ek0 |2 roughly matches the probability of a nuclear spin flip per neutralization/initialization event. (C) The similarity between the ionized and neutral eigenbases is quantified by the participation ratio (PR) [77], which is maximal (PR = 1) when the eigenbases differ maximally, and is minimal (PR = 1/D, where D is the system dimensionality) when the eigenbases are identical. At low Q, the participation ratio is nearly minimal (PR = 1/8, red dashed line). However, as Q increases, so does the participation ratio, which indicates that the ionized and neutral eigenbases start to deviate from one another

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8 Exploring Quantum Chaos with a Single High-Spin Nucleus

the hyperfine interaction is weak compared to the Zeeman interaction, and therefore does not significantly alter the eigenbasis after a tunneling event. The primary donor can additionally remain neutral while the second donor is manipulated, in which case the first electron readout is of the unscrambled second donor. In fact, a second hyperfine-coupled 123 Sb donor has likely already been observed in Appendix B, and the coherent control and readout of a second nuclear spin has been demonstrated for a hyperfine-coupled 29 Si donor [78]. A second approach is to use two quantum dots, where one of the two dots has a hyperfine coupling to a high-spin nucleus. In this case, the tunnel coupling between the quantum dots can be much higher than the nuclear spin transition frequencies. This enables precise timing of electron tunnel events, and hence any nuclear state scrambling can be accounted for and corrected. The coherent control and readout of a 29 Si nuclear spin, even after dozens of electron tunneling events, has been demonstrated using quantum dots [79]. A third solution is the adiabatic transferring of the electron from donor to the 28 Si/SiO2 interface, which maps neutral-donor eigenstates to ionized-donor eigenstates and vice versa. This can be achieved by the addition of an electrostatic gate above the donor that can attract its outer electron, a technique that is currently being developed in the context of achieving electrically active transitions and long-range coupling of donor nuclear spin qubits [75]. Once the electron is moved from the donor to the interface, spin-dependent tunneling of the electron from the interface to the SET island allows for readout without affecting the nucleus.

8.3.5 Realizing a Quantum Driven Top in the Rotating Frame The Hamiltonian of the periodically driven top described so far was defined in the laboratory frame. An alternative approach is to describe it in the rotating frame, defined by the frequency of an oscillating field. This results in a system ‘dressed’ by a continuous RF field, at a frequency that matches the nuclear Zeeman interaction strength ( f RF = γn B0 ). This is a well-established method that originates from quantum optics [80] and has recently been extended to microwave frequencies [81], including with the electron spin of the 31 P donor [82]. Here we analyze its application to the higher-dimensional nuclear spin of the heavier group-V donors. We consider a Hamiltonian for the ionized nucleus (A = 0) of the form Hˆ quantum,RF =γn B0 Iˆz + Q Iˆx2

+ γn B1,I cos (2π f RF t) + γn B1,Q cos (2π f t) sin (2π f RF t) Iˆy . (8.9) Here B1,I and B1,Q are the in-phase and quadrature amplitudes of an IQ-modulated radio-frequency drive at frequency f RF = γn B0 , with an additional amplitude modulation applied to the in-phase component of the drive, at a frequency f . Switching the

8.3 The Quantum Driven Top

149

system to the rotating frame4 and applying the rotating wave approximation (RWA) will effectively remove the static Zeeman energy, while reintroducing a linear term which scales with the strength of the continuous drive at frequency f RF = γn B0 (see Appendix G for a derivation). Now the Hamiltonian reads 1 1 1 Hˆ quantum,RWA = − γn B1,I Iˆy − Q Iˆz2 + γn B1,Q cos (2π f t) Iˆx , 2 2 2

(8.10)

which is, up to a trivial rotation, equivalent to the quantum driven top (Eq. (8.3)). Engineering a dressed system and considering this in the rotating frame has several important benefits for the experimental feasibility of our proposal. Firstly, the linear term in the new Hamiltonian, α = γn B1,I /2, is continuously (and rapidly, if desired) tunable all the way to zero, and up to a maximum set by the strongest attainable oscillating field strength (B1 ∼ 10 mT as assumed earlier), corresponding to α ∼ 30 kHz. This means that we can access a strongly chaotic regime, where the quadratic term β = Q I is comparable to the linear term, with quadrupole interaction strengths of only Q ∼ 10 kHz. Second, IQ modulation combined with amplitude modulation is a standard microwave control technique, which allows full and independent control over the strength of the periodic drive at frequency f between 0 and the maximum strength of the linear interaction 21 γn B1,I = 0 ∼ 30 kHz. This opens up a new parameter regime of very strong periodic drive, with increased size of classical chaotic regions (Fig. 8.4), which would be challenging to obtain in the laboratory frame. Lastly, the constraint of orthogonality between axis of quadrupole interaction and direction of oscillating magnetic field B1 , as imposed by the classical system (Eq. (8.2)), is relaxed in the rotating frame under the RWA (see Appendix G.3). This is important, since now the axis of the static Zeeman field B0 only needs to be perpendicular to the plane defined by the directions of quadrupole interaction and periodic drive, regardless of the relative angle between the latter two. This is easily achievable using a 3D vector magnet. Overall, moving to the rotating frame allows the exploration of a wider parameter space, but the actual timescale of dynamical phenomena will be scaled down by a factor ∼ 100 (for 21 γn B1,I ∼ 28 kHz at B1 = 10 mT vs γn B0 = 2.8 MHz at B0 = 0.5T ). For example, the period of dynamical tunneling (Fig. 8.9C for the case of the laboratory frame) will become ∼ 330 µs for the same choice of relative parameter strengths. This remains several orders of magnitude faster than the measured nuclear spin coherence times of ∼100 ms (Sect. 6.5), noting also that the technique of dressing a spin with a driving field often yields an extra order of magnitude in coherence time [82].

4

In contrast to the generalized rotating frame described in Sect. 2.7, here only the static Zeeman interaction is counteracted (Appendix G).

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8 Exploring Quantum Chaos with a Single High-Spin Nucleus

8.3.6 Summary In summary, these estimates suggest that the 123 Sb and 209 Bi donors in silicon are suitable candidates to implement the quantum driven top, due to their high spin quantum number (I = 7/2 and I = 9/2, respectively), low nuclear gyromagnetic ratio (γn = 5.55 MHz T−1 and γn = 6.96 MHz T−1 ), and, in the case of 123 Sb, low hyperfine coupling strength (A = 101.52 MHz). The additional advantage of 123 Sb is that its suitability for ion implantation has been demonstrated in this thesis, and is well documented [83]: after low-energy implantation and high-temperature rapid thermal anneal, the Sb atoms are fully activated, and the implantation damage to the silicon lattice is thoroughly repaired. Recent work [84] suggests that the implantation damage can be efficiently repaired also in the case of 209 Bi, although the electrical activation yield remains lower than that of Sb. Two approaches are presented to implement the quantum driven top in a high-spin nucleus, such as that of 123 Sb or 209 Bi. Both approaches use the quadrupole interaction as the quadratic interaction. The first approach uses the Zeeman interaction from a static magnetic field as the linear term, while the second approach dresses the system at the nuclear Zeeman frequency γn B0 , replacing the linear interaction strength by the drive strength. The main difference is that the former approach requires a significant quadrupole strength of Q  300 kHz, the latter approach works best with a lower Q in the range of tens of kHz. Considering that the quadrupole strength of the measured 123 Sb donor is likely around 50 kHz (Sect. 6.6), it would be perfectly suited for the rotating-frame Hamiltonian. This demonstrates that the parameter range can be reached where the equivalent classical driven top behaves chaotic throughout sizeable areas of its phase space. In what follows, we will concentrate our discussion on the use of 123 Sb as the model system to study quantum chaos in a single spin.

8.4 Quantum Dynamics and the Floquet Formalism The evolution of time-dependent periodic Hamiltonians, such as the quantum driven top, can be transformed to time-independent evolutions using the Floquet formalism. The Floquet operator Fˆ is equal to the time evolution operator Uˆ over one full period τ:  τ     ˆ ˆ Hˆ t  Uˆ t  , 0 dt  . F ≡ U(τ, 0) = 1 − i/ (8.11) 0

ˆ As a result, the Floquet operator Fˆ has the property |ψ(τ ) = F|ψ(0), irrespective of the state |ψ(0). A consequence is that |ψ(N τ ) = Fˆ N |ψ(0), and so once Fˆ is known, any state can be straightforwardly evolved over a discrete number of periods ˆ by repeated application of F. The Floquet operator can be decomposed into eigenstates |i  and corresponding eigenvalues λi , which all satisfy |λi | = 1 (Fˆ is unitary). All eigenvalues are therefore

8.4 Quantum Dynamics and the Floquet Formalism

151

of the form λi = exp (−ii τ/), where the angular frequency i is known as the quasienergy of the corresponding eigenstate. This enables decomposition of any initial state |ψ(0) into the Floquet eigenstates, and straightforward calculation of its state after evolution of N periods: |ψ(N τ ) =Fˆ N |ψ(0),

i |ψ(0)Fˆ N |i , =

(8.12)

i

=

i |ψ(0) exp (−ii N τ/)|i . i

The simulations in this chapter use this property to efficiently simulate the evolution of quantum states (Appendix H). Each Floquet eigenstate |i  accumulates a phase determined by its respective quasienergy i , and so superpositions of eigenstates will lead to interference effects. The resulting interference between the eigenstates leads to quasiperiodicity that causes partial revivals of initially localized states, as opposed to the exponential divergence of trajectories in the case of classical chaos. The finite number of eigenstates in the Floquet formalism therefore emphasizes the discrete nature of quantum mechanics. The inverse participation ratio (IPR) quantifies how many Floquet components significantly contribute to a given quantum state |ψ [77] IPR(|ψ) =

D

|ψ|i |4 ,

(8.13)

i=1

where D is the system dimensionality. The IPR is minimal (IPR = 1) when a quantum state is fully described by a single eigenstate, and is maximal (IPR = D) when the quantum state has an equal overlap with all eigenstates (|ψ|i |2 = 1/D ∀ i). Figure 8.7 shows the IPR of spin coherent states for the quantum driven top. A correspondence is observed between classically-chaotic regimes (Fig. 8.2A) and spin coherent states with a high IPR. This can be explained by the fact that states with a high IPR are a superposition of several interfering Floquet states. Since each Floquet state responds differently to perturbations, states with a high IPR are therefore generally more strongly affected by perturbations.

152

A

8 Exploring Quantum Chaos with a Single High-Spin Nucleus

B

Fig. 8.7 Inverse participation ratio of spin coherent states. The spherical surface is visualized as a sphere (A) and as a Hammer projection (B) [85]. Each pixel with polar coordinates (θ, φ) corresponds to the inverse participation ratio (IPR) of the spin coherent state |θ, φ. The IPR is calculated with respect to the Floquet eigenbasis of the quantum driven top with parameters I = 7/2, γn = 5.55 MHz T−1 , B0 = 0.5 T, Q = 800 kHz, B1 = 10 mT, f = 3.5 MHz, matching the classical chaotic driven-top parameters of Fig. 8.2A. Classically-chaotic regions are found to correspond to spin coherent states with a high IPR, reflecting the enhanced sensitivity of these spin coherent states to perturbations

8.5 Quantum Versus Classical Dynamics: A Comparison To illustrate the applicability of the 123 Sb system to the study of quantum chaos, we propose two types of experiments: one aimed at finding a correspondence between the classically chaotic driven top and its quantum counterpart, the other at demonstrating a violation of classical dynamics, exposing the true quantum nature of the system. In what follows, we focus on the system in the laboratory frame, as this puts the most stringent conditions on size and shape of the chaotic region in phase space of the classical equivalent system, however; the suggested experiments are equally well applicable to the system in the rotating frame.

8.5.1 Decoherence as a Precursor of Chaos Classical chaos is characterized by an extreme sensitivity to perturbations, and as a consequence, neighboring trajectories with slightly different initial coordinates rapidly diverge. This is in stark contrast with quantum dynamics, where the discrete nature of the energy spectrum results in quasiperiodic behaviour, leading to partial revivals of the initial quantum state instead (see Sect. 8.4 for details). However, quantum states evolving under slightly perturbed Hamiltonians do experience divergence. Since a quantum system is never truly isolated, interactions with its environment lead to unknown perturbations of the Hamiltonian, effectively entangling the system with its environment. The unknown nature of this process translates to the quantum state losing its purity, thus decohering into a mixed state. Crucial to the quantum driven top, certain initial quantum states are more prone to decoherence, while others remain relatively unperturbed. This behaviour appears related to the high sensitivity of certain classical states to perturbations, but is caused

8.5 Quantum Versus Classical Dynamics: A Comparison

153

here by the different time-evolution operator’s eigenstates that compose a quantum state (see Sect. 8.4 for details). Each eigenstate responds differently to perturbations, and so states that contain significant contributions from more of these eigenstates (i.e. that have a high inverse participation ratio) are more susceptible to phase errors. In line with this picture, Zurek and co-workers [30] predict that the rate at which different initial quantum states decohere provides a mapping to the chaotic or nonchaotic nature of the corresponding classical system, with chaotic classical regions corresponding to more rapidly decohering initial quantum states. The driven-top system can be used to verify this prediction, both through simulations and experiments. By evolving an initial state for a certain duration using the driven-top Hamiltonian (Eq. (8.3)), the resulting density matrix ρ provides information about the degree of decoherence through its purity Tr(ρ 2 ). Decoherence is simulated by randomly varying a Hamiltonian parameter during the state’s evolution, and calculating the state purity from the ensemble average of many final states, each obtained with a different randomized evolution (Appendix H.2). By sampling over all spin coherent states, a ‘purity map’ of the quantum driven top is obtained, which we compare to its classical counterpart (Fig. 8.8). The varied parameters are Q = 800 ± 4 kHz (Fig. 8.8C, D), B0 = 500 ± 1mT (Fig. 8.8E), and B1 = 10 ± 0.5 mT (Fig. 8.8F). Some details in the purity map differ depending on the varied parameter, such as varying B0 resulting in a high purity at the southern pole. Nevertheless, all purity maps share several common features, notably an enhanced purity at the two stable islands and top region, and a decreased purity surrounding them. These regions of high (low) purity correspond to the regular (chaotic) regions in the classical driven top (Fig. 8.8A, B). The simulations highlight a correspondence between the classically-chaotic regions and quantum regions of strong decoherence, and between classically-regular regions and quantum regions of weak decoherence. To experimentally verify these predictions, we aim to prepare spin coherent states (see Sects. 2.5 and 2.9 for details), evolve the system under the driven-top Hamiltonian, and finally reconstruct ρfinal using quantum state tomography [86, 87]. Repeating this for different initial spin coherent states allows for experimental reconstruction of the ‘purity map’, which can then be compared to the corresponding classical phase space.

8.5.2 Dynamical Tunneling In the absence of a periodic drive, trajectories of the classical top are closed orbits confined to distinct regions in a two-dimensional phase space (the surface of a sphere   due to  L  being a constant of motion). Upon addition of a periodic drive, this behaviour is largely kept intact, except near boundaries between the regular regions, where chaotic behaviour appears. The classically-separated regions of regular motion have an analogue in the corresponding quantum system, where they can be identified as regions of weak decoherence (Fig. 8.8). However, there is a fundamental difference

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8 Exploring Quantum Chaos with a Single High-Spin Nucleus

A

B

C

D

E

F

Fig. 8.8 Comparison between classical Poincaré map and quantum state purity of a periodically driven top. The spherical surfaces of constant angular momentum (A, C) are also visualized by Hammer projections [85] (B, D). (A, B) Classical: Stroboscopic map of one chaotic trajectory (orange) and six regular trajectories (blue) for N = 1000 sampling periods, with the same simulation  γ = 0.02α, f = 1.4α/2π . The chaotic region divides the parameters as Fig. 8.2A, i.e. β = α/| L|, three regular regions and contains regular trajectories within their respective regions. (C–F) Quantum: Purity of spin coherent states |θ, φ of an ionized 123 Sb atom for varying Q (C, D), B0 (E), or B1 (F). The nominal parameters are I = 7/2, γn = 5.55 MHz T−1 , B0 = 0.5 T, Q = 800 kHz, B1 = 10 mT, f = 3.5 MHz, matching the classical parameters. The purity Tr(ρ 2 ) at each spherical coordinate (θ, φ) is extracted from the density matrix ρ, which is obtained by evolving a spin coherent state |θ, φ for N drive periods while randomly perturbing the varied parameter once per drive period, and averaging over 200 such evolutions (see Appendix H for details of the calculation). To account for varying sensitivity to perturbations, the number of drive periods has been varied for the different parameters: N = 1000 for Q (C, D) N = 2000 periods for B0 (E), N = 10000 periods for B1 (F). In all cases, the resulting purity is qualitatively similar: the classical regular islands have a relatively high purity, and the classically chaotic areas surrounding these have a lower purity after the same evolution time. Variations between the purities are due to the Floquet components of the initial spin coherent states having differing sensitivities to each of the parameters

8.5 Quantum Versus Classical Dynamics: A Comparison

155

between the quantum and the classical case. In the classical system, the Kolmogorov, Arnold and Moser (KAM) theorem ensures that a system initially prepared within one of the regular regions will remain on a periodic orbit within such a region. The quantum system, however, cannot be precisely localized within a certain region, due to the uncertainty principle. The ‘leakage’ of the quantum wave function into a different regular region results in the phenomenon of dynamical tunneling, i.e. the tunneling of the quantum state between separate regular regions, in violation of the KAM theorem. Dynamical tunneling manifests itself in the quantum driven top as a periodic oscillation of a spin coherent state between the two classically-regular regions associated with the quadratic interaction (Fig. 8.9A). In contrast, a spin coherent state prepared within a classically chaotic region rapidly spreads out and shows no apparent revival to a spin coherent state (Fig. 8.9B, C). When viewed from the Floquet formalism, the two spin coherent states within each of the regular islands are an equal superposition of the two Floquet eigenstates. The difference between the two spin coherent states is that the relative phase between the Floquet eigenstates differs by π . As the system evolves, the relative phase accumulates, resulting in dynamical tunneling between the two islands at a frequency set by the difference in quasienergies of the two relevant Floquet states. Numerical simulations, conducted using Hamiltonian parameters appropriate for 123 Sb, clearly show the appearance of dynamical tunneling for spin coherent states prepared initially within the regions of classically regular periodic orbits (Fig. 8.9B, C). The predicted period of dynamical tunneling is ∼ 3 µs; this will increase by a factor ∼ 100 to ∼ 300 µs upon considering the system in the rotating frame (assuming γn B1,I /2 ∼ 30 kHz and Q is reduced by a factor 100 to ∼ 8 kHz). This is a crucial result, since this period is orders of magnitude shorter than the measured 123 Sb dephasing time of ∼0.1 s (Sect. 6.5), ensuring that the coherent dynamical tunneling oscillations can be observed in an experiment over unprecedented timescales.

8.5.3 Dependence of Dynamical Tunneling Rate on System Parameters We investigate how the dynamical tunneling rate is influenced by different system parameters. Dynamical tunneling arises naturally in the quantum system, since the uncertainty spread of a quantum state prevents it from being truly localized within a classical region of phase space. Therefore, a state prepared within one of the classically stable regions of the driven top will have a finite overlap of its wave function with the other classically stable region, and may tunnel back and forth between these two regions. Qualitatively, the dynamical tunneling rate can be understood as determined by the amount of such a wave function overlap. To gain a better understanding of the tunneling rate, we have numerically studied its dependence on different system parameters. First, as the spin quantum number

156

A

8 Exploring Quantum Chaos with a Single High-Spin Nucleus

B

C

Fig. 8.9 Time evolution of the quantum driven top. Two different initial spin coherent states are chosen in the directions of a classically chaotic region (orange) and a classically regular region (blue). Implementation in a I = 7/2 nuclear spin of an ionized 123 Sb donor in silicon with γn = 5.55 MHz T−1 , B0 = 0.5 T, Q = 0.8 MHz, B1 = 10 mT, f = 5 MHz. The equivalent implementation in the rotating frame, assuming γn B1,I /2 ∼ 30 kHz at a frequency f RF = γn B0 , corresponds to a factor ∼100 smaller values for Q and f , i.e. Q ∼ 8 kHz and f ∼ 50 kHz. (A) Classical angular-momentum trajectories with initial states corresponding to orientations of spin coherent states and parameters matching the quantum simulations (Table 8.1). Trajectories are visualized by both a three-dimensional spherical plot (top) and Hammer projection [85] (bottom), with enlarged black-outlined dots representing the two initial angular-momentum coordinates. (B) Husimi Q representation (Sect. 2.6) of the two spin coherent states (Sect. 2.5) at different moments in their evolutions. Color scale is constant across all panels, and varies between 0 (dark blue) and 1/π (bright yellow). Top (bottom) row corresponds to spin coherent state oriented in the direction of the classically chaotic (regular) region in phase space, specified by orange (blue). In both rows, the sequential Hammer projections show the evolution of an initial spin coherent state. The spin coherent state prepared in the classically chaotic region (top) displays a rapid dispersion over the phase space, while the classically regular spin coherent state (bottom) transfers back and forth between two classically regular regions. This property is known as dynamical tunneling, and is in stark contrast with classical dynamics, where trajectories cannot cross closed regions. (C) Overlap of the time-evolved state |ψ(t) with its initial state |ψ(0) for spin coherent states in classically chaotic (orange) and regular (blue) regions. Dynamical tunneling (blue) is revealed by a near-sinusoidal evolution of the overlap, returning to near unity. This is in contrast to the evolution of the classically chaotic spin coherent state, where the overlap quickly decreases and shows no near-unity revival. Note that the equivalent implementation in the rotating frame will effectively multiply the time axis by a factor ∼100 for the suggested parameters. As both in the laboratory and in the rotating frame implementation of the quantum driven top the dynamical tunneling time is multiple orders of magnitude smaller compared to the expected coherence time of order 0.1 s (Sect. 6.5), no decoherence effects are included in this simulation

8.5 Quantum Versus Classical Dynamics: A Comparison

A

157

B

Fig. 8.10 Dynamical tunneling frequency for varying system parameters. (A) Dynamical tunneling frequency versus spin quantum number I . The static magnetic field B0 = 0.5 T is kept constant and the drive is turned off (B1 = 0 T), while the quadrupole strength is scaled such that Q I = γn B0 = 2.8 MHz to ensure equal linear and quadratic contribution (Sect. 8.3.2). The dynamical tunneling frequency decreases exponentially with increasing I . (B) Dynamical tunneling frequency for varying quadrupole interaction strength (blue) and Zeeman interaction strength (orange). The spin quantum number is fixed at I = 7/2, B1 = 0 T, and γn B0 = 2.8 MHz for varying quadrupole interaction Q I (blue), and Q I = 2.8 MHz for varying Zeeman interaction γn B0 (orange)

I grows, the relative uncertainty spread of a spin coherent state on the sphere of radius I shrinks (Fig. 8.5), and the dynamical tunneling rate decreases accordingly (Fig. 8.10A). Second, the parameters B0 and Q define the relative distance of the corresponding classically stable regions. Upon increasing B0 , the two stable regions come together (Fig. 8.2B), which increases the overlap of coherent states prepared in each stable region, resulting in an increased tunneling rate (Fig. 8.10B). Increasing Q, on the other hand, separates the two stable islands towards opposite ends of the equator (Fig. 8.2C), thus decreasing the tunneling rate.

8.6 Conclusion and Outlook We have quantitatively described a proposal for the realization of a single quantum chaotic system, based upon the nuclear spin of a substitutional group-V donor in silicon. In particular, we have shown that, with realistically achievable parameters, the quantum version of the classically-chaotic driven top can be implemented in the I = 7/2 nucleus of a 123 Sb donor. In fact, the estimated Q = 52(2) kHz of the measured 123 Sb donor (Sect. 6.6) is ideal for the driven-top Hamiltonian in the rotating frame, were it not that the microwave antenna of the device is damaged. Finding another such donor in a future device with a working antenna would thus open up the whole spectrum of features of interest in the study of experimental quantum chaos, from state-dependent decoherence to dynamical tunneling. The experimental verification of these predictions would constitute the first observation of quantum

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chaos in an individual physical system. Such an achievement would reinvigorate the fundamental study of quantum-classical boundaries by providing a well-defined and exquisitely controllable experimental test bed. In terms of applications, one could envisage laying out and operating individually chaotic 123 Sb nuclei in the same types of multi-spin architectures that are being extensively studied in the context of quantum information processing with 31 P donors [75, 88–90]. Substituting a simple I = 1/2 spin with a multilevel system like 123 Sb could allow the study of quantum information processing where the information is encoded in an intrinsically chaotic system. This would be a different and complementary approach to the one taken, e.g. in superconducting systems, where it is the nature of the interaction between multiple qubits that produces a chaotic dynamics [34, 50]. Rather, it could constitute the quantum version of a type of analog computation that has started to show promise in the context of classical neural networks, where having individually chaotic elements can speed up the solution of complex problems [62].

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Chapter 9

Conclusions and Outlook

9.1 Summary In this thesis, we demonstrate full coherent control of the high-spin nucleus of an 123 Sb donor. However, rather than driving the nuclear spin via the conventional NMR, we find that the high-spin nucleus can be driven electrically via NER. This electric driving mechanism is mediated by the nuclear quadrupole interaction, which is found to couple electric fields to the nuclear spin. The nucleus is observed to be highly coherent, with measured coherence times on the order of 0.1 s. The microscopic origins of NER are attributed to a distortion of the covalent bonds between 123 Sb and the neighboring Si atoms, transducing electric fields into significant EFGs at the nucleus. The same effect causes spectral line shifts, an effect known as the linear quadrupole Stark effect (LQSE) [1, 2]. Both the LQSE and NER have been previously observed, but only in bulk samples of high-spin nuclei in polar crystals [3–6], and we have shown that they are also present in silicon. The effect of device strain on the static quadrupole interaction is also characterized through a combination of DFT and finite-element methods, and a good agreement is found with the experimental observations. The coherent control of an 123 Sb nucleus opens the door to experiments that explore quantum chaos. In particular, we simulate the dynamics of the nuclear spin under the quantum driven top Hamiltonian, whose classical equivalent is chaotic. We devise two experiments that probe the quantum-classical correspondence and the onset of chaos. These experiments can be realistically performed on an 123 Sb donor, as the measured 123 Sb already possesses the required Hamiltonian properties. Upon its realization, this would constitute the first-ever experiment on the time-resolved dynamics of quantum chaos in a single quantum system. These results demonstrate that the 123 Sb donor in silicon possesses the unique combination of high coherence, high dimensionality, and high-fidelity control and readout in an individual quantum system. The work in this thesis has set the stage for the next generation of 123 Sb experiments, which can range from practical applications

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Asaad, Electrical Control and Quantum Chaos with a High-Spin Nucleus in Silicon, Springer Theses, https://doi.org/10.1007/978-3-030-83473-9_9

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such as quantum computing to exploring foundational physics such as the quantumclassical transition. In the following sections, we compare the observed 123 Sb properties to other existing high-dimensional single quantum systems, and highlight potential directions that can be pursued with the 123 Sb donor in silicon.

9.2 Comparison with Other High-Dimensional Quantum Systems The 123 Sb donor that has been measured and discussed throughout this thesis is by no means the first high-dimensional single quantum system that has been controlled, and indeed experiments that exploit the high-dimensionality have been performed on a range of other systems. It is worth comparing the performance of some notable high-dimensional quantum systems to that of the 123 Sb donor in silicon. High-spin trapped ions and neutral atoms are systems whose physics most closely resemble that of high-spin donors in silicon. Since neutral atoms can operate in the low-field limit, the spins of the electron and nucleus can be combined (Sect. 3.2.4), thus accessing a larger Hilbert space. An example is 164 Dy, whose 16-dimensional Hilbert space (J = 8) is significantly higher than the eight-dimensional Hilbert space (I = 7/2) of an 123 Sb nucleus. However, as ensembles of neutral atoms are often used, the field inhomogeneities experienced by the atoms composing the ensemble often limits the coherence of the system. High-dimensional neutral atoms have been used to observe quantum chaos [7] and a metrological gain approaching the Heisenberg limit [8]. Trapped ions are one of the most advanced quantum systems to date, and one of the most promising candidates for quantum computing. Consequently, the high dimensionality of certain ion species are often considered as potential qudits to be used as a quantum processing element. For example, detailed qudit schemes have been developed for 137 Ba+ [9], which can host up to eight levels [9]. Superconducting transmons are another strong contender for the basis of a future quantum computer, and have been the platform where quantum supremacy has been demonstrated for the first time [10]. Transmons are high dimensional quantum systems, but for quantum computing only the lowest two states are used as qubit states; leakage to these higher states is a source of decoherence, and much effort has therefore been placed in mitigating leakage [11, 12]. However, the higher states can also be used as a resource, for instance to mediate a two-qubit gate [13], or as a potential qudit [14]. A detailed comparison of the first five transmon energy levels [15] has found a linear rise in decay rate with increasing energy levels, as well as a strong decrease in coherence. This places an upper bound on the number of energy levels that can be effectively used. Nevertheless, given the high-fidelity control operations and straightforward coupling to neighboring transmons, incorporating the high-dimensional nature of transmons could lead to exciting experiments.

9.2 Comparison with Other High-Dimensional Quantum Systems

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In recent work, Godfrin et al. have demonstrated the coherent control and readout of a high-spin Tb3+ ion (I = 3/2) embedded in a molecular magnet [16]. The four-dimensional Hilbert space of the Tb3+ ion was used to implement Grover’s algorithm. The superposition of the four nuclear states has also been used to measure the accumulation of geometric phases, create an iSWAP gate, and probe coherences between non-neighboring states [17]. Performing such interferometry experiments on a high-spin nucleus in silicon would be an interesting avenue to pursue, in part because the ability to drive m I = ±2 transitions should result in different geometric phase accumulations.

9.3 Further Characterization of the Quadrupole Coupling to Strain and Electric Fields We have found that the nuclear quadrupole interaction couples to both strain and electric fields, and have estimated their coupling strengths. Below we list several methods that can be employed to improve these coupling-strength estimates. Accurate estimates of the S-tensor components – which relate strain to EFG at the nucleus (Sect. 7.4.1) – are needed to extract the strain components from the nuclear spectrum with high precision. The method described in Sect. 7.4 combines DFT simulations with finite-element modeling of the device strain to determine the S-tensor components. While the resulting quadrupole splitting f Q is within a factor two of the observed value, other experimental techniques can be employed to verify and improve the S-tensor estimates. Importantly, the S-tensor components are fixed for a given donor species, donor charge state, host material, and crystal structure. These S-tensor components therefore need not be estimated from the spectrum of a single nuclear spin, but can instead be extracted from ensemble measurements where a precisely-controlled strain is applied. The electric-field response tensor R – which relates the electric field to the EFG at the nucleus – is fully characterized by a single component R14 due to the tetrahedral symmetry Td of the crystal (Sect. 7.3.2). The value of R14 has been estimated separately from the m I = ±1 Rabi frequencies, the m I = ±2 Rabi frequencies, and the spectral lineshifts (Sect. 7.3.3). That the three R14 estimates have a fairly large spread is not surprising, given the many steps involved to extract R14 , including electrostatic device simulations using finite-element calculations and the subsequent donor triangulation. The accuracy of the combined R14 estimate can be improved by also measuring the Rabi frequencies and spectral lineshifts for each of the other three donor gates. This can be even further improved by varying the orientation of the external magnetic field B0 , which changes the relevant electric-field components (Sect. 7.3.2). Once R14 is accurately known, the NER Rabi frequencies and spectral lineshifts due to different donor gates can be utilized to triangulate the donor more accurately

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than using capacitive coupling alone (Sect. 5.2). This is because each of the three measurements provides information about different electric-field components. Thus far, strain and electric fields are assumed to independently couple to the quadrupole interaction (Sect. 7.1). However, the tetrahedral symmetry Td at the 123 Sb lattice site is broken in the presence of strain, which might result in higher-order coupling terms that simultaneously depend on strain and electric fields. Such higherorder terms might be detected by adding a piezoelectric layer that can tune the strain at the donor. These higher-order couplings would then manifest themselves as a change in the NER Rabi frequencies or gate-dependent spectral shifts as the strain is varied. So far, little work has been done on how the quadrupole response differs for an ionized versus neutral donor, aside from the work by Franke et al. [18, 19]. This would be especially interesting for the electric-field R-tensor, as there are no measurements on the R-tensor of neutral donors. Finally, the electric modulation of the quadrupole interaction is not necessarily limited to group-V donors in silicon. It is therefore highly plausible that NER and LQSE can also be observed in other defects and host lattices other than silicon, such as defects in diamond and in SiC. Such a claim would need to be verified both experimentally and through simulations. The observation of NER and LQSE across different host materials would mean that the electric quadrupole modulation is a broadly-applicable control method for high-spin defects.

9.4 Strain Sensing The dependence of the quadrupole interaction on strain opens up the potential of utilizing group-V donors as a strain sensor in silicon. This could have important applications in the semiconductor industry, where significant strains are engineered in silicon to optimize the mobility of electrons and holes [20]. Current methods to characterize strain at the nanoscale either have a low spatial resolution, or require cross-sectional cuts that can alter the strain [21]. The ability to non-destructively measure strain at individual lattice sites would provide a wealth of information for the development of silicon-based transistors. A practical approach towards measuring strain in a device such as a silicon transistor could consist of adding a small amount of group-V dopants with I > 1/2 to the region of interest. The strain at a donor site can then be extracted by applying a magnetic field and measuring the nuclear spectrum of the donor in much the same way as is described in Sect. 6.1.1. However, for this strain sensing technique to be useful, the device should require as little features as possible. Since coherent control of the electron and nuclear spins is not necessary, an off-chip microwave antenna can be used to drive the electron and nuclear spins. If the device already contains a gate that is electrically tunable, no additional gates are needed to tune the donor electrochemical potential. The only modification to the device is then the addition of an electron readout mechanism, such as an SET or a quantum point contact.

9.4 Strain Sensing

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Furthermore, the donor positions can be triangulated by measuring the donor lever arms to the available gates (Sect. 5.2). This method could provide an accurate strain map, without first having to perform a cross-sectional cut of the device. Such information would provide valuable insights that could guide future device designs and improve the accuracy of device strain simulations. A potential limitation of using the donor as a strain sensor is that the quadrupole interaction is also affected by electric fields via the R-tensor. Electric fields can therefore shift the observed quadrupole splitting f Q , and thus introduce uncertainty to the extracted strain. However, f Q is unaffected by electric fields when the external magnetic field B0 is aligned with one of the crystallographic axes (Sect. 7.3.2). In this situation, f Q is solely due to strain, thereby mitigating the uncertainty caused by electric fields.

9.5 Quantum Computation The high-spin nucleus of an 123 Sb donor constitutes an eight-dimensional quantum system, making it an attractive candidate for quantum computing. It is the dimensional equivalent of three qubits, and so each of the three-qubit states can be mapped to one of the eight spin states [22]. Similarly, the single- and two-qubit gates can be decomposed into a series of pulses that apply the equivalent operation to the mapped spin states. This would allow the execution of basic three-qubit quantum algorithms on a single quantum system [23], such as Grover’s algorithm [16, 24] and the Deutsch-Jozsa algorithm [25]. One caveat is that the current nuclear spin readout scheme is only quantum non-demolition in the case of a neutral donor, and conditional on a negative readout outcome, i.e. no electron tunneling event [26]. This poses limitations on quantum algorithms that rely on feedback routines based on measurement outcomes. The eight-dimensional nuclear spin can also be employed as a qudit, which is the higher-dimensional equivalent of a qubit. The added dimensions of a qudit can be employed to significantly improve the efficiency of multi-qubit gates, such as the Toffoli gate [27, 28]. Error-correcting codes have also been developed that can correct phase-shift errors [29], possibly leading to enhanced coherence times. Furthermore, fault-tolerant stabilizer codes, such as the surface code, can be straightforwardly extended to qudit systems [30]. Recent results suggest that this leads to a higher maximum error threshold [31, 32]. Qudits also lead to improvements in magicstate-distillation protocols [33, 34], which introduces non-Clifford gates to stabilizer codes, a crucial ingredient for universal fault-tolerant quantum computing. Some of these algorithms require a qudit dimensionality lower than eight, which can be satisfied by utilizing a subset of the available spin states, or using a lower-spin donor such as 75 As or 121 Sb. These results suggest that qudits, such as the 123 Sb nuclear spin, could provide a viable path towards fault-tolerant quantum computing. The flip-flop transition – where the nuclear and electron spins are simultaneously flipped – has been observed in the 123 Sb donor (Sect. 5.5). This transition is caused by

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an electric modulation of the hyperfine interaction, and its observation is likely due to the damaged microwave antenna that enhances the electric fields (Sect. 6.3.1). The flip-flop transition has proven highly useful, as it provided a method to initialize the 123 Sb nuclear spin into a given state with high fidelity and without the need for NMR nor NER (Sect. 5.5.2). The fact that the flip-flop transition could be driven semicoherently, even though the antenna was not designed to maximize electric fields, is encouraging for the flip-flop qubit proposal [35], which relies on the coherent driving of the flip-flop transition. This would provide a scalable path towards quantum computing, and replacing 31 P by 123 Sb would result in a flip-flop qudit, which has the benefit of a larger Hilbert space. A recent result showed that lithographic quantum dots in silicon can be entangled with nuclear spins, and that the nuclear coherence can be preserved while shuttling the electron between different dots [36]. Electron spin qubits in silicon can be coherently controlled by electric fields with high speed and high fidelity [37]. Adding the ability to electrically control quadrupolar nuclei paves the way to quantum computer architectures that integrate fast electron spin qubits with long-lived nuclear quantum memories, while fully exploiting the controllability and scalability of silicon metal-oxide-semiconductor devices, without the complication of routing RF magnetic fields within the device.

9.6 Mutual Coupling of High-Dimensional Spins Even with eight dimensions, a single quantum system is of little use for applications such as quantum computing. The 123 Sb donor reaches its true potential when it can be coupled to other quantum systems. One method to couple 123 Sb donors is by operating them as flip-flop qudits. Their large electric dipole moments are expected to enable robust long-distance dipole-dipole interactions between donors [35]. The same electric dipole moment could also be used to reach strong coupling between an 123 Sb donor and a microwave resonator [38]. An alternative method that does not rely on operating the donor as a flip-flop qudit is to exploit the hyperfine coupling between the neutral 123 Sb donor and nearby spinful nuclei, such as 29 Si nuclei or other donor atoms. The coherent control and readout of a 29 Si nucleus mediated by a 31 P donor has already been demonstrated [39], and the same should be possible with 123 Sb. In fact, the ESR spectrum of the measured 123 Sb donor displays an additional eight-fold splitting, indicating a coupling to a second ionized 123 Sb donor (Appendix B). The coherent control of a second coupled 123 Sb nucleus would result in a 64-dimensional Hilbert space, the equivalent of six coupled qubits. This would already allow the execution of more complex quantum algorithms. The high coherence times of nuclear spins also open up the possibility of using the coupled nuclei as spin registers. Highly-coherent spin registers with up to 10 qubits have already been demonstrated with nitrogen-vacancy (NV) centers in diamond [40], and this method could also potentially be extended to donors in silicon.

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9.7 Quantum Chaos The experiment proposed in Chap. 8 implements the quantum equivalent of the classically-chaotic driven top in a single 123 Sb nucleus. Studying the quantum dynamics of the driven top in this highly coherent nucleus would provide valuable insights into the emergence of chaos, and it would constitute the first ever experiment of the dynamics of quantum chaos in a single quantum system. An exciting prospect for the quantum driven top is the measuring of OTOCs, which is a proposed method to characterize quantum chaos through operations that do not obey the usual time ordering. One demonstrated example of an OTOC is the Loschmidt echo [41], where the return fidelity after a reversal of the Hamiltonian provides an estimate of the Lyapunov exponent [42, 43]. One advantage of OTOCs over the purity experiment described in Sect. 8.5.1 is the possibility of extracting OTOC functions from the state populations alone. This would circumvent the need for measuring the full density matrix using quantum state tomography, which requires many measurement sequences and is sensitive to state preparation and measurement errors. A Loschmidt echo could potentially be implemented in the quantum driven top, and is the topic of ongoing theoretical work. The experiments on the quantum driven top can also be extended by observing the effects of weak measurements. Measurement-induced decoherence has been found to suppress dynamical tunneling in the kicked top [44], while for an open quantum system the choice of monitoring parameters have been found to either suppress or enhance chaos [45]. These results suggest that parameters such as the measurement strength can be used as a tuning knobs to control chaos. Variable-strength measurements have been demonstrated on 31 P by conditional tunneling of the donorbound electron [26]. Another approach to weak measurements would be to operate the donor as a flip-flop qudit and continuously monitor the state through a coupled resonator [38]. Understanding the interplay between measurement-induced decoherence and the emergence of chaos constitutes a crucial step towards unifying quantum and classical mechanics. The effects of quantum chaos are prevalent in the field of digital quantum simulations. For instance, one common approach to simulate a Hamiltonian is Trotterization. If a target Hamiltonian H = Hx + Hz cannot be implemented directly, but both Hx and Hz can be implemented, the time evolution operator U(t) under the Hamiltonian H can be approximated by a first-order Trotterization [46] U(t) = e−  Ht = e−  (Hx +Hz )t n  i i ≈ e−  Hx t/n e−  Hz t/n , i

i

(9.1)

where n is the number of Trotter steps. In the absence of control errors, a higher n leads to a more accurate Trotterized evolution, and hence a lower Trotter error. Recent work by Sieberer et al. [43] has studied the Trotterization of the magnetization in an Ising spin chain, and has found a correspondence between the number of Trotter

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steps n and the kick strength of the kicked top. In particular, a threshold value of n above which Trotter errors proliferate coincides with a high kick strength that results in quantum-chaotic behaviour in the kicked top. Understanding quantum chaos may therefore prove key to predicting the efficacy of digital quantum simulations. The effects of quantum chaos also extends to other domains of quantum computing. For example, the rapid exploration of the entire Hilbert space that is characteristic of quantum chaos makes these systems exponentially hard to simulate classically. It has therefore been suggested that the implementation of a quantum-chaotic circuit can serve as the first demonstration of quantum supremacy [47, 48], where a quantum computer outperforms a classical computer.

9.8 Quantum Metrology The field of quantum metrology uses quantum properties such as superposition and entanglement to enable high-precision measurements of physical quantities. The sensitivity of these measurements can be enhanced past the maximum sensitivity attainable using non-interacting spin-1/2 particles, known as the standard quantum limit. One route towards quantum-enhanced sensing is by creating genuine multipartite entanglement in an ensemble of usually identical quantum systems [49]. Another route is to create a superposition of a high-dimensional system, and here the 123 Sb donor in silicon is a prime candidate due to the ability to drive individual spin transitions, combined with the long coherence times (Sect. 6.5). Chalopin et al. [8] have recently demonstrated a metrological gain close to the Heisenberg limit using an ensemble of neutral isolated 164 Dy (J = 8) atoms, corresponding to a dimension of 2J + 1 = √ 16. By creating a superposition of maximally opposite spin states (|J  + | − J )/ 2, they demonstrated a metrological gain over a classical spin coherent state close to the Heisenberg limit. Such a fragile superposition is highly susceptible to decoherence, and indeed the observed 58(4) µs coherence time of this state was lower with respect to a spin coherent state by a factor roughly equal to the metrological gain. This is where the features of a high-spin donor in silicon could shine, as the combination of high-fidelity initialization, control, and readout, together with long coherence times should allow the creation of a maximally opposite superposition nuclear spin state whose metrological gain very closely approaches the Heisenberg limit, and whose coherence can be maintained over far longer timespans. A recent theoretical study by Fiderer and Braun [50] has found that quantumchaotic states could exhibit a significant metrological gain while being less susceptible to decoherence than superpositions of maximally-opposite states. In their study of the kicked top, the highest metrological gain was observed at the boundaries between the classically regular and chaotic regimes. The quantum-chaotic driven top proposed in Chap. 8 is an excellent candidate to test this chaos-enhanced quantum sensing, as the spin state of the 123 Sb nucleus can be prepared in the classically regular or chaotic

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regime, or right at the boundary. The observed metrological gain and decoherence can then be compared to that of the maximally-opposite spin state, thus testing the efficacy of chaotic quantum sensors.

9.9 Spin-Mechanical Coupling A fascinating prospect raised by our results is the possibility to coherently couple a single nuclear spin to a mechanical resonator. This would require that the dynamical strain caused by the zero-point fluctuations of the mechanical beam results in a nuclear Rabi frequency exceeding the inhomogenous linewidths of both the electron and the mechanical resonator. Fortunately, both systems can have exceptionally narrow linewidths, in the range of a few Hz. Below we provide an estimate of the zero-point strain in a silicon doubly-clamped mechanical oscillator, following the calculation presented in Ref. [51] (see also Ref. [52] for a similar calculation in the case of a singly-clamped beam). Consider a mechanical beam of length L, width w and thickness t, clamped at both its extremities. We assume that the long axis of the beam coincides with the [110] crystallographic direction. For Si, the Young’s modulus is E = 188 GPa for stress along [110], and the mass density is ρ = 2300 kg/m3 . The moment of inertia of the beam is I = wt 3 /12 and the wave number of the fundamental resonance mode is k0 = 4.73/L. From this, one derives the (angular) frequency of the fundamental mode as:   E I t E = 4.732 2 (9.2) 2π f 0 = k02 ρwt L 12ρ In practice, one will need to choose f 0 to match the nuclear Larmor frequency γn B0 (we neglect the small quadrupole shift f Q in this context). Therefore, f 0 and the resulting choice of beam geometry are not all free parameters. Choosing for example to fix the thickness t of the beam (typically to the lowest value allowed by fabrication), we can derive the length L necessary to yield the mechanical resonance that matches the nuclear precession frequency: √ L = 4.73 t

√

E/12ρ 2π γn B0

(9.3)

Next, we calculate the zero-point mechanical strain zpf in such a beam. We assume that the nucleus is placed in the middle of the beam (z = L/2), as close as possible to the surface (where the strain is maximum), and find: 

√  · w −1/2 L −3/2 ≈ 1.23 E −5/8 ρ 1/8 w −1/2 t −3/4 (2π γn B0 )3/4 (9.4) zpf ≈ 5 √ ρE

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This equation shows that, at a constant frequency, zpf is maximized by making the beam as thin and narrow as possible, which also minimizes the length (Eq. (9.3)). It also shows that zpf depends on frequency (and thus on field) to the power of 3/4. Now we can use our models, benchmarked against the LQSE experimental data on the 123 Sb nucleus, to estimate the spin-elastic coupling. In other words, we analyze a hypothetical NAR experiment [53] where a dynamical strain δ at the nuclear site is caused by the zero-point strain zpf of the resonator. We assume that the length of the beam, the z-axis, coincides with the [110] crystallographic axis of Si. We apply a magnetic field along the [52] crystal axis. Considering the m I = 5/2 ↔ 7/2 transition, the Rabi frequency is given by Rabi,NAR f 5/2↔7/2 = α5/2↔7/2

where α5/2↔7/2 = obtain

√

 eqn S44 (δx x − δzz )2 + 2(δ yz − δx y )2 , (9.5) 2I (2I − 1)h

63. Taking the value S44 = 6.1 × 1022 Vm−2 (Sect. 7.4.2), we

Rabi,NAR ≈ 1.93 × 108 Hz · f 5/2↔7/2

 (δx x − δzz )2 + 2(δ yz − δx y )2 .

(9.6)

Identifying the dynamical strain component along the beam, δzz , with the zero-point strain of the mechanical beam, we find a simple linear relation Rabi,NAR ≈ 1.93 × 108 Hz · zpf . f 5/2↔7/2

(9.7)

Rabi,NAR can be identified as the spin-mechanical In the context of cavity-QED, f 5/2↔7/2 vacuum Rabi splitting, 2g. The interesting strong-coupling regime is achieved when g exceeds both the phonon loss κ and the qubit dephasing γ . In the notation and Rabi,NAR  γ , m , where γ = language of relevance to our system, this means f 5/2↔7/2 3.3 Hz is the experimentally observed nuclear spin inhomogeneous linewidth, and

m = f 0 /Q m is the resonator linewidth, determined by its mechanical quality factor Qm . For the purpose of achieving strong coupling, the resonator design must be optimized considering both zpf and m . Typical Si resonators have a constant product Q m · f 0 ∼ 1013 Hz determined by material properties [54]. Therefore, m ∝ f 02 , 3/4 whereas g ∝ f 0 (Eq. (9.4)), indicating that the best chance to achieve strong coupling is by reducing the operating frequency until the lower bound m ∼ γ is reached (γ is usually independent of frequency). Taking for example w = 40 nm, t = 20 nm, and placing the 123 Sb nuclear spin in a static field B0 = 1.5 T, the resulting beam would need to have a resonance frequency f 0 = 8.33 MHz, a length L = 4.71 µm, and would produce a zero-point Rabi,NAR ≈ 1.2 Hz. strain zpf ≈ 0.6 × 10−8 , resulting in f 5/2↔7/2 Assuming that the resonator has a quality factor Q m ≈ 106 at this frequency (consistent with Q m · f 0 ∼ 1013 Hz), i.e. m = 8.3 Hz, we thus see that the spincavity coupling g is within less than a factor 10 of the dephasing rates.

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Importantly, our experiment shows that the challenge in reaching strong coupling would not be set by the nuclear spin coherence, but rather by the zero-point strain and quality factor of the mechanical resonator, both of which can be significantly improved from the values used in the simple estimate above. Combining novel high-Q m resonators with designs that explicitly maximize the zero-point strain, could bring the goal of coherent coupling between a single nuclear spin and a mechanical oscillator tantalizingly within reach. Achieving the singlephonon limit will be very challenging ( f 0 ∼ 10 MHz is equivalent to ∼ 0.5 mK phonon energy), but interesting physics can already be explored in hot cavities [55].

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Appendix A

Finite Element Model Parameters

See Table.

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Appendix A: Finite Element Model Parameters

Table A.1 Simulation parameters used in the COMSOL device model. Parameters taken directly from the COMSOL material library are not listed here. The top section of the table lists the different thermal expansion coefficients. Thermal deformation in the two cool-down steps is captured by the average coefficient of thermal expansion in this temperature range for all materials present. The bottom section lists all applied gate voltages, see Fig. 7.2 for labeling of the gates Parameter Symbol Value Thermal expansion 850 ◦ C to 400 ◦ C Silicon [1] α¯ Silicon oxide [2] α¯ α-Quartz [3] α¯ ⊥ α¯  Thermal expansion 400 ◦ C to 200 mK Silicon [4] α¯ Silicon oxide [2] α¯ α-Quartz [3, 5] α¯ ⊥ α¯  Aluminum [6, 7] α¯ Aluminum oxide [1] α¯ Applied gate voltages Top gate (TG) VTG Right barrier (RB) VRB Left barrier (LB) VLB Donor gate front left (DFL) VDFL Donor gate front right (DFR) VDFR Donor gate back left (DBL) VDBL Donor gate back right (DBR) VDBR Plunger gate VPL Source VSRC

4.198 × 10−6 K1 0.4 × 10−6 K1 23.62 × 10−6 K1 13.33 × 10−6 K1 2.28 × 10−6 K1 0.446 × 10−6 K1 14.186 × 10−6 K1 7.824 × 10−6 K1 21.43 × 10−6 K1 4.98 × 10−6 K1 1.881 V 0.3421 V 0.4293 V 0.4823 V 0.3915 V 0.4177 V 0.4059 V −0.0032 V 0.0002 V

Appendix B

Hyperfine-Coupled 123 Sb Nucleus

The neutral donor’s outer electron can also have a hyperfine coupling to other nuclei in the vicinity, resulting in additional splittings of each ESR spectral line. Though significantly weaker, these couplings can still range up to several MHz for nearby nuclei. Repeated ESR spectrum scans of the |7/2 ESR spectral line shows jumps in the ESR frequency (Fig. B.1B). Superimposing all the ESR spectra exposes eight distinct peaks, separated by a hyperfine splitting of 390 kHz. The approximately equal splitting between the eight peaks strongly suggests that the second nucleus is an additional ionized 123 Sb donor, located a few lattice sites away [8]. The hyperfine-coupled nucleus exhibits several peculiar features that differentiate it from the primary donor. The most striking feature is that the nucleus seems to have a strong preference for the four nuclear spin states with the lowest ESR frequency, and regularly flips between them. There does not seem to be a preference for higher nuclear spin states, which can be explained by a much weaker hyperfine interaction, and hence flip-flop decay. Finally, the nucleus flips more frequently than the primary donor nucleus, which could be due to the hyperfine interaction acting as a (strong) perturbation to the nuclear Zeeman interaction, resulting in a high nuclear flipping probability. The electron coupling to this second 123 Sb nucleus complicates coherent control of the electron. However, the electron is primarily used as a readout method for the nuclear spin (Sect. 6.1.1), and therefore chirp pulses can adiabatically invert the electron spin regardless of the second nuclear spin. Additionally, coupled nuclei can be harnessed as an additional coupled quantum system [9, 10], which in this case results in a combined 128-dimensional (8 × 8 × 2) Hilbert space, the equivalent of seven qubits. While controlling a quantum system with such a large Hilbert space is a challenging feat, it would allow the execution of state-of-the-art quantum computing algorithms.

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Appendix B: Hyperfine-Coupled 123 Sb Nucleus

180 A

1.00 0.75 0.50 0.25 0.00

|−7/2 |−5/2 |−3/2 |−1/2 |1/2 |3/2 |5/2 |7/2

B

0.9

200 0.8 Measurement iteration

175 150 125 100

0.7 0.6 0.5 0.4

75

0.3

50

0.2

25

0.1

0 39.65 39.6505 39.651 39.6515 39.652 39.6525 39.653 39.6535 Microwave frequency (GHz)

0.0

Fig. B.1 Additional hyperfine splitting from a second coupled 123 Sb donor. Repeated singleESR of nuclear spin m = 7/2 tone ESR spectrum scans around a single ESR resonance frequency f 7/2 I B show discrete jumps of the ESR frequency. Superimposing all ESR spectra A reveals eight equidistantly-spaced ESR peaks, separated by 390 kHz. The eight-fold splitting of a single ESR peak suggests a hyperfine-coupled second 123 Sb nucleus

Appendix C

Slope in Δm I = ±2 Rabi Frequencies

Both the m I = ±1 and m I = ±2 NER Rabi frequencies should be symmetric about the center transition (Table 6.1). While this symmetry has been largely observed (Figs. 6.6I and 6.7I), the m I = ±2 Rabi frequencies exhibit a small negative slope with increasing frequency. This slight asymmetry either has a physical cause not captured by the model Hamiltonian of the ionized nucleus, or is caused by an experimental artifact external to the donor, such as a frequency-dependent attenuation of the gate line in this frequency range. To discriminate between these two possible causes, we reduced the static magnetic field B0 from 1.496 T to 1.376 T (Decreasing transition frequencies by 20 f Q ), and remeasured the Rabi frequencies of the outer transitions |3/2 ↔ |7/2 and |−7/2 ↔ |−3/2 (Fig. C.1A) keeping the driving amplitude constant. The two measured Rabi frequencies are both higher than their counterparts at higher magnetic field, thereby excluding that the effect is associated with specific nuclear transitions. Furthermore, by scaling the Rabi frequencies by their coefficients βm I −2↔m I (Table 6.1), all Rabi frequencies follow a straight line with a gradient of 14.1(3) Hz MHz−1 (Fig. C.1B). This is strongly suggestive of a dependence of the observed Rabi frequencies on the electric drive frequency, but not due to an underlying microscopic cause. The likely cause is a frequency dependence of the gate line attenuation, as room-temperature measurements of identical transmission lines have shown variations up to 1 dB MHz−1 in the few-MHz regime. Although the electric drive amplitude is kept constant at the gate line input, this frequency-dependent attenuation causes the drive amplitude to vary at the donor site. This interpretation is further confirmed by the lack of a slope in the Rabi frequencies when driving with the antenna (Sect. 6.3.2), as the antenna transmission line (designed to operate up to 40 GHz) does not have any significant frequency dependence in the few-MHz regime.

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Appendix C: Slope in Δm I = ±2 Rabi Frequencies

Fig. C.1 Rabi frequencies for varying resonance frequencies. A Measured Rabi frequencies for m I = ±2 transitions. The electric drive amplitude applied to the DFR donor gate line input (i.e. before cable transmission losses) is kept constant for all measurements. While the Rabi frequencies are supposed to be symmetric about the center, the measured values at B0 = 1.496 T exhibit a small decrease with increasing electric drive frequency. The outer two transitions have been remeasured at a lower magnetic field 1.376 T and hence at a lower drive frequency. Both Rabi frequencies are higher than their counterparts at higher magnetic field. B Rescaled Rabi frequencies accounting for the Rabi frequency coefficient βm I ↔m I −2 (Table 6.1). The Rabi frequency of each transition m I −2 ↔ m I is scaled down by βm I −2↔m I /β3/2↔7/2 . All scaled Rabi frequencies accurately follow a straight line with slope −14.1(3) Hz MHz−1 , indicating a dependence of the electric drive amplitude on the electric drive frequency, which affects the Rabi frequency. This effect is not observed for m I = ±1 transitions. Since the electric drive amplitude applied to the gate line is kept constant, this effect is likely an experimental artifact caused by a non-uniform transmission profile of the gate line in the frequency regime of the m I = ±2 transitions

Appendix D

Spectral Shift at Charge Transition

A charge stability diagram of the vicinity of the donor charge transition shows a second charge transition at lower donor gate voltages. This charge transition is crossed gate at sufficiently high negative bias voltages VDC exceeding −45 mV (Fig. D.1A). gate As VDC is decreased past this charge transition, the nuclear transition frequencies experience a discrete shift, measured at 1.43 kHz for f 5/2↔7/2 (Fig. D.1B). This disgate crete shift is measured on top of the Stark shift directly caused by VDC (Sect. 6.4). The 123 Sb nucleus must be affected by the presence/absence of the electron corresponding to the charge transition. It is unlikely that this coupling is caused by the hyperfine interaction, as this should decrease the transition frequency upon ionization by an amount A/2, where A is the hyperfine interaction. Additionally, the ESR spectrum would likely have shown a splitting due to the J-coupling with this ancillary electron, and this has not been observed. Instead, the electric field emanating from this ancillary electron most likely couples to the quadrupole interaction via the electric-field response tensor (Sect. 7.3.1), which is the same mechanism that underlies NER (Sect. 6.2) and the LQSE (Sect. 6.4). When nuclear transitions were driven while tuned close to this charge transition gate (VDC ≈ −45 mV), the nuclear spin was nearly always found to have a random final spin state. This effect was not observed when the NER drive was turned off. This could be explained by the RF electric drive rapidly loading/unloading the ancillary electron corresponding to the charge transition. Similar to the neutralization/ionization of the donor (Sect. 3.2.4), each such loading/unloading event has a nonzero probability to flip the nuclear spin. After many such events, the nuclear spin is effectively scrambled.

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A

Appendix D: Spectral Shift at Charge Transition

B

0.52 1.0

0.50 0.48

0.8

read -10 mV

0.6

0.46 0.44

-45 mV

0.42

-65 mV

0.40 0.3

0.4 0.2 0.0

0.4

6

-10 mV -15 mV -20 mV -25 mV 4 -30 mV -35 mV -40 mV -45 mV 2 -50 mV -55 mV -60 mV -65 mV

0

8.441

8.442

8.443

Electric drive frequency (MHz) Fig. D.1 Discrete spectral shift when crossing charge transition. A Charge stability diagram for DFL versus DFR donor gates. The nuclear spectrum is measured while the NER DC bias voltage gate VDC (colored dots) is decreased past a second charge transition (white dashed). B Nuclear gate gate spectrum of |5/2 ↔ |7/2 for varying VDC . As VDC is decreased past the charge transition, a discrete shift of 1.43 kHz is observed, on top of the quadrupole Stark shift. This shift is likely caused by the donor experiencing a quadrupole coupling to the electric fields emanating from the displaced electron

Appendix E

DFT Simulation Details

DFT simulations have been employed by A. Baczewski to calculate the effect of strain on the quadrupole splitting (Sect. 7.4). This section provides ancillary simulation details. For all supercell calculations we use the Projector Augmented-Wave (PAW) formalism [11] with a plane wave basis, as implemented in the Vienna Ab-Initio Simulation Package (VASP) [12–14]. Within the PAW formalism, the all-electron KohnSham orbitals, and their associated density, can be accessed via an explicit linear transformation on smoother pseudo orbitals that can be efficiently represented in a plane wave basis. As such, we can accurately compute properties that depend sensitively on the charge density and local potential near nuclei, while also making use of the frozen core approximation to reduce the number of orbitals that must be explicitly included in a given calculation. One such property is the EFG, which has been shown to be accurately represented within the PAW formalism when compared to both experiments and calculations using the linear augmented plane wave (LAPW) formalism [15]. Because we are studying an ionized Sb donor, we do not need to represent its hydrogenic bound state in our calculations and we can thus work with smaller supercells than are required for neutral shallow defects [16]. That we do not need to represent states near the conduction band edge also mitigates concerns about the impact of the band gap problem on our results. We thus find the semilocal PerdewBurke-Ernzerhof (PBE) construction of the generalized gradient approximation to the exchange-correlation functional [17] to be adequate. Our calculations are carried out using a plane wave cut-off of 500 eV for the orbitals and 1000 eV for the augmentation charges, with a 3 × 3 × 3 Gamma-centered sampling of the first Brillouin zone for 64 atom supercells, and a Gamma-only sampling for 512 atom supercells. Even though we do not need to represent the neutral donor wave function, finitesize effects still impact our results because our supercell calculations are non-cubic and have a net positive charge. It is well-known that total energy and forces in such calculations are slow to converge with system size. Dipole corrections that would otherwise account for finite size errors are only implemented for cubic supercells in © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Asaad, Electrical Control and Quantum Chaos with a High-Spin Nucleus in Silicon, Springer Theses, https://doi.org/10.1007/978-3-030-83473-9

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Appendix E: DFT Simulation Details

the software package that we are using. The values of the EFG computed for the 64 and 512 atom supercells at a given strain are within 10% of one another, and at this level of accuracy suffices to demonstrate a linear trend across different strains. In the worst case, the difference between the values of the slope obtained regressing on a linear model with a zero y-intercept and a nonzero y-intercept is 0.1%. Our calculations were carried out using 4 electron PAW potentials for Si and 15 electron PAW potentials for Sb. To assess the primary chemical contribution to the EFG, we repeated the calculations with a 5 electron PAW potential for Sb and found that the EFG decreased by 5% for both models of uniaxial strain and 2.5% for shear strain. This small change confirms that the EFG is generated primarily by the contribution to the local potential due to the charge density in the sp3 -like Sb-Si bonding orbitals, and not a strain-induced distortion of the on-site contribution to the local potential due to the Sb donor’s d electrons.

Appendix F

Classical Equations of Motion for the Driven Top

F.1

Derivation of Classical Equations of Motion

The general expressions for the equations of motion of the driven top are given here. The starting point is Hamilton’s equation of motion in Poisson bracket formulation:  d L ˆ + ∂L ,  H} = { L, dt ∂t

(F.1)

 T with L = L x L y L z the angular momentum vector, Hˆ the Hamiltonian, and ˆ the Poisson bracket relation between L and H. ˆ Angular momentum conser H} { L,  ˙ vation implies ∂ L/∂t = 0. Introducing L i ≡ dL i /dt, the equations of motion are ˆ L˙ x = {L x , H}, ˆ L˙ y = {L y , H}, ˆ L˙ z = {L z , H}.

F.2

(F.2)

Equations of Motion for the Classical Driven Top

The equations of motion given above can be applied to the classical driven-top Hamiltonian (Sect. 8.2). As an example, we derive an expression for L˙y :

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Asaad, Electrical Control and Quantum Chaos with a High-Spin Nucleus in Silicon, Springer Theses, https://doi.org/10.1007/978-3-030-83473-9

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Appendix F: Classical Equations of Motion for the Driven Top

L˙ y ={L y , αL z + β L 2x + γ L y cos (2π f t)} =α {L y , L z } +β{L y , L 2x } + γ cos (2π f t) {L y , L y }       =0

=L x

=αL x − β {L x , L y } L x − β L x {L x , L y }       =L z

(F.3)

=L z

=αL x − 2β L x L z , where the product rule for Poisson brackets is used in the third line and whenever Poisson brackets are computed, the relation {L i , L j } = ijk L k , with ijk the LeviCivita symbol, is used. Similarly, equations for L x and L z can be derived, resulting in the system of equations L˙ x = − αL y + γ L z cos (2π f t) , L˙ y = αL x − 2β L x L z , L˙ z = − 2β L x L y − γ L x cos (2π f t) .

(F.4)

Appendix G

Quantum Driven Top in the Rotating Frame and the Rotating Wave Approximation

The technique of ‘dressing’ a quantum spin state relies on applying a microwave tone with a frequency matching the dominant linear Zeeman interaction term in the system. Upon transforming the system to the rotating frame,1 and applying the rotating wave approximation (RWA), the original linear Zeeman interaction term disappears, and an effective linear interaction with a strength set by the microwave amplitude appears. We derive the effective ionized nuclear spin Hamiltonian of a donor in silicon using this approach. The starting point is the Hamiltonian proposed in Eq. (8.9), Hˆ quantum,RF = γn B0 Iˆz + Q Iˆx2 (G.1)  + γn B1,I cos (2π f RF t) + γn B1,Q cos (2π f t) sin (2π f RF t) Iˆy .

G.1

Transforming Spin Operators to the Rotating Frame

The transformation of Hˆ quantum,RF to a frame rotating with angular velocity ωRF = 2π f RF is given by

Hˆ quantum,RF = R (−ωRF t) Hˆ quantum,RF R (ωRF t) − f RF Iˆz ,

(G.2)

where R is the rotation operator corresponding to a basis rotation over an angle φ around the z axis: 1

In contrast to the generalized rotating frame described in Sect. 2.7, here only the static Zeeman interaction is negated. For a static Hamiltonian, any quantum state would remain static in the generalized rotating frame, regardless of the static interactions. However, in this rotating frame, any additional interactions such as the quadrupole interaction are not accounted for, and would therefore evolve the system in the rotating frame.

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Appendix G: Quantum Driven Top in the Rotating Frame … ˆ

ˆ

R (φ) = e−iφ Iz = e−iωRF t Iz .

(G.3)

The remaining task is to transform the (squared) spin operators of the original Hamiltonian to the rotating frame. Using the series expansion of the matrix exponent, commutator rules of spin operators and recognizing sine or cosine series in the expansion, one can derive the following identities for rotated spin operators: R (−φ) Iˆz R (φ) = Iˆz , R (−φ) Iˆy R (φ) = sin φ Iˆx + cos φ Iˆy ,

(G.4)

R (−φ) Iˆx R (φ) = cos φ Iˆx − sin φ Iˆy . With the same strategy, albeit less trivially, one can derive for the squared spin operators the following identities: R (−φ) Iˆz2 R (φ) = Iˆz2 , 1 1 R (−φ) Iˆy2 R (φ) = − Iˆz2 + sin 2φ{ Iˆx , Iˆy } 2 2

 1 1 − cos 2φ Iˆx2 − Iˆy2 + I (I + 1) , 2 2 1 1 2 2 R (−φ) Iˆx R (φ) = − Iˆz − sin 2φ{ Iˆx , Iˆy }, 2 2

 1 1 + cos 2φ Iˆx2 − Iˆy2 + I (I + 1) , 2 2

(G.5)

where { Iˆx , Iˆy } denotes the anticommutator of Iˆx and Iˆy .

G.2

Hamiltonian Under the Rotating Wave Approximation

Using the results of the previous section, it is straightforward to arrive at the rotating frame version of Eq. (8.9), which is given by: 1 1

Hˆ quantum,RF = − Q Iˆz2 − Q sin 2ωRF t{ Iˆx , Iˆy } 2 2 1 1 2 2 + Q cos 2ωRF t( Iˆx − Iˆy ) + Q I (I + 1) 2 2  1 + γn B1,I sin 2ωRF t + γn B1,Q cos ωt(1 + cos 2ωRF t) Iˆx 2  1 γn B1,I (1 − cos 2ωRF t) + γn B1,Q cos ωt sin 2ωRF t Iˆy , − 2

(G.6)

Appendix G: Quantum Driven Top in the Rotating Frame …

191

with ω = 2π f the drive frequency to create the quantum driven top, and ωRF = 2π f RF , with f RF = γn B0 the drive frequency of the rotating frame (exactly canceling the Zeeman interaction term γn B0 Iˆz ). Applying the RWA now reduces to neglecting all terms involving oscillatory factors at frequency 2ωRF . Further ignoring the 1 irrelevant static energy offset Q I (I + 1), the Hamiltonian reduces to Eq. (8.10): 2 1 Hˆ quantum,RWA = − γn B1,I Iˆy 2 1 1 − Q Iˆz2 + γn B1,Q cos (2π f t) Iˆx . 2 2

(G.7)

Upon applying two trivial rotations, first by an angle π/2 around the y axis, followed by an angle π/2 around the x axis, the original quantum driven top Hamiltonian of Eq. (6.5) is recovered, demonstrating equivalence between the laboratory and rotating frame approach to creating a quantum driven top.

G.3

Relative Angle Between Quadrupole Interaction and Periodic Drive

The Hamiltonian Hˆ quantum,RF of Eq. (G.1) still contains strong constraints on the relative directions of the different terms, as the linear Zeeman interaction, quadratic quadrupole interaction, and periodic drive are all orthogonal to each other. In a realistic experiment, the angle θ between the principal axis of the quadrupole interaction and the direction of the periodic driving field will be an intrinsic and uncontrollable device property. Under the RWA, however, the constraint of orthogonality between quadrupole interaction and periodic drive may be relaxed. Defining the x, y plane as the one containing the principal axis of the quadrupole coupling and the periodic driving field, the periodic drive term in Eq. (G.1) may be rewritten as 

[γn B1,I sin (2π f RF t + ϕ) + γn B1,Q cos (2π f t) cos (2π f RF t − ϕ)] cos θ Iˆx + sin θ Iˆy ,

(G.8) where a phase shift by an angle ϕ is included in the drive. Upon choosing ϕ to be equal to the angle θ , in the rotating frame, the only oscillatory factors containing the phase θ are all terms in 2ωRF , which are neglected under the RWA. One then recovers as the remaining terms the desired combination − 21 γn B1,I Iˆy + 21 γn B1,Q cos (2π f ) Iˆx . This implies that one may correct for the angle θ between quadrupole interaction and drive axis straightforwardly by including this angle as a phase shift in the periodic drive.

Appendix H

Simulation Details for Chaotic Dynamics

H.1

Classical Simulations

The classical driven-top simulation results (Sect. 8.2) were obtained using the ordinary-differential-equation (ODE) solver SUNDIALS [18]. Communication between MATLAB and Sundials was through self-written C code that was optimized for the driven-top system. The combination of SUNDIALS and the intermediate C code resulted in a computational speedup of over an order of magnitude compared to the native MATLAB ODE solvers. As chaotic dynamics are highly sensitive to perturbations, stringent error tolerances were chosen to ensure a high degree of accuracy in the computation of the trajectories. The most computationally expensive simulations were those to determine the percentage of phase space that is chaotic (Figs. 8.3 and 8.4). Both color maps consist of 25 × 25 logarithmically-spaced points, each of which corresponds to a particular parameter set. For each parameter set, a total of 2000 initial angular-momentum coordinates were chosen uniformly distributed over the phase space. To determine whether the dynamics of an initial coordinate is chaotic, a neighboring point with distance 10−8 was chosen, and both were evolved for a fixed duration of 100/α. Whether or not a trajectory is chaotic is determined by measuring the distance between the two points over time, and fitting this to an exponential curve. Chaos is characterized by an exponential sensitivity to perturbations, and so the trajectory is categorized as chaotic if its exponent is above a certain threshold. This procedure is repeated for each of the 2000 initial conditions, resulting in the percentage of phase space that is chaotic. Some trajectories, especially near a chaotic-regular boundary, can display an initial exponential divergence but nevertheless behave regularly over sufficient evolution time. These cases, although uncommon, can result in the trajectory being wrongly categorized as chaotic, and we expect a small uncertainty in the percentages of Figs. 8.3

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Asaad, Electrical Control and Quantum Chaos with a High-Spin Nucleus in Silicon, Springer Theses, https://doi.org/10.1007/978-3-030-83473-9

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194

Appendix H: Simulation Details for Chaotic Dynamics

and 8.4 It should furthermore be noted that the fitted exponent is not necessarily equal to the Lyapunov exponent, as the inter-trajectory distance may have increased sufficiently to be limited by the finite size of the phase space. Although the exponent is then an underestimate of the Lyapunov exponent, it will certainly be above the chaotic threshold, thereby correctly characterizing the trajectory as chaotic.

H.2

Quantum Simulations

ˆ which can The quantum driven-top system is evolved using the Floquet operator F, be approximated through segmentation as Fˆ ≈

N

e− N H( it

ˆ

N −k N t

),

(H.1)

k=1

which becomes an equality in the limit N → ∞. In the simulations, a fixed value of N = 1000 is used, as results showed that the Floquet operator did not significantly change upon further increasing N . Additionally, the SUNDIALS ODE solver was ˆ which was found to be nearly identical to Fˆ computed using the used to compute F, above method. Decoherence of spin coherent states under influence of a perturbed driven-top Hamiltonian (Sect. 8.5.1) was simulated by randomly fluctuating a Hamiltonian parameter during its evolution and averaging over many such evolutions. To this end, Floquet operators were calculated for 30 values of the varied parameter, uniformly distributed within three standard deviations of its mean value. The varied parameters are Q = 800 ± 4 kHz (Fig. 8.8C, D), B0 = 500 ± 1 mT (Fig. 8.8E), B1 = 10 ± 0.5 mT (Fig. 8.8F). For the evolution, a sequence of Floquet operators was chosen through random sampling of this Floquet-operator set using a Gaussian distribution. This sequence was then applied to all initial spin coherent states, and this randomized process was repeated for 300 such sequences. For each spin coherent state, the final density matrices were averaged, resulting in a mixed state ρ with corresponding purity Tr(ρ 2 ). We note that in all cases the correspondence between classical phase space being regular or chaotic and quantum state purity decaying slow or fast is not exact. Rather the general structure of the underlying classical phase space is recovered; i.e. classically stable points correspond to quantum states being less sensitive to fluctuations.

About the Author

Serwan Asaad is a postdoctoral research fellow studying nonlocal physics and nonabelian statistics in Charlie Marcus’ group at the Niels Bohr Institute, University of Copenhagen. He completed his Ph.D. at the University of New South Wales (UNSW) in Sydney, where he worked in the group of Andrea Morello on high-spin nuclei in silicon. He received the Malcolm Chaikin award for this work.

Patent A. Morello, S. Asaad, and V. Mourik. 2019. “Nuclear spin quantum processing element and method of operation thereof.” International Patent Application PCT/PCT/AU2019/000029, filed March 2019. Patent Pending. Peer reviewed journal articles S. Asaad* , V. Mourik* , B. Joecker, M. A. I. Johnson, A. D. Baczewski, H. R. Firgau, M. T. Ma˛dzik, V. Schmitt, J. J. Pla, F. E. Hudson, K. M Itoh, J. C. McCallum, A. S. Dzurak, A. Laucht, and A. Morello. “Coherent electrical control of a single high-spin nucleus in silicon.” Nature 579.7798, pp. 205–209 (2020). S. B. Tenberg, S. Asaad, M. T. Ma˛dzik, M. A. I. Johnson, B. Joecker, A. Laucht, F. E. Hudson, K. M. Itoh, A. M. Jakob, B. C. Johnson, D. N. Jamieson, J. C. McCallum, A. © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 S. Asaad, Electrical Control and Quantum Chaos with a High-Spin Nucleus in Silicon, Springer Theses, https://doi.org/10.1007/978-3-030-83473-9

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S. Dzurak, R. Joynt, and A. Morello. “Electron spin relaxation of single phosphorus donors in metal-oxide-semiconductor nanoscale devices.” Physical Review B vol. 99, no. 20, p. 205306 (2019). V. Mourik,2 S. Asaad* , H. R. Firgau, J. J. Pla, C. Holmes, G. J. Milburn, J. C. McCallum, and A. Morello. “Exploring quantum chaos with a single nuclear spin.” Physical Review E vol. 98, no. 4, p. 042206 (2018). S. Asaad* , C. Dickel* , N. K. Langford, S. Poletto, A. Bruno, M. A. Rol, D. Deurloo, L. DiCarlo. “Independent, extensible control of same-frequency superconducting qubits by selective broadcasting.” npj Quantum Information vol. 2, p. 16029 (2016). A. Bruno, G. de Lange, S. Asaad, K. L. van der Enden, N. K. Langford, L. DiCarlo. “Reducing intrinsic loss in superconducting resonators by surface treatment and deep etc.hing of silicon substrates.” Applied Physics Letters vol. 106 no. 18, p. 182601 (2015). Articles submitted for publication P. D. Blocher, S. Asaad, V. Mourik, M. A. I. Johnson, A. Morello, K. Mølmer. “Measuring out-of-time-ordered correlation functions without reversing time evolution.” arXiv:2003.03980 (2020). C. Adambukulam, V. K. Sewani, H. G. Stemp, S. Asaad, M. T. Ma˛dzik, A. Morello, A. Laucht. “An ultra-stable 1.5 tesla permanent magnet assembly for qubit experiments at cryogenic temperatures.” arXiv:2010.02455 (2020).

2

These authors contributed equally.

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