Long Distance Entanglement Between Quantum Memories (Springer Theses) 9789811979385, 9789811979392, 9811979383

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Springer Theses Recognizing Outstanding Ph.D. Research

Yong Yu

Long Distance Entanglement Between Quantum Memories

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.

Yong Yu

Long Distance Entanglement Between Quantum Memories Doctoral Thesis accepted by University of Science and Technology of China, Hefei, China

Author Dr. Yong Yu Hefei National Laboratory for Physical Sciences at the Microscale University of Science and Technology of China Hefei, China

Supervisor Prof. Xiao-Hui Bao Hefei National Laboratory for Physical Sciences at the Microscale University of Science and Technology of China Hefei, China

ISSN 2190-5053 ISSN 2190-5061 (electronic) Springer Theses ISBN 978-981-19-7938-5 ISBN 978-981-19-7939-2 (eBook) https://doi.org/10.1007/978-981-19-7939-2 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 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 Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Supervisor’s Foreword

The quantum internet delivers an upgraded version of our current internet, which enables exciting functions like secure communication, networked quantum computation, and enhanced quantum metrology, among many others. Entanglement between remote quantum memories is the building block of such a network, which, however, has only been achieved at limited scales due to scientific and technical challenges. In order to realize entanglement between remote quantum memories, this thesis focuses on the system of cold atomic ensembles and performs in-depth experimental investigations. The author first reviews the background and state of the art and gives a crash course on the quantum network and cold atom system, which are helpful for beginners in this research field. Afterwards, the author introduces two state-ofthe-art techniques enabling remote entanglement. One is a quantum memory with intrinsic retrieval efficiency of 0.9. The other is a quantum frequency conversion module with an end-to-end conversion efficiency of 0.34. Finally, the author presents two remote entanglement experiments: one through the two-photon scheme with 22-km fibre transmission; the other through the single-photon scheme with 50-km fibre transmission. Beyond these, the author also introduces an experiment regarding verifying a quantum memory through the measurement-device-independent method and proposes some proposals to improve the cold atomic ensemble-based quantum memories. The research described in this thesis paves the way for atomic ensemble-based quantum networks. It may become a valuable resource for a broad readership to study efficient quantum memory, remote entanglement, quantum networks, etc. Anhui, China September 2022

Xiao-Hui Bao

v

Abstract

A quantum network means a network connecting quantum processors located in different places and transferring quantum states among them. It could provide some revolutionary functions, such as global secure communication, distributed quantum computing, etc. It relies on the entanglement of remote quantum memories over long distances. In this thesis, we try to extend the distance by using quantum memories based on cold atom ensembles. The development of remote entanglement is hindered by low brightness of atom–photon entanglement and high transmission loss in fibre channel. We first use a ring cavity to enhance the atom–photon interaction, thence the retrieval efficiency is promoted. Besides, the cavity serves as a filter itself. Hence, no extra filter loss is introduced. Compared with our previous results, the brightness of atom– photon entanglement is one order of magnitude higher. Then by using quantum frequency conversion, we shift the atomic wavelength from 795 nm to 1342 nm in the telecommunications band. The transmission attenuation drops from 3.5 dB/km to 0.3 dB/km. Next, we create remote entanglement by photon interference. There are two main schemes in remote entanglement creation: two-photon interference scheme and single-photon interference scheme. Generally, the two-photon interference scheme requires less on phase stability. However, it offers a lower entanglement probability. The single-photon interference scheme offers a higher entanglement probability. Nevertheless, it requires a more stable phase environment. We realize entanglement first via two-photon interference over 22 km of field-deployed fibres, and second via single-photon interference over 50 km of coiled fibres. To achieve remote singlephoton interference, we design and perform two-stage phase stabilization. The phase uncertainty of 50 km is suppressed to about 50 nm. Though two memories are located in the same laboratory now, our experiment could be extended to nodes physically separated by similar distances with some minor amendments. In addition, we make other two considerations about our quantum memory. One is about the verification of a quantum memory. We, for the first time, demonstrate the verification of a quantum memory via the measurement-device-independent scheme. Besides, we use a small atomic ensemble to generate high-quality single photons vii

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Abstract

with the help of the Rydberg blockade effect, and a quantum random number generator to ensure a random polarization preparation. Our approach takes into account manoeuverability while ensuring safety, thus has strong practicality. The other one is that we propose some protocols to enhance the performance of current quantum memory. One part is about achieving deterministic atom–photon entanglement and deterministic entanglement swapping inside a quantum memory. The other part is about spin-wave manipulation in a DLCZ quantum memory. We introduce how to achieve a long-lifetime and multimode storage of a qubit. Our work of 22 km and 50 km entanglement between quantum memories by two different schemes paves the way towards a quantum repeater-based quantum network. By combining with techniques such as Rydberg-based deterministic entanglement, long-lifetime storage, etc., it will greatly promote the experimental research of quantum repeaters and full quantum networks. Keywords Quantum memory · Cold atoms · Quantum frequency conversion · Remote entanglement · Quantum network

Acknowledgements

The six years in USTC have been one of my fastest growing periods, from knowing nothing about research to being able to run an experiment independently. I must thank many people for their help, concern, and support. First, I would like to thank my supervisor, Prof. Xiao-Hui Bao. He is very knowledgeable and rigorous in the research, giving us ample freedom to explore. Under his careful guidance, I grow rapidly. In addition, he pays great attention to building our ‘physical pictures’, hoping that we can recognize the simple principles behind complex phenomena, which has benefited me a lot. He is also very easy-going and always encouraging, which has rekindled my confidence repeatedly. In addition, he is also a ‘full stack scientist’. He has deep knowledge of various abilities involved in the experiment, including programming and electronics, among others, which I admire very much. Thanks to Prof. Jian-Wei Pan, the Director of the Research Department. Prof. Pan is one of the leaders in the field of quantum information. It is fortunate to stand on the shoulders of giants and start a research career. Prof. Pan has created excellent scientific research conditions, enabling us to carry out our work without distractions. At the same time, he also advocates a free academic atmosphere. He organizes frequent academic exchanges so that we have the opportunity to experience the collision of academic ideas in different fields. Thanks to Prof. Qiang Zhang for his support and encouragement in the long distance entanglement experiment. Prof. Zhang is humorous. Usually, a few simple words from him makes me happier and more confident. I want to thank Prof. Feihu Xu for his theoretical support in the MDI experiment and his guidance and concern for my studies. I want to thank Prof. Christoph Simon at the University of Calgary, who gave me helpful advice and sincere encouragement. Thanks to the senior colleagues in our lab, Yan Jiang and Bo Jing. Although Yan was approaching graduation when I entered the laboratory, the atmosphere in the group was relaxed and lively because of her, making me quickly integrate into the group. Bo gave me a lot of guidance when I started, and many of my basic experimental skills were taught by him. At the same time, I would like to thank

ix

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Acknowledgements

Sheng-Jun Yang, Jun Li, Xu-Jie Wang, and Ming-Ti Zhou. Sheng-Jun’s down-toearth work style is worth learning all the time. Jun is willing to communicate, and many ideas come from his discussions. Xu-Jie and Ming-Ti gave me much help in the early days of my Ph.D. Thanks to Peng-Fei Sun. He was responsible for the Rydberg part of the MDI experiment. Thanks to his excellent work, the experiment progressed smoothly. Thanks to Xi-Yu Luo, most of my work during Ph.D. was basically done in cooperation with him. Thanks to Chao-Wei Yang, Jian-Long Liu, Ren-Zhou Fang, Zi-Ye An, Chao-Yang Wang, Xiao-Yi Lai, Bin Wang, Bo-Wei Lu, and other colleagues in our small group. The failures and successes we have experienced together are unforgettable memory. Thanks to Dr. Fei Ma. He made the frequency conversion waveguide in the long distance entanglement experiment and gave many valuable suggestions in other parts of the experiment. Thanks to Yu-Zhe Zhang for his assistance in the experimental theory part of MDI. Thanks to Dr. Zheng-Da Li for constructive discussions in the MDI experiment. Thanks to Prof. Jun Zhang and his team of single-photon detectors. All the siliconbased detectors used in the experiment were improved and maintained by them. Thanks to Yu-Qiang Fang. He is always patient in helping me adjust the detector to meet various experiment needs. Thanks to Bing Bai, the quantum random number generator in the MDI experiment was provided by him. Thanks to Prof. Xiao Jiang and his electronics team. Professor Jiang’s team provided many electronic devices for the experiment. Prof. Jiang is happy to discuss the specific needs of the experiment with us and support the experiment with the most simple and effective solutions. Thanks to Yi Hu for the signal identification and timing control system, Na Li and Hao Min for the RF signal source, and Ming-Qi Huang for his support in the DFB laser. Thanks to Ming-Yang Zheng, Yao Quan, and Jing-Li Fan from Jinan Institute of Quantum Technology for their help with frequency conversion waveguides, fibre lasers, amplifiers, and many others. Thanks to Prof. Li-Xing You and Wang Zhen from the Shanghai Institute of Microsystems, as well as Jing-Hao Shi, Yan-Yang Jiang, Wei-Jun Zhang, and ChengJun Zhang for their help and guidance on SNSPDs. Thanks to Prof. Teng-Yun Chen, Prof. Kai Chen, Prof. You-Jin Deng, Prof. ZhenSheng Yuan, Prof. Bo Zhao, Prof. Yu-Ao Chen, Prof. Chao-Yang Lu, Prof. Xi-Lin Wang, Prof. Jing-Yun Fan, Prof. Ping Xu, Prof. Han-Ning Dai, and other professors for their care and guidance. Thanks to everyone in the group for helping me, including but not limited to Yang Liu, Qi-Chao Sun, Guo-Liang Shentu, Ming-Jia Shangguan, Ya-Li Mao, Yi-Zheng Zhen, Wen-Fei Cao, Yu-Ming He, Yu He, Jin-Peng Li, Zhao-Chen Duan, Meng-Da Li, Yong-Guang Zheng, Yu-Meng Yang, Yue-Fei Wang, Wen-Hao Jiang, Xuan-Kai Wang, Wei-Yong Zhang, Xing Ding, Ying-Qiu Mao, Yang-Dan Jiang, Xing-Yang Cui, Chao Yu, Shu Xu, Xiang-Xiang Sun, Bao-Zong Wang, Hong-Tai Xie, Mi Zou, Bi-Xiao Wang, Hao-Tao Zhu, An Luo, Ying Liu, Wan Lin, Hui Wang, Han-Sen

Acknowledgements

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Zhong, Bi-Ying Wang, Yi-Han Luo, Yu-Hao Deng, Ming-Han Li, Liang Huang, Shuai Zhao, Cheng Guo, De-Zhi Zhu, among many others. Thanks to the secretaries of the big group for bringing convenience to the daily procedures. Thanks to Secretary Wen Li of the Department of Modern Physics for her help in student affairs. Thanks to Prof. Hai Wang and Tian Tian from my postgraduate class for their help. Thanks to my parents for nurturing and educating me over the years; they gave me the best growth environment, taught me by example, and silently supported my choice. Finally, I would like to thank my girlfriend, Mrs. Hui Liu. She has given me countless help both in life and in my study. She is very tolerant of my shortcomings and cheers me up when the research is not going well. I cannot imagine life without her. Hefei, China June 2020

Dr. Yong Yu

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Fundamental of Quantum Information Processing . . . . . . . . . . . . . . . 1.2 Entanglement-Based Cryptography . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Ekert 91 Protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Quantum Teleportation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Quantum Repeater . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Quantum Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Structure of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 5 5 6 6 8 8 10 11

2 Principles of Quantum Memories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 EIT Quantum Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Property Manipulation of the Optical Medium . . . . . . . . . . . . 2.1.2 EIT Quantum Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 The Dynamic Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 DLCZ Quantum Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 DLCZ Quantum Memory in Free Space . . . . . . . . . . . . . . . . . 2.2.2 Cavity Enhanced DLCZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Rydberg Blockade Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Rydberg Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Rydberg Blockade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15 15 16 18 19 21 22 24 26 26 28 28

3 A High-Efficiency Quantum Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Preparation of Cold Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Doppler Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Magneto-Optical Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Polarization Gradient Cooling . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4 Energy Diagram for Atomic Cooling . . . . . . . . . . . . . . . . . . . . 3.2 Energy Scheme for Quantum Memory Experiments . . . . . . . . . . . . . 3.3 The Ring Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31 31 31 33 35 36 36 39 xiii

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3.3.1 Cavity Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Cavity Locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Quantum Memory Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Quantum Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Storage Efficiency and Lifetime . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Entanglement Benchmarking . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Comparison of the State-of-the-Art Quantum Memories . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39 41 44 44 46 46 48 49

4 Quantum Frequency Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Nonlinear Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Coupled Wave Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Quasi Phase Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Periodically Poled Lithium Niobate Waveguides . . . . . . . . . . . . . . . . 4.2.1 Generation of Periodic Poling . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Waveguide Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Integrated QFC Chip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 QFC Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 QFC Setup and Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51 51 51 53 55 57 59 59 61 61 62 64

5 Remote Entanglement via the Two-Photon Scheme . . . . . . . . . . . . . . . . 5.1 Comparison of Two Entanglement Schemes . . . . . . . . . . . . . . . . . . . . 5.2 Experiment Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 The Degree-of-Freedom Conversion of Photon Qubit . . . . . . . . . . . . 5.4 Field Deployed Fibres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Optimization of the Experiment Condition . . . . . . . . . . . . . . . . . . . . . 5.6 Results for Remote Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Imperfection Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67 67 70 71 73 74 75 76 78

6 Remote Entanglement via the Single-Photon Scheme . . . . . . . . . . . . . . 6.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Phase Locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Phase Locking Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Suppression of Fast Phase Variation in Long Fibre Situation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Phase Uncertainty in PPLN Waveguide Chips and Atomic Ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Benchmarking the Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Experimental Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Imperfection of Photon Interference . . . . . . . . . . . . . . . . . . . . 6.4.2 Write-out Photon Mismatch . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.3 Phase Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.4 Decrease of SNR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81 81 82 82 85 85 87 92 92 92 93 93

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6.5 Lasers in Outdoor Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6.5.1 Phase of Control Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6.5.2 Test Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 6.5.3 Statistics of Phase Fluctuation . . . . . . . . . . . . . . . . . . . . . . . . . 99 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 7 Measurement-Device-Independent Verification of a Quantum Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 The Theory of Measurement-Device-Independent Verification . . . . 7.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Rydberg Single Photon Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 The EIT Quantum Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Optimization of the Readout Profile . . . . . . . . . . . . . . . . . . . . . 7.4.3 Phase Locking and the Losses in the Setup . . . . . . . . . . . . . . . 7.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Further Improvement of Atomic Ensemble Quantum Memories . . . . 8.1 Entanglement Creation and Swapping via Rydberg Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Quantum Circuits Representations for Operations in Rydberg Collective States . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Deterministic Entanglement Generation . . . . . . . . . . . . . . . . . 8.1.3 Intra-atom Entanglement Swapping . . . . . . . . . . . . . . . . . . . . . 8.2 Raman Transition-Based Spinwave Operations . . . . . . . . . . . . . . . . . 8.2.1 Long-Lived Storage of Qubits . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 The Configuration of Raman Beams in Ring Cavity Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.3 Spinwave Echo-Based Multimode Quantum Memory . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

101 101 104 106 108 108 110 115 116 118 121 121 122 123 124 125 126 128 129 132

9 Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

Chapter 1

Introduction

Quantum information science is the intersection of physics and information science that expands the understanding of information science by processing information using the principles of quantum mechanics [1]. For example, the quantum computer, first proposed by Feynman [2], is considered the game-changer of the current cryptography and information processing due to its promising computing power and the potential to tackle complex problems [3, 4]. Experimentally, the most successful quantum computer system so far is superconducting circuits [5–9], on which Google has already achieved quantum superiority for specific problems in 2019 [10], while quantum computers based on trapped ions [11–14], neutral atom arrays [15–18], photon-interference [19, 20] and other platforms are also following. With the fast growth of quantum computers, the traditional RSA protocol encrypting information under computational complexity is facing a big challange [21, 22]. Nevertheless, in the meantime, a window of quantum cryptography is again opened by quantum mechanics, in which the uncertainty principle and no-cloning theorem secure the information. The first scheme of quantum key distribution was proposed by C. Bennet and G. Brassard in 1984 [23]. A lot of theoretical [24–27] and experimental [28–31] works have been done subsequently to improve the scheme in terms of the key rate, communication distance, ease of use, and security. Arguably, quantum key distribution is quantum information science’s first functional and practical application. However, point-to-point quantum communication has fundamental limitations in the distance of a few hundred kilometres due to the exponential growing loss of photons in optical fibres. And the effort of using the classical relay amplification technique will violate the information because of the no-cloning theorem [32]. To break the limitation of long-distance quantum communication, Briegel et al. introduced the concept of quantum repeater [33] in 1998, which could, in principle, reduce the loss of long-distance transmission from exponential to polynomial levels. And later, the concept of quantum networks [34, 35] was proposed based on which many powerful functions, such as distributed quantum computing, can be provided. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Yu, Long Distance Entanglement Between Quantum Memories, Springer Theses, https://doi.org/10.1007/978-981-19-7939-2_1

1

2

1 Introduction

1.1 Fundamental of Quantum Information Processing In classical information, the bit is the unit of information and can take the value 0 or 1. The corresponding unit of information in quantum information is the quantum bit (qubit).1 As allowed by quantum mechanics, the qubit can not only take the value 0 or 1, but can also be in the superposition of 0 and 1, |ψ = α|0 + β|1,

(1.1)

where α and β are complex numbers and satisfy |α|2 + |β|2 = 1. Thus the state of a qubit can be determined by three free real parameters, among which the global phase has no observable effect and is usually unimportant. Therefore we usually write a qubit with two parameters, |ψ = cos

    θ θ |0 + sin eiφ |1. 2 2

(1.2)

In this way, any qubit state can be represented by a point on the sphere of a unit ball, as shown in Fig. 1.1, known as the Bloch sphere or the Poincaré sphere. On the sphere, apart from |0 and |1, we have additionally marked several commonly used maximal superposition states as follows, 1 1 |± = √ (|0 ± |1) , |±i = √ (|0 ± i|1) . 2 2

(1.3)

Most of the advantages of quantum information over classical information arise from the nature that single qubits can stay in a superposition state and that multiple qubits can be entangled with each other [36]. A famous example of entanglement is the four maximally entangled two-qubit states, i.e. the Bell states, 1 1 |±  = √ (|00 ± |11) , | ±  = √ (|01 ± |10) . 2 2

(1.4)

Qubits can be encoded on different particles in different physical systems. In some protocols, a logical qubit consists of several physical qubits. In this thesis, we mainly use two types of the physical system for qubit encoding, single photons and neutral atoms. Photons are the energy quanta of light. They can not be cloned, thus effectively preventing eavesdropping. Photons are generally weakly coupled to the environment and always propagate at the speed of light, making them ideal carriers of information. Photons also have many degrees of freedom available for qubit encoding. In this thesis, we will use polarization encoding and time-bin encoding. In the polar1

Quantum information can also be encoded in higher dimensional Hilbert space as a quantum dit (qudit).

1.1 Fundamental of Quantum Information Processing

3

Fig. 1.1 The Bloch sphere of a qubit

ization encoding, |0 and |1 are encodes as two orthogonal polarization states. For instance, horizontal polarization |H  ≡ |0 and vertical polarization |V  ≡ |1 is a commonly used encoding. A few other commonly used maximum superposition states are diagonal polarization |D ≡ |+, anti-diagonal polarization |A ≡ |−, right circular polarization |R ≡ |+i, and left circular polarization |L ≡ |−i. In experiments, polarization states are manipulated by the combination of λ/2 waveplates (HWP), λ/4 waveplates (QWP) and measured by polarizing beamsplitters (PBSs). In the time-bin encoding, |0 and |1 are encoded as two nearby time bins. Usually we define the early bin |E as |0 and late bin |L as |1. The atom we use in this thesis is an isotope of the alkali atom rubidium (Rb), 87 Rb. As an alkali atom, 87 Rb has an outermost electron with the principal quantum number of n = 5. Its ground and first excited states have the same principal quantum and the orbital quantum numbers L = 0 and L = 1, respectively. The transitions between them are called the D lines [37]. The outermost electron has an electron spin of S = 1/2, which couples to the orbital angular momentum and forms a total angular momentum J = L + S, ranging in |L − S| ≤ J ≤ L + S. For the ground state, J = 1/2, the electron configuration is written as 52 S1/2 ; and for the first excited state, two total angular momentum J = 1/2 and J = 3/2 are allowed, written as 52 P1/2 and 52 P3/2 , respectively. Transition lines 52 S1/2 → 52 P1/2 and 52 S1/2 → 52 P3/2 correspond to wavelengths of 795 nm and 780 nm, and are called the D1 and D2 lines, respectively. In atomic physics, the coupling between the electron orbital momentum and spin momentum is called the fine structure. For 87 Rb, its nuclei has a nuclear spin I = 3/2, which couples to J to form a new total angular momentum F = J + I, ranging in |J − I | ≤ F ≤ J + I . For 52 S1/2 and 52 P1/2 , F takes the values 1 and 2, and for 52 P3/2 , F takes the values 0, 1, 2, and 3. We call

4 Fig. 1.2 Energy diagram of the ground state and the first excited state of 87 Rb atom. Data from [37]

1 Introduction F=3 266.7 MHz

52P3/2

F=2 156.9 MHz

F=1 72.2 MHz

F=0 780.241 nm 384.230 THz

F=2

52P1/2

814.5 MHz

F=1

794.979 nm 377.107 THz

F=2

52S1/2 6.83 GHz

F=1

the nuclear spin-assisted splitting the hyperfine structure. In the zero magnetic field case, the hyperfine levels degenerate, while in the presence of a magnetic field, they split in energy due to the Zeeman effect. The energy level structure of the ground and first excited states of 87 Rb is shown in Fig. 1.2. There are two common ways of encoding qubits in atoms. One method is to encode |0 and |1 as a pair of internal states. For example, we can encode on a pair of Zeeman sublevels, |0 ≡ |52 S1/2 , F = 1, m F = −1 and |1 ≡ |52 S1/2 , F = 1, m F = +1. Another method is to encode on the Fock state of a specific level, i.e. |1 for one population on the state and |0 for no population on the state.

1.2 Entanglement-Based Cryptography

5

1.2 Entanglement-Based Cryptography Many quantum communication schemes use remote quantum entanglement as a key resource. Hence we give a brief overview of the role of entanglement as a resource in these schemes.

1.2.1 Ekert 91 Protocol A. Ekert first proposed using entanglement as a resource for quantum key distribution in 1991, known as the Ekert 91 protocol [38]. Let us suppose a situation where two people who want to communicate, Alice and Bob, each hold a qubit which is in a√maximally entangled state with the qubit of their partner | +  ≡ (|01 + |10)/ 2. For each trail of communication, Alice randomly chooses a quantum axis in a1 , a2 and a3 to make a projection measurement of her qubit. At the same time, Bob also randomly chooses a quantum axis in b1 , b2 and b3 to perform projective measurements on his bits. For simplicity, we choose an , bm (n, m = 1, 2, 3) both (n−1)π on the equatorial plane of the Bloch sphere, e.g. an = |0 + eπ· 4 |1 and bm = π· mπ |0 + e 4 |1. After repeating the above operation many times, everyone announces their measurement base publicly via a classical channel but keeps measurement results privately. It is easy to see that there are nine combinations of the measurement base of two people, which can be categorized into two groups two people choose the same base, and two people choose different bases. In the case of Alice and Bob choosing the same bases, they must get the same results due to the nature of the maximally entangled state. In this way, they share a set of keys without anyone knowing. Next, they can check the security of the keys by using the other case that they choose different bases. They can construct the following S parameter, S = E(a1 , b1 ) − E(a1 , b1 ) + E(a3 , b1 ) + E(a3 , b3 ).

(1.5)

It can be seen that this is the S parameter in the CHSH-type Bell inequality [39], where E is the expectation of the measurement, defined as E(an , bm ) = P++ (an , bm ) + P−− (an , bm ) − P+− (an , bm ) − P−+ (an , bm ). (1.6) P±± is the corresponding probability of different combinations of the results. They can announce part of the results and calculate the S parameter. S > 2 means that the state they have in hand is nonclassical and can be used for key distribution. Eavesdropping will inevitably break the nonclassicality of the entanglement. When |S| < 2, they can immediately become aware that the nonclassicality is too low for distributing keys and cease the key distribution.

6

1 Introduction

1.2.2 Quantum Teleportation Supposing we send an arbitrary qubit |ψ1 = α|0 + β|1 to a remote position, the simple way is to encode the qubit on a photon and transmit it. However, this way, we risk the photon being intercepted and resend unconsciously. On the contrary, quantum teleportation [40] helps us to send the qubit securely. + 23 ≡ (|01 + If Alice √ and Bob share a pair of maximally entangled states | + |10)/ 2 beforehand, we can write the joint state of |ψ1 and | 23 as a product state |ψ1 ⊗ | + 23 . Let us rewrite the product state as follows, 1 |ψ1 ⊗ | + 23 = [|+ 12 (β|0 + α|1)3 + |− 12 (−β|0 + α|1)3 2 + | + 12 (α|0 + β|1)3 + | + 12 (α|0 − β|1)3 ].

(1.7)

If we perform a joint Bell state measurement (BSM) on photon 1 and photon 2, they will be projected to one of four Bell states. Conditioned on the BSM result, the state of photon 3 is determined, which is different from the initial state |ψ1 with a simple unitary operation depending on the BSM results. Bob can easily recover the initial state on photon 3 based on the BSM result given by Alice. In the protocol, Alice and Bob only communicate the BSM result, from which the |ψ1 cannot be inferred. In addition, any attempt to attack the remote entanglement can be monitored in real-time by the Bell test introduced in the Ekert 91 protocol.

1.3 Quantum Repeater As remote entanglement plays a crucial role in quantum communication, the next question is how we can create remote entanglement efficiently. A direct method generates entangled photons through a spontaneous parametric downconversion (SPDC) source and distributes two entangled photons. Up to now, the longest distance for entanglement distribution in optical fibres is 300 km [41], and the distance for freespace entanglement distribution with the help of satellites is up to 1200 km [42]. However, at these extreme distances, the success rate of entanglement distribution is already low and not very practical. Limited by the exponential growth of fibre loss, the former record cannot be primarily improved. For the satellite-assisted entanglement distribution, on the one hand, the 1200 km distance is still far from the requirement for global-scale quantum communication. On the other hand, due to its high cost, it suits more as the backbone of a network instead of a general solution. To solve this problem, Briegel et al. proposed the quantum repeater protocol [33, 43]. We give a simple example to illustrate the basic idea of the quantum repeater protocol. Taking an example of distributing entanglement between two remote nodes A and D, the success probability of direct distribution of entangled photons is p L ∝ e−L/L 0 . As a new strategy, we can divide the whole distance to two equal parts,

1.3 Quantum Repeater

7

A − B and C − D, and create entanglement | +  AB and | + C D , respectively, in each segment, whose joint quantum state can be expressed as, | +  AB ⊗ | + C D =

1 + |  BC | +  AD − | −  BC | −  AD 2  + |+  BC |+  AD − |−  BC |−  AD

(1.8)

We only need to do a joint Bell state measurement for the photons at B, C to project the photons at A, D to one of the four maximally entangled states. This process is called entanglement swapping. We see that the success probability of entanglement distribution in each segment is p L/2 ∝ e−L/2L 0 , which is the square root of p L . Nevertheless, for doing the entanglement swapping, we need both entanglements being successfully established in each segment. For those cases where the entanglement in one segment is established but the other not will have to be abandoned. To save these cases, we can implement a quantum memory in each node. Thus every successfully established entanglement can be stored to wait for the other part to be ready for the entanglement swapping. Intuitively, we wait roughly two times the time for distributing the entanglement in a segment. Analysis [43] shows that the success probability of the whole process is a linear function of p L/2 , P ∝ 23 p L/2 , which is much higher than the direct distribution. For longer distances A − Z , as shown in Fig. 1.3, we can divide the whole distance to more segments, A − B, C − D,... , Y − Z . A step-by-step connection is used to gradually extend the entanglement from short to long distances.

(a) Entanglement creation QM

QM

QM

A

B

C

QM ... QM

D

...

W

QM

QM

X

Y

QM

Z

(b) First entanglement swapping QM

A

QM ... QM

D

... W

QM

Z

(c) Last entanglement swapping QM

QM

A

Z

Fig. 1.3 Schematic of the quantum repeater. Figure from [43]

8

1 Introduction

1.4 Quantum Network Given the massive success of the internet, people wish to construct a quantum version of the internet, the quantum network [35], in which different nodes are connected via quantum communication. In addition to the primary feature of secure communication, a fully developed quantum network also provides many other valuable functions. The development of quantum networks can be coarsely divided into the following stages according to their functions. • Quantum cryptography network: Current functional quantum networks are all in this stage, including DARPA [44] in Massachusetts, USA; SECOQC [45] in Vienna, Austria; SwissQuantum QKD [46] in Geneva, Switzerland; and BeijingShanghai trunk line [47] in China. The quantum networks at this stage rely on point-to-point key distribution and do not have the ability to communicate unconditionally securely between non-adjacent nodes, which is realized by using the trusted relays. Beyond this, functions such as location confirmation [48] can be demonstrated. If deterministic measurements are used, the device-independent scheme can be implemented [49]. • Quantum memory network: In this stage, quantum memories with sufficiently high efficiency and fidelity are equipped in each node, such that we can run the quantum repeater protocol on the networks and build remote entanglement with complex topology. Besides the ability to perform secure communication between any pair of nodes, the quantum network at this stage also provides the functions like remote clock synchronization [50], extending the baseline length of astronomical telescopes [51], etc. • Quantum computer network: Further improvements of quantum networks are mainly in the implementation of error correction and improving the network’s scale and complexity, which are based on a sufficient number of qubits and high enough gate fidelity. With these improvements, we can perform distributed quantum computing[52], run algorithms such as quantum elections [53], and fast Byzantine consensus [54].

1.5 State of the Art Quantum memory is the basis of quantum networks. In the past two decades, different types of quantum memories based on various physical systems and protocols have been developed. In order to realize quantum memories, systems with discrete energy levels well-isolated from the environment are required. Thus, as listed in Table 1.1, the main platforms for quantum memory research include atom and atomlike systems, among which include natural atoms like laser-cooled neutral atoms [43, 55], electrically trapped ions [56], rare-earth ions doped in crystals [57], and warm atoms [58]; defects in solids like nitrogen-vacancy (NV) centers and silicon-vacancy

1.5 State of the Art

9

Fig. 1.4 Roadmap of the quantum network. Figure from [35]

(SiV) centers in diamonds [59, 60] and in silicon carbide (SiC) [61]; and artificially ‘atoms’ like quantum dots (QD) [62] (Fig. 1.4). The above-mentioned systems can be divided into two main categories in terms of the number of atoms studied: ensemble systems and single-particle systems. Ensemble systems consist of a large number of atoms, without distinguishing the contribution of each atom and are generally easier to prepare and manipulate and therefore are studied earlier. In contrast to single-particle systems, there are also various collective effects present in large numbers of atoms that make interaction with light easier. Protocols such as photon echo [57] that exploit interatomic interference can only be implemented in ensemble systems. Single particle systems are relatively more difficult to prepare. For example, isolation of single atoms from large numbers of cold atoms requires complex techniques like micro trap [63]. In addition, the interaction of individual particles with light is always weak. In order to enhance this weak interaction, high-quality optical cavities need to be built to achieve strong light-atom coupling. However, the single-particle system also has a big advantage in that, as an isolated two-level system, it behaves nonlinearly. For instance, a single atom is inherently a good single photon source because multiple excitations are not allowed. In addition, when the cavity-atom coupling steps into the strong coupling regime, gate operations between photons and atoms can be achieved, building a bridge to the quantum computer. Based on the above quantum memory systems, many experiments have shown the entanglement among two or three spatially separated quantum memories, though the entanglement distance in most of the experiments is rather limited. The only

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1 Introduction

Table 1.1 Leading platforms of quantum network developments Platform wavelength (nm) Groups Atomic ensemble Single atom Solid-state defects Trapped ions Rare-earth ions Quantum dots

780/795:Rb, 852:Cs

Kimble, Pan, Kuzmich, Laurat Rempe, Weinfurter, Vuleti´c 637:NV, 738:SiV, Hanson, Awchalom, 1079:VV@4H-SiC Lukin, Wrachtup 422:Sr+ , 854:Ca+ , Blatt, Monroe, Keller, 935:Yb+ Lucas 606:Pr3+ , 883:Nd3+ , Gisin, Reidmatten, 985:Yb3+ , 1532:Er3+ Tittel, Faraon 910:InAs, 950:InGaAs Imamolu, Yamamoto, Lodahl

References [64–66] [67–69] [49, 70, 71] [72–74] [75, 76] [77, 78]

proof-of-principle kilometre-scale entanglement experiment is achieved in NV centre system [49], with a 1.3 km physical separation between two quantum memories and a 1.7 km fibre transmission. One of the important reasons currently limiting the entanglement distance is the fibre transmission loss. The fibre telecommunication band is 1300–1600 nm, with a loss of only 0.2–0.3 dB/km. However, most well-developed quantum memory systems work at wavelengths in the visible and near-infrared bands as listed in Table 1.1. In order to tackle this problem, one idea is to work with atoms with inherent telecommunication transitions, such as the first transition line of Erbidium ions, Er3+ [79, 80] and one of transition between Rubidium excited states [81]. A more general approach is to build communication band interfaces to existing quantum memories with the help of nonlinear optics-based quantum frequency conversion [82], which has been applied for atoms [83–85], trapped ions [86, 87], NV centers [88], quantum dots [78] and many others.

1.6 Structure of the Thesis This thesis is divided into eight chapters. • This chapter: Introduction. I will briefly review the background and the state of the art of the quantum network. • Chapter 2: Principle of quantum memories. I will explain the principle of two protocols in cold atom-based quantum memory, the DLCZ protocol and the EIT protocol. I will also explain the Rydberg blockade mechanics, which serves as nonlinearity in the cold atomic ensemble. • Chapter 3: Cold atom-based high-efficiency quantum memory. I will introduce the preparation of a cold atomic ensemble. Based on this, I will introduce the building and the characterization of a cavity-enhanced quantum memory.

References

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• Chapter 4: Quantum frequency conversion. I will explain the principle of nonlinear optics-based quantum frequency conversion. Moreover, I will introduce the fabrication and characterization of the PPLN waveguide in our experiments. • Chapter 5: Remote entanglement via the two-photon scheme. Based on the quantum memory and the quantum frequency conversion module, we create a remote entanglement between two quantum memories via the two-photon scheme and 22 km deployed fibre transmission. • Chapter 6: Remote entanglement by single-photon interference. With the development of the long fibre phase locking technique, we create a remote entanglement between two quantum memories via the single-photon scheme and 50 km fibre transmission. • Chapter 7: Measurement-device-independent verification of a quantum memory. Using the high-quality single photons from a Rydberg atomic ensemble and the quantum random number generators, we certify a quantum memory via the measurement-device-independent scheme. • Chapter 8: Improvements of quantum memories. I will discuss some possible ideas to further improve the performance of the atomic ensemble-based quantum memories. • Chapter 9: Conclusion and outlook.

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1 Introduction

68. Nölleke C, Neuzner A, Reiserer A et al (2013) Efficient teleportation between remote singleatom quantum memories. Phys Rev Lett 110(14):140403 69. Ritter S, Nölleke C, Hahn C et al (2012) An elementary quantum network of single atoms in optical cavities. Nature 484(7393):195 70. Bernien H, Hensen B, Pfaff W et al (2013) Heralded entanglement between solid-state qubits separated by three metres. Nature 497(7447):86–90 71. Humphreys PC, Kalb N, Morits JPJ et al (2018) Deterministic delivery of remote entanglement on a quantum network. Nature 558(7709):268–273 72. Moehring DL, Maunz P, Olmschenk S, et al (2007) Entanglement of single-atom quantum bits at a distance. Nature 449(7158):68–71 73. Slodiˇcka L, Hétet G, Röck N et al (2013) Atom-atom entanglement by single-photon detection. Phys Rev Lett 110(8):083603 74. Stephenson L, Nadlinger D, Nichol B et al (2020) High-rate, high-fidelity entanglement of qubits across an elementary quantum network. Phys Rev Lett 124(11):110501 75. Zhong T, Kindem JM, Bartholomew JG et al (2017) Nanophotonic rare-earth quantum memory with optically controlled retrieval. Science 357(6358):1392–1395 76. Kindem JM, Ruskuc A, Bartholomew JG et al (2020) Control and single-shot readout of an ion embedded in a nanophotonic cavity. Nature 580(7802):201–204 77. Delteil A, Sun Z, Gao WB, et al (2015) Generation of heralded entanglement between distant hole spins. Nat Phys 12(3):218–223 78. De Greve K, Yu L, McMahon PL et al (2012) Quantum-dot spin-photon entanglement via frequency downconversion to telecom wavelength. Nature 491:421–425 79. Yin C, Rancic M, de Boo GG et al (2013) Optical addressing of an individual erbium ion in silicon. Nature 497(7447):91–94 80. Raha M, Chen S, Phenicie CM et al (2020) Optical quantum nondemolition measurement of a single rare earth ion qubit. Nat Commun 11(1):1–6 81. Radnaev AG, Dudin YO, Zhao R et al (2010) A quantum memory with telecom-wavelength conversion. Nat Phys 6:894–899 82. Kumar P (1990) Quantum frequency conversion. Opti Lett 15(24):1476 83. Ikuta R, Kobayashi T, Kawakami T et al (2018) Polarization insensitive frequency conversion for an atom-photon entanglement distribution via a telecom network. Nat Commun 9(1):1997 84. Maring N, Farrera P, Kutluer K et al (2017) Photonic quantum state transfer between a cold atomic gas and a crystal. Nature 551(7681):485 85. van Leent T, Bock M, Garthoff R et al (2020) Long-distance distribution of atom-photon entanglement at telecom wavelength. Phys Rev Lett 124(1):010510 86. Bock M, Eich P, Kucera S et al (2018) High-fidelity entanglement between a trapped ion and a telecom photon via quantum frequency conversion. Nat Commun 9(1):1998 87. Walker T, Miyanishi K, Ikuta R et al (2018) Long-Distance single photon transmission from a trapped ion via quantum frequency conversion. Phys Rev Lett 120(20):203601 88. Dréau A, Tcheborateva A, Mahdaoui AE et al (2018) Quantum frequency conversion of single photons from a nitrogen-vacancy center in diamond to telecommunication wavelengths. Physical Review Applied 9(6):064031

Chapter 2

Principles of Quantum Memories

The physical platform being used in this thesis is the cold atomic ensemble. Several quantum memory protocols can be implemented, including electromagnetically induced transparency (EIT) [1], Duan-Lukin-Cirac-Zoller (DLCZ) [2] scheme, and also controlled reversible inhomogeneous broadening inversion (CRIB) [3], and photonic gradient echo memory (GEM) [4]. In this chapter, we will introduce the first two protocols, which will be experimentally realized in the following chapters. In addition, we will also briefly introduce some basics of the Rydberg interaction [5], which was utilized to tackle the higher-order excitation problem in the ensemblebased quantum memories.

2.1 EIT Quantum Memory Electromagnetically induced transparency (EIT), a phenomenon in which an otherwise opaque medium becomes transparent after being irradiated with specific conditions of intense light (i.e., electromagnetic induction), was first discovered by Harris in 1989 [6, 7]. For its on-demand manipulation of the properties of the optical medium, the EIT has soon attracted a great deal of attention. The mechanism behind the phenomenon is that the strong light modulates the refractive index of the medium, based on which the refractive index of the medium, that is, the speed of light in the medium, can be controlled. Based on the understanding of the machanism, experiments such as slow light [8], stopping of light [9], superluminal light [10] and many others have been demonstrated. Here we briefly introduce EIT and how it can be used to achieve quantum memory.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Yu, Long Distance Entanglement Between Quantum Memories, Springer Theses, https://doi.org/10.1007/978-981-19-7939-2_2

15

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2 Principles of Quantum Memories

Fig. 2.1 Energy scheme (a) and the optical path (b) of an EIT experiment

2.1.1 Property Manipulation of the Optical Medium The energy level and the optical beam arrangement of an EIT experiment are shown in Fig. 2.1. In the -type energy level of the atoms, |g, |s are the two ground states and |e is the excited state. All atoms are prepared in state |g. There are a control and a probe beam involved in the experiment. The strong control beam with frequency ωc coupling |s ↔ |e transition with Rabi frequency c and detuning c , modulates the medium and a weak probe beam with frequency ω p coupling |g ↔ |e transition with Rabi frequency  p and detuning c probes the property of the medium. Without the control beam, the atoms absorb the probe beam strongly and form an opaque medium. With the control light on, we can write the Hamiltonian for the light-atom interaction as  Hint = − [ p (t)σˆ eg ei p t + c (t)σˆ es eic t ] + h.c. 2

(2.1)

where σˆ μν = |μ ν|, is the flip operator for atoms. In order to understand phenomena such as transparency and slow light, we can calculate the first-order dielectric constant χ (1) of the atoms under the control beam modulation. When exposed to light, the medium is induced to produce an electric dipole of  eri /V  P(t) = − i

N = [dge ρeg e−iωeg t + dse ρes e−iωes t + c.c.] V

(2.2)

where d is the transition dipole, which can be written as a scalar since we are considering the one-dimensional case where the control beam and the probe beam are on the colinear configuration. N and V are the number of atoms involved and the light field quantized volume, respectively. The N /V is the atomic density, which we represent as . The density matrix elements in Eq. 2.2 can be calculated by solving the following Lindblad equation [11],

2.1 EIT Quantum Memory

17

   i dρ 1 † = − [Hint , ρ] + σ ji σi j , ρ , i j σi j ρσ ji† − dt  2 i, j

(2.3)

where i, j take the values of three states |g, |s and |e. In our assumptions only the excited state |e has spontaneous radiation, so that eg and es correspond to nonzero terms, and in addition considering the decoherence of |e and |s with respect to the ground state, we define their decoherence rates as γs ≡ ss and γe ≡ ee . Terms other than the decay and decoherence rates are 0. For operational convenience, we define the following three quantities γeg ≡ eg + es + γe , γes ≡ eg + es + γs , and γsg ≡ γs . Substituting the initial state ρgg  1, the relationship between the nondiagonal elements can be found under the rotating coordinate system as follows, ic ei p t ρgs , γes + 2ic ic eic t =− ρge , γsg + 2i(c −  p )

ρes = ρgs

ρeg =

(2.4)

i p ei p t ic eic t + ρsg . γeg + 2i p γeg + 2i p

We define single-photon detuning  =  p and two-photon detuning δ =  p − c . The first-order electric susceptibility of the medium at the detected optical frequency reads, |dge |

χ (1) (−ω p , ω p ) = 0   4δ(|c |2 − 4δ) − 4γsg2 ×   (2.5) |c |2 + (γeg + 2i)(γsg + 2iδ)2 8δ 2 γeg + 2γsg (|c |2 + γsg γeg ) . +i |c |2 + (γeg + 2i)(γsg + 2iδ)|2 The variation of the real and imaginary parts of χ (1) with ω p is given in Fig. 2.2. We can see that the imaginary part drops to 0 at the 0 detuning point, which is the reason for the transparency of the medium. And at the same time, the real part changes, and we know that the refractive index of the medium is given by the real part of the dielectric constant as n=

1 + Re[χ (1) ].

(2.6)

If the probe beam is pulsed, the group velocity in the medium is vg =

∂ω p c . |δ=0 = ∂k p n + ω p (dn/d p)

(2.7)

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2 Principles of Quantum Memories

Fig. 2.2 The real and imaginary parts of the medium electric susceptibility as a function of the detuning. Figure from [12]

Therefore, by adjusting the light intensity of the control beam, we can achieve slow light and superluminal.

2.1.2 EIT Quantum Memory Since the EIT effect enables the slowing down of the speed of light, a natural idea is whether it is possible to extend this concept to reduce the speed of light to 0 and achieve the stopping of light. Let us start with an intuitive example, when the twophoton resonance condition is fulfilled (δ = 0), we get the three eigenstates of the interacting Hamiltonian in Eq. 2.1, |+  = sin θ sin φ|g + cos φ|e + cos θ sin φ|s, |−  = sin θ sin φ|g + cos φ|e + cos θ sin φ|s,

(2.8)

|  = cos θ |g − sin θ |s, 0

where the mixing angles θ and φ read,  p , tan 2φ = tan θ = c

2p + 2c 

.

(2.9)

We can see that the | 0  state has no excited states involved and therefore does not emit photons. Thus it is called the “ark state”. When c   p , sin θ → 0, cos θ → 1, the dark state approaches state |g. On the contrary, when c  p , the dark state approaches state |s. Suppose we dynamically modulate the intensity of the control field. In that case, the dark state can be coherently shifted from |g to |s while the probe beam field is absorbed, enabling light storage. To describe this phenomenon more precisely and include a quantized probe field, we rewrite Eq. 2.1 by replacing the probe field with its annihilation operator aˆ and

2.1 EIT Quantum Memory

19

include the statistics of all atoms as follows,

  i g aˆ σˆ eg − c (t)e−iωc t σˆ esi + h.c., 2 i=1 N

Hint = −

(2.10)

 ωp is the vacuum Rabi frequency. Again, the system still has dark where g = dge 20 V states, the simplest of which is, | 0 , 1 = cos θ (t)|G, 1 − sin θ (t)|S, 0.

(2.11)

More generally, we have dark states, | 0 , n =

n  k=0

 n! (− sin θ )k (cos θ )n−k |S k , n − k, k!(n − k)!

where the mixing angle θ is

√ g N . tan θ (t) = (t)

(2.12)

(2.13)

The atomic states in the solution are expressed in terms of the collective state as follows, N 1  |G = |g1 , g2 , . . . , g N , |S = √ |g1 , . . . , si , . . . , g N . N i=1

(2.14)

|S is the collective state of atoms, and the higher order |S k  has a similar definition. Note that Eq. 2.11 answers how to build a quantum memory. We can start by preparing all atoms at the ground state. By adiabatic1 turning off the control beam, the probe photon is coherently converted to a collective excitation in the atom, also known as a spin wave. Conversely, adiabatically turning on the control light converts the spin wave back to a single photon, i.e. the readout process.

2.1.3 The Dynamic Process To have a better understanding of the kinetic evolution of the photon in the medium, we can also take the spatial dimension into account by correcting the Hamiltonian in Eq. 2.10 to

1

Some following analysis [13, 14] shows that the adiabatic condition is actually very loose. In the experiment, even turning the control light off in the fastest way barely affects the storage efficiency.

20

2 Principles of Quantum Memories

Fig. 2.3 The dynamics of the EIT process. Plotting after [15]

Hint = −



[deg σˆ egj Eˆ (+) (z j ) + σˆ esj c (z j , t)ei(kc z j −ωt) ] + h.c.

(2.15)

j

where the light field operator and the atomic flip operator can be written as 

ˆ (+)

E

 ω ω p ˆ p E(z) exp i (z − ct) , 20 V c  ω μν j (z − ct) . = σ˜ μν (t) exp −i c

(z, t) = j σˆ μν

(2.16)

In calculations, it is simpler to consider the evolution of operators under the Heisenberg picture. We define here √ two atom relevant operators: the atomic collective polarization operator Pˆ = N σˆ ge (z, t) and the spin-wave annihilation operator √ Sˆ = N σˆ se (z, t), so that the dynamical processes of an EIT quantum memory can be described by E, P and S operators as, √ ˆ , (∂t + c∂z ) Eˆ = ig√ N Pn(z)L/N √ ˆ ˆ ˆ ˆ ∂t P = −(γ + i) P + ig N E + i S + 2γ Fˆ P , √ ∂t Sˆ = −γs Sˆ + i∗ Pˆ + 2γs FˆS .

(2.17)

Under general experimental conditions, the Langevin noise operators Fˆ P and FˆS approximate 0. Another way of describing this process is to consider the light field and atomic operators jointly, defining the following two operators, √ ˆ t) − sin θ (t) N σ˜ gs (z, t)eikz , ˆ = cos θ (t)E(z,  √ ˆ t) − cos θ (t) N σ˜ gs (z, t)eikz , ˆ = sin θ (t)E(z, 

(2.18)

where k = k p − kc . It can be seen that this is the field operator for the mixing of the optical and atomic fields, where the mixing angle is still given by equation 2.13. The system evolution can then be described as,

2.2 DLCZ Quantum Memory

21



 ∂ ∂ ∂ ˆ = −θ˙  ˆ − sin θ cos θ c , ˆ + c cos2 θ  ∂t ∂z ∂z   sin θ ∂ ∂ sin θ ˆ = ˆ − cos θ ) ˆ + i √ FS .  + γ tan θ (sin θ  2 g N ∂t ∂t g N

(2.19)

In the adiabatic approximation (which can be fulfilled in most experimental condiˆ operator can be written as tions), then the evolution of the  

 ∂ ∂ ˆ = 0, + c cos2 θ  ∂t ∂z

(2.20)

corresponding to the propagation of a wave of constant amplitude with propagation speed v = vg (t) = c cos2 θ (t).  ˆ and  ˆ into different plane wave modes  ˆ = k ˆ = ˆ k (t)eikz ,  Decomposing   ikz ˆ k k (t)e , one can find the following commutative relations, ˆ k,  ˆ k† ] ≈ [ ˆ k,  ˆ †k ] ≈ δk,k , [ ˆ k,  ˆ †k ] ≈ 0. [

(2.21)

The annihilation operators for two quasiparticles obey the bosonic statistics, corresponding to a type semi-particles, polariton. The dark states we introduced earlier can be obtained from the dark state polariton production operators as follows, 1 ˆ k† )n |g | 0 , n k  = √ ( n!

(2.22)

Equation 2.20 is its equation of motion. The motion of a typical dark-state polariton, the light field and spin-wave modes during the EIT storage are shown in Fig. 2.3. The process of EIT storage can then be considered as a photon being converted into a dark state polariton upon incidence and moving through the medium in this form. At the same time, the control light regulates the velocity of the dark state polariton. When the control light is turned off, the dark state polariton velocity is zero, known as stopping light.

2.2 DLCZ Quantum Memory There is a crucial point in the quantum repeater protocol: the distribution of entanglement at the fundamental level must be heralded so that we can decide on the next step based on whether the entanglement is successfully established: if it is established successfully, we wait for the entanglement swapping; if it is not, we try again to establish the entanglement. Otherwise, the acceleration effect that the scheme has would not be realized. The most straightforward idea to achieve this is to add

22

2 Principles of Quantum Memories

a heralding function to the memory so that when an external photon is inputted, the memory can herald whether the photon state was successfully stored without destroying it. But the EIT memory that we have just described, along with Atomic Frequency Comb (AFC) [16], Controllable Reversible Inhommohenous Broadening (CRIB) [17], Gradient Echo Memory (GEM) [18], and other “write-read” memory schemes do not have such capabilities. In order to achieve the heralding storage, strong interactions between cavities and atoms need to be introduced, exploiting the non-linear effects given by cavity quantum electrodynamics [19, 20] to enable non-destructive measurements of photons. The Vuleti´c group at MIT [21] and Lukin group at Harvard [22] have initially implemented this heralded functionality in the atomic system and the diamond SiV colour centre system, respectively. Lukin group has used it to demonstrate a boost to the key rate of the quantum key distribution [22]. However, this approach is generally challenging and requires a good match between the entangled light source and the memory in terms of wavelength, bandwidth, and other aspects to work well. In 2001, Lu-Ming Duan et al. proposed another scheme for achieving quantum repeaters, known as the DLCZ scheme [2]. Instead of using an external entanglement source, this scheme directly generates entanglement between a quantum memory and a photon so that there is no problem matching the entanglement source and the memory. Instead, the entanglement between the two memories can be established after making a joint Bell state measurement of the photons in two such pairs of entanglement. The result of the Bell state measurement can then be used as a heralding signal for establishing the entanglement. In this section, we describe the basic principle of the DLCZ-type quantum memory, its improvement with the help of optical cavities and the entanglement establishment between quantum memories.

2.2.1 DLCZ Quantum Memory in Free Space The principle of this scheme is shown in Fig. 2.4. All atoms are prepared in the ground state (|g) in an -type energy level structure. We use a write beam coupling |g ↔ |e to induce the spontaneous Raman scattering of a ground state atom. The spontaneous scattering is probabilistic, and the direction of the scattered photon is random. Assuming that we collect a photon at a certain angle and call it a write-out photon, the state of the atomic ensemble is: N 1  i(kw −kwo )·ri |Sk  = √ e |g1 g2 . . . si . . . g N , N i=1

(2.23)

where kw and kwo are wave vectors of the write and write-out mode, respectively and ri is the position of the ith atom. We can see that this state has the same form as the collective excited state in the EIT of Eq. 2.14. We often call a two-energy system in an atom a spin. If the time evolution factor is considered, the form of Eq. 2.23 is

2.2 DLCZ Quantum Memory

23

Fig. 2.4 The write a and read b process of the DLCZ-type quantum memory

very similar to a plane wave. Therefore it is also called a spin wave, and kw − kwo is the spin wave vector. The creation of a spin wave can be described by the creation operator Sk† = |Sk  G|. Taking the write-out field into account, the complete write process can be described as  √ | = 1 + χ Sk† aw† + o(χ ) |G, 0,

(2.24)

where χ is the probability of collecting a write-out photon in a given direction. When χ is small, the higher order terms can be ignored. We can describe the entanglement between the atomic spin wave and the write-out photon as |  |0a |0 p +

√ χ |1a |1 p ,

(2.25)

where |0 and |1 are Fock state (particle number state) representations, representing the number of particles in a photon or spin wave, and a or p in the subscripts indicate that this is an atomic or photonic state, respectively. To test the entanglement, we can read out the spin wave into a photon by illuminating the atomic cloud with a read beam coupling |s ↔ |e such that the population on the |s state returns to the |g state and a read-out photon is emitted accordingly. Assuming that the wave vectors of the read beam and read-out fields are kr and kr o , respectively, then we have Sk ar† |Sk , 0 = |G ⊗

N 1  i(kw −kwo +kr −kr o )·ri † e a |0. N i=1

(2.26)

For a single atom, there is no summation sign. Therefore the read-out photon will emit to any possible direction in the probability p. But for atomic ensembles, the radiation intensity in the kr o direction appears N times enhanced if kw − kwo + kr − kr o = 0, i.e., the phase matching condition, is satisfied. Assuming that we collect a solid angle 2  λ , the readout efficiency in the collective enhancement direction is of d = π πw 0 Pd = (N d) · p.

(2.27)

24

2 Principles of Quantum Memories

Considering that the real atoms usually have more energy levels, the above equation needs to be multiplied by the branching ratio ξeg . The read-out efficiency of this scheme is the ratio of the probability of detecting a photon in the read-out direction to the full space, C Pd = . (2.28) ηr = Pd + 4π · p C +1 The parameter C reads, C=

ξeg λ2 d N ξeg λ2 ξeg λ2 n L = . ≈ 4π 4π σ 4π 2 w02

(2.29)

where d = σ · n L is the optical depth (OD) of the atomic ensemble. During the read process, the read-out photon will have to transmit in the atomic cloud before being collected. Due to EIT induced by the strong read beam, the atomic cloud become transparent to the read-out photons in the ideal case. For a more general case, we can take the transmission loss into account with the imaginary part of the electric susceptibility of Eq. 2.5,

ηt = exp −|kwo |LIm(χ (1) ) .

(2.30)

Therefore the total read-out efficiency is, η = ηr ηt .

(2.31)

Note that the read-out efficiency does not increase monotonically with the OD. The loss will outweigh the collective enhancement when the OD is too high. Hence, an intermediate OD must be chosen to achieve the best read-out efficiency.

2.2.2 Cavity Enhanced DLCZ Alternation of the optical field around atoms with optical cavities is a common tool in quantum optics [23]. A good example is the strong coupling in cavity quantum electrodynamics (cQED), referring to the situation where the cavity-atom coupling strength is much greater than the decay rate of both atom and the cavity. When the strong coupling condition is fulfilled, many exciting phenomena occur, such as the enhancement of the radiative light field of the atom towards the cavity mode with significant probability and the strong nonlinear effects of the system when the number of photons in the cavity is small. Some properties of the atom, such as the upper energy level lifetime, are also changed as a result. Here we consider a weak-coupling scenario. Assuming a cloud of atoms is placed in a Fabry-Pérot cavity, as shown in Fig. 2.5, where the cavity mirror on the left is a total reflector, and the cavity mirror on the right has a reflectivity of R2 = r 2 .

2.2 DLCZ Quantum Memory

25

Fig. 2.5 Schematic of the atom-cavity interaction. Figure from [24]

√ The cavity fineness can be easily calculated in definition as F = π r /(1 − r ). Considering the atomic radiation in this case, there are several ways for the emission to go outside the cavity. It may come out directly without reflection, with one or more rounds of reflection in the cavity. We can write down the possibility of each way as follows, √ √ p0 = p · t, √ √ p1 = p · r · t, .. . (2.32) √ √ n pn = p · r · t, .. . where pn represents the probability of going n rounds before leaving the cavity, and t 2 = 1 − r 2 is the transmittance. The possibility of emission at the cavity output is given by summing up all the possibility amplitudes as, ∞  √ 2  pout =  pn  = n=0

pt 2 . 1 − r2

(2.33)

If the reflectivity of the right cavity mirror R2 is close to 1, we get pout / p  2F/π . Therefore when atoms are in the cavity, their radiation intensity along the cavity mode is enhanced by a factor of 2F/π 2 , which benefit for both the write process and the read process of DLCZ memory. For the read process, the cavity enhances the readout efficiency, which is undoubtedly an advantage. For the write process, a lower write beam intensity is required to achieve the same excitation rate compared with the free space case. This reduces the difficulty of filtering residual noise from the write beam. In addition, a lower write beam intensity helps to reduce generating the write-out photons outside the cavity mode, which contribute to stochastic noise during the read process.

2

The real cases are more complex. The dissipation introduced by atoms will decrease the cavity enhancement factor. More detailed derivations can be found in the [23, 25].

26

2 Principles of Quantum Memories

2.3 Rydberg Blockade Mechanism Although DLCZ-type quantum memory has promising applications, one of its insurmountable weaknesses is that it is a probabilistic entanglement source and the higherorder excitations are unavoidable. In the experiment, we must suppress this noise by sacrificing the entangling probability. The fundamental suppression of higher-order excitations requires nonlinear interactions that make it forbidden to excite more atoms simultaneously. In atomic physics, a commonly used nonlinear interaction is the Rydberg blockade mechanism [5]. We briefly explain the basic principles of this mechanism in this section.

2.3.1 Rydberg Interaction The Rydberg state [26] is the quantum state when the outermost electron of an atom is excited to a very high principal quantum number (generally n > 20). Because the radius of the outermost electron orbital is large (∼ n 2 a0 ), the Rydberg state has a small overlap with the ground state orbital, leading to a long lifetime3 . When two Rydberg atoms are close, they have a strong dipole-dipole interaction. Their non-interacting eigenstates mix, giving rise to energy level shifts. Assuming two Rydberg atoms are R apart (disregarding the case where the electron orbitals have overlap, i.e. |R|  n 2 a0 ), the electrostatic interactions between them are mainly dipole-dipole interactions, d1 · d2 (d1 · R) (d2 · R) Vˆdd = − . 3 R R5

(2.34)

To calculate Eq. 2.34, we should, in principle, take all possible paired states of two atoms into account. Instead, a simple analysis [27, 28] including only two neighbouring atomic states can give us a primary picture of the interaction. As shown in Fig. 2.6, initially populated state is |n, 1 = | p and two dipolecoupled states are |n, 0 = |s and |n + 1, 0 = |s . The pair states |ss  and |s s are almost degenerate with | pp with a small energy difference between them . The Hamiltonian of the subset (| pp, |ss ) can be written as (with the energy of | pp set to 0):  ⎡ ⎤  pp|Vˆdd |ss  ⎦. H = ⎣ (2.35) ss |Vˆdd | pp 0 Ignoring the angular term in Eq. 2.34, the diagonal elements in the Hamiltonian read,

3

According to Fermi golden rules, the decay rate is proportional to the overlap between initial and the target states. At 0 K, the typical lifetime of Rydberg states is around 100 µs.

2.3 Rydberg Blockade Mechanism

27

Fig. 2.6 Rydberg interaction between to neighbor atoms. Figure from [27]



  p|d1 |s  p|d2 |s  d1 d2 ˆ pp|V |ss = = 3 . R3 R

(2.36)

The eigenenergies of the new system can be solved as,  E± = − ± 2





 2

2

 +

d1 d2 R3

2 .

(2.37)

If  = 0, the energy shift of state | pp is E pp =

d1 d2 . R3

(2.38)

This case is known as the Föster resonance, where the energy shift scales as the reciprocal of the cubic of distance. Hence we can define C3 ≡ d1 d2 . For the more general cases of  = 0, we can expand Eq. 2.37 as, E pp

 =− + 2



 1 (C3 )2 + + ... 2  R6

 =

(C3 )2 / . R6

(2.39)

This is the van der Waals (vdW) interaction energy. We can define C6 ≡ (C3 )2 /, which is estimated to be C6 ∝ n 11 . To obtain stronger cubic interactions, we can apply an electric field to meet the Föster resonance condition.

28

|rr Energy

Fig. 2.7 The energy level of the two-atom state as a function of their distance. The shaded part indicates the Rydberg blockade region. Figure from [27]

2 Principles of Quantum Memories

|gr

|gg Rb

R

2.3.2 Rydberg Blockade In the two-atom model, the pair state | pp is shifted due to the Rydberg interaction. A consequence is that only one atom can be excited to the Rydenerg state when we try to drive the |g ↔ |r  transition. This conclusion stands for the multi-atom system if the strong Rydberg interaction is fulfilled. We can prepare a Rydberg collective state as N 1  |R = |g1 g2 . . . ri . . . g N . (2.40) N i=1 The strong Rydberg interaction means the Rydberg state shift is larger than the excitation laser linewidth ν. Thus we define the Rydberg blocking radius Rb as E(Rb ) = ν.

(2.41)

In addition, another remarkable feature of the Rydberg interaction is that it makes the frequency of Rabi oscillations between the collective ground state and the Rydberg √ excitation become N times that of the single-atom case, indicating a many-body interaction (see Fig. 2.7).

References 1. Fleischhauer M, Imamoglu A, Marangos JP (2005) Electromagnetically induced transparency: Optics in coherent media. Rev Mod Phys 77(2):633 2. Duan LM, Lukin MD, Cirac JI et al (2001) Long-distance quantum communication with atomic ensembles and linear optics. Nature 414(6862):413–418 3. Alexander AL, Longdell JJ, Sellars MJ et al (2006) Photon echoes produced by switching electric fields. Phys Rev Lett 96(4):043602

References

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4. Hétet G, Longdell J, Sellars M et al (2008) Multimodal properties and dynamics of gradient echo quantum memory. Phys Rev Lett 101(20):203601 5. Saffman M, Walker TG, Mølmer K (2010) Quantum information with Rydberg atoms. Rev Mod Phys 82(3):2313–2363 6. Harris SE (1989) Lasers without inversion: interference of lifetime-broadened resonances. Phys Rev Lett 62(9):1033 7. Harris S E (1997) Electromagnetically induced transparency. In: Quantum electronics and laser science conference. Optical Society of America, QTuB1 8. Hau LV, Harris SE, Dutton Z et al (1999) Light speed reduction to 17 m per second in an ultracold atomic gas. Nature 397(6720):594–598 9. Liu C, Dutton Z, Behroozi CH et al (2001) Observation of coherent optical information storage in an atomic medium using halted light pulses. Nature 409(6819):490–493 10. Wang Liej J, Kuzmich A, Dogariu A (2000) Gain-assisted superluminal light propagation. Nature 406(6793):277–279 11. Breuer HP, Petruccione F, et al (2002) The theory of open quantum systems. Oxford University Press on Demand 12. Li Jun (2017) Single quantum state preparation and manipulation based on Rydberg blocking mechanism in cold atom ensemble. University of Science and Technology of China 13. Matsko AB, Kocharovskaya O, Rostovtsev Y et al (2001) Slow, ultraslow, stored, and frozen light. Adv Atomic Molecul Opti Phys 46:191–242 14. Fleischhauer M, Lukin MD (2002) Quantum memory for photons: dark-state polaritons. Phys Rev A 65(2):022314 15. Fleischhauer M, Lukin MD (2000) Dark-state polaritons in electromagnetically induced transparency. Phys Rev Lett 84(22):5094 16. Afzelius M, Simon C, De Riedmatten H et al (2009) Multimode quantum memory based on atomic frequency combs. Phys Rev A 79(5):052329 17. Moiseev S, Kröll S (2001) Complete reconstruction of the quantum state of a single-photon wave packet absorbed by a doppler-broadened transition. Phys Rev Lett 87(17):173601 18. Hétet G, Hosseini M, Sparkes BM et al (2008) Photon echoes generated by reversing magnetic field gradients in a rubidium vapor. Opti Lett 33(20):2323–2325 19. Steck DA (2007) Quantum and atom optics 20. Reiserer A, Rempe G (2015) Cavity-based quantum networks with single atoms and optical photons. Rev Mod Phys 87(4):1379–1418 21. Tanji H, Ghosh S, Simon J et al (2009) Heralded single-magnon quantum memory for photon polarization states. Phys Rev Lett 103(4):043601 22. Bhaskar MK, Riedinger R, Machielse B et al (2020) Experimental demonstration of memoryenhanced quantum communication. Nature 580(7801):60–64 23. Tanji-Suzuki H, Leroux ID, Schleier-Smith MH, et al (2011) Interaction between atomic ensembles and optical resonators: classical description. In: Advances in atomic, molecular, and optical physics, vol 60. Elsevier, pp 201–237 24. Bao XH (2010) Quantum information with entangled photons and cold atomic ensembles 25. Jong B (2018) Quantum network with multiple cold atomic ensembles. Springer Nature 26. Gallagher TF (2005) Rydberg atoms, vol 3. Cambridge University Press 27. Amthor T (2008) Interaction-induced dynamics in ultracold rydberg gases—mechanical effects and coherent processes. PhD thesis, Universität Freiburg 28. Singer K, Stanojevic J, Weidemüller M et al (2005) Long-range interactions between alkali Rydberg atom pairs correlated to the ns-ns, np-np and nd-nd asymptotes. J Phys B: Atomic, Molecul Opti Phys 38(2):S295

Chapter 3

A High-Efficiency Quantum Memory

The physical platform we use in this thesis is an isotope of the alkali atom rubidium, 87 Rb. It has only one electron in its outermost layer and is known as a hydrogenlike atom with a simple energy-level structure. This chapter briefly describes the preparation of the atomic system and the basic results of cavity-enhanced DLCZ storage using this system.

3.1 Preparation of Cold Atoms In the 1980s, Steven Chu [1], W. Phillips [2], and collaborators first achieved the laser trapping and cooling of atoms from room temperature to hundred µK. After that, cold atom technology developed rapidly, and Bose-Einstein condensate was achieved in the 1990s. Today, extremely low temperatures of the order of pK can be achieved, and laser cooling and trapping of atoms have become a standard technique in experiments such as atomic clocks, precision measurements, and quantum information. In this thesis, magneto-optical trap and polarization gradient cooling techniques are used to prepare atomic ensembles of ∼ 10µK temperature. We briefly describe the cold atom preparation process in this section. We first prepare a high vacuum environment [3] of 4 × 10−10 mbar using a three-stage vacuum pumping with a scroll pump, a turbo pump, and an ion pump, which ensures that the experiment is not affected by background gases. Then we admit 87 Rb gas to the vacuum chamber for the following experiments. By applying currents to a metal dispenser preassembled in the vacuum chamber, 87 Rb gas is released due to the heat-induced chemical reaction.

3.1.1 Doppler Cooling The principle of laser cooling [4] is slowing atoms down by applying photon radiation pressure on them. Considering a toy model depicted in Fig. 3.1, a group of hot two© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Yu, Long Distance Entanglement Between Quantum Memories, Springer Theses, https://doi.org/10.1007/978-981-19-7939-2_3

31

32

3 A High-Efficiency Quantum Memory

Fig. 3.1 Energy scheme (a) and atomic momentum variation (b) of Doppler cooling

level atoms is cooled in one dimension by a laser red-detuned from the |g ↔ |e transition. Atoms staying still or moving along the laser direction do not interact with the laser due to the detuning . On the contrary, some atoms moving towards the laser will be excited by absorbing a laser photon because the Doppler blue shift compensates for the detuning. As a result, the excited atoms decay spontaneously and emit photons in random directions, meaning that the decay photons have an average momentum of 0. These atoms accept a momentum p = k opposite to its moving direction during the excitation and, averagely, lose no momentum during the decaying, which results in deceleration, i.e. cooling of atoms1 . The force of the laser rendering on atoms is called the radiation pressure, and its magnitude is the momentum exerted by each interaction multiplied by the average rate of action, Fsc = k Rsc .

(3.1)

Rsc is the spontaneous scattering rate. For an atom moving at speed v, Rsc reads Rsc =

I /Isat  . 2 1 + 4( − kv)2 / 2 + I /Isat

(3.2)

If we send laser beams in front and back, we can restrict the thermal motion of the atom in both directions, resulting in a continuous loss of kinetic energy and a cooling effect. The radiation pressure created by the two laser beams reads, FD =

Fsc+

+

Fsc−

 = k 2



I /Isat 1+

4(−kv)2 2

+ I /Isat

−

I /Isat 1+

4(+kv)2 2

+ I /Isat

 . (3.3)

Cooling process is quantized. The energy quanta being pumped is E = p 2 /2M. Hence the cooling limit is Tre = 2E/k B , which is called the recoil limit. For 87 Rb, the recoil limit is Tre ≈ 360 nK.

1

3.1 Preparation of Cold Atoms

33

When kv  , we can simplify the force to the form FD = −αv, where α = −4k 2

1+

I /Isat 2 4 / 2 +

(3.4)

I /Isat

.

(3.5)

Note that a velocity-dependent viscous force is formed by the lasers, creating a cooling effect on the atoms (|v| < /k) within a certain velocity range, so Doppler cooling is also known as optical molasses. The cooling power can be written as, (dE/dt)cool = Fv = −αv 2 .

(3.6)

It is worth mentioning that Doppler cooling also heats the atom. This is because even  2  though the average momentum of the spontaneous photons is 0, their variance pre has  a nonzero value. As a result, each cooling process also transfers the energy 2 /2M to the atom. The heating rate reads, of pre  2 pre Rsc . = 2M 

(dE/dt)heat

(3.7)

When the cooling and heating reach equilibrium, i.e. (dE/dt)cool = (dE/dt)heat , we can derive the Doppler cooling limit temperature as TD =

 . 2k B

(3.8)

For 87 Rb atoms, cooling with the D2 line, the limiting temperature is about TD  143 µK.

3.1.2 Magneto-Optical Trap In Doppler cooling, the radiation pressure of the cooling light is only a function of the velocity, which cannot confine atoms spatially. Instead, we need a positiondependent force to trap atoms. To do this, we can introduce a gradient magnetic field, which produces a space-dependent Zeeman energy shift. Thus atom will feed different radiation pressure at different positions. The technique of trapping atoms based on this principle is called magneto-optical trap (MOT) [5, 6]. Taking the one-dimensional case as an example, we consider atoms with 0 and 1 as the total angular momentum quantum numbers J . The excited state has three degeneracies without a magnetic field and removes the degeneracy with a magnetic field. The direction of the magnetic decides whether Zeeman sublevels M J = +1

34

3 A High-Efficiency Quantum Memory

b

a

!"#

%$B

!"#

%$Fig. 3.2 The principle (a), and the schematic (b) of a magneto-optical trap. Figure from [5]

and M J = −1 are lifted or lowered. Based on this, the device can be set up as shown in Fig. 3.2a. The polarization of the left and right beams of cooling light is σ + and σ − , respectively. The magnetic field has a spatial distribution of B(z) = βz.

(3.9)

Atoms at the z = 0 position do not undergo an energy shift and do not interact with the two beams of light. Atoms at position z > 0, with M J = −1 state being lowered, absorb σ − photons and are pushed to the left; conversely, the atom at position z < 0, with M J = +1 state being lowered, absorbs σ + photons and are pushed to the right. Taking the exciting state shift by the magnetic field into account, we can correct Eq. 3.3 to FMOT

 = k 2



I /Isat 1+

4(−kv+gμ B βz/2 )2 2

+

I Isat

−



I /Isat 1+

4(+kv−gμ B βz/2 )2 2

+

I Isat

. (3.10)

Similarly, when kv  , the equation can be simplified to FMOT = −αv −

αβ z. k

(3.11)

Generalizing the 1D model to 3D, we can realize laser cooling and trapping. The gradient magnetic field in three-dimensional space can be produced by three pairs of anti-Helmholtz coils shown in Fig. 3.2b. The actual atomic energy levels are more complex. For instance, leakage from excited states to other steady states (so-called “dark states”) always exists in addition

3.1 Preparation of Cold Atoms

35

to the designated ground state. Additional pump lasers are required to pump atoms back to the two-level system of interest.

3.1.3 Polarization Gradient Cooling The MOT technique was experimentally confirmed soon after being theoretically proposed, but it was found that the temperature of the cooled atoms could easily break the Doppler limit TD . Cooling beyond the Doppler limit is due to a sub-Doppler cooling mechanism called polarization gradient cooling[7, 8]. This mechanism is also used in our experiments to reduce the temperature of the atomic ensembles. We briefly introduce its principle in this section. For an atom with a dipole moment d, its potential in the electric field E is U = −d · E.

(3.12)

Atoms will tend to stay at lower energies, meaning that their electric dipole d orientation will be parallel to the direction of the electric field and will generally reach this state by radiation. The counter propagating cooling beams from the same laser will interfere in space, forming a standing wave. The polarization (magnitude or direction) of the standing wave light field varies periodically in space, forming a periodic potential field for the atoms. Taking an example of two orthogonal linear polarized (lin ⊥ lin) counterpropagating cooling beams, the polarization of standing wave varies from linear to circular and to linear again in space. The spin state of M J = 1/2 and the spin state of M J = −1/2 feel different periodic potentials as shown in Fig. 3.3. Supposing an atom starts from the lowest potential, the thermal motion, whether to which direction, will lift its potential. When the atom at higher potential is excited, it tends to decay to the lower potential state. During this cycle, the energy provided by thermal motion is transferred to the emitted photon, leading to a deceleration of the atom. This process will keep happening, resulting in a continuous cooling of atoms. Because the atom is lifted up to the summit and rolled down to the valley to and fro, this technique is called the Sisyphus cooling. According to the analysis, the equilibrium temperature of the Sisyphus cooling is approximately kB T 

2 . 8||

(3.13)

One can see that the cooling temperature is proportional to the light intensity (the square of the Rabi frequency) and inversely proportional to the detuning. In the experiment, we usually first cool down atoms to near the Doppler limit with a regular MOT, then turn off the gradient magnetic field and cool the atoms further by means of decreasing the laser intensity and increasing laser detuning.

36

3 A High-Efficiency Quantum Memory

Fig. 3.3 The schematic of Sisyphus cooling. Figure from [9]

3.1.4 Energy Diagram for Atomic Cooling In this thesis, we choose a cooling laser (Cooler) coupling the D2 transition line, |52 S1/2 , F = 2 ↔ |52 P3/2 , F = 3 with a 20 MHz red detuning and a repumping laser (Repumper) coupling |52 S1/2 , F = 1 ↔ |52 P3/2 , F = 2. In the last 3 ms of the cooling process, we turn off the Repumper light and the gradient magnetic field, increase the cooling laser detuning and reduce the cooling laser intensity to benifit from the Sisyphus cooling. After the preparation of the atomic cloud, we can measure related parameters such as the size, density and optical thickness OD by absorption imaging. Our imaging laser couples |52 S1/2 , F = 2 ↔ |52 P3/2 , F = 3. To measure the atomic temperature, we release the cooled atoms from MOT and measure their volume expansion after a period of free fall due to the thermal motion, known as the time of flight method. The frequencies for all cooling lasers are shown in Fig. 3.4. The cooling process’s time sequence is shown in Fig. 3.5. All the control lasers are locked to the spectral absorption line of rubidium atoms by the saturation absorption method [10] and shifted by acoustic, optical modulators (AOMs), which also act as fast switches. Table 3.1 lists typical parameters for the atomic ensemble in our experiment, and more details can be found in [3].

3.2 Energy Scheme for Quantum Memory Experiments In the experiment, we use the D1 line of 87 Rb atoms for quantum memory experiments. Specifically, ground state |g ≡ |52 S1/2 , F = 1, m F = −1, excited state |e ≡ |52 P1/2 , F = 2, m F = 0 and two |s states: |↑ ≡ |52 S1/2 , F = 2, m F = −1, |↓ ≡ |52 S1/2 , F = 2, m F = +1 are used. After MOT, atoms are equally distributed on five Zeeman sublevels of |52 S1/2 , F = 2. As shown in Fig. 3.6, we gather all

3.2 Energy Scheme for Quantum Memory Experiments

37

20 MHz

F=3

52P3/2

F=2 F=1

Depumper

Repumper

Image

~780 nm

Cooler

F=0

F=2

52S1/2 F=1 Fig. 3.4 Energy level diagram for cooling lasers in the experiment

Fig. 3.5 Time sequence of the laser cooling

atoms on |g ≡ |52 S1/2 , F = 1, m F = −1 by a Depumper laser and a σ − polarized σ -pump laser, where |g is the only dark state in this configuration. The pumping process takes around 3 ms. The energy level scheme for quantum memory experiments is shown in Fig. 3.7. The write beam is polarized as σ + , is coupled to |g ↔ |e, and has a 40 MHz red detuning with the upper energy level. The Rayleigh scattering photons, coming from atoms falling back to the F = 1 energy levels, are filtered out. Moreover, the atoms experiencing Rayleigh scattering will not contribute to the read process. The

38

3 A High-Efficiency Quantum Memory

Table 3.1 Typical specification of the atomic ensemble Radius OD Number of atoms Density 360 µm

4.5

1.4 × 107

7 × 1010

Temperature cm−3

10 µK

Fig. 3.6 Schematic for the atomic state initialization

Fig. 3.7 Energy scheme of DLCZ quantum memory experiments

Raman scattering photons, stemming from atoms dropping to the F = 2 energy levels, have three polarization components, σ ± and π , corresponding to Zeeman sublevels m F = ±1 and 0, respectively. Only σ ± polarization exists when observing along the magnetic axis, and they have the same scattering coefficients. Thus, when we collect a write-out photon along the magnetic axis, the joint state of atom and photon can be written as  1  |  = √ |↑ σ −  + |↓ σ +  . 2

(3.14)

3.3 The Ring Cavity

39

where ↑ and ↓ refer to the spin waves containing the corresponding states (which also apply to all cases thereafter without other notes). This is an entanglement between the photon polarization and the atomic internal state. Read beam couples |52 S1/2 , F = 2 and |52 P1/2 , F = 2, polarized to σ − , such that |↑ and |↓ are both coupled to the excited states and fall back to the |g state, producing a readout photon. They both have more than one decay channel. In √ contrast to the√write process, |↑ and |↓ have different dipole matrix element of 1/2 er  the excited states back and 1/12 er , respectively, from  to the ground state in  the read process, where er  = J = 1/2 | er | J = 1/2 . Thanks to the collective enhancement condition, the decay channel back to the |g state is largely enhanced so that the readout imbalance between two spin waves is largely mitigated. Other than the readout in |↑/|↓ basis, one needs the capability of readout in |↑ ± |↓ basis to characterize the qubit. We achieve this by performing a basis transformation inside the atom [11] by applying two Raman beams coupling |↑ and |↓ as shown in Fig. 3.7, resulting in a Rabi oscillation between |↑ and |↓. By choosing the intensity and width of the Raman beams, a π/2 pulse can be achieved, |↑ → |↑ + |↓, |↓ → |↑ − |↓.

(3.15)

The Raman beam is detuned coupled to |52 P1/2 , F = 1 and |52 P1/2 , F = 2 at the same time, which enables the exploitation of both channels.

3.3 The Ring Cavity We build ring cavities around the atoms to enhance the light-atom interaction. In contrast to Fabry-Perot cavities, the light field in ring cavities is travelling wave fields which makes the interaction less position-dependent. More importantly, the forward and backward modes in ring cavities are emitted in different spatial directions and can be distinguished [12], which, in Fabry-Perot cavities, would mix and significantly reduce the storage lifetime [13].

3.3.1 Cavity Geometry The ring cavity geometry is shown in Fig. 3.8. It consists of three mirrors, where HR1 and HR2 are the two highly reflective mirrors with reflectivity > 99.9% and PR is a partially reflective mirror with reflectivity r 2  80% that serves as the output mirror of the cavity. The theoretical fineness of the cavity is

40

3 A High-Efficiency Quantum Memory

Fig. 3.8 The geometry of the ring cavity. Figure from [15]

F=

√ π r = 28. 1−r

(3.16)

The cavity length needs to enable the write-out and read-out photon on resonance with the cavity simultaneously. Thus the cavity length L is an integral multiple of these two wavelengths. ωr o − ωwo = n

c = 6.834682 GHz + 80 MHz, L

(3.17)

where n is a positive integer. Hence we have, L = n × 4.34 cm.

(3.18)

We choose L = 65.1 cm so that the free spectral range is νFSR  461 MHz and the linewidth of the resonance peak is νFSR  16.5 MHz. F

(3.19)

According to our energy level scheme, neither the write nor the read beam is on resonance with the cavity, so the ring cavity serves as a filter for the read and write beam noise. In our experiments, the ring cavity provides about 18 dB filtering for the Rayleigh noise and the leakage of the read and write beam [3, 14]. In addition to an appropriate cavity length, it is also important to eliminate light field distortion during propagation. Hence we place two lenses of the same focal

3.3 The Ring Cavity

41

length f 1 = f 2 = f for transforming the light field. Two lenses are placed symmetrically in geometry so that the structure of the cavity is an axisymmetric isosceles triangle. Due to the symmetry constraint, w0 and w0 must be the minimum waist positions of the Gaussian light field. Let us consider the case of the left side. Assuming the distances from the lens at the positions w0 and w0 be z 0 and z 0 , respectively, we have the following equations of the Gaussian light field transformation, z 0 − f = z0 − f (z 0 −  2 w0 = w0 (z 0 −

f2 , f )2 + z 2R f2 . f )2 + z 2R

(3.20)

In our setup, z 0 = L 1 /2 + L 2 , z 0 = L 3 . Obviously, we have a constraint relationship L1 L + L2 + L3 = . 2 2

(3.21)

We choose a lens of f = 250 mm so that w0 = 90 µm, positioned at the focal length of the lens, and the other side of the lens is collimated, with no strict requirement for the minimum beam waist position. In addition, the light field undergoes a chiral transformation when reflected by the mirrors, which decomposes the light field into horizontal and vertical polarizations with a phase shift of size π attached. Our ring cavity has three mirrors, resulting in a chiral flip for circular-polarized photons after one round of travelling. To compensate for this, we place a set of waveplates in a sandwich structure QWP-HWP-QWP in the cavity, which adds the π phase between the horizontal and vertical components of the circularly polarized photons, equivalent to the fourth mirror. In addition, the waveplates can also compensate for other minor phase shifts from lenses and other components.

3.3.2 Cavity Locking In order to make the ring cavity work stably, we need to stabilize it actively. We place a partially reflective mirror with reflectivity 99% in the readout optical path to introduce a locking beam. The locking signal is detected by a fast photodiode behind the high HR2 . We change the cavity length with the help of a piezoelectric ceramic on HR1 . To realize high-speed feedback, we use the Pound-Drever-Hall (PDH) method [16] to generate a ramped error signal for locking.

42

3 A High-Efficiency Quantum Memory

Write-out Locking

EOM

BPP

PR

99:1

HR1

PZT

QWP HWP

RF

phase shifter

Read-out Lens

φ

HR2 mixer PD

low pass PID controller Fig. 3.9 Schematic of the ring cavity locking

The diagram of the locking scheme is shown in Fig. 3.9. Supposing a monochromatic laser beam with an optical field of E in = E 0 e−πωt is incident into the cavity, the cavity output is E out = F(ω)E in , where F(ω) is the response function and |F(ω)|2 is the transmission spectrum of the cavity. We use an EOM to add a high-frequency modulation with a frequency of  = 21.4 MHz to the locking field. The incident optical field becomes E in = E 0 eπ(ωt+β sin t)

 ≈ E 0 J0 (β)eπωt ± J1 (β)eπ(ω±)t .

(3.22)

Here, we expand the light field in Bessel functions, where Jn is an n order Bessel function, which holds when the modulation depth β is small. At this point, there are at least three frequency components in the optical field, and the cavity output becomes

 (3.23) E out ≈ E 0 F(ω)J0 (β)eπωt ± F(ω ± )J1 (β)eπ(ω±)t .

3.3 The Ring Cavity Fig. 3.10 The cavity transformation (red) and the error signal (green) in ramping (a) and locking modes (b). Figure from [3]

43

a

b

The signal observed on the photocell reads Pout =Pc |F(ω)|2 + Ps |F(ω + )|2 + Ps |F(ω − )|2



 + 2 Pc Ps  F(ω)F ∗ (ω + ) − F ∗ (ω)F(ω − ) cos(t) 

 +  F(ω)F ∗ (ω + ) − F ∗ (ω)F(ω − ) sin(t) + 2.c.

(3.24)

The first three terms are the DC terms, which have the intensity of the carrier frequency of Pc = J02 (β)|E 0 |2 , and two sidebands of Ps = J1 (β)2 |E 0 |2 , respectively. The fourth is an interference term between the carrier frequency and the sidebands, oscillating at . The fifth is an interference term between the two sidebands, oscillating at 2. After mixing this signal with an  local oscillator and low-pass filtering, only the terms with frequency  can be retained, and their coefficients are the error function. Subsequently, the feedback control is performed using a proportionalintegral-derivative (PID) controller. The error signal and cavity transmission in ramping and locking mode are shown in Fig. 3.10. In addition, to avoid the noise introduced by the locking beam, we use a BPP to convert the locking beam to a higher-order Ermey-Gaussian (Hermite-Gaussian) mode, TEM lm , where l = 0, m = 1, which in theory does not couple to single-mode fibres. The higher-order mode has a different Gouy phase than the fundamental mode,

44

3 A High-Efficiency Quantum Memory

 φ(z) = (l + m + 1) arctan

z zR

.

(3.25)

This leads to a phase difference in the locking beam and the signals, resulting in a different resonance frequency. We compensate for this by setting a frequency offset to the locking beam.

3.4 Quantum Memory Characterization 3.4.1 Quantum Correlation The DLCZ quantum memory is a probabilistic entanglement source, as described in section 2.2. Here we characterize our DLCZ-type quantum memory by measuring two parameters. The first one is the cross-correlation between write-out and read-out photons. The measurement setup is shown in Fig. 3.11 without using the beamsplitter. The following statistics are performed [17], (2) gcr oss =

Pwo,r o , Pwo Pr o

(3.26)

where Pwo (Pr o ) is the probability of write-out (read-out) photon and Pwo,r o is the probability of the their coincidence. If the Cauchy-Schwarz inequality in the classical correlation  (2) 2 (3.27) R = gcr oss /gwo,wo gr o,r o ≤ 1, is violated, it can be proved that there are nonclassical correlations between the write-out and read-out photons. Where gwo,wo and gr o,r o are the auto-correlation functions of the write-out and read-out photons, respectively. They generally have (2) values smaller than 2; thus we use gcr oss > 2 to determine the nonclassical correlation between the write-out and read-out photons. Another parameter is the auto-correlation function [18] between two outputs of the beamsplitter in Fig. 3.11 triggered by a successful detection of the read-out photon,

BS read-out

write-out

wo

ro wo’ Fig. 3.11 Setup for the quantum correlation measurement

3.4 Quantum Memory Characterization

45

a 0.5 0.4 0.3 0.2 0.1 0.0 0.00

0.02

0.04

0.06

0.08

Excitation probability pw

b 20

g(2) cross

15

10

5

0 0.000

0.005

0.010

0.015

0.020

0.025

0.030

Excitation probability pw

Fig. 3.12 Auto-correlation (a), and cross-correlation (b) functions as functions of the write-out probability

(2) gauto =

P(wo, wo | wo) , P(wo | r o)P(wo | r o)

(3.28)

where wo and wo refer to two modes of the write-out field after the beamsplitter, P(wo | r o) (P(wo | r o)) is the probability of the wo (wo ) mode in the condition of the read-out click and P(wo, wo | r o) is the probability of coincidence of wo and wo modes in the condition of the read-out click. Sometimes the auto-correlation (2) function gauto is called the α value. The results of these two parameters as functions of the write-out probability are shown in Fig. 3.12. A lower write-out probability will bring greater nonclassical

46

3 A High-Efficiency Quantum Memory

Table 3.2 Typical read-out efficiencies and lifetimes of quantum memories ηret,↑ ηret,↓ τm (μs) Memory A Memory B

0.333 ± 0.011 0.339 ± 0.015

0.204 ± 0.008 0.218 ± 0.010

71.19 ± 0.09 65.29 ± 0.08

statistics. To balance the quantum memory performance and the entangling rate, we usually choose the write-out probability between 0.01 and 0.02.

3.4.2 Storage Efficiency and Lifetime To access the best read-out efficiency, in addition to the phase-matching condition and a good cavity enhancement, it is also essential to remove the influence from the ambient magnetic field, which includes the laboratory’s geomagnetic and residual magnetic fields instrument. In order to do this, we place three pairs of Helmholtz coils are placed around the atoms. To find the best compensation magnetic fields, we run the DLCZ quantum memory without applying the bias field, and the Zeeman states pumping, i.e., atoms are evenly initialized to all Zeeman sublevels of |52 S1/2 , F = 2. The read-out signal, in this case, contains the contribution of all sublevels. Any residual ambient magnetic field will cause interference between channels, resulting in a time-dependent oscillation of the read-out signal. Minimizing the oscillation leads to the cancellation of the ambient magnetic field [3]. In Table 3.2, we list typical read-out efficiencies and storage lifetimes for two quantum memories used in this thesis.

3.4.3 Entanglement Benchmarking Before verifying the atom-photon entanglement, we characterize the Raman π pulse, which we use for basis rotation. By mapping them to two polarization modes of the read-out photon, we measure the Rabi oscillation between two spin waves. Figure 3.13 shows the read-out efficiency of two read-out polarization modes conditioned by two write-out Polarization modes as a function of the Raman pulse width. We fit the oscillation by exponential decaying sinusoidal functions:    2π t e−γ t . P(t) = A + C cos T

(3.29)

Based on the parameters obtained from the fitting, we can estimate the average π/2 operational fidelity as Fπ/2 = e−γ T /4 ≈ 0.97.

3.4 Quantum Memory Characterization

a

47

0.30

Probability

0.25 0.20 0.15 0.10 0.05 0.00 0

500

1000

1500

2000

2500

3000

2500

3000

Raman width (ns)

b 0.12

Probability

0.10 0.08 0.06 0.04 0.02 0.00 0

500

1000

1500

2000

Raman width (ns)

Fig. 3.13 Conditional read-out efficiency as a function of the Raman pulse width

Fig. 3.14 Tomogrphy of the atom-photon entanglement

48

3 A High-Efficiency Quantum Memory

This way, we can do the complete set of local measurements of the writ-out photon and spin wave qubit. On this basis, we perform a two-body tomography. Figure 3.14 shows the density matrices of atom-photon entanglement generated from two quantum memories. By comparing the density matrix with the target Bell state, we get the fidelity of the two entangled states as 0.930(6) and 0.933(6).

3.5 Comparison of the State-of-the-Art Quantum Memories In Table 3.3, we compare the state-of-the-art quantum memories in different physics systems. The three figure-of-merits we compare are the atom-photon entanglement probability, the atom qubit verification efficiency and the experiment repetition rate because these factors mainly affect the long-range entangling rate. • Atom-photon entanglement probability: All platforms except the atomic ensemble in the table are single-particle physics systems. In principle, single-particle systems work deterministically in creating entanglement, while the atomic ensemble works probabilistically. But in practice, only the Weinfurter group’s single neutral atom platform and the Blatt group’s trap ion platform outperform the Pan group’s atomic ensemble platform. This is because most single-particle platforms suffer from poor photon collection efficiency. However, the technical challenge of achieving a strong cavity coupling system will be overcome someday, but the probabilistic nature of the atomic ensemble system cannot be circumvented. Therefore, the atomic ensemble system’s future must be developing Rydberg blockade techniques, enabling deterministic entanglement generation. • Atom qubit verification efficiency: NV centre and trapped ion platforms have a well-defined cyclic transition, which enables deterministic fluorescence detection. In an atomic ensemble platform, though the intrinsic readout efficiency can be reached with collective enhancement, the overall detected readout efficiency is around 35%, which limits the scaling of the entanglement. One possible way of achieving deterministic detection is to perform fluorescence detection [19], which is challenging for complex energy-level structures in neutral atoms. Another choice is making use of the Rydberg blockade mechanism. Because the Rydberg excitation will shift the energy level of other atoms around, we can predict the population of Rydberg excitation by measuring whether the other atom is energy shifted. Ionization of Rydberg atoms is also a feasible detection method [20]. • Experiment repetition rater: In quantum repeater protocol, the repetition rate upper bound is set by the communication time. Nevertheless, most of the platforms are limited by their system initialization time. Every system has its advantages and weakness. Although the atomic ensemble is not the best in a particular dimension, it is still among the best on average. Taking the three-node entanglement experiment performed in this platform as an example [14],

3.5 Comparison of the State-of-the-Art Quantum Memories

49

Table 3.3 Comparison of the state-of-the-art quantum memories Group Platform AtomAtomic Repeatation Counts per photon qubit rate hour entangleverification ment efficiency probability H. Weinfurter G. Rempe C. Monroe R. Blatt M. Lukin R. Hanson A. Imamoglu Y. Yamamoto J. Pan J. Pan

Single Atom Single Atom Trapped ion Trapped ion NV center NV center Quantum dot Quantum dot Atomic ensemble Atomic ensemble

Ref.

1.0E-03

6.0E-01

5.0E+04

1.0E-02

[21]

1.2E-01

9.0E-03

5.0E+03

6.0E-03

[22]

4.0E-03 5.7E-02 1.0E-06 4.5E-04 1.4E-04

1 1 1 1 2.0E-03

4.7E+05 6.7E+02 1.0E+05 2.0E+04 1.9E+07

36 111 9.0E-11 1.6E-03 3.8E-10

[23] [24] [25] [26] [27]

1.0E-03

1.0E-03

2.0E+07

2.0E-08

[28]

2.3E-03

1.5E-01

1.0E+04

5.0E-04

[29]

2.0E-02

3.0E-01

3.0E+05

6

[14]

The shadowed row refers to the result in this thesis. The counts refer to the simulation results of the three-node entanglement experiment [14]. Table from [3] with slight modifications

simulation shows that the atomic ensemble system gives one of the highest count rates per hour, and this is an important basis for the long-distance entanglement experiment performed in Chaps. 5 and 6.

References 1. Chu S, Hollberg L, Bjorkholm JE et al (1985) Three-dimensional viscous confinement and cooling of atoms by resonance radiation pressure. Phys Rev Lett 55(1):48 2. Migdall AL, Prodan JV, Phillips WD et al (1985) First observation of magnetically trapped neutral atoms. Phys Rev Lett 54(24):2596 3. Jong B (2018) Quantum network with multiple cold atomic ensembles. Springer Nature 4. Lett PD, Phillips WD, Rolston S et al (1989) Optical molasses. JOSA B 6(11):2084–2107 5. Foot C, Foot D, Foot C (2005) Oxford master series in physics: atomic physics. OUP Oxford 6. Bagnato V, Lafyatis G, Martin AG et al (1987) Continuous stopping and trapping of neutral atoms. Phys Rev Lett 58(21):2194 7. Dalibard J, Cohen-Tannoudji C (1989) Laser cooling below the Doppler limit by polarization gradients: simple theoretical models. JOSA B 6(11):2023–2045 8. Ungar PJ, Weiss DS, Riis E et al (1989) Optical molasses and multilevel atoms: theory. JOSA B 6(11):2058–2071 9. Luschevskaya E, Golubev A (2014) Current progress in laser cooling of antihydrogen. arXiv preprint arXiv:1406.7521

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3 A High-Efficiency Quantum Memory

10. Jiang Xiao (2009) Laser Frequency Stabilization and Phase Locking Technology in Cold Atom Quantum Storage. University of Science and Technology of China 11. Jiang Y, Rui J, Bao XH et al (2016) Dynamical zeroing of spin-wave momentum to suppress motional dephasing in an atomic-ensemble quantum memory. Phys Rev A 93(6):063819 12. Bao XH, Reingruber A, Dietrich P et al (2012) Efficient and long-lived quantum memory with cold atoms inside a ring cavity. Nat Phys 8(7):517–521 13. Simon J, Tanji H, Thompson JK et al (2007) Interfacing collective atomic excitations and single photons. Phys Rev Lett 98(18):183601 14. Jing B, Wang XJ, Yu Y et al (2019) Entanglement of three quantum memories via interference of three single photons. Nat Photo 13(3):210–213 15. Yang Shengjun (2015) Experimental Research on Cold Atom Quantum Memory. University of Science and Technology of China 16. Black ED (2001) An introduction to pound-Drever-Hall laser frequency stabilization. Am J phys 69(1):79–87 17. Kuzmich A, Bowen W, Boozer A et al (2003) Generation of nonclassical photon pairs for scalable quantum communication with atomic ensembles. Nature 423(6941):731–734 18. Grangier P, Roger G, Aspect A (1986) Experimental evidence for a photon anticorrelation effect on a beam splitter: a new light on single-photon interferences. EPL (Europhy Lett) 1(4):173 19. Fuhrmanek A, Bourgain R, Sortais YRP et al (2011) Free-space lossless state detection of a single trapped atom. Phys Rev Lett 106:133003 20. Ortegel N (2016) State readout of single Rubidium-87 atoms for a loophole-free test of Bell’s inequality. lmu 21. Hofmann J, Krug M, Ortegel N et al (2012) Heralded entanglement between widely separated atoms. Science 337(6090):72–75 22. Ritter S, Nölleke C, Hahn C et al (2012) An elementary quantum network of single atoms in optical cavities. Nature 484(7393):195 23. Hucul D, Inlek IV, Vittorini G et al (2015) Modular entanglement of atomic qubits using photons and phonons. Nat Phys 11(1):37 24. Stute A, Casabone B, Schindler P et al (2012) Tunable ion-photon entanglement in an optical cavity. Nature 485(7399):482–485 25. Togan E, Chu Y, Trifonov A et al (2010) Quantum entanglement between an optical photon and a solid-state spin qubit. Nature 466(7307):730–734 26. Bernien H, Hensen B, Pfaff W et al (2013) Heralded entanglement between solid-state qubits separated by three metres. Nature 497(7447):86–90 27. Delteil A, Sun Z, Gao W b, et al (2015) Generation of heralded entanglement between distant hole spins. Nat Phys 12(3):218–223 28. De Greve K, Yu L, McMahon PL et al (2012) Quantum-dot spin-photon entanglement via frequency downconversion to telecom wavelength. Nature 491:421–425 29. Yuan ZS, Chen YA, Zhao B et al (2008) Experimental demonstration of a BDCZ quantum repeater node. Nature 454(7208):1098–1101

Chapter 4

Quantum Frequency Conversion

The 87 Rb atoms used in our experiments emit write-out photons at a wavelength of 795 nm, which have a transmission loss in the fibre of about 3.5 dB/km. We use quantum frequency conversion techniques (QFC) to coherently convert the photon wavelength to 1342 nm (O-band, loss ∼ 0.3 dB/km) to reduce the loss. This chapter briefly explains the principles of quantum frequency conversion and presents the periodic lithium niobate waveguide used in our experiments and its characterization.

4.1 Principle 4.1.1 Nonlinear Optics The nonlinear optical process happens when the light propagates in medium [1]. Light as an electromagnetic wave polarizes the medium; in turn, the polarized dielectric effect back on the light field and changes some properties of the light. In a dielectric, we describe the electric field inside the dielectric by the electric displacement field D, (4.1) D = 0 E + P, where 0 is the vacuum dielectric constant, E is the applied electric field, and the polarization strength P is a function of the electric field strength, P = 0 (χ (1) E + χ (2) E2 + χ (3) E3 + . . . ) = 0 χ (1) E + PNL .

(4.2)

In general, χ (n) is an n + 1 order tensor that represents the n order polarizability of the medium. χ (1) is the linear polarizability, describing linear optical phenomena such as refraction; χ (2) is second-order polarizability, describing three-wave mixing; and χ (3) is third-order polarizability, describing four-wave mixing. For convenience, we sometimes use PNL to represent all the nonlinear terms. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Yu, Long Distance Entanglement Between Quantum Memories, Springer Theses, https://doi.org/10.1007/978-981-19-7939-2_4

51

52

4 Quantum Frequency Conversion

Fig. 4.1 Second order optical nonlinear interaction. Figure from [1]

According to the theory of electromagnetics, the equations for the propagation of light in a medium can be derived from Faraday‘s law equations in the differential form of Maxwell‘s set of equations as follows, ∇ ×E=−

∂ B. ∂t

(4.3)

By applying curl operation to both sides of the equation, the left side reads, ∇ × ∇ × E = ∇(∇ · E) − ∇ 2 E.

(4.4)

Usually we have ∇ · E ≈ 0. And for a non-magnetic medium with no current, the right side of the equation reads, ∇×

∂ ∂2 ∂ B = μ0 ∇ × H = −μ0 2 (0 E + P). ∂t ∂t ∂t

(4.5)

We can get the nonlinear wave equation as follow, − ∇2E +

 (1) (ω) ∂ 2 1 ∂ 2 NL E = − P , c2 ∂t 2 0 c2 ∂t 2

(4.6)

in which we make use of the refractive index n 2 = 1 + χ (1) , and the relationship between the dielectric constant, the magnetic dielectric constant, and the speed of light in vacuum 0 μ0 = 1/c2 . Equation 4.6 is a wave equation with a source, where the nonlinear term on the right side of the equation is its source. When this term is neglectable small, the equation degenerates to a wave equation with no source, which describes the linear propagation of light in the medium. In this thesis, we consider only second-order nonlinear effects. Considering the case in Fig. 4.1, in which the two monochromatic waves propagate in the same direction along z, and the light field is polarized to the same linear polarization, we can replace the vectors in Eq. 4.6 with scalars. The total incident light field reads, E = E 1 e−iω1 t + E 2 e−iω2 t + c.c. We can calculate P (2) by its definition,

(4.7)

4.1 Principle

53

P (2) =0 χ (2) E 2 =0 χ (2) [E 12 e−iω1 t + E 22 e−iω2 t + 2E 1 E 2 e−i(ω1 +ω2 )t +

2E 1 E 2∗ e−i(ω1 −ω2 )t

+ c.c.] + 20 χ

(2)

[E 1 E 1∗

+

(4.8) E 2 E 2∗ ].

Conventionally, we write these terms as, P (2) =



P(ωn )e−iωn t ,

(4.9)

n

where P(ωn ) is defined as follows: P(2ω1 ) = 0 χ (2) E 12 (SHG), P(2ω2 ) = 0 χ (2) E 22 (SHG), P(ω1 + ω2 ) = 20 χ (2) E 1 E 2 (SFG), P(ω1 − ω2 ) = 20 χ

(2)

E 1 E 2∗

(DFG),

P(0) = 20 χ

(2)

(E 1 E 1∗

+ E 2 E 2∗ ) (OR).

(4.10)

The abbreviation in parentheses after each term indicates its corresponding secondorder nonlinear effect, SHG for second-harmonic generation, SFG for secondfrequency generation, DFG for differential-frequency generation, OR for optical rectification, which refers to the light-induced potential difference between the two sides of the medium. Usually, however, it is difficult for these phenomena to occur efficiently in a natural medium. The reason behind this is the phase mismatch in the medium.

4.1.2 Coupled Wave Equations To observe the effect of the phase mismatch, we take the sum-frequency process as an example and calculate the amplitude of the sum-frequency light, i.e. the coupled wave equation. For simplicity, we start with a source-free case in a loss-free nonlinear medium. Then the solution for Eq. 4.6 reads, E 3 (z, t) = A3 ei(k3 z−ω3 t) + c.c,

(4.11)

where the wave number of the sum frequency light k3 and the refractive index of the medium at its wavelength n 3 are k3 =

n 3 ω3 2 , n 3 =  (1) (ω3 ). c

(4.12)

54

4 Quantum Frequency Conversion

Obviously, the amplitude A3 is a constant for the source-free case. To include the source, we make A3 a function of z A3 (z), and substitute it to Eq. 4.6. Then we get the linear terms as, (4.13) P˜3 (z, t) = P3 e−iω3 t + c.c. where1 .

P3 = 40 deff E 1 E 2 = 40 deff A1 A2 ei(k1 +k2 )z .

(4.14)

Since we are now considering the one-dimensional case, we replace the Laplace operator ∇ 2 in Eq. 4.6 with d2 /dz 2 and get,  d2 A

3 dz 2

+ 2ik3

dA3  (1) (ω3 )A3  i(k3 z−ω3 t) − k32 A3 + e + c.c. dz c2

−4deff ω32 A1 A2 ei[(k1 +k2 )z−ω3 t] + c.c. = c2

(4.15)

Because k32 =  (1) (ω3 )A3 /c2 , the third and fourth terms on the left side of the equation can be eliminated. The above equation holds equally well with the complex conjugate term removed, and the time-containing terms on both sides are equal so that it can be simplified to −4deff ω32 dA3 d2 A 3 = + 2ik A1 A2 ei(k1 +k2 −k3 )z . 3 dz 2 dz c2

(4.16)

In practice, the first term in the left side of the equation is small enough to be ignored, which is called the slowly varying amplitude approximation,    2     d A3    k3 dA3  .   dz   dz 2 

(4.17)

Then the equation can be further simplified as dA3 2ideff ω32 = A1 A2 eikz . dz k3 c2

(4.18)

1

From here on, we follow the usual convention of replacing the second-order nonlinear coefficients 1 (2) χi(2) jk with the symbol di jk , which obeys di jk = 2 χi jk . Although there are 27 combinations for the three subscripts, not everyone is independent, thanks to the crystal symmetry. Conventionally, we reduce the last two subscripts to a new one, l, by jk : 11 22 33 23, 32 31, 13 12, 21 l: 1 2 3 4 5 6 For selected directions, one can also define an effective nonlinear coefficient deff , which is a linear combination of a few tensor elements, so that the set of tensor equations can be reduced to a single equation. A more specific explanation can be found in the textbook [1].

4.1 Principle

55

where k is the difference in momentum between the sum frequency and the fundamental frequency light. This equation is called the coupled wave equation. Because of the symmetry, we have two similar equations for A1 and A2 . Equation 4.18 is a trivial first-order differential equation. We can get the electric field intensity of the sum frequency light by integrating the right part of the equation, 

0

A3 = L

2ideff ω32 eik L − 1 2ideff ω32 ikz A A e = A A . 1 2 1 2 c2 c2 ik

(4.19)

The electromagnetic wave energy is given by the time averaging of the Poynting vector, (4.20) Ii = 2n i 0 c|Ai |2 , i = 1, 2, 3. Thus we get, 2  2 ω34 |A1 |2 |A2 |2  eik L − 1  8n 3 0 deff I3 =2n 3 0 c|A3 | =  k  k32 c3   k 8d 2 ω2 I1 I2 L . = eff 3 2 L 2 sinc2 n 1 n 2 n 3 0 c 2 2

(4.21)

It can be seen that the sum frequency light intensity is a periodic function about L with period L coh = 2π/k. This characteristic length is called the coherence length.

4.1.3 Quasi Phase Matching It can be seen that in Eq. 4.21, the sum frequency light intensity will monotonically increase with the interaction distance only when k = 0, i.e., the phase matching condition is satisfied. However, the speed of light in the medium varies for different wavelengths, which is known as dispersion. Hence, in general, the phase matching condition can not be naturally fulfilled, and the sum-frequency light intensity will oscillate with the growth of the interaction distance, staying far from optimal conversion efficiency. We can understand the picture of the phase mismatch by imagining the nonlinear medium as a source of sum-frequency light, consisting of many point sources. The sum-frequency light that we observe comes from the interference of many point sources. On the one hand, the point sources are linked by the signal light. Sumfrequency light inherits the phase from the source light; hence the phase difference between two point sources separated by z is φ1 = (k1 + k2 )z. On the other hand, when the sum frequency light from two point sources interferes, they accumulate a propagation phase difference of φ2 = k3 z. In the end, the total phase difference when two sum-frequency lights interfere is φ = φ1 − φ2 = kz. It is evident that the sum of many point sources when k = 0 shows a messy interference. One

56

4 Quantum Frequency Conversion

Fig. 4.2 The polarization of the lithium niobate crystal in the natural (a) and periodical poling (b) cases. Figure from [1]

straightforward idea to tackle the phase mismatch issue is to reduce the medium size so that the medium approximates a point source without further interference. However, it has to be ensured that the optical power density in the medium is high enough, which is technically challenging. A more common solution is to fulfil the phase matching condition. One of the first phase matching methods that were proposed and implemented is birefringent phase matching [2]. This scheme takes advantage of the relatively significant difference in refractive indices between different crystallographic axes of birefringent crystals. It selects a suitable angle that can fulfil the phase matching condition. However, this method has large limitations, on the one hand, because not all combinations of frequencies of interest can find a suitable birefringent phase matching condition; on the other hand, when the matching condition is fulfilled, the polarization usually does not match the crystal axis, so that the maximum nonlinear coefficients d33 cannot be used. A universal phase-matching method is the quasi-phase matching technique, the basic idea of which is to compensate for the phase difference between the fundamental frequency light and the sum frequency light by artificially introducing modulation. In Eq. 4.21, we can see that when L ≤ L coh /2, the sum frequency light intensity is continuously enhanced, indicating that for a small region with the scaling of L coh /2, the point sources are showing constructive interference. However, when L coh /2 < L ≤ L coh , the sum frequency light intensity starts to decrease with distance and eventually drops to 0, indicating that there is a phase mismatch between two adjacent L coh /2 regions. We can avoid destructive interference if we can adjust the relative phase between two small regions. The theoretical scheme of quasi-phase matching was first proposed in 1962 by Bloembergen et al. [3]. However, limited by the fabrication imprecision, it was not experimentally achieved until the 1990s s with the maturation of techniques such as the applied electric field polarization. As shown in Fig. 4.2, in the quasi-phase matching scheme, we periodically modulate the nonlinear coefficients of the medium. In practice, we change the nonlinear coefficient by transposing their poling. The secondorder polarizability is a third-order tensor. Under the corresponding transformation,

4.2 Periodically Poled Lithium Niobate Waveguides

57

the nonlinear coefficients of our interest deff have an extra negative sign in the same reference system. By assuming a modulation period of , the effective nonlinear coefficients become, deff (z) = d0 sign[cos(2π z/)] = d0

∞ 

G m ex p(ikm z).

(4.22)

m=−∞

Here we expand it with the Fourier series, where G m = (2/mπ ) sin(mπ ) is the weight coefficient of each component and km = 2π m/ is the wave vector of the monochromatic wave of that component. We first consider only the component of m = 1, which is the case of the first-order approximation. Then we have,   2π 2d0 exp i z . deff (z) = π 

(4.23)

Substituting this into the coupled wave equation of Eq. 4.18 gives 2id0 ω32 dA3 = A1 A2 ei(k+2π/)z . dz k3 c2

(4.24)

Provided that an appropriate period  is chosen such that k + 2π/ = 0, i.e., phase matching, the frequency light can be converted efficiently. In the point source model, we introduce extra phases among point sources to mitigate the phase mismatch. Note that we have only focused on the component of m = 1, whose weight in the nonlinear coefficient expansion Eq. 4.22 is only G 1 = 2/π . For other m = 1 cases, the conditions  + km = 0 still stand, hence the sum-frequency process cannot occur efficiently. The sum-frequency light behaves the phase mismatch oscillation in these cases. The typical performance of the QPM is shown in Fig. 4.3. Compared to perfect phase matching, its equivalent nonlinear coefficient is about (2/π )2 ≈ 40.5%. However, with proper growth of the dielectric length, the same conversion efficiency of 100% can be achieved in principle. As a comparison, in the case of lithium niobate [4], quasi-phase matching can utilize its maximum nonlinear coefficient d33 = 27 pm/V and equivalent nonlinear coefficient d Q = 2d33 /π ≈ 17.2 pm/V for quasi-phase-matching. In contrast, though birefringent quasi-phase matching gives an ideal phase matching, it can only utilize a smaller nonlinear coefficient of d31 = 3.8 pm/V, and the conversion efficiency for the same length is only 1/20 of the former.

4.2 Periodically Poled Lithium Niobate Waveguides The nonlinear medium chosen for our experiments is the lithium niobate (LiNbO3 , LN) [5], which is a ferromagnetic material [6] with a Curie temperature of about

58

4 Quantum Frequency Conversion

Fig. 4.3 The amplitude of the second-harmonic signal as a function of the interaction length in the perfect phase matching (a), quasi-phase-matching, and mismatching cases. Figure from [1]

Fig. 4.4 Process of the periodic poling generation. Figure [7]

1210 ◦ C, below which it is in the ferroelectric phase. In the ferroelectric phase, the positively charged Nb5+ and Li+ ions do not coincide with the negatively charged O2− centre, leading to polarising spontaneously. In our experiments, we start from a lithium niobate wafer, generate periodic polarization on the wafer, fabricate a waveguide structure on the upper side, and finally encapsulate it as a plug-and-play device. This part of the work was mainly done by Dr Fei Ma [7] and his colleagues at the Jinan Institute of Quantum Technology. We give a brief description of the main fabrication process in this section.

4.2 Periodically Poled Lithium Niobate Waveguides

59

4.2.1 Generation of Periodic Poling We use lithium niobate wafers of 3 in. (76.2 mm) in diameter and 0.5 mm in thickness, grown by the Czochralski method. The wafer is growing along the Z-axis of the crystal, which has the most significant nonlinear coefficient. We apply an electric bias to some regions to generate the periodic poling. The flowchart of the whole process is shown in Fig. 4.4, which consists of: 1. Spin a layer of photoresist on the +Z surface of the wafer and bake the wafer to make the photoresist adhesive. 2. Optically expose the region to be pole reverted, remove the exposed photoresist chemically and bake the wafer again to make the remaining photoresist a good insulating layer. 3. Appy a high voltage between the front and back surfaces of the wafer. The exposed region is reverted, yet the other region keeps unchanged because of the insulating photoresist. 4. To confirm the result conforms with the design, we can etch the wafer in hydrogen fluoride. Due to a factor of 1000 difference in etch rate for the +Z and the −Z axis, the periodical poling become visible under microscope.

4.2.2 Waveguide Fabrication While feasible, the direct use of periodically polarized crystals for frequency conversion has several drawbacks. First, the beam needs to be focused on the crystal to ensure a relatively good nonlinear action. Focusing causes the beam to diverge rapidly, with an effective action length of only twice the Rayleigh distance, making achieving the desired conversion efficiency difficult. There is also a need to overlap the signal light well with the pump light, and the optical path can be adjusted and maintained with a significant amount of effort. Compared with the bulk crystal, waveguide structures [8] make the light largely confined, increasing the power density. At the same time, it also benefits from integration and scalability. We use proton exchange to generate waveguides on periodically polarized crystals with the following steps. 1. Sputtering a layer of SiO2 thin film via the physical vapour deposition. 2. Spinning a layer of photo-resist on top and bake the wafer. 3. Exposing the waveguide pattern, removing the exposed photo-resist chemically and baking the wafer to stabilize the remaining photo-resist. 4. Wet etching the exposed SiO2 to form the profile of waveguides. 5. Dicing the wafer to a few waveguide chips. 6. Applying the proton exchange, annealing and reverse proton exchange to form the waveguides.

60

4 Quantum Frequency Conversion

Fig. 4.5 Waveguides fabrication process. Figure from [7]

The most critical steps in waveguide fabrication are proton exchange, annealing and reverse proton exchange. We briefly explain how they work here. Proton exchange (PE) is to replace lithium ions in the crystal with hydrogen ions (i.e., protons). The refractive index of the replacement region rises by about 0.1 compared to the lithium niobate background, forming a waveguide structure. However, the protons are mainly concentrated on the surface of the crystal, and the crystal loses its nonlinearity in this region. Then we anneal the crystal to diffuse the protons at the surface to a deeper level so that the concentration of protons is diluted and the nonlinearity is gradually restored in this region. Although the protons diffuse into the deep layer a bit after annealing, the maximal density still appears at the surface. We then perform the reverse proton exchange (RPE) [9], which is to use the high-temperature melt containing lithium ions to exchange with the protons at the surface so that the concentration of protons at the surface decreases, the highest refractive index point appears in the middle of the crystal, and the symmetry is greatly improved. Meanwhile, the internal transmission efficiency of the waveguide and the coupling with the fibre efficiency of the waveguide and the coupling with the fibre are both improved to some extent. In addition, it is essential to note that the introduction of protons increases the refractive index of the transverse magnetic mode (TM mode) and at the same time decreases the refractive index of the transverse electric mode (TE mode), meaning that our structure forms waveguides only for the TM mode. These two modes correspond to the two linear polarizations, so our waveguide chip supports only single-polarization transmission (Fig. 4.5).

4.3 QFC Scheme

61

Mode Filter S-band

Taper

{

{

{ Pump Signal

{

{

{ Taper

Directional Coupler

QPM Gratings

Fig. 4.6 Schematic of the integrated QFC chip

4.2.3 Integrated QFC Chip We designed and fabricated a more compact and functional integrated QFC chip in experiments. The scheme of the chip is shown in Fig. 4.6, consisting of two waveguides. One is a straight waveguide containing a periodic polarization region, which serves as the signal light input and frequency conversion. The other is a curved waveguide for the pump light input. The main reason for this design is that the wavelengths of the signal light and the pump light are very different, and the best coupling efficiency can be achieved by coupling the two beams through the two waveguides separately. Two mode filters with different widths at the leftmost end of the chip are designed for the two wavelengths separately, after which the optical field is adiabatically coupled to the waveguide by a tapered waveguide structure (Taper). In order to introduce the pump light from the curved waveguide into the straight waveguide, an S-shaped section (S-band) is designed to move closer to the straight waveguide gradually. The pump light is evanescently coupled into the straight waveguide using a Directional Coupler [10].

4.3 QFC Scheme There are two commonly used transition lines for 87 Rb atoms, the D1 line with a wavelength of 795 nm and the D2 line with a wavelength of 780 nm. These two wavelengths correspond to a fibre loss of about 3.5 dB/km. The communication band window with the lowest loss in fibre is the C-band (conventional band): 1530– 1565 nm, where the lowest loss point is at 1550 nm, with a typical loss value of 0.2 dB/km; followed by the O-band (original band): 1260–1360 nm, with the lowest loss point at 1310 nm, with a typical loss value of 0.3 dB/km. It can be seen that the 87 Rb transition lines have shorter wavelengths than the target fibre communication bands. We need to convert a high-energy photon to a low-energy one. Therefore we choose the differential frequency effect for frequency conversion, and the following energy conservation conditions need to be fulfilled,

62

4 Quantum Frequency Conversion

1 1 1 − = . λ1 λ2 λ3

(4.25)

We consider the conversion schemes as follows: 1. 2. 3. 4.

D1 line to C band, λ1 = 795 nm, λ3 = 1550 nm, hence λ2 = 1632 nm; D1 line to O band, λ1 = 795 nm, λ3 = 1310 nm, hence λ2 = 2022 nm; D2 line to C band, λ1 = 780 nm, λ3 = 1550 nm, hence λ2 = 1570 nm; D2 line to O band, λ1 = 780 nm, λ3 = 1310 nm, hence λ2 = 1927 nm.

As described in chap. 3, our existing cavity-enhanced quantum storage is designed based on D1 lines, so we prioritise schemes 1 and 2. In order to get a high conversion efficiency, the pump optical power needs to be about in the order of a hundred milliWatts. At around 2000 nm, the mature industrial solution exists for high-power, narrow linewidth commercial laser products, while it is the opposite at 1600 nm. Our test shows that the transmission loss in the O-band keeps constant. Therefore we modified scheme 2 slightly and chose the standard industrial 1950 nm laser as the pump laser. In addition, a commercial solution exists for the 1570 nm pump laser in option 3 because it is close to the Erbium transition lines in fibre. In the meantime, the target wavelength of 1550 nm has a lower transmission loss, making this scheme more favourable. In fact, the Riedmatten group [11–13] is using this scheme. We did not choose this scheme based on several considerations. First, modification of our existing D1 line-based quantum memory is costly, and the 0.1dB/km difference is not significant in the distance range of several tens of kilometres that we are currently interested in, which makes scheme 3 not cost-effective. In addition, the high-power pump laser will introduce a significant broad-spectrum noise in the medium, from the amplified spontaneous emission of the laser and from the Raman noise in the medium, both of which have a much higher density around the laser frequency. In scheme 3 there is only a 2.5 THz difference between λ2 and λ3 , while in scheme 3 the frequency difference is 69.7 THz. Therefore, as the first attempt at combining quantum memories and the quantum frequency conversion, we prefer scheme 2 with less setup modification and simple noise filtering.

4.4 QFC Setup and Characterization We package the QFC chip front end directly with a 780HP single-mode fibre from Nefern for 795 nm signal light and a 1550 nm single-mode fibre for 1950 nm pump light. Free space coupling is used for the back end in favour of noise filtering. In addition, the nonlinear conversion efficiency in the waveguide is temperature-dependent. Thus we stable the waveguide temperature at ±0.1 ◦ C using an external temperature controller. The noise generated by the frequency conversion process has two main parts: the 1950 nm pump light itself, and its frequency doubles and triples with wavelengths

4.4 QFC Setup and Characterization

1950nm Pump 795nm Signal

63

SNSPD

Lens

PPLN-WG Att.

PC

Temp Control

DM

LP BP

Fig. 4.7 The test setup for the QFC characterization

of 975 nm and 650 nm, respectively. The other is the Raman noise generated by the 1950 nm pump light in the waveguide, which is from the photon-assisted Raman Scattering in the waveguide. The spectral distribution of Raman noise is almost uniformly distributed in far-off-resonance regions. In our experiments, we use two dichromatic mirrors (DM) to first filter out the high-power 1950 nm pump laser noise and then use a long pass filter (LP) with a cutoff wavelength of 1150 nm to filter out frequency doubling and tripling noise. For the broad Raman noise, we used a bandpass filter (BP) with a bandwidth of 5 nm to filter it out. In addition, this BP filter has a large optical density (OD > 6) in the 1950 nm band to further filter out the residual 1950 nm noise. In our experiments, we found that the pump laser has amplified spontaneous emission (ASE) broad-spectrum noise, which cannot be filtered after the conversion. To eliminate the ASE noise, we customize a fibre filter with a high cutoff at 1342 nm band and plug it directly after the pump laser. Then we characterize the waveguide performance by a test setup as shown in Fig. 4.7. We first measure the conversion efficiency and noise as functions of the pump power. The conversion efficiency in the figure includes only the waveguide part, without filtering and fibre coupling. The collection efficiency of the collimating lens is estimated to be 93%. The transmission of the filters are, DM mirror ∼ 98%, long-pass filter ∼ 98%, and band-pass filter ∼ 95.5%. In total, we have a transmission of about 80% to 84% (different waveguides have different Raman noise strengths, and sometimes two bandpass filters are required). The coupling efficiency from free space to the fibre is ∼ 60% (following tests show that the coupling efficiency can be improved to ∼ 75% by using a smaller focal length fibre collimator in combination with an anti-reflection coating for the fibre). Thus, for the QFC chip with the best conversion efficiency of 70%, the end-to-end efficiency of the whole QFC module is 33%, which has the best performance in all telecommunication interfaces for quantum memories to out knowledge (after this work was published, the group of H. Weinfurtur in Germany achieved an end-to-end efficiency of greater than 50% [14]). In addition, we are also interested in whether the statistical properties of single photons are changed after the conversion. Hence we perform the Hanbury-BrownTwiss experiment before and after conversion for the write-out photon triggered by the readout photon (photons from the DLCZ memory introduced in chap. 3). A comparison of the unconverted (blue) and converted (red) results at the write-out photon probability χ = 0.057 is shown in Fig. 4.8b. It can be seen that the frequency conversion does not affect the statistical property of the write-out photon.

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4 Quantum Frequency Conversion

Fig. 4.8 QFC characterization. a The conversion efficiency and the signal-to-noise ratio (SNR) as functions of the pump laser power. b The autocorrelation function of a single photon before (blue) and after (red) the QFC

References 1. Boyd RW (2019) Nonlinear optics. Academic press 2. Maker P, Terhune R, Nisenoff M et al (1962) Effects of dispersion and focusing on the production of optical harmonics. Phys Rev Lett 8(1):21 3. Armstrong JA, Bloembergen N, Ducuing J et al (1962) Interactions between light waves in a nonlinear dielectric. Phys Rev 127:1918–1939 4. Webjorn J, Laurell F, Arvidsson G (1989) Fabrication of periodically domain-inverted channel waveguides in lithium niobate for second harmonic generation. J Lightwave Technol 7(10):1597–1600 5. Weis R, Gaylord T (1985) Lithium niobate: summary of physical properties and crystal structure. Appl Phys A 37(4):191–203 6. Känzig W (1957) Ferroelectrics and antiferroeletrics. In: Solid State physics, vol 4. Elsevier, pp 1–197 7. Ma Fei (2018) Research on the core device of quantum communication based on periodically poled lithium niobate waveguide. University of Science and Technology of China 8. Roussev RV, Langrock C, Kurz JR et al (2004) Periodically poled lithium niobate waveguide sum-frequency generator for efficient single-photon detection at communication wavelengths. Opti Lett 29(13):1518–1520 9. Parameswaran KR, Route RK, Kurz JR et al (2002) Highly efficient second-harmonic generation in buried waveguides formed by annealed and reverse proton exchange in periodically poled lithium niobate. Opti Lett 27(3):179–181 10. Chou M, Hauden J, Arbore M, et al (1998) 1.5-μm-band wavelength conversion based on difference-frequency generation in LiNbO 3 waveguides with integrated coupling structures. Opti Lett 23(13):1004–1006

References

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11. Fernandez-Gonzalvo X, Corrielli G, Albrecht B et al (2013) Quantum frequency conversion of quantum memory compatible photons to telecommunication wavelengths. Opti Exp 21(17):19473–19487 12. Farrera P, Maring N, Albrecht B et al (2016) Nonclassical correlations between a C-band telecom photon and a stored spin-wave. Optica 3(9):1019–1024 13. Maring N, Farrera P, Kutluer K et al (2017) Photonic quantum state transfer between a cold atomic gas and a crystal. Nature 551(7681):485 14. van Leent T, Bock M, Garthoff R et al (2020) Long-distance distribution of atom-photon entanglement at telecom wavelength. Phys Rev Lett 124(1):010510

Chapter 5

Remote Entanglement via the Two-Photon Scheme

In Chap. 3, we prepared the entangled states between the atomic ensembles and the photons, and in Chap. 4, we coherently converted the photon frequency to the fibre telecommunication band. The next step is to transmit photons from the two atomic ensembles to an intermediate node and establish entanglement between the two atomic ensembles by photon interference. In this chapter, we use a two-photon interference scheme to establish entanglement between two atomic ensembles connected by a 22 km field fibre.

5.1 Comparison of Two Entanglement Schemes There are two types of photon interference schemes for entanglement swapping, the single-photon scheme and the two-photon scheme. In the two-photon scheme, the qubit is encoded on the polarization or time-bin of a photon, and a two-photon interference is performed. In the single-photon scheme, the qubit is encoded as the 0 and 1 photon in the Fock space, and a single-photon interference is performed. Each of these two approaches has advantages and disadvantages. Here we give a comparison of these two schemes. The two-photon interference-based Bell state measurement was first proposed by H. Weinfurter in 1994 [1], was soon applied to quantum teleportation experiments [2], and is now one of the most commonly used tools in quantum information processing. Figure 5.1 shows a two-photon interference-based Bell state measurement setup for polarization qubits. First, we select only the cases in which each output of the first PBS has a photon so that these two photons have the same polarization state, both √ |H  or both |V . The two Bell states |±  = (|H H  ± |V V )/ 2 can be further distinguished by using a pair of |± = |H  ± |V  basis PBSs performing joint σ X measurements.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Yu, Long Distance Entanglement Between Quantum Memories, Springer Theses, https://doi.org/10.1007/978-981-19-7939-2_5

67

68

5 Remote Entanglement via the Two-Photon Scheme

The two-photon interference scheme has no requirement for phase stability of the photons. Thus only the independent frequency stabilization of lasers in each node is required. According to the analysis, the requirement for phase stability of the fibre is less than L coh /10 (L coh is the coherence length of the laser) [3]. Since the scheme requires one photon to be produced by each of the two atomic systems simultaneously, the entangling probability is a function of χ 2 . These two photons also have to reach the intermediate node simultaneously, and the equivalent photon propagation distance is the length of the fibre between the two nodes L. Together with the 50% intrinsic efficiency of the linear optics-based Bell state measurement, its theoretical entangling probability is Pent = 0.5χ 2 exp[−L/L att ] (L att is defined as the distance of 1/e propagation efficiency). The two-photon interference scheme was initially considered incompatible with DLCZ-type quantum repeaters. Because in DLCZ quantum memory, the secondorder excitation will occur in a probability of χ 2 /2, producing two write-out photons. The accidence of two second-order photons arriving at the middle station simultaneously will result in a fake response in the Bell state analyzer with a probability of 1/2. It is easy to estimate the probability of the fake signal as 0.5χ 2 exp[−L/L att ], the same as Pent . Hence a successful Bell state detection, in fact, heralds a mixture of a maximally entangled state and higher-order excitations in one ensemble. Furthermore, the high-order excitations will accumulate through the following entanglement swappings. In 2007, Zhao et al. [3, 4] and Jiang et al. [5] independently proposed that the higher-order excitation components can be removed by using an alternative type of Bell state analyzer, as shown in Fig. 5.1b. The single-photon interference scheme for remote entanglement was first mentioned in [6, 7], among others. Furthermore, it was introduced to the atomic ensemblebased quantum repeater by Lu-Ming Duan et al., who developed a whole set of methods for establishing a quantum repeater using this scheme [8]. We discussed the atom-photon entanglement generated in the DLCZ memory in Sect. 2.2. Following this, we can prepare two pairs of such entanglements and interfere with their writeout field through a beam splitter as shown in Fig. 5.1c. The operation of the BS can be described as (5.1) a L† → d R† + id L† , a †R → d L† + id R† . a.

b.

c. L

R

BS

PBS

dL

dR

PBS ±

Fig. 5.1 Entanglement swapping setup through the two-photon interference (a, b) and the singlephoton interference (c)

5.1 Comparison of Two Entanglement Schemes

69

where a L† and a †R are the photon generation operators at the left and right incident ports, respectively, and d L† and d R† are the photon generation operators at the left and right detection ports. In turn, we can deduce that the incident light modes corresponding to the left and right detection ports are, d L† → a †R − ia L† , d R† → a L† − ia †R .

(5.2)

The photon states at the inputs corresponding to the left and right detectors’ click read, 1 (a †R − ia L† )|vac = √ (|0 L 1 R  − i|1 L 0 R ) 2 (5.3) 1 (a L† − ia †R )|vac = √ (|1 L 0 R  − i|0 L 1 R ) 2 These are the two orthogonal Bell states in the Fock state space, so we can use the device to perform an entanglement swapping, entangling two distant atomic ensembles. In this process, only one photon physically reaches the middle node, so we call this a single photon interference. This photon may come from either the left or the right atomics ensemble. The BS eliminate the which-path information of the arriving photon. Since only one photon is generated throughout this scheme, the entangling probability Pent is a linear function of the excitation rate χ . The fibre transmission loss in this scheme is the square root of that of the whole length L. Analysis [9] shows that the theoretical entangling rate of this scheme is Pent = 2χ exp[−L/(2L att )]. However, after the successful establishment of entanglement at neighbouring nodes, the vacuum part of the entanglement will accumulate in the following entanglement swappings steps. In order to eliminate the vacuum parts, Duan et al. [8] proposed to use two parallel chains of atomic ensembles. One can filter out the vacuum component by combining the Fock state entanglement on each chain. The single-photon scheme has a higher entangling rate than the two-photon scheme under the same condition, but it also has some problems. The first one is that the event that both ensembles are excited and generate photons is detrimental to the final entangled state. Unlike the higher-order excitation issue in the two-photon scheme, which can be addressed by introducing nonlinearity like Rydberg interactions, this problem is of the single-photon scheme itself. It cannot be solved even if an ideal single photon source is used, so we have to have the excitation rate χ low enough. Second, single-photon interference is very sensitive to the photon phase, including the initial phase of the writing laser during the write process and the path phase during photon transmission. Analysis shows that the phase jitter of the fibre needs to be below λ/10 [3]. The scheme requires not only high-frequency coherence but also good phase correlation for the lasers in the two nodes, and these requirements are not a small challenge at the current state of the art. In Table 5.1, we compare two schemes. The two-photon scheme generally has a lower entangling probability but a lower experimental difficulty, while the single-photon interference scheme has a higher entangling probability but a greater experimental difficulty. In this chapter, we

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5 Remote Entanglement via the Two-Photon Scheme

Table 5.1 Comparison between single-photon and two-photon schemes Scheme Two-photon [3] Single-photon [8] Entangling probability, Pent Fiber stability Laser locking Noise source Noise suppression

0.5χ 2 exp[−L/L att ] ∼ L coh /10 Independent frequency locking Vacuum state Two types of BSMs

References

[10–12] [16–18]

2χ exp[−L/(2L att )] ∼ λ/10 Phase locking Higher-order excitation Two chains of atomic ensembles [13–15] [19–21]

first use the easier-to-implement two-photon scheme to establish long-range entanglement between two atomic ensembles. The more difficult single-photon interference scheme will be tried in the next chapter to extend the entanglement distance further.

5.2 Experiment Setup The setup for the two-photon scheme experiment is shown in Fig. 5.2, where we build two quantum memories described in Chap. 3 in nodes A and B. After the atom-photon entanglement is established in each node, we convert the write-out photon from 795 nm to 1342 nm nm and send them to the middle station for the joint Bell state measurement. Before the frequency conversion, we convert the photons from polarization encoding to time-bin encoding. Time-bin encoding, on the one hand, circumvents the polarization selection of the PPLN waveguides; on the other hand, it is more favourable for long-distance transmission due to its decoherencefree characteristic. The Bell state measurement of the time-bin encoding photons is implemented by a 50:50 beamsplitter. Assuming that the two output ports of the BS are Da and Db , the two Bell states that it can resolve, and the results of its measurements are | +  → ({E, Da } & {L , Da }) || ({E, Db } & {L , Db }) , | −  → ({E, Da } & {L , Db }) || ({E, Db } & {L , Da }) .

(5.4)

The content in curly brackets indicate one of the four types of detector clicks; two click event linked by the logical symbol and and enclosed by parentheses indicate a coincidence; the || symbol indicates the or relationship. On the successful measurement of one of the above two Bell states, the atoms in both nodes are projected to, 1 | ± TPI = √ (|↑ A |↓ B ± |↓ A |↑ B ) . (5.5) 2

5.3 The Degree-of-Freedom Conversion of Photon Qubit

71

SNSPD

Middle Station Da PPLN-WG LP BP

Db BS

PBS

DM

QWP HWP

PC Node-A

Node-B

Fig. 5.2 Schematic of the remote entanglement generation between atomic ensembles. Two quantum memory nodes (Node-A and Node-B in one lab) are linked by fibres to a middle station for photon measurement. In each node, a 87 Rb atomic ensemble is put inside a ring cavity. All atoms are prepared in the ground state at first. We first create a local entanglement between an atomic ensemble and a write photon by applying a write pulse (blue arrow). Then the write-out photon is collected along clockwise (anticlockwise) cavity mode and sent to the QFC module. With the help of a PPLN waveguide chip (PPLN-WG) and a 1950 nm nm pump laser (green arrow), 795 nm writeout photon is converted to the telecom O band. The combination of a half-wave plate (HWP) and a quarter-wave plate (QWP) helps to couple with the TM-polarized mode of the waveguide. After noise filtering, two write-out photons are transmitted through long fibres, interfered in a BS and detected by two SNSPDs with efficiencies of about 50% at a dark count rate 100 Hz. The effective interference in the middle station heralds two ensembles entangled. Fibre polarization controllers (PCs) and polarization beamsplitters (PBSs) before the interference BS is designed for actively compensating polarization drifts in the long fibre. To retrieve the atom state, we apply a read pulse (red arrow) counter-propagating to the write pulse. With the help of a phase match of spin-wave and cavity enhancement, the atomic state is retrieved efficiently into the anticlockwise (clockwise) mode of the ring cavity

5.3 The Degree-of-Freedom Conversion of Photon Qubit In our quantum memory scheme (see Chap. 3), the photon qubit is encoded on polarization. However, as described in Chap. 4, our waveguides prepared by reverse proton exchange are strongly polarization selective, allowing only one linearly polarization mode. In order to perform the frequency conversion for two polarization modes, a straightforward approach is to separate the two modes into two paths and convert them separately with two crystals. This approach requires two waveguides for each photon and double the pumped optical power and may have to sacrifice some of the conversion efficiency to balance two paths. Another approach is to convert the polarization encoding qubit into a time-bin qubit by a setup shown in Fig. 5.3, where we map two polarization modes to two paths of an asymmetric Mach-Zehnder interferometer (AMZI) and unify their polarization via a fast switching Pockesl cell. This approach is more resource economy than the previous one, and the time-bin photon is also more favourable for long fibre transmission. Two time-bin modes are equally polarized and very close in the time domain (150 ns in our experiments). Hence the phase change of the fibre is negligible on this time scale. In contrast, active feedback must be added if polarisation coding is used to suppress the phase jitter between the two polarization modes.

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5 Remote Entanglement via the Two-Photon Scheme

AMZI

kels

Poc

P

QW

P

HW

795nm

mp

Pu

DM LP BP

1342nm Fig. 5.3 Setup for polarization photon QFC. Two PBSs and a coiled polarization maintaining (PM) delay fibre constitutes an AMZI. Two orthogonal polarization components (|/|) of 795 nm photon are separated in the time domain after the AMZI, and the polarization information is actively erased by a Pockels cell. Then the time-bin encoded photon is sent to the QFC module

The long arm of the AMZI consists of a 30 m long polarization-maintaining fibre, and the write-out optical signal is coupled to the slow axis of the fibre. We know from Chap. 3 that the write-out photon width is ∼ 50 ns, corresponding to a fibre length of 10 m so that the two time-bin modes are entirely distinguishable. In order to make the two time-bin modes phase stable, we perform phase locking between the long and short arms of the AMZI. The phase locking scheme is shown in Fig. 5.4, where we placed two AOMs, one before and one after the AMZI. During the entangling trial, both AOMs are off, and the without light can pass through them. During the MOT loading, two AOMs are on. A locking beam is introduced to the AMZI from AOM1 and taken out from AOM2 . Since the polarization of the phase-locked light passing through the two arms of the AMZI is orthogonal, we place a polarizer with its axis at 45◦ to interfere with two modes. The linewidth of the phase locking beam is ∼1 MHz, corresponding to a coherence length of ∼300 m. Therefore the two parts passing through the long and short arms are still coherent. The locking signal is measured by a fast photodiode. After low-pass filtering, the measured signal is fed to a PID module. The feedback signal is applied to a piezoelectric ceramic to change the long arm length slightly. The phase locking light has the same frequency as the ring cavity locking light to avoid influencing atoms. Note that although two idler ports exist for the AMZI, they cannot be used for phase locking. Because the laser beam injected from these two ports will couple to the fast axis of the PM fibre, which has no phase correlation with the slow axis for the signal photons.

5.4 Field Deployed Fibres

73

Fig. 5.4 The setup for the phase locking of the AMZI

5.4 Field Deployed Fibres There are six G.652 fibres deployed by China Unicom between the east campus of University of Science and Technology of China (USTC, N 31◦ 50 6.96

, E 117◦ 15 52.07

) with the Software Park (N 31◦ 51 6.01

, E 117◦ 11 54.72

) in the HighTech Zone of Hefei City. The crow-fly distance between two places is about 6.4 km, and the length of each fibre is about 11 km. We choose two fibres with the lowest loss of about 4 dB in the O-band for write-out photon transmission. Both quantum memories are located in the same laboratory in the USTC, and the middle station is a location in the Software Park. We use another independent fibre other than the six fibres deployed by China Mobile for building a local area network (Fig. 5.5). The main control unit is placed in the laboratory in USTC. Hence we send back the click signal of SNSPDs to the USTC for data analysis. We do this by using a homemade photo-electro conversion module. On the sender side, the SNSPD output is used to drive a 1550 nm DFB laser, generating a corresponding laser pulse. On the receiver side, a commercial photodiode is used to convert the laser pulse back to an electronic pulse (Fig. 5.6).

N Software Park 11km 500m

Fig. 5.5 Birds-eye view of the remote entanglement experiment over the field fibre

USTC

74

Probability

a 0.3 0.2 0.1 0.0 0.0

0.5

b Noise(Hz)

Fig. 5.6 Benchmarking the field-deployed fibres. a Probability distribution of the reflectivity for the polarization filtering PBS (shown in Fig. 5.2 after long fibres), with active compensation. The data shown was recorded once per second and accumulated during 24 h. b Background noise in SNSPD during 24 h

5 Remote Entanglement via the Two-Photon Scheme

1.0 1.5 2.0 Reflectivity(%)

2.5

3.0

300 200 100 18:00

00:00

06:00 Time

12:00

18:00

Another question of concern in the deployed fibre experiments is the polarization stability of fibres. The Bell state analyzer at the middle station works based on indistinguishable polarizations of two photons. To ensure this, we use PM fibre-based beam splitter for photon interference and a PM fibre-based PBS on each input port of the beamsplitter for polarization filtering. Although polarization filtering ensures photon polarization indistinguishability, it works at the cost of losing efficiency when polarization drifts a lot in the fibre. To minimize the loss, we perform active polarization feedback by monitoring photon counts in the idle port of each PM PBS and minimizing the count rate of each of the two idle ports by adjusting the electric polarization controller (EPC) before the PBS. Since the polarization perturbation is relatively slow, we perform the feedback every second. The counting and feedback are performed using the multi-functional module PCIe-7833R from National Instruments.

5.5 Optimization of the Experiment Condition We first perform a simplified entanglement experiment between the two memories without frequency conversion and the long fibre transmission. We measure the entanglement fidelity as a function of the write-out probability. The results are shown in Fig. 5.7 with an optimal fidelity F = 0.842 ± 0.080 when χ = 2%. The DLCZ memory is a probabilistic source. Similar to the spontaneous parametric down-conversion source [22], a main source of the infidelity is the accidental coincidence introduced by higher-order as well as other unwanted modes of photons. Let us try to deduct their contribution and observe the result. In the experiment, the four-body coincidence rate from the expected part reads,

5.6 Results for Remote Entanglement

1.00 0.75 Fidelity

Fig. 5.7 Average fidelity of the remote entanglement |Ψ ± t pi generated locally as a function of χ. Blue (square) dots refer to the measurement result. Red (triangle) dots show the corrected results through deduction of accidental coincidences (see Supplementary Information)

75

0.50 0.25 0.00 0

2

Pexp = 0.5 pw2 ηr et ,

4

6 χ (%)

8

10

(5.6)

where pw is the measured write-out probability, equaling the intrinsic write-out probability multiplied by the optical path efficiency and the detection efficiency. We consider the following two main noisy coincidences. The first happens after a successful Bell state measurement; in the read process, one of the real read-out photons is missing; instead, an irrelevant photon is detected in that read-out photon mode. The second one happens when a higher-order excitation occurs and contributes a fake Bell state measurement success, an irrelevant photon is detected on the readout mode of the idle quantum memory. The probability of these two types of noises reads,   pw 2 Pacc = 0.5 pw2 ηr et (1 − ηr et ) pr × 2 + 0.5 √ [2ηr et (1 − ηr et )] pr × 2, (5.7) 2 where pr is the measured read probability. The noise has no quantum correlation, so it contributes randomly to each possible outcome in the fidelity measurement. We subtract their contribution to the fidelity and plot the result as the red points in Fig. 5.7, where we can see that the corrected fidelity does not vary with the write-out probability χ .

5.6 Results for Remote Entanglement Based on the above efforts, we perform the remote entanglement experiment with the long-deployed fibre transmission. When the frequency conversion and the long fibre transmission are introduced, without changing any other setting, the experimental count rate is lowered to 1–2 coincidences per hour. Therefore we increase the writeout probability to χ = 3.8%, which increases the count rate by a factor of about four.

5 Remote Entanglement via the Two-Photon Scheme

Normalized coincidences

76

1.00 + |ψtpi

0.75 0.50 0.25 0.00

|ψtpi

0

2

4

6

δt (µs) Fig. 5.8 Normalized coincidences measured in the |± = |↑ ± |↓ basis for the two atomic qubits. The Raman pulse in Node-A is applied slightly later than Node B with an offset of δt, which induces a linearly changing phase in |Ψ ±  and results in the observed oscillations. Parallel correlations (|+|+ or |−|−) of |Ψ +  (blue squares) and |Ψ −  (red triangles) are shown. Solid (red) and dashed (blue) lines correspond to the fitting results. The 5.4 μs oscillation period agrees with Zeeman splitting between |↑ and |↓. This plot is based on 2.9 × 104 heralding events during a total measurement time of 487 h over 30 d. The error bars represent one standard deviation

To measure the entanglement fidelity, we measured the visibility of two different sets of observables. For the | +  state, the visibility V1 of σz σz is 68.4 ± 0.075; for the | −  state, V1 = 0.635 ± 0.075. By performing a Raman π/2 operation on both atomic ensembles before reading, the visibility V2 of σx σx can be measured. Changing the delay δt of the Raman beam on both sides introduces a phase shift to the remote entanglement. We change δt and observe the oscillation of coincidence when measuring σx σx . The results of the parallel coincidence (|+|+ or |−|−) oscillations as functions of δt are shown in Fig. 5.8 for | +  and | − . By fitting the results we get that for | + , V2 = 0.574 ± 0.064, and for | − , V2 = 0.647 ± 0.066. This plot was measured for 30 d, with an effective measurement time of 487 h, during which a total of 2.9 × 104 entanglement heralding signals were recorded. Based on the above visibility measurements, we can calculate the entanglement fidelity. For | + , F  (1 + V1 + 2V2 )/4 = 0.708 ± 0.027 and for | − , and F = 732 ± 0.038.

5.7 Imperfection Analysis The homogeneity of two photons is critical to the remote entanglement. In order to check the homogeneity, we use the read-out photon as the trigger and perform the Hong-Ou-Mandel (HOM) experiment on the write-out photons from two systems. We scan the photon homogeneity by changing the polarization of one photon with an HWP. A HOM dip is observed as shown in Fig. 5.9. From this, we know the

5.7 Imperfection Analysis

77

Fig. 5.9 Hong-Ou-Mandel result of triggered write-out photons

HOM visibility is Vwo = Nmin /Naverage = 0.082, Nmin and Naverage representing the minimum counts at an angle of 0◦ and the average counts at an angle of 90◦ , respectively. In an ideal HOM experiment, the interference of two photons can be described by the light field generation operator. aU† b†D → (aU† + ia †D )(b†D + ibU† ),

(5.8)

where a † , b† represent the corresponding write-out photon generation operators from the two atomic ensembles and U , D represent the two paths of BS. Let us write b† by a † and the generation operator of its orthogonal subspace a˜ † as b† = αa † + β a˜ † . Then Eq. 5.8 can be written as aU† b†D → α(aU† + ia †D )(a †D + iaU† ) + β(aU† + ia †D )(a˜ †D + i a˜ U† ),

(5.9)

where the term with coefficient α corresponds to perfect HOM interference and gives no coincidence, and the term with coefficient β corresponds to the interference of two distinguishable photons and gives coincidences with a probability 1/2. Accordingly, the HOM visibility can be calculated as V = β 2 /2. Let us consider the Bell state measurements again. The Bell state measurement corresponds to the following two-body projection, ± ∓ ∓ Sˆ ± = (1 − λ)| ± pp   pp | + λ| pp   pp |,

(5.10)

√ where  ± = (|E L ± |L E)/ 2 are two Bell states and the coefficient λ is used to measure the measurement imperfection. The first term corresponds to an ideal Bell state measurement that gives the correct result. The second term gives the wrong result, and this setup does not give the response of the other two Bell states. We represent the two-body Bell state in terms of the annihilation generating operator as follows,

78

5 Remote Entanglement via the Two-Photon Scheme † † † † | ± pp  = a E b L ± a L b E |vac   = α(a †E a L† + a L† a †E ) ± β(a †E a˜ L† + a L† a˜ †E ) |vac.

(5.11)

One can see that the first term corresponds to an ideal two-photon interference, while the second term corresponds to the interference of two distinguishable photons. Comparing Eqs. 5.10 and 5.11 we get λ = β 2 /2 = V . Based on this, we can calculate the entangled state between atoms after the Bell state measurement and its fidelity as follows, AB ρaa =

A B T r pp [(1aa ⊗ Sˆ ± )ρap ρap ] , A ρB ] T r [(1aa ⊗ Sˆ ± )ρap ap

± ± AB  aa |)ρaa ]. F = T r [(|aa

(5.12) (5.13)

A B and ρap are two atom-photon entanglements from two atomic ensembles, respecρap tively, and the subscripts a and p represent atoms and photons. Note that the measured result of the HOM is also affected by higher-order excitation instances, which are already taken into account in the state tomography measurements, so we correct the

= 0.063. Based on this, the remote entanglement fidelity can HOM result to Vwo be estimated as F = 0.835, which is basically in agreement with the experimental results.

References 1. Weinfurter H (1994) Experimental bell-state analysis. EPL (Europhys Lett) 25(8):559 2. Pan JW, Bouwmeester D, Weinfurter H et al (1998) Experimental entanglement swapping: entangling photons that never interacted. Phys Rev Lett 80(18):3891–3894 3. Zhao B, Chen ZB, Chen YA et al (2007) Robust creation of entanglement between remote memory qubits. Phys Rev Lett 98(24):240502 4. Chen ZB, Zhao B, Chen YA et al (2007) Fault-tolerant quantum repeater with atomic ensembles and linear optics. Phys Rev A 76(2):022329 5. Jiang L, Taylor JM, Lukin MD (2007) Fast and robust approach to long-distance quantum communication with atomic ensembles. Phys Rev A 76:012301 6. Cabrillo C, Cirac JI, García-Fernández P et al (1999) Creation of entangled states of distant atoms by interference. Phys Rev A 59:1025–1033 7. Bose S, Knight PL, Plenio MB et al (1999) Proposal for teleportation of an atomic state via cavity decay. Phys Rev Lett 83:5158–5161 8. Duan LM, Lukin MD, Cirac JI et al (2001) Long-distance quantum communication with atomic ensembles and linear optics. Nature 414(6862):413–418 9. Sangouard N, Simon C, de Riedmatten H et al (2011) Quantum repeaters based on atomic ensembles and linear optics. Rev Modern Phys 83:33–80 10. Hensen B, Bernien H, Dréau AE et al (2015) Loophole-free bell inequality violation using electron spins separated by 1.3 kilometres. Nature 526(7575):682–686 11. Yuan ZS, Chen YA, Zhao B et al (2008) Experimental demonstration of a BDCZ quantum repeater node. Nature 454(7208):1098–1101

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12. Hofmann J, Krug M, Ortegel N et al (2012) Heralded entanglement between widely separated atoms. Science 337(6090):72–75 13. Chou CW, de Riedmatten H, Felinto D et al (2005) Measurement-induced entanglement for excitation stored in remote atomic ensembles. Nature 438(7069):828–832 14. Chou CW, Laurat J, Deng H et al (2007) Functional quantum nodes for entanglement distribution over scalable quantum Networks. Science 316(5829):1316–1320 15. Humphreys PC, Kalb N, Morits JPJ et al (2018) Deterministic delivery of remote entanglement on a quantum network. Nature 558(7709):268–273 16. Nölleke C, Neuzner A, Reiserer A et al (2013) Efficient teleportation between remote singleatom quantum memories. Phys Rev Lett 110(14):140403 17. Moehring DL, Maunz P, Olmschenk S et al (2007) Entanglement of single-atom quantum bits at a distance. Nature 449(7158):68–71 18. Stephenson L, Nadlinger D, Nichol B et al (2020) High-rate, high-fidelity entanglement of qubits across an elementary quantum network. Phys Rev Lett 124(11):110501 19. Slodiˇcka L, Hétet G, Röck N et al (2013) Atom-atom entanglement by single-photon detection. Phys Rev Lett 110(8):083603 20. Delteil A, Sun Z, Gao WB et al (2015) Generation of heralded entanglement between distant hole spins. Nat Phys 12(3):218–223 21. Kalb N, Reiserer AA, Humphreys PC et al (2017) Entanglement distillation between solid-state quantum network nodes. Science 356(6341):928–932 22. Tittel W, Brendel J, Zbinden H et al (1998) Violation of Bell inequalities by photons more than 10 km apart. Phys Rev Lett 81(17):3563

Chapter 6

Remote Entanglement via the Single-Photon Scheme

In Chap. 5, we established a remote entanglement using the two-photon scheme. To improve the entangling rate and extend the entanglement distance, in this chapter, we try to long-range entanglement by using the single photon scheme.

6.1 Experimental Setup In the write process of our DLCZ quantum memory scheme, if only the |σ −  component of the write-out photon is selected, we prepare an atom-photon entanglement in the Fock space as described by Eq. 2.25. Performing the single-photon interference of the write-out field of two atomic ensembles on a beamsplitter in the middle station, which erases the path information of the photon, we prepare two atomic ensembles to an entangled state in the Fock space as,  1  | ± SPI = √ |0 A |1 B ± eiφ |1 A |0 B , 2

(6.1)

where φ is the phase difference between the two write-out fields at the time of interference. Since the single photon scheme requires only one photon arriving at the middle station for building remote entanglement, the entangling rate is Pent ∼ 2χ . In the meantime, the effective photon transmission length is half of the total length, so √ the total transmission loss is η L/2 = η L . These two factors make the single photon scheme outperform the two-photon scheme in terms of the entangling rate. However, the single photon scheme also poses many challenges to the experimental techniques, especially the demand of keeping the phase φ stable during the experiment. The phase fluctuation, on the one hand, comes from the laser frequency fluctuation, and on the other hand, it comes from the photon propagation phase fluctuation in long fibres. In order to solve the former problem, the frequency and phase locking between remote lasers are needed. In this thesis, we did not focus on this challenge and just multiplexed the same lasers for both quantum memories. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Yu, Long Distance Entanglement Between Quantum Memories, Springer Theses, https://doi.org/10.1007/978-981-19-7939-2_6

81

82

6 Remote Entanglement via the Single-Photon Scheme WG1

t

LTel1

ou

rite

W

PBSw

L

W rit e

BSRO

ro2

Re ad

LP1

LR1

L ro1

Dc Dd

L WO1

MOT1

LW1

ad Re t -ou

LW2

PD1

Da

Pump LR2

BSR

BSP

BSWO

LP2

MOT2

PD2

Db

L

WO 2

LTel2

WG2 PBS

BS

HWP

FBS

FPBS

FC

PC

PD

APD

Fig. 6.1 Setup for single photon scheme-based remote entanglement experiment

To solve the later problem, we design and implement a two-level phase locking scheme, ensuring a stable phase environment for building the remote entanglement. The experimental setup is shown in Fig. 6.1. The write process part is essentially the same as in the two-photon scheme, except that the qubit conversion process is no more needed. The local measurement for atomic qubits in the Fock space is lacking. To verify the atomic entanglement, we interfere with the read-out mode to detect the entanglement phase. The details are described in Sect. 6.3.

6.2 Phase Locking 6.2.1 Phase Locking Scheme One of the difficulties of the single photon experiment is to assure phase is stabilized between write beams in write process, read beams in retrieval process, pump lasers in QFC process, write-out fields along long fiber channels and read-out fields in measurements. Here we prove that all these problems can be solved by stabilizing two interferometers. As depicted in Fig. 6.1, we label the distance from PBSW to MOT1 (MOT2 ) as L W 1 (L W 2 ), distance from BSR to MOT1 (MOT2 ) as L R1 (L R2 ) and distance from MOT1 (MOT2 ) to BSRO as L R O1 (L R O2 ). Similarly, we denote the distance from MOT1 (MOT2 ) to the front facet of WG1 (WG2 ) as L W O1 (L W O2 ), the distance from BSP to the front facet of WG1 (WG2 ) as L P1 (L P2 ), and the distance from the end facet of WG1 (WG2 ) to BSWO as L T el1 (L T el2 ). Still, in case of the imperfection of manufacturing, we assume the length of two PPLN waveguide chips are different, as L W G1 and L W G2 .

6.2 Phase Locking

83

First, we observe the phase evolution in the entanglement creation process. Starting from write beam being split in PBSW , we can write the phase of two write beams at two atomic ensembles as: φW i =

LWi , i = 1, 2. λW

(6.2)

λW is the wavelength of write beams. Subsequently, they interact with atomic ensembles, and Raman scattering happens. Afterwards, the phase of write-out fields and atoms evolve independently. We can write the phase of two write-out fields at the front facets of two waveguides as: φW Oi =

LWi L W Oi − φM O T i + , i = 1, 2. λW λW O

(6.3)

λW O represents the wavelength of write-out fields, and φ M O T i refers to a timedependent phase of atoms. Next, quantum frequency conversion happens, and the pump beams’ phase will also be introduced, considering the write-out field propagating. Assuming the conversion process always happens in the front facet of waveguides (proof of plausibility in Sect. 6.2.3), we can write the phase right before the BSWO as: φW Oi =

LWi L W Oi L W Gi + L T eli L Pi − φM O T i + + − , i = 1, 2. λW λW O λT el λP

(6.4)

λT el and λ P represent the wavelength of the telecom field after the conversion and pump beam, respectively. Similarly, we could write the phase of two read-out fields before the BSRO as: φ R Oi =

L Ri L R Oi + φM O T i  + , i = 1, 2. λR λR O

(6.5)

The difference between φ M O T i and φ M O T i  comes from the phase evolution of a certain energy level of the atom. Because the interval time between write and read process is fixed, it is obvious that φ M O T i  − φ M O T i = constant.

(6.6)

The phase condition to be fulfilled is: φW 1 + φ R1 = φW 2 + φ R2 + 2nπ,

(6.7)

where n is an integer. We find a solution to Eq. 6.7 as below, L R1 LW2 L R2 LW1 + = + + 2n  π, λW λR λW λR

(6.8)

84

6 Remote Entanglement via the Single-Photon Scheme

Write-Read:

QWP QWP

Write-out-Read-out: MOTA HWP MOTB

Write PBS Locking beam

PD

Pizeo

Read Feed back

Write-out

Read-out

Fiber

Phase shifter

BS

Pump

MOTA

BS

99:1BS

PPLN Ring cavity

Locking PBSbeam MOTB

Fig. 6.2 Configuration of phase stabilization. Phase stabilization includes two interferometers, i.e. write-read and write-out-read-out

L R O1 L W O1 L W G1 + L T el1 L P1 + + − λR O λW O λT el λP L R O2 L W O2 L W G2 + L T el2 L W G2 = + + − + 2m  π, λR O λW O λT el λP

(6.9)

where n  and m  refer to two integers. Equations 6.8 and 6.9 refer to two MachZehnder interferometers as depicted in Fig. 6.2. The first one takes PBSW and BSR as two beamsplitters of the interferometer and covers paths of write and read beams. Therefore we introduce a 795 nm locking beam from the idle port of Read-BS to detect the interference signal in the second output of the BSW and feed it back to a piezoelectric ceramic. A combination of two half-wave plates (HWPs) and a quarterwave plate (QWP) in the sandwich configuration introduce a relative phase θ between the phase-locking laser and the write beam without changing their polarization. The second interferometer takes BSWO and BSRO as two beamsplitters and covers paths of write-out and read-out photons meanwhile include the frequency conversion modules and two several-kilometre-long fibre coils. We introduce another locking Reflectivity = 99 : 1) and detect the interbeam from BSRO using an unbalanced BS ( Transmission ference signal in the BSWO in the same way. The frequency of this beam is far detuned from the resonance point of the cavity; thus, it neither enters the cavity nor interacts with the atoms. In this method, we assume the zero scale of our atomic ensemble. There still exists a little uncertainty of phase difference introduced by the non-zero scale of the ensemble. Nevertheless, we will prove that it is small enough in Sect. 6.2.3. The two ends of the second interferometer, BSWO and BSRO , cover the full path of the write and read-out, including the pump light for frequency conversion and the long fibre section included, and we introduced the phase-locked light from a PBS placed in the optical path at the back end of BSRO and the BSWO rear end with a partial reflector (reflectivity : transmittance = 99 : 1) to derive the phase-locked light, detect the interference signal with a photocell and feed it back to the piezoelectric ceramic. In the above derivation, we assume that the atomic cluster is a dimensionless point, and later we will prove that this assumption is reasonable.

6.2 Phase Locking

85

Long fiber part PD1

PD2

From QFCs BSWO

CWDM DFB

FBS 99:1BS PS1

PS2

To SNSPDs

Fig. 6.3 Configuration of assistant phase stabilization in long fibre situation

6.2.2 Suppression of Fast Phase Variation in Long Fibre Situation Although the phase stabilization method described in Sect. 6.2.1 works well in short fibre situations, it cannot sustain a good stabilization in long fibre situations. This is because, in our former stabilization protocol, phase stabilization only works in MOT loading phases but is turned off during each 2 ms experimental phase. Hence the phase randomly fluctuates during these periods. The fluctuation is small enough in the short fibre case but too significant to be accepted in the long fibre case. Therefore, we introduce an extra phase stabilization for the long fibre part as depicted in Fig. 6.3. A 1550 nm nm laser beam is led into the interferometer right behind the QFC modules and led out after BSWO with the help of coarse wave demultiplexing modules (CWDMs). Because its wavelength is far enough from our 1342 nm nm signal in the spectrum, this 1550 nm nm beam and the stabilization process could run continuously. Nevertheless, for the same reason, its stabilization result does not represent the phase we care about. It can only serve as assistance to help us to suppress the phase fluctuation when the phase locking laser is off.

6.2.3 Phase Uncertainty in PPLN Waveguide Chips and Atomic Ensembles In the DFG process, we always have the energy conservation principle 1 1 1 = + . λSignal λPump λTelecom

(6.10)

Supposing DFG happens x from the front facet of the PPLN waveguide chip as depicted in Fig. 6.4a, then the phase accumulation of optical field is

86

6 Remote Entanglement via the Single-Photon Scheme

Fig. 6.4 a Transition point uncertainty in a PPLN waveguides chip. b Exciting position uncertainty in MOT

φ=

x λSignal

=x ·( =

x

−

λPump

1

λPump L ,

+

+

L−x λTelecom

1 λTelecom

)−

x λPump

+

L−x λTelecom

(6.11)

λTelecom

which means we can always assume DFG is happening at the front facet of the PPLN waveguides chip. In the former discussion about interferometers such as Eqs. 6.8 and 6.9, we assumed the atomic ensemble as a point with no scale and was settled at a certain point. Then we can ignore the wavelength change after this point and use one laser to stabilize the phase. Now we consider a real ensemble with diameter D as depicted in Fig. 6.4b, which is around 100 µm in our system. Here, two lasers are around λ = 795 nm with δ = 6.8 GHz difference. First, we assume the location where spontaneous Raman scattering happens in the left and right edges of the ensemble. The phase difference is θ = 2π · (D/λ1 − D/λ2 ) δ = 2π D · ≈ 0.014 = 0.81◦ . c

(6.12)

It is plausible to consider that this location obeys uniform distribution in this regime; we can easily know the standard deviation of phase difference S as  S=

θ 2 ≈ 4 × 10−3 = 0.24◦ . 12

(6.13)

So, theoretically, in the perfect stabilizing condition, only 0.24◦ uncertainty of phase will be introduced.

6.3 Benchmarking the Entanglement

87

6.3 Benchmarking the Entanglement The rotation operation is unavailable for atomic qubits in Fock space; therefore, a complete set of local qubit measurements is also missing. Given this, we follow the method in the literature [1] and consider two atomic ensembles jointly. After reading out the spin waves, we consider only the case where there is at most one photon in the read-out light field of each atomic ensemble. Then the density matrix of the read-out light field reads ⎛

p00 1 ⎜ 0 ρ= ⎜ P⎝ 0 0

0 p01 d∗ 0

0 d p10 0

⎞ 0 0 ⎟ ⎟. 0 ⎠ p11

(6.14)

This density matrix is written under the Fock state representation |n A |m B , where {n, m} = {0, 1} is the photon number in the read-out field. P = p00 + p01 + p10 + p11 is the normalization factor, where p00 , p01 , p10 and p11 can be obtained directly by photon statistics. The parameter d represents the coherence between the two components in the entangled state, which can be obtained by interfering with the readout optical field at a beamsplitter, similar to the write-out entanglement exchange process. Changing the path phase allows us to observe an oscillation in the signal. By fitting the interference, we get the interference visibility Vθ , and further get d Vθ ( p01 + p10 )/2. Based on this, the Concurrence of the entanglement can be calculated [2], √ (6.15) C = max(0, 2|d| − 2 p00 p11 )/P. C >0 is a sufficient condition for witnessing the entanglement, and it is a monotonic entanglement witness that only decreases under local operation and classical communication (LOCC). By analysis, the assumption of ignoring the components with more than one photon number in each read-out field is equivalent to a LOCC. Thus the C that we measured is a lower bound for the actual value. We measured the entanglement concurrences for four cases, namely 1. 2. 3. 4.

without QFC, L = 10 m; with QFC, L = 10 m; with QFC, L = 10 km; with QFC, L = 50 km.

Figure 6.5 shows the counts oscillation in the read-out signal as functions of the phase θ for aforementioned cases 1, 3 and 4. The rest of the results are listed in Table 6.1. For case 4, with QFC and 50 km fibre transmission, C = 0.088 ± 0.002 (0.077 ± 0.003), meaning that the entanglement is witnessed. Next, we exclude the influence of loss and inefficiency in the read-out process. The ideal read-out is a unit matrix mapping spin waves to read-out photons. The lossy read-out equals the following operator,

88

6 Remote Entanglement via the Single-Photon Scheme

Fig. 6.5 Characterization of the remote entanglement via SPI. When the atomic modes are retrieved as optical modes for interference, the normalized photon count in one output mode of the fibre BS oscillates as a function of the relative phase θ between the two optical modes. Da heralded events are shown in a, and Db heralded events are shown in b. Blue squares, red triangles and green dots refer to L=10 m, 10 and 50 km separately. Sinusoids with corresponding colours (solid, dashed and dotted in shape) show the fitting results. The result of 50 km is based on 1.7 × 105 heralding events during a total measurement time of 6 h over two days. The error bars represent one standard deviation

1→

 √ η1 + 1 − ηa,

(6.16)

where a is the annihilation operator and η is the efficiency of the read-out process. This allows us to establish the following relationship. ⎛

⎞ ⎛ ⎞⎛ ⎞ 1 1 − η 1 − η (1 − η)2 p00 d00 ⎜ ⎟ ⎜ d01 ⎟ ⎜ 0 η 0 η(1 − η) ⎟ ⎟ ⎜ ⎟ ⎜ p01 ⎟ . ⎜ ⎝ d10 ⎠ = ⎝ 0 0 ⎠ ⎝ p10 ⎠ η η(1 − η) 2 d11 p11 0 0 0 η

(6.17)

6.3 Benchmarking the Entanglement

89

Table 6.1 Raw data of the SPI experiment d00 d01 Non-conv. 10 m

0.6683 ± 0.0049 0.6588 ± 0.0052 0.6728 ± 0.0051 0.6963 ± 0.0049 0.7043 ± 0.0029 0.7084 ± 0.0032 0.7089 ± 0.0039 0.7287 ± 0.0042

Da Db

Conv.10 m

Da Db

Conv.10 km Da Db Conv.50 km Da Db

0.1664 ± 0.0032 0.1728 ± 0.0035 0.1623 ± 0.0034 0.1560 ± 0.0033 0.1470 ± 0.0019 0.1463 ± 0.0021 0.1480 ± 0.0026 0.1394 ± 0.0028

d10

d11

Vθ

0.1599 ± 0.0032 0.1624 ± 0.0034 0.1607 ± 0.0033 0.1440 ± 0.0032 0.1444 ± 0.0019 0.1409 ± 0.0021 0.1391 ± 0.0026 0.1282 ± 0.0027

0.0053 ± 0.0005 0.0061 ± 0.0006 0.0041 ± 0.0005 0.0037 ± 0.0005 0.0043 ± 0.0003 0.0044 ± 0.0004 0.0040 ± 0.0004 0.0038 ± 0.0004

0.8288 ± 0.0085 0.8168 ± 0.0078 0.8068 ± 0.0187 0.8194 ± 0.0215 0.6944 ± 0.0128 0.7037 ± 0.0040 0.6767 ± 0.0065 0.6787 ± 0.0125

Table 6.2 Concurrence and estimated Fidelity in different situations in SPI experiment. Craw , C p and Ca refer to concurrence calculated by raw data, subtracting detection and optical loss and subtracting retrieval loss, respectively. Fest : estimated fidelity of PME state Not conv./10 m Conv./10 m Conv./10 km Conv./50 km

Da Db Da Db Da Db Da Db

Craw

Cp

Ca

Fest

0.151 ± 0.003 0.147 ± 0.003 0.155 ± 0.006 0.144 ± 0.006 0.092 ± 0.004 0.090 ± 0.001 0.088 ± 0.002 0.077 ± 0.003

0.484 ± 0.008 0.486 ± 0.007 0.484 ± 0.015 0.438 ± 0.016 0.311 ± 0.009 0.303 ± 0.003 0.297 ± 0.005 0.257 ± 0.008

0.678 ± 0.012 0.711 ± 0.012 0.665 ± 0.020 0.581 ± 0.020 0.428 ± 0.013 0.416 ± 0.008 0.407 ± 0.008 0.348 ± 0.011

0.786 0.774 0.781 0.788 0.701 0.704 0.692 0.690

By solving this equation, we get the results net of the effect of the optical path and detector efficiency C p and net of the effect of the read process efficiency Ca , the latter directly reflecting the interatomic entanglement, and all the results are shown in Table 6.2. In the next step of the DLCZ protocol, another EME state ρC D is introduced and leads to a PME state, as shown in Fig. 6.6. So it will be helpful to estimate the Fidelity of the PME state with current data. For simplicity of analysis, we rewrite the density matrix of two ensembles in another complete set of bases {|00, | + , | − , |11} as

90

6 Remote Entanglement via the Single-Photon Scheme

Oringinal state ρAB

Reference state ρCD Fig. 6.6 Illustration of next step in the DLCZ protocol Table 6.3 All combinations in PME preparation process with ρ AB and | +  + | |00 | −  | +  |11 Probability Coef.

Right Wrong

p00 0 0

p+ 1/2 0

p− 0 1/2

p11 1/2 1/2

1 + − − ( p00 |0000| + p+ | + p  p | + p− | p  p | + p11 |1111| + A.D.), P (6.18) √ where | ±  = (|01 ± |10)/ 2 and A.D. refers to anti-diagonal terms. It is easy to get p± = (1 ± V )( p01 + p10 )/2. Four ensembles are retrieved into read-out fields simultaneously, and a BS combines two fields from one side. By registering the coincidence that only one detector clicks on each side, we capture the PME state and perform communication. We consider this process and suppose an ideal state ρC D = | +  + | first for simplicity. Here we list four items of ρ AB ⊗ ρC D and their contributions in next step in Table 6.3. The vacuum part contributes nothing to the final results due to the no coincidence it gives. Coefficient 1/2 from | +  is an intrinsic success probability of this protocol. Because of the converse phase, | −  gives rise to false coincidences. Owing to two read-out fields existing simultaneously, |11 always gives coincidences. Moreover, it contributes equally to the right and wrong parts thanks to randomness. We calculate the fidelity of the PME state as o ρ rAB =

Fρ,+ =

p+ + p11 Right = Right + Wrong p+ + p− + 2 p11

(6.19)

Considering more realistic situation that ρC D = ρ AB . We list all outputs in Table 6.4. Then the fidelity of the PME state could be calculated similarly as above. We show the simulation results in Table 6.2. Besides, the estimated fidelity Fest hardly varies whether we subtract the loss, implying a good filtering result for the vacuum part of the DLCZ protocol.

p+ p− 0 1/2

|00 |11

Probability p00 p00 p00 p+ p00 p− p00 p11 p+ p00 p+ p+ Coef. Right 0 0 0 1/2 0 1/2 Wrong 0 0 0 1/2 0 0

|00 | − 

| +  | − 

|00 | + 

| +  | + 

|00 |00

| +  |00

Table 6.4 All combinations in PME preparation process with two ρ AB s | −  |00

| −  | + 

p+ p11 p− p00 p− p+ 1/2 0 0 1/2 0 1/2

| +  |11 p− p− 1/2 0

| −  | − 

|11 |00

|11 | + 

|11 | − 

|11 |11 p− p11 p11 p00 p11 p+ p11 p− p11 p11 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2

| −  |11

6.3 Benchmarking the Entanglement 91

92

6 Remote Entanglement via the Single-Photon Scheme

6.4 Experimental Analysis We consider the influence from the following four factors: (1) imperfection of photon interference; (2) mismatch of write-out photons; (3) phase-locking instability; and (4) decrease of SNR.

6.4.1 Imperfection of Photon Interference We consider an imperfect single photon interference process similar to the TPI case. Because the write-out and read-out fields both interfere, we express them as two operators: ± = λ| ±  ± | + (1 − λ)| ∓  ∓ |, Sˆwo

Sˆr±o = λ | ±  ± | + (1 − λ )| ∓  ∓ |.

(6.20)

It is easy to get λ = Vwo and λ = Vr o = 0.074 similar to the TPI case. Vθ corresponds to the measurement result is: Vθ =

± F F ρ A ρB ] T r [( Sˆr±o − Sˆr∓o ) Sˆwo . ± T r [ Sˆwo ρ A ρ B ]

(6.21)

ρ AF and ρ BF refer to the Fock state entanglement between the write-out and read-out field, which is not known in our experiment. Hence we feed into an ideal maximally entangled state as (6.22) ρ AF = ρ BF = | +  + |.

Fig. 6.7 Hong-Ou-Mandel experiment result of the read-out photons

6.4 Experimental Analysis

93

SNSPD counts

400

300

200

100

0

0

25

50

75

100

125

Relative arriving time (ns)

Fig. 6.8 Time domain waveform of the write-out field from two different MOTs. Dots for counts in SNSPD and curves for their fittings

After subtracting multiple excitations relatedly counted in write-out and read-out HOM, the simulation result shows that for this ideal state, Vθ = 0.827, which is similar to the non-conversion case as listed in Table 6.2.

6.4.2 Write-out Photon Mismatch In the entanglement building process, the write-out fields need to be calibrated to have the same arriving time. We adjust the difference of two optical paths’ lengths to achieve it. Via accumulating counts in SNSPD, we construct the shape of write-out fields and compare them. As depicted in Fig. 6.8, there exist a mismatch of 2.10 and 1.45 ns for 10 and 50 km situation, which will bring 5.8 × 10−3 and 3.0 × 10−3 decrease to Vθ separately.

6.4.3 Phase Instability We probe the phase instability in the long fibre situation by monitoring the phase stabilized laser. We separately deduce 8.3◦ and 13.4◦ fluctuation for 10 and 50 km situations through Gaussian fit. Figure 6.9 shows the statistic results of 50 km. Regarding phase fluctuation as a small disturbance we consider disturbed state ρ AB , + −  = |01 + eiδθ |10 and |δθ = in which | +  and | −  are transferred to |δθ |01 − eiδθ |10. Taking phase disturbance into account, we have

94

6 Remote Entanglement via the Single-Photon Scheme

Fig. 6.9 The phase distribution of a 50 km fibre. Blue and yellow correspond to the locking and unlocking cases, respectively. The dashed line is the Gaussian fitting for the locking data

Vθ =

+ +  − − δθ |ρ AB ) − tr(|δθ δθ ||ρ AB ) max − min tr(|δθ = + +  − − max + min tr(|δθ δθ |ρ AB ) + tr(|δθ δθ ||ρ AB ) +∞ ( p+ − p− ) · −∞ f (δθ ) cos(δθ )dδθ = p+ + p− + p11 = Vθ · C ph,

(6.23)

+∞ where C ph = −∞ f (δθ )cos(δθ )dδθ is a coefficient introduced by phase fluctuation, in which f (δθ ) is the Gaussian probability density function of phase distribution. Through calculation, we get C ph = 0.989 and 0.973 for 10 and 50 km situation.

6.4.4 Decrease of SNR Along with the write-out field being attenuated in the long-distance situation, the noise of phase stabilization and dark counts of SNSPD introduce more disturbance. In 10 and 50 km situations, we get about 15 : 1 and 4.5 : 1 signal to noise (SNR) ratio in SNSPD between with and without write-out field input. First, in consideration of the definition of Vθ ≡ (max − min)/(max + min), random noise will contribute to max and min term equally. Therefore, we have disturbed the visibility pmax + pnoise − pmin − pnoise Vθ = pmax + pnoise + pmin + pnoise (6.24) cv = Vθ · . cv + cn

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c refers to the coincidence probability with subscript denoting its source, v to the valid combination, and n to unwanted ones from noise. They could be estimated as: cv ≈ 2 pwo ηret , cn ≈ 2 pn · 2 pr .

(6.25)

pwo is the tested probability of write-out photons in SNSPSD. pn = pwo /S N R is the noise in SNSPD. pr is the probability of read-out photons. After calculations, we know that Vθ suffers 0.003 and 0.01 decrease for 10 and 50 km situations. Second, the extra noises contribute to more vacuum parts in the remote entangled state. Considering the non-zero term in concurrence estimation √ Vθ ( p01 + p10 ) − 2 p00 p11 C = . p00 + p01 + p10 + p11 +

(6.26)

The extra vacuum part will decrease the numerator and increase the nominator simultaneously. p00 could be estimated as p00 ( p01 + p10 ) ·

1 . SN R

(6.27)

Feeding this estimation into Eq. 6.26 we can calculate C + as 0.57 and 0.45 for 10 and 50 km situation. This result implies that the extra noise is the main reason for the decrease of C in a long fibre situation.

6.5 Lasers in Outdoor Applications In the current experiment, two atomic ensembles are placed nearby in one laboratory. For simplicity, two ensembles share control beams (i.e. write, read and pump beams) from the same lasers. Moreover, in the phase stabilisation process of the SPI experiment, auxiliary beams to detect the phase difference between two paths are also split from one laser. In TPI experiments, lasers located in different nodes could be easily locked to an absolute frequency standard, such as absorption spectrum or ultra-stable cavity. Nevertheless, in SPI experiments, the frequency and the phase of lasers in different nodes need to be synchronised. There must be a question of whether it is necessary to share these control and auxiliary beams among distant nodes or actively lock their relative phase in outdoor application. Here we propose a practical protocol to deal with phase fluctuation in outdoor applications. Furthermore, we did a brief experimental test to simulate some scenarios in future experiments. The result supports the feasibility of our protocol.

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6.5.1 Phase of Control Lasers Single-photon interference is also utilised in recently developed twin-field quantum key distribution (TF-QKD) [3], which brings similar questions about phase. Based on their solution [4], we propose a practical protocol to cancel phase fluctuation and uncertainty in SPI outdoor experiments. There are three main steps in DLCZ protocol: heralded entanglement creation in primary segments, a series of entanglement swappings and converting two effective maximally entangled (EME) states to a polarisation maximally entangled (PME) state. Without loss of generality, we observe three typical cases here to represent three steps, respectively. First, in the entanglement creation step, we put only one laser in each node serving as a write beam as depicted in Fig. 6.10a. In node-B, write beams for two ensembles are split from one laser. Suppose each laser has an initial phase φ A , φ B and φC , respectively. Besides, we assume all lasers work in the same frequency ωw . When Da1 and Da3 click, the state of the four ensembles becomes | ABL B R C = (eiφ A |1 S 0 S  ABL + eiφ B eiθ AB |0 S 1 S  ABL ) ⊗ (eiφ B |1 S 0 S  B R C + eiφC eiθ BC |0 S 1 S  B R C )eiωw t ,

(6.28)

where t is the time difference between entanglement M O T A − M O TBL and M O TBR − M O TC creation. θ AB and θ BC are the phase of the long fibres. Besides, the pump beam in the QFC process brings an extra phase (initial phase of pump laser) between |0 S  and |1 S , which is similar to the write beam. Therefore we consider the pump laser as a part of the write laser and do not list it in the equation. Here, the first step of our protocol is that, right before or after each trial of entanglement creation, every memory node sends a series of strong reference pulses to the middle station to estimate the phase difference between two nodes, for instance, φ AB = φ B + θ AB − φ A for node A and B and φ BC = φC + θ BC − φ B for node B and C. Then the second step is to send the estimation results back to memory nodes. Next we perform entanglement swapping as in Fig. 6.10b. Spin-waves in M O TBL and M O TBR are mapped onto read-out optical fields with the help of read beams in frequency ωr . Since they are placed in one node, read beams are split from one laser with the initial phase ψ B . After retrieval, the state of two ensembles and two optical fields becomes | ABL B R C = (eiφ A |1 S 0r o  ABL + ei(φ B +ψ B ) eiθ AB |0 S 1r o  ABL ) ⊗ (ei(φ B +ψ B ) |1r o 0 S  B R C + eiφC eiθ BC |0r o 1 S  B R C )eiωw t .

(6.29)

After two read-out fields interfere at middle BS, entanglement is swapped to M O T A and M O TC as | AC = (eiφ A |1 S 0 S  AC + eiφC ei(θ AB −θ BC ) |0 S 1 S  AC )eiωw t .

(6.30)

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Generally, during each quantum swapping process, the initial phase from the middle laser is eliminated, and the phase of the final state only comes from the long fibre and laser in the end nodes. Finally, through entanglement creation in basic segments and a series of swaps, we get a pair of entanglements M O T AU − M O TZU and M O T AD − M O TZD . We convert these two EME states to a PME state. Mapping all spin-waves onto read-out fields, we get | AU Z U A D Z D = (ei(φ A +ψ A ) |1r o 0r o  AU Z U + ei(φ Z +ψ Z ) eiθ AZ |0r o 1r o  AU Z U ) 





⊗ (ei(φ A +ψ A ) |1r o 0r o  A D Z D + ei(φ Z +ψ Z ) eiθ AZ |0r o 1r o  A D Z D ) (6.31) 

· eiωw t eiωr t , where φ A (ψ A ) and φ Z (ψ Z ) refer to the initial phase of the write (read) laser in node-A and Z. The prime on some terms indicates a discrepancy from the primefree one, arising from too much interval time larger than the coherence time of the laser or too rapid fibre fluctuation. t  is the time difference between entanglement M O T AU − M O TZU and M O T AD − M O TZD generation. When we register only the coincidences of two-side detectors, the effective part contributing to the final result is 







| AZ = (|1r o 0r o  AZ + ei(φ Z +θ AZ −φ A ) ei(φ Z +θ AZ −φ A ) |0r o 1r o  AZ )eiωw t eiωr t . 



= (|1r o 0r o  AZ + ei(φ AZ +φ AZ ) |0r o 1r o  AZ )eiωw t eiωr t . (6.32) The last step of our protocol is to calculate φ AZ and φ AZ from φ AB , φ AB , φ BC , …and compensate it to state in Eq. 6.32 via a phase modulator. We can see that only the first step is challenging. It requires a fast estimation of phase and a slow phase variation to meet phase consistency between detection and experiment moments. In our situation, the interval time between two trials is about 5 µs. Supposing detecting phase difference 2 µs before each trial, we need an estimation fast enough, which is already achieved in [4], and a phase variation slow enough in this time scale.

6.5.2 Test Result Based on the existing condition of our lab, we perform two tests to simulate the situation when two nodes are physically separated. First, we test the phase stability between two independent lasers as shown in Fig. 6.10d. We lock two 795 nm lasers (TOPTICA DLpro) to two ultra-stable cavities (Stable Laser System ATF6010-4) separately. Their locking points are around 170 MHz apart, and linewidths are less than 5 kHz. After shifting their frequency with two acoustic optical modulators, we interfere with two lasers with the help of a beamsplitter. Then we detect and record its result via a high-speed photodiode (Thorlabs PDA8GS; 9.5 GHz Bandwidth) and an

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a.

MOTBL

MOTA D

b.

1 a

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2 a

3 a

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c.

Db1 Db2

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Fig. 6.10 Three main steps in DLCZ protocol. a Heralded entanglement creation in basic segments. b Entanglement swapping. c Converting two EME states to a PME state. d Test of phase fluctuation between two independent lasers

oscilloscope (Keysight DSO-X 4054A; 500 MHz Bandwidth, 5 GHz sampling rate) as depicted in Fig. 6.10d. Choosing different sampling intervals δt, we can record the phase fluctuation ϕ = ϕ(t + δt) − ϕ(t) as shown in Fig. 6.11a (similar to Allan deviation in frequency). We can see that in the time scale we care about (δt =2 µs), its value is 0.03π . This means if we detect the relative phase between two lasers in two nodes as ϕ0 , 2 µs before each trial of the single-photon-interference experiment, the phase contributes to the entangled state is ϕ = ϕ0 ± 0.03π ≈ ϕ0 ± 5.4◦ , which is acceptable. Furthermore, we can further suppress this value by simultaneously performing this detection before and after each trial. Next, we observe the stability of fibres. We did similar statistics to the 50 km fibre interferometer in the SPI experiment. Due to two arms sharing one laser, only fibre fluctuation contributes to this result. In Fig. 6.11b, we can see that when δt =2 µs, ϕ < 0.005π both in locking and unlocking case, which is negligible compared with laser phase fluctuation.

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Fig. 6.11 a Deviation of the phase difference between two independent lasers in different sampling intervals. b Deviation of the phase difference between two long fibres in different sampling intervals

6.5.3 Statistics of Phase Fluctuation In the phase measurement, we need to infer the phase from intensity, which is directly detected. However, the mapping from phase to intensity is not one-to-one correspondence but periodical. This may bring a few problems with this measurement. First, one cannot infer an absolute phase. However, this is not critical because only the phase difference between two adjacent sampling points is essential. Second, the larger the phase difference, the bigger the odds it is misrecognized. For instance, a phase difference larger than π would be recognized as a small one. Third, near the extremum of intensity fringe, i.e. phase around nπ , even a small phase difference would lead to misjudgment.

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Fig. 6.12 Simulation result of test accuracy A ≡ stat /real along with the variation of real

The second and third questions may lead to underestimation of real results. To ensure a trustworthy result, we briefly analyze the influence of the above questions. Supposing the phase at moment t is ϕ(t), it would be distributed evenly in phase space. After a period δt has passed, due to random fluctuation, it is predictable that the phase in t + δt is obeys an Gaussian distribution N (ϕ(t), ϕ) with standard deviation ϕ centered at ϕ(t). We simulate this process by random sampling in a computer. In given phase ϕreal , we first randomly choose ϕ(t) in phase space and give a series of phase points obeying N (ϕ(t), ϕreal ). Then we calculate the intensity of each phase and derive it back to a phase which is possibly different from the original one. Finally, do statistics of this phase and average it for all ϕ(t) to give a statistical result ϕstat . By defining test accuracy as A ≡ stat /real , we show the simulation result in Fig. 6.12. Apparently, in Fig. 6.11, results after ϕ > 0.1 are untrustworthy. For ϕ ≤ 0.03π , accuracy of test result is larger than 0.97.

References 1. Chou CW, de Riedmatten H, Felinto D et al (2005) Measurement-induced entanglement for excitation stored in remote atomic ensembles. Nature 438(7069):828–832 2. Gühne O, Tóth G (2009) Entanglement detection. Phys Rep 474(1–6):1–75 3. Lucamarini M, Yuan ZL, Dynes JF et al (2018) Overcoming the rate-distance limit of quantum key distribution without quantum repeaters. Nature 557(7705):400–403 4. Liu Y, Yu ZW, Zhang W et al (2019) Experimental twin-field quantum key distribution through sending or not sending. Phys Rev Lett 123(10):100505

Chapter 7

Measurement-Device-Independent Verification of a Quantum Memory

In Chaps. 5 and 6, we demonstrate remote entangled states between quantum memories. As quantum researchers, we build quantum memories entirely by ourselves; all experimental equipments are under our control. Considering that the quantum repeater and quantum memories become commercial products with technical development, quantum memories and other quantum devices are probably provided by third-party suppliers. In this case, the ability to certify these devices’ quantumness is important to information security. In this chapter, we consider such a typical scenario: Alice, as a buyer of quantum memory, verifies the quantumness of a quantum memory provided by Bob, the seller.

7.1 The Theory of Measurement-Device-Independent Verification There are different criteria for a good quantum memory in different application scenarios. Yet there exist some general measures as follows, • Storage efficiency η: the ratio of readable photon and the input photon. • Fidelity F: The likeliness of the output quantum state with the input quantum state. • Storage lifetime τ : The characteristic time when the storage efficiency drops to 1/e of the initial value. For a quantum memory, fidelity is the most critical metric; a quantum memory with low efficiency and a short lifetime can still fit some applications. However, a memory with a fidelity lower than the classical limit F = 50% cannot be regarded as a quantum memory but rather an entanglement-breaking channel [1]. There are two commonly used verification schemes in previous experiments; one is process tomography. The flowchart of this scheme is shown in Fig. 7.1a. Alice first prepares a quantum state ψ, sends it to Bob and asks him to store it in the quantum memory, and after a period of storage, asks Bob to read out the quantum © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Yu, Long Distance Entanglement Between Quantum Memories, Springer Theses, https://doi.org/10.1007/978-981-19-7939-2_7

101

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Fig. 7.1 Quantum memory verification via process tomography (a) and Bell test (b). Figure from [6]

a

b

state and send back the quantum state ψ  = N (ψ). Alice then performs quantum state tomography to this state. By preparing different input states ψ and repeating the above process, Alice can reconstruct the process matrix of the memory N and thus make a judgement. However, this scheme strongly relies on the security of the device. For the state preparation part, Bob can perform a photon number splitting attack when the single photon source is not perfect [2, 3], which can, of course, be defended against with the decoy-state method [4]. In terms of the experiment, the more vulnerable part is the detection part. The commonly used avalanche photodetector is vulnerable to the blinding attack [5], under which Bob has complete control over the detector’s behaviour: to respond or not to respond and what kind of output it gives. Then Bob can use the intercept-and-resend scheme to forge a “quantum memory”. Inside the “quantum memory”, Bob directly measures a randomly chosen basis right after the qubit is input and gets the result R. When Alice chooses a measurement basis, Bob eavesdrops on this message and compares it with the basis he chooses. If their bases are the same, Bob manipulates Alice’s detector to output a counterfeit result R. Otherwise, he makes the detector output nothing, pretending to be a failed storage. Alice is equivalent to doing a tomography on unprocessed qubits for the same bases case. She will always conclude an identity process, i.e. a quantum memory with unitary fidelity and infinite long lifetime. If Alice chooses measurement in an entirely random fashion, Bob has a probability of 1/3 to guess the same basis as her, faking a memory with the corresponding efficiency. If Alice chooses the basis, not random or passive, Bob could know her basis before his measurement. Thus he can always give the output, making the memory efficiency up to 100%. Another test is the Bell test, as shown in Fig. 7.1b, Alice prepares an entangled photon pair ϕ R A , sends one for storage and another not, and performs a joint Bell test on the stored and unstored photons; if one of the photons undergoes memory storage N , then the two-photon state becomes ϕ˜ R B = 1 ⊗ N (ϕ R A ). The Bell test results will reflect the influence of the memory. An unqualified memory will fail the Bell test. We know that the Bell test is only based on the assumption of local

7.1 The Theory of Measurement-Device-Independent Verification

103

Fig. 7.2 Schematic of the MDI quantum memory verification. Figure from [6]

realism, so the violation of Bell inequality gives a good indication of the existence of non-classical correlations; hence this method is also known as the self-testing [7– 9]. In order to ensure a reliable Bell test, some loopholes must be closed, one of which is the localization loophole [10], which requires two measurement setups that are space-like separated. For a long-lived quantum memory, we need to have a sufficiently long fibre delay for the unstored photon. Another is the efficiency loophole [11, 12]. According to the analysis [13], the total system efficiency (including storage, transmission and measurement) in the experiment needs to be  83%, which is challenging in experiments. In addition, violation of Bell inequality requires entanglement fidelity F > 78% [14], which is excessive demand, making many quantum memories fail the test. One may relax the technical overheads by trusting some parts of the verification components. A similar idea was implemented in the field of QKD by designing a measurement-device-independent (MDI) scheme [15, 16]. It excludes the most common attacks towards measurement setup while maintaining a moderate experimental feasibility level. The MDI-QKD has been very successful and improves practical QKD security significantly. Furthermore, the MDI scheme has successfully advanced the field of entanglement witness [17–19]. In the case of memory verification, it is also very natural to trust the state preparation process. Rosset et al. laid out a theoretical framework on how to verify a quantum memory via MDI [6]. The flowchart of the MDI scheme is shown in Fig. 7.2. In the scheme, Alice, the client, verifies a quantum memory N afforded by an untrustworthy supplier, Bob, by playing a semi-quantum signalling game with him. In each round, Alice sends two quantum questions ξx and ψ y , according to the random number x and y, sequentially to Bob and asks him for a classical answer b. In the language of quantum optics, ξx and ψ y are two photonic qubits. Bob performs a joint BSM towards them and gives the result b. Preparing a photon in ξx (ψ y ) could be viewed as a virtual process that preparing an entanglement between this photon and a virtual photon in state |+  = √ (|00 + |11)/ 2, then projecting the virtual photon onto ξx (ψ y ). The BSM could, therefore, be viewed as an swapping. It makes two virtual photons onto   entanglement an entangled state |+  + . With one photon under the storage of the memory,   the entangled state between two virtual photons becomes JN = (N ⊗ 1)|+  + ,

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which is exactly the Choi matrix of the memory [20]. Hence we can verify a non-EB memory when we witness two virtual photons entangled, i.e. W  = T r [JN W ] > 0, where W is the entanglement witness operator [14].

7.2 Experimental Setup The experiment setup is shown in Fig. 7.3. The quantum memory we verified is based on the two-channel electromagnetically induced transparent (EIT) mechanism [21] in a laser-cooled 87 Rb atomic ensemble. In the beginning, all atoms are prepared on the ground state |g ≡ |5S1/2 , F = 2, m F = +2, which is not degenerate with other sublevels due to a slight bias magnetic field. A signal photon, with a qubit encoded on its polarization, is spatially split by a polarizing beamsplitter (PBS). In each spatial mode, polarization is rotated to σ + for atoms to couple transition |g ↔ |e ≡ |5P1/2 , F = 1, m F = +1. Each signal beam and the control beam have ◦ the waist 90 µm and 300 µm, overlapping well in the atomic region with 3 angle. The lifetime of the EIT memory is determined by the spin-wave wavelength and the temperature of the atomic ensemble. For two spatial modes, storage efficiency ηs = 0.269 and 0.285, and memory lifetime τm = 58.2 and 56.6 µs, showing a good conformity between them. On such a time scale, the phase between the two spatial modes of the signal light is almost constant. The main noise source is the decrease of the signal-to-noise ratio when the readout signal becomes weaker. Thus we can describe the EIT quantum memory as a depolarization model as follows, 1 N (ρ) = νρ + (1 − ν) . 2

(7.1)

According to the Choi theory [20], any quantum mechanical operation can correspond to a two-particle state. For our depolarization model, the Choi state is   1 JN = ν|+  +  + (1 − ν) . 4

(7.2)

The entanglement criterion corresponding to this state is,   1 W = |+  +  − . 2

(7.3)

Accordingly, we can design the test scheme as follows: ξx and ψ y are randomly selected from the set {|H , |V , |+, |R}. Bob can distinguish two Bell states |+  and |−  with his Bell state analyzer, and b takes the values + and − correspondingly. Regardless of the undetected instances, we have the payoff functions as follows,

7.2 Experimental Setup

105

Fig. 7.3 Experimental setup and energy level scheme (grey shaded area). MOT-A and B are two laser-cooled 87 Rb atomic ensembles, serving as the single-photon source and the quantum memory, respectively. MOT-A is further loaded into a small dipole trap to confine its dimension. For both MOT-A and B, a slight bias magnetic field is applied along the y direction to remove the degeneracy of Zeeman sublevels, and all atoms are prepared in the ground state |g initially. In MOT-A, only one atom is excited to Rydberg state |r  by beam Pump I and II, forming a Rydberg superatom. It is soon retrieved into a signal photon by the Read beam. The photon is dynamically switched to two paths with/without the quantum memory via a Pockels cell. In each path, an encoder consisting of two Pockels cells controlled by a QRNG encodes a polarization qubit onto this photon. The first photon is sent into MOT-B to store. This is achieved by mapping it onto a spin-wave on |s with the help of the Control beam. After a duration of storage, it is retrieved and interferes with the second photon in the BSM device. In MOT-B, two polarization modes of a photon are mapped onto two space modes before storing and recombined after retrieving. Two space modes are actively phase-locked by adjusting the PZT. DM: dichroic mirror. PBS: polarizing beamsplitter. QWP: quarter-wave plate. HWP: half-wave plate. SPCM: single-photon counting module

⎛

wx+y

⎞ 0 − 21 − 21 21 ⎜− 1 0 − 1 1 ⎟ 2 2 2 ⎟ =⎜ ⎝− 1 − 1 1 0 ⎠ , 2 2 1 1 0 −1 2 2

⎛

wx−y

⎞ 0 − 21 21 − 21 ⎜− 1 0 1 − 1 ⎟ 2 2 2⎟ =⎜ ⎝ 1 1 −1 0 ⎠ . 2 2 − 21 − 21 0 1

(7.4)

The photon qubit preparation process is shown in the left half of Fig. 7.3, where we generate single photons using another atomic ensemble MOT-A. MOT-A is a small 87 Rb atomic ensemble loaded via an optical dipole trap, in which at most one atom can be excited to the Rydberg state due to the Rydberg blockade, forming a Rydberg superatom [22]. By reading out this superatom, we get a good single photon, which minimizes the possibility of photon number splitting attacks. Afterwards, we dynamically switch the photon to optical paths with or without storage using a Pockels cell and a subsequent PBS. We set up a pair of Pockels cells in each optical path to prepare the photon polarization state. The scheme for polarization state preparation is shown in Fig. 7.4. The first Pockels works as a dynamic QWP. By controlling the voltage applied on the Pockels cell, we can choose whether change the input

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1st Pockels

2nd Pockels

OFF

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Fig. 7.4 Schematic of polarization states preparation

polariton from |H  to |R or not. The second Pockels cell works as a dynamic HWP. By applying the voltage on it, we change |H  to |D or |R to |L. We get the four polarisation states we need with the unitary transform by the following QEP. The working status of the Pockels cells is controlled by a quantum random number generator (QRNG), which generates the random numbers slightly before they are used.

7.3 Rydberg Single Photon Source The Rydberg single-photon source was mainly set up by Peng-Fei Sun. We give a brief description of its basics here. As described in Sect. 2.3, the Rydberg interaction only works within a small blockade radius Rb , which limits the ensemble size. To

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107

increase the atom numbers to a limited size to gain more collective enhancement, we use the dark magneto-optical trap technique to increase the cluster density. Two main reasons for limiting the atomic density in the standard MOT exist. One is the spontaneous light pressure, which refers to the fact that the spontaneously radiated photons from the atoms are absorbed again by neighbour atoms during cooling, creating a repulsive pressure between the atoms. As the density increases, the spontaneous radiation pressure also increases, eventually limiting further increases in density. The other one is a light-assisted collision, which means that the atoms gain a larger cross section in the cooling process because there are some populations at the excited state with larger cross sections. Limited by these two mechanisms, the atomic density in a standard MOT is typically 3 × 1010 ∼ 5 × 1010 cm−3 . The method to solve these limitations is the dark MOT, whose idea is to transfer a part of atoms to the dark state, minimizing their contribution to the light pressure and the light-assisted collision. There are two versions of dark MOT, the spatial dark MOT and the temporal dark MOT. In the spatial dark MOT, we block the central part of the Repumper beam after a period of standard MOT, resulting in the central region atoms being transferred to the dark state. In the temporal dark MOT, we increase the cooler beam detuning and lower the Repumper beam power, resulting in a lower excited state population. In our experiments, we use both types of dark MOT. After a standard MOT of 30 ms, we first gradually turned off the Repumper I beam for the standard MOT by 3 ms and simultaneously turned on the hollow core Repumper II beam and the Depumper beam for the spatial dark MOT, and keep them running for 64 ms. In the end, the Cooler beam detuning is being increased, and the Repumper II beam power is being reduced for another 3 ms, which work both for the temporal dark MOT and the sub-Doppler cooling. The time sequence for involved laser beams is shown in Fig. 7.5.

Fig. 7.5 Time sequence for the Rydberg single photon source

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After the dark MOT, we loaded the atom ensemble into an optical dipole trap, which was realized by a 1064 nm laser with a power of about 2.5 W. The size of the dipole trap is about 25 µm in the direction perpendicular to the magnetic field and about 5 µm in the direction along the magnetic field. The density of atoms in the dipole trap is tested by the absorption to be about 4.13 × 1011 cm−3 , and the temperature of the atoms in it is estimated to be 24.3 µK. The atomic ensemble is initialized at |g ≡ |5S1/2 , F = 2, m F = 2 and excited via the intermediate state |e ≡ |5P1/2 , F = 1, m F = 1 by 795 nm + 475 nm twophoton resonance to the Rydberg state |r  ≡ |81S1/2 , m J = +1/2. The 795 and 475 nm pumping beams have beam widths at the ensemble of 7 µs and 7.7 µm, respectively, and the Rabi frequencies of 2.5 MHz and 7 MHz, respectively. The 795 nm pumping beam has a 40 MHz detuning to the intermediate state |e, preventing population on state |e. The Rydberg excitation is read out by a read beam on resonance with |r  ↔ |e transition. We first optimized the pulse shape of the readout photon by varying the readout light power. The optimized shape of the readout photon is shown in Fig. 7.6a, fitting well to a Gaussian profile with a pulse width of 58.95 ns. The probability of having a photon is about 3.85% when measured directly at the output collimator. By deducting the fibre coupling loss and the detection inefficiency, we obtain the single photon efficiency of Pph = 6% (defined as the probability of getting a single photon before the path-switching Pockels cell). To test its single-photon nature, we performed a Hanbury-Brown and Twiss experiment. The results in Fig. 7.6b shows the autocorrelation function g (2) (0) = 0.03 ± 0.02, demonstrating a good single-photon source.

7.4 The EIT Quantum Memory 7.4.1 Specifications The quantum memory we verified is based on the two-channel electromagnetically induced transparent (EIT) mechanism [21] in a laser-cooled 87 Rb atomic ensemble. In the beginning, all atoms are prepared on the ground state |g ≡ |5S1/2 , F = 2, m F = +2, which is not degenerate with other sublevels due to a slight bias magnetic field. A signal photon, with a qubit encoded on its polarization, is spatially split by a polarizing beamsplitter (PBS). In each spatial mode, polarization is rotated to σ + for atoms to couple transition |g ↔ |e ≡ |5P1/2 , F = 1, m F = +1. Each signal beam and the control beam have the waist 90 µm and 300 µm, overlapping well in the ◦ atomic region with 3 angle. In storing phase, the control beam coupling |e ↔ |s dynamically converts the signal photon into a spin-wave on |s and is switched off. After a duration of storage, two spatial modes of the signal photon are simultaneously retrieved by turning the control beam on and combined to a polarization-encoded one again.

7.4 The EIT Quantum Memory

a 150 Counts in 3min

Fig. 7.6 The temporal profile (a) and the second order autocorrelation function g (2) of the Rydberg single photon (b)

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5

10

In order to test the consistency of the two spatial modes, we conducted the following three tests. Firstly, we tested the absorption of the two spatial modes, M1 and M2, using continuous weakly coherent light as the probe beam. Figure 7.7a shows the transmission of the probe beam, where we get an optical depth of 9.75 (9.62) for the spatial mode M1 (M2). A similar test in the presence of a control beam is shown in Fig. 7.7b. The EIT spectrum is measured with the control beam Rabi frequency of ∼ 16 MHz. Lastly, we perform a complete EIT storage with weak coherent pulses having the same temporal profile as the Rydberg single photons. Figure 7.8 shows the storage efficiency ηm of spatial modes M1 (red) and M2 (blue) as a function the storage time t. The initial readout efficiencies ηm0 of modes M1 and M2 are 0.269 and 0.250, respectively, and the storage lifetimes τm are 58.2 µs by fitting equation 2 2 ηm = ηm0 e−t /τm . All three tests show that the two spatial modes of the signal light have a good consistency.

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7 Measurement-Device-Independent Verification of a Quantum Memory

a

1.0

Transmission

0.8

0.6

0.4

0.2

0.0

- 30

- 20

- 10

0

10

20

30

10

20

30

Detune (MHz)

b 1.0

Transmission

0.8

0.6

0.4

0.2

0.0

- 30

- 20

- 10

0 Detune (MHz)

Fig. 7.7 The transmission of two spatial modes without (a) and with (b) control beam. The orange and blue points correspond to spatial modes M1 and M2, respectively

7.4.2 Optimization of the Readout Profile To have a good two-photon interference between the stored and the unstored photon, which to the best of our knowledge has never been shown in previous experiments, we need a good homogeneity between them. In EIT quantum memory, the profile of the output photon is largely affected by the control field. In most EIT quantum memory experiments [23–27], the shape of the control light is a step function, i.e. control is always on before the probe photon enters the atomic ensemble and turns off immediately after the photon enters. The readout is also done in a similar fashion

7.4 The EIT Quantum Memory

111

Fig. 7.8 The efficiency-time curve of the EIT quantum memory

in that the control beam is immediately turned on to a constant value at a specified moment. This is easy to operate and has less impact on the storage efficiency, but it makes the readout photon shape somehow distorted. Gorshkov et al. carefully studied the dynamic process in EIT storage [28–31] and proposed a method [31, 32] to improve the storage efficiency by optimizing the control light pulse shape. In the optimal case, the readout process is the time inverse of the write process. Thus the readout photon has the same profile as the write signal. From Sect. 2.1, we know that the main physical process in the EIT quantum memory is essentially a transform from the probe photon to the spin wave. Let us consider the dynamic process described by Eq. 2.17. For simplicity, we write the  equations in the motion coordinate

z    t = t − z/c and with the unitless space-time  ˜ coordinates t = γ t , z˜ = 0 dz n z /N . Similar treatments are applied to some of ˜ = /γ , ˜ = /γ . We also reduce the functions and physical quantities involved: √ the scale of the light field by a factor of c/(Lγ ); then the equations can be written as, √ ∂z˜ E = i d P, √

)P + i dE + i (t˜)S, (7.5) ∂t˜ P = −(1 + i ∂t˜ S = i ∗ ( t)P. At the initial moment, we have the light field as the incident probe field Ein (z, t) and the atoms all in the ground state with no spin wave or excited state components. Thus we have the boundary conditions as follows E(0, t˜) = Ein (t˜), P(˜z , 0) = 0, S(˜z , 0) = 0.

(7.6)

˜ t˜) that allows the probe field The question now is what is a control beam profile ( to be mapped onto as many spin wave modes as possible, i.e. the maximization of

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7 Measurement-Device-Independent Verification of a Quantum Memory

the storage efficiency,

 ηs =

1

)|2 . d z˜ |S( z, T

(7.7)

0

This is a problem of the optimal solution of a function with constraints. It can be solved using the Lagrange multiplier method. We first define Lagrange multipliers ¯ z , t˜), P(˜ ¯ z , t˜), S(˜ ¯ z , t˜) and their evolution equations are for three operators E(˜ √ ¯ ∂ z˜ E¯ = √ i d P, ¯ . ∂t˜ P¯ = P¯ + i dE + i S, ∂t˜ S¯ = i P¯ The initial conditions are,

¯ t˜) = 0, E(1, ¯ z, T

) = 0, P(

(7.8)

(7.9)

¯ z, T

) = S(

). S( z, T ˜ t˜) (usually a constant function We start with a guessed control beam waveform ( is already a good starting point). By solving Eqs. 2.17 and 7.9, we can first solve for a set of operators with their corresponding Lagrange multipliers. Substituting S(˜z , t˜) into Eq. 7.7, we get the current storage efficiency. If it is far from the ideal, the following correction can be made to the control beam waveform, ˜ t˜) → ( ˜ t˜) −

(

1 λ



1

  ¯ z , t˜)S ∗ (˜z , t˜) . d z˜ Im S¯ ∗ (˜z , t˜)P( z, t˜) − P(˜

(7.10)

0

The optimal value can then be gradually approached by repeating the above steps using the updated control waveform, where λ is an optimization step that determines the convergence rate of the optimization, which is usually taken to be 0.1. For the case in our experiments, taking the Gaussian profile of the incident light (see Sect. 7.3), the optimized control light shape is obtained as shown in the inset of Fig. 7.9b. We take an arbitrary function generator AFG-3252 from Tektronix to modulate this waveform to the amplitude of the control beam AOM driving signal. A comparison of the readout photon shapes obtained from the step-type control beam and the optimized control beam is shown in Fig. 7.9. It can be seen that the photon shape in the step control case has some distortion, and the photon shape in the optimized case fits the Gaussian function well, with a width of 60.06 ns, basically the same as the unstored case. Also, we are interested not only in the similarity of the photon profiles but also in their homogeneity in the two-photon interference. Yet the Hong-Ou-Mandel experiment is not efficient enough as a test because of a minimal two-photon coincidence rate Pph . Therefore we use single-photon interference for the test. We set up a MachZehnder interferometer as shown in Fig. 7.10a. The interferometer’s two arms are EIT storage and a 36 m fibre delay line, respectively. By setting the storage time the same as the fibre delay, one can measure the homogeneity of two modes in the

7.4 The EIT Quantum Memory

113

Fig. 7.9 The profile of readout photon with step profile (a) and optimized profile (b) control beam. The corresponding profiles of the control beam are shown in the insets

single-photon interference fashion. Figure 7.10b shows the normalized counts for one of the interferometer outputs as a function of phase between the two arms. We get fitted interference visibility of V = 0.875 ± 0.018. From the interference visibility, we can infer the fidelity of the Bell state measurements. We first examine the interference visibility V in terms of photon homogeneity. After the first beamsplitter, we have a †D → aU† + ia †D .

(7.11)

a † is the creation operator of a Gaussian shape photon. Subscript U and D indicate up and down space mode. We model the distortion of photon during storage by replacing creation operator a † with b† = αa † + β a˜ † , where a˜ † is a creation operator in the orthogonal space of a † . We can calculate the state after the second beamsplitter as

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7 Measurement-Device-Independent Verification of a Quantum Memory

Fig. 7.10 Characterization of photon distortion during storage. a A Mach-Zehnder interferometer detects homogeneity between stored and unstored photons. One arm is the EIT quantum memory, and the other arm is a 36 m fibre delay line. An HWP surrounded by two QWPs is inserted in the delay line arm. The relative phase between two arms will vary along with the angle δϕ of the HWP. b The oscillation of counts in port L along with the angle δϕ of the HWP

aU† + ib†D       → a †D + iaU† + ieiφ α aU† + ia †D + ieiφ β a˜ U† + i a˜ †D

(7.12)

=(1 − αeiϕ )a †D + i(1 + αeiϕ )aU† − eiφ β a˜ †D + ieiφ β a˜ U† . By varying the phase φ, we can observe the oscillation of counts in two ports. The oscillation visibility is given by V =

[(1 + α)2 + β 2 ] − [(1 − α)2 + β 2 ] Nmax − Nmin = α. = Nmax − Nmin [(1 + α)2 + β 2 ] + [(1 − α)2 + β 2 ]

(7.13)

N with subscript max (min) represents maximal (minimal) counts of the oscillation pattern. We then consider the influence of inhomogeneity upon BSM. Our BSM apparatus is shown in Fig. 5.1a. The first PBS is to post-select two photons both in horizontal or vertical polarization. Two PBS in |+/− = |H  ± |V  basis discriminate |+  from |− . Considering an imperfect interference, we may misrecognize two Bell states. Thus we can model a realistic BSM as     S ± = (1 − λ)|±  ∓  + λ|∓  ∓  .

(7.14)

We write these two bell states by creation operators as |±  = a †H b†H ± aV† b†V |vac   = α(a †H a †H + aV† aV† ) ± β(a †H a˜ †H + aV† a˜ V† ) |vac,

(7.15)

where the subscript represents the polarization of the photon. The first term corresponds to two homogeneous photons interfering on |+/− basis PBSs, i,e, an ideal

7.4 The EIT Quantum Memory

115

BSM. The second term corresponds to two distinguishable photons interfering on |+/− basis PBSs. This would lead to a random coincidence, i.e. Bell states being misrecognized by 1/2 probability. Therefore we get λ = β 2 /2 = (1 − V 2 )/2.

7.4.3 Phase Locking and the Losses in the Setup The phase locking laser between two spatial modes is introduced through the idle port of the first PBS and led out through the idle port of the second PBS. The phaselocking laser is far detuned from the atomic levels. The locking is performed during the atom cooling phase and paused during the experiment. To test the performance of phase locking, we follow the experiment time sequence and measure the interference between two modes with a weak coherent beam. Figure 7.11 shows the oscillation of the normalized counts in one output, where we get fitted visibility of 0.983, demonstrating good phase locking. We calibrate the losses in the setup as shown in Fig. 7.12, where all numbers are with the unit of µW. The two spatial modes of the EIT are labelled in red and blue. We briefly summarize the result as follows. 1. The efficiency of the polarization encoding for the memory-free path (AB): 0.505; 2. The overall efficiency of the EIT memory path, ηopt (AE): 0.188; 3. The optical efficiency of the EIT memory, M1: 0.347, M2: 0.339; 4. The efficiency of the Bell state analyzer, from the memory-free path: 0.714; from the memory path 0.680.

1.0

Normalized Counts

0.8

0.6

0.4

0.2

0.0

0

10

20

30

40

Degree

Fig. 7.11 Phase locking result for the EIT interferometer. The normalized counts on one output as a function of the degree of QWP in one arm of the interferometer

116

7 Measurement-Device-Independent Verification of a Quantum Memory 500

630

780(A)

394(B)

18m From Rydberg

480

room room Breadboard 1006 1005

Breadboard

250

155 150

266 240 EOM

PBS

HWP

360(C)

102 100

125 122 (D) E

QWP

FC

Flange

Etalon

276 85

340

278 282 290 84 79 88

Fig. 7.12 Measurement of losses in the setup. Numbers refer to the measured powers, with units of µW

7.5 Results We run the measurement scheme with different storage times from 5 µs to 55 µs, and the results are shown in the blue data points in Fig. 7.13a. To eliminate the influence of optical and detection losses, we normalize P(b|x y) by only considering detected events. We always have W  > 0 witnessing a quantum memory within the memory lifetime. As the storage time, t, increases, W  decreases because of decoherence. In the case of long storage times, the larger statistical error comes from, the smaller amount of data. There are two reasons leading to this result. One is that the retrieval efficiency of the quantum memory gradually drops. The other is that the period of one cycle increases, leading to a lower repetition rate. To verify the validation of the MDI results, we performed two simulations and compared them with the measured result. In the first simulation, we modelled the quantum memory as a pure depolarizing channel as described in Eq. 7.1. To determine the normalized noise strength p, which is closely related to the signal-to-noise ratio (SNR) as (1 − p)/ p = SNR, we measured the parameters of our system, including the efficiency and lifetime of the quantum memory, background noise, as well as optical efficiency, listed Table 7.1. This ‘depolarizing simulation’ only accounts for depolarizing noise and should set an upper bound for the measured results. We performed a process tomography of the quantum memory to include more decoherence mechanisms other than depolarising. In this model, the storage process is described by a quantum dynamic map, E(ρ) =

3  3 

E˜ m ρ E˜ n† χmn .

(7.16)

m=0 n=0

We choose E˜ 0 = 1, E˜ 1 = σx , E˜ 2 = −iσ y and E˜ 3 = σz , where σx , σ y and σz are Pauli matrices. Then the element with coefficient χ00 represents a perfect quantum memory, and the other elements represent different kinds of decoherence. With the

7.5 Results

117

Fig. 7.13 MDI measurement and simulation results. MDI verification results of the quantum memory. The blue circles are the tested results. The error bars represent one standard deviation. The black dashed line is the ‘depolarizing simulation.’ The green shadow is the one standard deviation region of the ‘tomography simulation.’ Pink shading indicates the entanglement-breaking regime. b. Noise strength as a function of t. The black dash line is p in the depolarizing model. The green points indicate the overall noise strength 1 − χ00 in the tomography model Table 7.1 Experiment parameters Pph ηopt ηdet 0.060

0.108

0.70

0 ηm

τm (µs)

Pnoise

0.269 0.250

58.2 56.6

8.57 × 10−5

knowledge of the process matrices χ of the quantum memory, we performed the second simulation, the ‘tomography simulation’. Figure 7.13b depicts the noise strength p in the depolarizing model and the overall noise strength 1 − χ00 in the tomography model as functions of the storage time t. For the depolarizing model, we can observe that the normalized noise strength p increases from 0.054 at t = 0 µs to 0.146 at t = 60 µs. This trend can be explained by the decrease in storage efficiency over time. Though the input signal strength and depolarizing noise keep constant, a lower retrieval efficiency will lead to a lower SNR, resulting in a bigger p. In the tomography model, the overall noise strength 1 − χ00 grows at the same pace as p and is slightly higher than p. This confirms our hypothesis that the dominant noise in this system is the depolarizing noise. Results from the ‘depolarizing simulation’ and ‘tomography simulation’ are shown as the black dash line and green shadow in Fig. 7.13a, respectively. In both simulations, the influence of inhomogeneity of stored and unstored photons is included. We can see that when t ≤ 30 µs, the measured results are in good agreement with the two simulation results; when t > 30 µs, the measured and ‘tomography simulation’ results become lower than the ‘depolarizing simulation’ result, indicating that other noise possibly gradually starts to dominate. The measured and ‘tomography simulation’ results are consistent with the margin of error throughout the measurement interval.

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References 1. Horodecki M, Shor PW, Ruskai MB (2003) Entanglement breaking channels. Rev Math Phys 15(06):629–641 2. Huttner B, Imoto N, Gisin N et al (1995) Quantum cryptography with coherent states. Phys Rev A 51(3):1863 3. Brassard G, Lütkenhaus N, Mor T et al (2000) Limitations on practical quantum cryptography. Phys Rev Lett 85(6):1330 4. Wang XB (2005) Beating the photon-number-splitting attack in practical quantum cryptography. Phys Rev Lett 94(23):230503 5. Gerhardt I, Liu Q, Lamas-Linares A et al (2011) Experimentally faking the violation of Bell’s inequalities. Phys Rev Lett 107(17):170404 6. Rosset D, Buscemi F, Liang YC (2018) Resource theory of quantum memories and their faithful verification with minimal assumptions. Phys Rev X 8(2):021033 7. Mayers D, Yao A (1998) Quantum cryptography with imperfect apparatus. In: Proceedings 39th annual symposium on foundations of computer science (Cat. No. 98CB36280). IEEE, pp 503–509 8. Mayers D, Yao A (2003) Self testing quantum apparatus. arXiv preprint quant-ph/0307205 9. Magniez F, Mayers D, Mosca M, et al (2006) Self-testing of quantum circuits. In: International colloquium on automata, languages, and programming. Springer, pp 72–83 10. Brunner N, Cavalcanti D, Pironio S et al (2014) Bell nonlocality. Rev Mod Phys 86(2):419 11. Garg A, Mermin ND (1987) Detector inefficiencies in the Einstein-Podolsky-Rosen experiment. Phys Rev D 35(12):3831 12. Eberhard PH (1993) Background level and counter efficiencies required for a loophole-free Einstein-Podolsky-Rosen experiment. Phys Rev A 47(2):R747 13. Larsson JÅ (1998) Bell’s inequality and detector inefficiency. Phys Rev A 57(5):3304 14. Gühne O, Tóth G (2009) Entanglement detection. Phys Rep 474(1–6):1–75 15. Lo HK, Curty M, Qi B (2012) Measurement-device-independent quantum key distribution. Phys Rev Lett 108(13):130503 16. Braunstein SL (2012) Pirandola S side-channel-free quantum key distribution. Phys Rev Lett 108(13):130502 17. Branciard C, Rosset D, Liang YC et al (2013) Measurement-device-independent entanglement witnesses for all entangled quantum states. Phys Rev Lett 110(6):060405 18. Verbanis E, Martin A, Rosset D et al (2016) Resource-efficient measurement-deviceindependent entanglement witness. Phys Rev Lett 116(19):190501 19. Li ZD, Zhao Q, Zhang R et al (2020) Measurement-device-independent entanglement witness of tripartite entangled states and its applications. Phys Rev Lett 124(16):160503 20. Jiang M, Luo S, Fu S (2013) Channel-state duality. Phys Rev A 87:022310 21. Fleischhauer M, Imamoglu A, Marangos JP (2005) Electromagnetically induced transparency: optics in coherent media. Rev Mod Phys 77(2):633 22. Dudin YO, Kuzmich A (2012) Strongly interacting Rydberg excitations of a cold atomic gas. Science 336(6083):887–889 23. Chanelière T, Matsukevich D, Jenkins S et al (2005) Storage and retrieval of single photons transmitted between remote quantum memories. Nature 438(7069):833–836 24. Chen YH, Lee MJ, Wang IC et al (2013) Coherent optical memory with high storage efficiency and large fractional delay. Phys Rev Lett 110(8):083601 25. Park KK, Cho YW, Chough YT et al (2018) Experimental demonstration of quantum stationary light pulses in an atomic ensemble. Phys Rev X 8(2):021016 26. Hsiao YF, Tsai PJ, Chen HS et al (2018) Highly efficient coherent optical memory based on electromagnetically induced transparency. Phys Rev Lett 120(18):183602 27. Jiang N, Pu YF, Chang W, et al (2019) Experimental realization of 105-qubit random access quantum memory. NPJ Q Inf 5(1):1–6 28. Gorshkov AV, André A, Lukin MD, et al (2007) Photon storage in -type optically dense atomic media. I cavity model. Phys Rev A 76(3):033804

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Chapter 8

Further Improvement of Atomic Ensemble Quantum Memories

In previous chapters, we built a ring cavity-enhanced efficient quantum memory and used it to build long-range quantum entanglement. Nonetheless, the current memory is still facing some challenges, such as the low atom-photon entanglement probability and limited storage time (all the long-lived quantum memories achieved on the same platform [1–3] serve for single spinwave mode, but not the qubit). This chapter focuses on some theoretical considerations and scheme design, seeking to improve the quantum memory performance with the least modification for the current setup. Two main topics are discussed. On the one hand, we discuss the deterministic atomphoton entanglement creation and intra-atom Bell state analyzers with the help of Rydberg interactions. On the other hand, we discuss the long-lived and multimode storage in DLCZ -type quantum memories.

8.1 Entanglement Creation and Swapping via Rydberg Interactions Rydberg interactions can be used as a powerful weapon in quantum information processing, and it has been previously discussed how collective Rydberg states can help quantum computing [4]. The basic idea is to prepare Rydberg collective states deterministically via the Rydberg blockade, transfer them to ground state spin waves, encode qubits with the Fock states of the ground state spin waves, and implement gate operations between different qubits via Rydberg interactions. However, the experimental implementation of this scheme is challenging in many aspects: firstly, the qubit initialization is serial and hence inefficient; secondly, in order to encode more qubits, atoms with rich round state levels are required, e.g. holmium atoms [5]; most critically, it is challenging to implement multi-qubit gate operations in the atomic ensembles. Instead, the atomic array system [6–8] is more experimentally favourable for realizing Rydberg quantum computing.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Yu, Long Distance Entanglement Between Quantum Memories, Springer Theses, https://doi.org/10.1007/978-981-19-7939-2_8

121

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8 Further Improvement of Atomic Ensemble Quantum Memories

Nevertheless, the requirement of the quantum network is essentially different: the qubit quantity and initialization speed are not the main limitations. Instead, the low probability of atom-photon entanglement and inefficient entanglement swapping trouble more, where collective Rydberg state can help to address. Here we propose two schemes for deterministic atom-photon creation and efficient entanglement swapping with the collective Rydberg state in atomic ensembles.

8.1.1 Quantum Circuits Representations for Operations in Rydberg Collective States The system we consider consists of a ground state |g in which a large number of atoms reside, two meta-stable states |s and |s   for qubit encoding, two Rydberg states |g and |g  , and an intermediate excited state |e for Rydberg collective state preparation and readout. We first prepare a Rydberg excitation by a Raman operation between |g and |r  as qubit |1r , whereas collective ground state as |0r , N 1  |0r  = |g1 . . . g N , |1r  = √ |g1 . . . si . . . g N . N i=1

(8.1)

We can see that different qubits encoded with different Rydberg states share the same |0. As shown in Fig. 8.1, the π and π/2 rotation between |g and |r  correspond to the σx and Hadamard gate operations, respectively, of the qubit. Applying a pulse resonant with |e − |r  transition, |1r  is read out to a photon, yet |0r  is read out to no photon. Thus this process can be treated as a swap operation between a qubit and a vacuum field. 1 One can transfer the Rydberg population to one of the metastable states to avoid blocking the second qubit preparation and increase the qubit lifetime, which we call the memory gate. After preparing the two qubits, one can implement CNOT gates between them. Four different two-qubit states vary as follows during the CNOT operations, |0s 0s  −→|0r 0s  −→|0r 0r  −→|0r 1s  −→|0r 1s  −→|0s 1s   |0s 1s  −→|0r 1s  −→|0r 1r  −→|0r 0r  −→|0r 0s  −→|0s 0s   |1s 0s  −→|1r 0s  −→|1r 0r  −→|1r 0r  −→|1r 0s  −→|1s 0s   |1s 1s  −→|1r 1s  −→|1r 1s  −→|1r 1s  −→|1r 1s  −→|1s 1s  

(8.2)

The quantum circuit is an abstract language of quantum information processing [9]. Writing qubit operations in the quantum circuits fashion can, on the one hand, help us think abstractly and, on the other hand, learn and borrow good ideas from other platforms. 1

This is not a reciprocal process; thus not a real swap gate but an equivalence.

8.1 Entanglement Creation and Swapping via Rydberg Interactions σx

Gate r

Energy Level

r π

Memory

π

π/2

g

X

CNOT r

r

s g

Symbol

Hadamard

123

g

1,5 s

3

SWAP r’

r

2,4

e

s’ g

1~5:π

g

H

Fig. 8.1 Gate operations in Rydberg collective states

8.1.2 Deterministic Entanglement Generation Although Rydberg collective state |1r  can be read out to a photon, the |0r  can be In the swap operation, |1r  is read out to a single photon, yet |0r  is read out to no photon, meaning that an atom qubit is swapped to a Fock state-encoded photon qubit, which is not favourable for qubit characterization. Dr Jun Li proposed a scheme to read out a random atom qubit α|0r  + β|1r  to a time-bin photon. As shown in Fig. 8.2a, we introduce an auxiliary qubit in the |0r   and entangle the target qubit and the auxiliary qubit via a CNOT gate (α|0r  + β|1r ) ⊗ |0r   → α|0r 1r   + β|1r 0r  .

(8.3)

Since the auxiliary qubit start from a known initial state |0r  , we can use a simplified version of the CNOT gate by applying a π pulse coupling g ↔ r  as shown in Fig. 8.2b. The two atomic qubits form a logical qubit. Reading out two collective Rydberg states sequentially, we get two complementary Fock-state photons, forming a timebin encoded photonic bit. With a similar idea, we can prepare deterministic entanglement between a timebin photon and an atom qubit. Starting from the atomic ground √ state, we can first prepare the target qubit to a superposition state (|0r  + |1r ) / 2 using a Hadamard operation. With the help of a CNOT gate between the target and the auxiliary qubits, √ two bits can be prepared to the maximally entangled state (|0r 1r   + |1r 0r  )/ 2. Then we read out the target qubit to a Fock state photon. By applying a CNOT gate between the auxiliary qubit and the target qubit, we can get the target qubit involved again. Next, we read out the auxiliary qubit as well, at which point the system becomes a three-body entangled state:  1  √ |0r 1 Ep 0 Lp  + |1r 0 Ep 1 Lp  . 2

(8.4)

124

8 Further Improvement of Atomic Ensemble Quantum Memories

Fig. 8.2 Deterministic atom-photon entanglement scheme

The superscript of the Fock state of the light field indicates its time first (E) after (L). By encoding the two Fock states of the light field as two time-bins of a single photon, we get 1 (8.5) √ (|0r , E + |1r , L). 2 The entanglement generation process is, in principle, deterministic; the scheme also works with a more straightforward experimental configuration than the previous semi-deterministic entanglement scheme [10]. According to the current experimental parameters, the cavity-enhanced Rydberg state readout efficiency is > 10%, which is an order of magnitude higher than the cavity-enhanced DLCZ quantum memory in Chap. 3, and essentially immune to higher-order excitations. Furtherly, after a round of the scheme, we can repeat it many times from the second step. As shown in Fig. 8.2d, a many-body GHZ state between an atom and multiple photons is prepared, where the superscripts of E and L indicate the number of photons. This can also be expanded to the cluster state, which is very useful in optical quantum calculations if the Hadamard operation for atoms is added at the appropriate locations [11–13].

8.1.3 Intra-atom Entanglement Swapping We know that the theoretical upper limit of linear optics Bell state measurements is 50% [14], which drops the total entangling rate by at least half for every step of entanglement swapping in the quantum repeater. The total entangling rate decreases exponentially with the number of entanglement swappings, which is a very unfavourable factor. If the nonlinear interactions between qubits are exploited, it is, in principle,

8.2 Raman Transition-Based Spinwave Operations a

125

b Mx Alice

r

Bob

BSM

Mz BSM

r’

s

s’ g

Middle

Fig. 8.3 Quantum repeater with the intra-atom Bell state measurement. a A BSM is realized by using a CNOT gate and two local measurements. b The schematic of the intra-atom BSM

possible to perform entanglement swapping deterministically. For example, as shown in Fig. 8.3a, an entanglement swapping can be completed with an efficiency of 100% by measuring two qubits on the Z basis and X basis after a CNOT gate between them. As described in Sect. 8.1.1, we have at least two qubits available and a CNOT gate operation between them in the collective Rydberg state system. The scheme is shown in Fig. 8.3b, with the Rydberg atomic ensemble as an intermediate node in a quantum repeater, where the two atomic qubits are responsible for establishing connections to the Alice node and the Bob node, respectively. Taking the atomic qubit on the left as an example, we first prepare a collective Rydberg state |1r  and partially read out the qubit, preparing a Fock state entanglement between the atom and the photon α|0r 1 p  + β|1r 0 p . We use this atom-photon entanglement to build remote entanglement with Alice via the single photon scheme. We repeat this process until the remote entanglement is heralded and built and then transfer the Rydberg qubit to a memory qubit, waiting for the successful entanglement established with Bob. A similar attempt is made with the atomic bit on the right, and when entanglement is successfully established on both sides, we can perform the intra-atom Bell state measurement.

8.2 Raman Transition-Based Spinwave Operations In Chap. 3, we used Raman transitions to manipulate the atomic internal states and achieve atomic qubit rotations. At that time, the two beams of circularly polarized Raman beams were, in fact, two components of a linearly polarized beam. The momentum of the two beams cancelled each other out after the operation, with no effect on the wave vector of the spin wave. Suppose we choose different angles of two Raman beams, in addition to the flip of the atomic state. In that case, the wave vector of the spin wave can also be changed, resulting in many exciting and valuable operations.

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8 Further Improvement of Atomic Ensemble Quantum Memories

8.2.1 Long-Lived Storage of Qubits In Chap. 3, we used two-channel Raman scattering to store qubits in the internal state of the atomic ensemble, but the 70 μs storage time is even shorter than the remote entanglement creation time, limiting its further applications. Meanwhile, most previous long-lifetime storage experiments have only involved long-lived storage of a single spin wave [1, 2, 15] and have not addressed the qubit storage. Here we consider how to extend the storage lifetimes of both spin wave modes of an atomic qubit simultaneously and without compromising their coherence. One of the critical factors limiting the quantum memory lifetime is the atomic motion, including the random thermal motion and the free fall of the atoms after the MOT is turned off. Our experiments’ write-out and read-out modes are at a radius of ∼ 100 μm at the atomic ensemble. If the atoms escape this region, they cannot be read out. According to estimation, for an atom ensemble with a temperature of about T = 10 μK, the storage lifetime limits given by the free fall and the thermal motion are 4 ms and 3 ms, respectively, which are in general agreement with the experimental results [16]. When we consider the storage time less than the millisecond time scale, there is another limiting factor, the dephasing of the spin wave caused by the atomic motion, which weakens the collective enhancement and leads to readout inefficiency [1]. For the atomic ensemble prepared with MOT, atoms barely collide on the time √ scale of interest. Considering atoms with uniform linear motion at a velocity v = k B T /M ≈ 3 cm/s, the spin wave phase becomes ϕi (t) = ks · r(t) = ks · (r0 + vi t).

(8.6)

The atomic thermal motion obeys the Boltzmann distribution, and atoms move randomly in speed and directions; the longer the time, the greater the phase ϕi different between each other; as a result, the readout efficiency decreases. One solution to this problem is to make the write beam co-linear with the write-out photon so that ks ≈ 0 and ϕi do not vary with time and dephasing involved. However, the co-linear configuration is also the noisiest configuration with no spatial filtering involved. Previously, Jiang et al. solved this problem using the spinwave freezing technique [3, 17], where the thermal motion-limited lifetime can be reached at any spatial configuration. The idea of the scheme is shown in Fig. 8.4, where the |s state is transferred, immediately after the spin wave is generated, to the |s   state by a Raman π pulse. In the Raman transition, the atom absorbs a photon with the momentum of k+ while simultaneously emitting a photon with the momentum of k− . Thus the spin wave phase becomes ϕi (t) = (ks + k+ − k− ) · (r0 + vi t).

(8.7)

8.2 Raman Transition-Based Spinwave Operations

127

Fig. 8.4 The principle and setup of the spinwave freezing. a Energy levels and Raman beam configuration. |g and |s   are selected as a pair of clock states. b Time sequence of the experiment. The write and read pulses have a duration of 100 ns. The Raman π pulses have a duration of about 2 μs. The first Raman π pulse is turned on immediately after the write process. T is the storage duration. c Experimental layout. The write beam couples the transition |F = 1 ↔ |F  = 2 with a detuning of −10 MHz and a power of 1 μW. The read pulse resonantly couples the |F = 2 ↔ |F  = 2 transition with a power of 100 μW. The waist of the write or read beam is 200 μm. For the write-out or read-out detection mode, the waist is 90 μm. A bias magnetic field is applied to define the quantization axis and split the Zeeman levels. In order to freeze-out the motion-induced decoherence, intersection angle between the two Raman beams is arranged to be exactly the same as θs. The write-out and read-out photons are first polarization filtered before single-mode fiber coupling and then frequency filtered with etalons and detected with single-photon counters (not shown). The inset: Momentum relationships for write, write-out and Raman beams. Figure from [3]

By choosing an appropriate k+ − k− = ks , the spin wave wavevector can be erased to achieve the same outcome of the co-linear configuration, and the spin wave vector can be restored by repeating the Raman transition before reading out. However, the method cannot be directly applied to qubit storage. For spin wave, |S, the method still works, but for spin waves |S   it obtains exactly the opposite momentum as k− − k+ . Instead of working to freeze the spin wave, this will speed up its dephasing. Note that the phase of the spin wave refers to the relative phase between the |g state and the |s (|s ) state. Therefore we can manipulate the |g state for the same purpose. With the Raman flip from |g to |g  , each flipped atom gains momentum k+ − k− and the spin wave becomes

128

8 Further Improvement of Atomic Ensemble Quantum Memories N 1  ik·ri  i(k+ −k− )·r j  e e |g1 . . . si . . . g N  √ N i=1 j =i

=

N  i=1

ei(k+ −k− )·r j

N 1  i(k−k+ +k− )·ri  ×√ e |g1 . . . si . . . g N . N i=1

(8.8)

The first multiplier contains the global phase, which has no observable effect and can be neglected. We can see that the spinwave wave vector gains a new momentum k− − k+ . It can be seen that attaching a momentum to all |g state atoms is equivalent to attaching an opposite momentum to the |s state atoms2 . We only need to swap k+ and k− for two Raman beams for achieving the same spin wave freezing result. Since this operation is applied on |g state, it works for both spin waves |S and |S  .

8.2.2 The Configuration of Raman Beams in Ring Cavity Setup By zeroing the wavevector of the spin wave via the spin wave freezing technique, we can, in principle, reach the thermal motion-limited lifetime. Nevertheless, in this case, the direction of two Raman beams is fixed (considering only the in-plane case), i.e. one overlapped with the write beam mode and the other overlapped with the writeout beam mode. Nevertheless, the ring cavity on the write-out beam mode hinders the implementation of the Raman beam. Here we find another geometrical configuration for Raman beams here. Assuming there is an angle θ between the write and the writeout beam, as shown in Fig. 8.5a, we fix one of the Raman still overlapped with the write beam and optimized the angle ϕ for the other one as shown in Fig. 8.5b. Though the wavevector cannot be completely zeroed, we can calculate the residual wavevector and give an equivalent write-out angle θ  after the Raman compensation. During optimization, we neglect the small frequency difference among several beams, i.e. |kw | = |kwo | = |k+ | = |k− | = 2π/795 nm. The optimization result at θ = 2◦ is shown in Fig. 8.6. Unsurprisingly, the equivalent write-out angle is zeroed when ϕ = θ . We get a suboptimal solution that θ  = 0.07◦ when setting ϕ = −θ , i.e., the Raman beam symmetry to the write-out beam along the write beam direction. The storage lifetime at this point is [1], τ=

1 = 3.4 ms. |kw | |v| sin θ 

This value is close to the thermal motion-limited lifetime.

2

Interestingly, this behavior resembles the holes in the solid-state physics.

(8.9)

8.2 Raman Transition-Based Spinwave Operations

a

WO(cavity)

129

c

MOT Write

θ Ks

b

MOT

Kr Raman2 φ Raman1

Fig. 8.5 Beam configurations for qubit quantum memories

Fig. 8.6 Optimization of Raman beams for the cavity-enhanced quantum memory

8.2.3 Spinwave Echo-Based Multimode Quantum Memory In Sect. 8.2.1, we choose the Raman beam momentum k+ − k− = ks to zero the spinwave wavevector to obtain a long lifetime. Nonetheless, this is not the only option; we can let the spin wave evolve freely for a while and apply the Raman beams with k+ − k− = 2ks at the moment T . The phase of the spin wave at t > T becomes, (8.10) ϕi (t) = ks · (r0 + vi T ) − ks · [r0 + vi (t − T )] . It can be seen at t = 2T , ϕ  (t) resumes its initial value, and the spin wave can be read out collectively enhanced. Moreover, when 0 < t < 2T or t > 2T , the spin wave is out of phase and thus cannot be read out efficiently. We call the process the spinwave echo.

130

8 Further Improvement of Atomic Ensemble Quantum Memories

Fig. 8.7 The phase evolution in the spinwave echo-based multimode quantum memory

The spinwave echo has two characteristics: firstly, the spinwave preparation and echoes are symmetric in time along the moment of Raman beam operation; secondly, the spinwave can be read out efficiently only at the echo moment but not at the rest of the time. We can establish a one-to-one mapping between the spinwave generation and its readout moments based on these two points. This mapping can be used as multimode storage in the DLCZ quantum memory, as shown in Fig. 8.7. This is a first-in-last-out (FILO) memory, similar to gradient echo storage. In previous storage experiments, we generally set the angle between the write and write-out beams to be small, making the spin wave wavelengths large to slow down its dephasing. However, in the current multimode storage scheme, we want each mode to decoherence quickly so that the next mode can be created soon. According to experiments, the time domain width of each write-out photon is about 100 ns, and the time scale for spinwave decoherence is > 50 μs if a 3◦ small angle is used. To speed up the decoherence, we can do the opposite, as shown in Fig. 8.9, using countercolinear configuration, i.e. |ks | ≈ 2|kw |. The spin wave lifetime can be estimated as about τs ≈ 2 μs. According to previous spinwave freezing experiments, the estimated storage lifetime is about τlife ≈ 1 ms. In principle, the number of modes that can be stored is, τlife /2 ≈ 125. (8.11) 2τs The 1/2 on the numerator accounts for that the write and read processes in our scheme being symmetric with respect to the Raman light operation, each taking half the time. Factor 2 on the denominator is the redundant time for each newly prepared spinwave mode to decoherence until the next one is prepared, in case of crosstalk. Figure 8.8 shows a schematic for the 4-mode scenario. Next, let us consider how to achieve k+ + k− = 2ks . Due to the counter-colinear configuration we choose, the wavevector of the spinwave is too big to be reverted with a single Raman π pulse. Instead, we can set up the Raman beams as shown in Fig. 8.9.

8.2 Raman Transition-Based Spinwave Operations

131

Fig. 8.8 Simulated relative readout efficiency as a function of time in a four-mode quantum memory

Fig. 8.9 The beam configuration for the multimode quantum memory

After one π pulse, the wavevector is zeroed, and then we swap two Raman beams and apply the Raman π pulse again, resulting in a reverse spinwave. In addition, to avoid the overlap of one of the Raman beams with the cavity-enhanced write-out mode, we can change the θ  while keeping its angle bisector parallel with the wavevector, such that the spinwave will echo slower, but the scheme still works. Lastly, we consider the noise in the scheme. In practice, all spinwave modes will be read out when the read beam is applied, though usually not collectively enhanced in a specific direction. But this non-enhanced reading will contribute noise to the desired mode. Meanwhile, the noise will increase linearly as the mode number increases. Simon et al. [18] proposed an idea to solve this problem, and Heller et al. [19] verified the scheme in gradient-echo memory recently. The general idea of this scheme is to suppress the noise by using the cavity. In the DLCZ quantum memory, there are two main sources of read-out noises as follows, • The unwanted spinwave modes correlated with write-out photons other than the collecting direction are read out inefficiently to the read-out mode. • In the cavity-enhanced memory, the cavity also enhances the unwanted spinwave modes.

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8 Further Improvement of Atomic Ensemble Quantum Memories

In order to tackle these two noises, the scheme proposes, • In the write process, one suppresses the unwanted spinwave modes, i.e. write-out photons to other directions, with the help of a cavity on the specific write-out mode. • In the read process, one removes the cavity to reduce its enhancement for the unwanted spinwave modes. In the work of Heller et al., for a ten-mode memory, the cross-correlation function (2) gwo,r o between the write-out and read-out photons was reduced from 16 to 8 only, with about 7.3 times the suppression effect on noise. In order to realize the scheme in the DLCZ system, a key challenge is that to keep an efficient read process for the cavity-free configuration, a large enough optical depth of the atomic ensemble is needed.

References 1. Zhao B, Chen YA, Bao XH et al (2009) A millisecond quantum memory for scalable quantum networks. Nat Phys 5(2):95 2. Yang SJ, Wang XJ, Bao XH et al (2016) An efficient quantum light-matter interface with sub-second lifetime. Nat Photonics 10(6):381 3. Jiang Y, Rui J, Bao XH et al (2016) Dynamical zeroing of spin-wave momentum to suppress motional dephasing in an atomic-ensemble quantum memory. Phys Rev A 93(6):063819 4. Saffman M, Walker TG, Mølmer K (2010) Quantum information with Rydberg atoms. Rev Mod Phys 82(3):2313–2363 5. Saffman M, Mølmer K (2008) Scaling the neutral-atom Rydberg gate quantum computer by collective encoding in holmium atoms. Phys Rev A 78(1):012336 6. Barredo D, Lienhard V, De Leseleuc S et al (2018) Synthetic three-dimensional atomic structures assembled atom by atom. Nature 561(7721):79–82 7. Omran A, Levine H, Keesling A et al (2019) Generation and manipulation of Schrödinger cat states in Rydberg atom arrays. Science 365(6453):570–574 8. Graham T, Kwon M, Grinkemeyer B et al (2019) Rydberg-mediated entanglement in a twodimensional neutral atom qubit array. Phys Rev Lett 123(23):230501 9. Nielsen MA, Chuang I (2002) Quantum computation and quantum information. American Association of Physics Teachers 10. Li J, Zhou MT, Yang CW et al (2019) Semideterministic entanglement between a single photon and an atomic ensemble. Phys Rev Lett 123(14):140504 11. Walther P, Resch KJ, Rudolph T et al (2005) Experimental one-way quantum computing. Nature 434(7030):169–176 12. Lindner NH, Rudolph T (2009) Proposal for pulsed on-demand sources of photonic cluster state strings. Phys Rev Lett 103(11):113602 13. Economou SE, Lindner N, Rudolph T (2010) Optically generated 2-dimensional photonic cluster state from coupled quantum dots. Phys Rev Lett 105(9):093601 14. Calsamiglia J, Lütkenhaus N (2001) Maximum efficiency of a linear-optical Bell-state analyzer. Appl Phys B 72(1):67–71 15. Radnaev AG, Dudin YO, Zhao R et al (2010) A quantum memory with telecom-wavelength conversion. Nat Phys 6:894–899 16. Bao XH, Reingruber A, Dietrich P et al (2012) Efficient and long-lived quantum memory with cold atoms inside a ring cavity. Nat Phys 8(7):517–521

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17. Rui J, Jiang Y, Yang SJ et al (2015) Operating spin echo in the quantum regime for an atomicensemble quantum memory. Phys Rev Lett 115(13):133002 18. Simon C, de Riedmatten H, Afzelius M (2010) Temporally multiplexed quantum repeaters with atomic gases. Phys Rev A 82:010304 19. Heller L, Farrera P, Heinze G et al (2020) A cold atom temporally multiplexed quantum memory with cavity-enhanced noise suppression. arXiv Prepr. arXiv:2003.08418

Chapter 9

Conclusion and Outlook

Quantum networks are one of the most highly anticipated long-term goals in quantum information science. Many blueprints regarding its architecture have been drawn [1– 8] and open-source scripts have been developed [9] to simulate its operation. In contrast, its experimental realization is more challenging and less developed. While not as demanding as the high fidelity requirements for qubit operations in quantum computing, quantum networks are demanding in another dimension: sufficiently long coherent storage for qubits and well-separated entanglement. The various systems have been greatly improved over the years in terms of coherent storage. In the rareearth ion system, a coherence time as long as six hours is realized [10]; In the cold atom system, coherent storage of 0.22 s is achieved [11]. Nevertheless, in terms of entanglement distance, experimental progress has been slow, with the longest entanglement distance currently being only 1.3 km. The limiting factor includes the low probability of the entanglement source, low entanglement verification efficiency, high photon transmission loss, and mismatching between multiple memories, among many others. The primary motivation of this thesis is to try to push the entanglement distance limit using the cold atomic system. The basis for achieving this is the cavity-enhanced quantum memory with a high probability atom-photon entanglement source introduced in Chap. 3, which has an order of magnitude improvement than our previous non-cavity-enhanced version. In the meantime, by using the quantum frequency conversion module introduced in Chap. 4, we convert the 795 nm single photons from the cavity-enhanced DLCZ memory to 1342 nm nm in the telecommunication band with an efficiency of 33% and without losing its coherence. Based on these, in Chap. 5, we entangled two quantum memories using the two-photon scheme via 22 km fielddeveloped fibre transmission. As a comparison, the transmission of 795 nm photons in a 22 km fibre is 2 × 10−8 , while the transmission of 1342 nm nm photons in the same fibre is 0.22, with seven order of magnitudes improvement. Nevertheless, the entangling rate of the two-photon scheme is a quadratic function of the atom-photon entanglement probability. The DLCZ memory is probabilistic by © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Yu, Long Distance Entanglement Between Quantum Memories, Springer Theses, https://doi.org/10.1007/978-981-19-7939-2_9

135

136

9 Conclusion and Outlook

Table 9.1 Comparison of two-node remote entanglement experiments Experiment Two-photon (Chap. 5) Single-photon (Chap. 6) Physical separation Overall fiber length, L Entanglement probability, Pent Entanglement quality Entanglement creation time, Tent Quantum link efficiency, ηlink Assumed memory lifetime, τm

NV center [12]

0.6 m 22 km 1.46 × 10−6

0.6 m 50 km 3.85 × 10−4

1.3 km 1.7 km 6.4 × 10−9

F = 0.732 ± 0.038

C = 0.407 ± 0.008

F = 0.92 ± 0.03

75 s

0.65 s

1.3 × 103 s

2.9 × 10−3

0.34

4.6 × 10−4

0.22 s [11]

0.22 s [11]

0.6 s [13, 15]

nature, and we have to run it in the low-probability regime to avoid the higher-order mode excitation, limiting the entangling rate in the two-photon scheme. To further extend the entanglement distance, in Chap. 6 , we adopt the single-photon scheme. In the single-photon scheme, the entangling rate is a linear function of the atomphoton entanglement probability, and the equivalent transmission distance is halved. Regardless of the technical challenges, we can extend the entanglement to ∼150 km.1 The single-photon scheme imposes high requirements on the phase controlling of both the lasers and the fibres. We have designed a phase-locking scheme by using two lasers in the same frequency and the different frequency as the single photon for longtime-scale phase calibration and short-time-scale phase following, respectively. With this scheme, we successfully limit the optical path length difference of two 25 km fibres to about 50 nm, enabling the remote entanglement establishment through the single-photon scheme. Table 9.1 compares the performance of the two-photon experiment, single-photon experiment, and the NV center experiment [12]. The single-photon experiment has the highest entangling rate Pent despite the longest distance. In the long-fibre case, the propagation delay results in a maximal repetition rate of Rrep = C/L, where C ≈ 2 × 10−8 is the speed of light in the fibre. Thus the heralded entanglement creation time is estimated as Tent = (Rrep Pent )−1 . The entanglement link efficiency [13, 14] is defined as the ratio of the memory lifetime τm (the state-of-the-art results are used for estimation) to the entanglement generation time Tent , ηlink =

τm . Tent

(9.1)

2 η , where p is the measured In the two-photon scheme, the entangling rate is Pent  0.5 pw L w   2 p η  . Let write-out probability. In the single-photon scheme, the entangling rate is Pent w L /2  Pent = Pent , we can estimate the count rate in the single-photon scheme, thus estimating the upper bound for the entanglement distance.

1

References

137

ηlink  0.83 [13] is an important milestone in quantum network construction because it means that the entanglement generation process can be considered deterministic. We can see that the single-photon experiment is close to this goal. In order to approach the goal of building a quantum network, some shortcomings in the current experiment must be fixed. First, the lifetime of the quantum memory in the current experiments does not meet the requirement of running the quantum repeater protocol. With the protocol we proposed in Sect. 8.2, we expect to improve the qubit storage lifetime to the order of ms. In order to push the memory lifetime to the second scale, a 3D optical lattice needs to be implemented, leaving some technical challenges. Second, the two memories in the current experiment are still placed in the same lab with less than 1 m spatial separation, which needs to be spatially separated in the next step. Especially for the single-photon scheme, as described in Sect. 6.5, technical challenges in laser phase synchronization remains. Third, limited by the probabilistic nature of DLCZ memories, the trade-off between higher-order excitations and the atom-photon entanglement probability is inevitable. Introducing Rydberg interactions can fundamentally eliminate the higher-order excitations, which we have done some experimental works [16, 17] and drawn some theoretical framework as described in Sect. 8.1. In addition to entanglement between distant memories, this thesis covers some other relatively independent parts. One part is the experiment presented in Chap. 7 to verify an EIT quantum memory with a measurement device-independent scheme. A similar method can be applied to other quantum memories and processors as a standard test procedure. One of the weaknesses in the current experiment is the low brightness of the Rydberg single-photon source, which makes verifying the long-lifetime case difficult. Improving the readout efficiency will be one of the most important directions of the Rydberg system. The other part is some schemes for improving the performance of atomic ensemble-based quantum memories as described in Chap. 8, which I hope will be helpful for future experiments.

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8. Dahlberg A, Skrzypczyk M, Coopmans T et al (2019) A link layer protocol for quantum networks. In: Proceedings of the ACM special interest group on data communication, pp 159173 9. DiAdamo S, Nözel J, Zanger B et al (2020) QuNetSim: a software framework for quantum networks. arXiv Prepr. arXiv:2003.06397 10. Zhong M, Hedges MP, Ahlefeldt RL et al (2015) Optically addressable nuclear spins in a solid with a six-hour coherence time. Nature 517(7533):177–180 11. Yang SJ, Wang XJ, Bao XH et al (2016) An efficient quantum light-matter interface with sub-second lifetime. Nat Photonics 10(6):381 12. Hensen B, Bernien H, Dréau AE et al (2015) Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres. Nature 526(7575): 682–686 13. Humphreys PC, Kalb N, Morits JPJ et al (2018) Deterministic delivery of remote entanglement on a quantum network. Nature 558(7709):268–273 14. Monroe C, Raussendorf R, Ruthven A et al (2014) Large-scale modular quantum-computer architecture with atomic memory and photonic interconnects. Phys Rev A 89(2):022317 15. Bar-Gill N, Pham L, Jarmola A et al (2013) Solid-state electronic spin coherence time approaching one second. Nat Commun 4(1):1743 16. Li J, Zhou MT, Jing B et al (2016) Hong-Ou-Mandel interference between two deterministic collective excitations in an atomic ensemble. Phys Rev Lett 117(18):180501 17. Li J, Zhou MT, Yang CW et al (2019) Semideterministic entanglement between a single photon and an atomic ensemble. Phys Rev Lett 123(14):140504