CLASSICAL AND QUANTUM PARAMETER ESTIMATION THEORY FOR OPTICAL SPECTROSCOPY AND IMAGING ANG SHAN ZHENG NATIONAL UNIVERSITY OF SINGAPORE 2016 CLASSICAL AND QUANTUM PARAMETER ESTIMATION THEORY FOR OPTICAL SPECTROSCOPY AND IMAGING ANG SHAN ZHENG (B.Eng.(Hons.), NUS) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2016 DECLARATION I hereby declare that this thesis is my original work and it has been written by me in its entirety. I have duly acknowledged all the sources of information which have been used in the thesis. This thesis has also not been submitted for any degree in any university previously. Ang Shan Zheng August 5, 2016 Acknowledgement Before venturing any further, I want to thank my supervisor, Dr. Mankei Tsang, forhisguidancethroughoutthisresearchandhisgenerosityinfund- ing my study and research. Special thanks also goes to my colleagues, Mr. Shilin Ng, Dr. Ranjith Nair, Dr. Xiao-Ming Lu, and Dr. Shibdas Roy for countless hours of aca- demic discussions and inspirations. I am grateful for the pleasant research environment in our research group, and I would like to thank our former group members, Dr. Adam Chaudhry, Mr. Soham Saha, Ms. Dan Li, and Dr. Andy Chia. I also thank all my collaborators, Dr. Glen I. Harris, Pro- fessor Warwick P. Bowen, Dr. Trevor A. Wheatley, Dr. Hidehiro Yonezawa, Professor Akira Furusawa, and Professor Elanor H. Huntington. Without their efforts, experiments described in this thesis would not have been pos- sible. Foremost, I would like to thank my lovely wife ChinHui for the patience she has had with me throughout this journey and the love and encourage- ment she has given me to succeed. I wish to thank my sister Siao Wen and my parents in Malaysia for their support and trust in pursuing my interests. i ii Contents Acknowledgement i Table of Contents vi Summary vii List of Tables ix List of Figures xii List of Symbols xiii List of Abbreviations xv 1 Introduction 1 1.1 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . 2 2 Theoretical Background 5 2.1 Background on Measurement Theory . . . . . . . . . . . . . 5 2.1.1 Classical measurement theory . . . . . . . . . . . . . 5 2.1.2 Quantum measurement theory . . . . . . . . . . . . . 8 2.2 Background on Parameter Estimation Theory . . . . . . . . 15 2.2.1 Parameter estimation theory . . . . . . . . . . . . . . 15 2.2.2 Examples of estimators . . . . . . . . . . . . . . . . . 17 2.2.3 Cram´er-Rao (CR) lower bound . . . . . . . . . . . . 20 2.2.4 Quantum estimation theory . . . . . . . . . . . . . . 23 iii 2.2.5 Quantum Cram´er-Rao (QCR) bound . . . . . . . . . 24 3 Parameter Estimation with Optomechanical Systems 29 3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.2 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.3 Theoretical Model . . . . . . . . . . . . . . . . . . . . . . . . 31 3.4 Review of CR Lower Bound . . . . . . . . . . . . . . . . . . 33 3.4.1 Fisher information . . . . . . . . . . . . . . . . . . . 36 3.5 Parameter Estimation Algorithms . . . . . . . . . . . . . . . 36 3.5.1 Averaging . . . . . . . . . . . . . . . . . . . . . . . . 36 3.5.2 Radiometer . . . . . . . . . . . . . . . . . . . . . . . 37 3.5.3 EM algorithm . . . . . . . . . . . . . . . . . . . . . . 38 3.6 Experimental Data Analysis . . . . . . . . . . . . . . . . . . 42 3.6.1 Procedure . . . . . . . . . . . . . . . . . . . . . . . . 42 3.6.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.7 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4 Estimation of Spectral Parameters with Quantum Dynam- ical Systems 49 4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.2 Quantum Metrology . . . . . . . . . . . . . . . . . . . . . . 50 4.2.1 Review of parameter estimation theory . . . . . . . . 50 4.2.2 Extended convexity of QCR bound . . . . . . . . . . 51 4.2.3 Tighter bounds . . . . . . . . . . . . . . . . . . . . . 53 4.3 Continuous Optical Phase Modulation . . . . . . . . . . . . 57 4.3.1 Homodyne detection . . . . . . . . . . . . . . . . . . 58 4.3.2 Spectral photon counting . . . . . . . . . . . . . . . . 59 4.3.3 Ornstein-Uhlenbeck spectrum analysis . . . . . . . . 63 4.4 Experimental Data Analysis . . . . . . . . . . . . . . . . . . 66 4.4.1 Procedure . . . . . . . . . . . . . . . . . . . . . . . . 69 iv 4.4.2 Data recalibration . . . . . . . . . . . . . . . . . . . 71 4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5 Imaging 75 5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.2 Source and System Model . . . . . . . . . . . . . . . . . . . 77 5.3 Quantum Limit on Two-source Localization . . . . . . . . . 80 5.3.1 Review of the QCR bound . . . . . . . . . . . . . . . 80 5.3.2 Quantum Fisher information (QFI) matrix for two- source localization . . . . . . . . . . . . . . . . . . . 81 5.3.3 Comparison to direct imaging . . . . . . . . . . . . . 87 5.4 Super Localization by Image Inversion (SLIVER) Interfer- ometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.4.1 Detection probabilities . . . . . . . . . . . . . . . . . 94 5.4.2 Fisher information matrix for SLIVER . . . . . . . . 97 5.5 Two-dimensional Spatial-mode Demultiplexing (SPADE) . . 101 5.6 Monte-Carlo Analysis of SLIVER and SPADE . . . . . . . . 104 5.6.1 Monte-Carlo analysis of SLIVER . . . . . . . . . . . 104 5.6.2 Monte-Carlo analysis of SPADE . . . . . . . . . . . . 106 5.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 6 Concluding Remarks 111 6.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.2 Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 A Super-resolution Microscopy 129 B Recent Advances in Quantum Theory of Two-source Local- ization 131 B.1 Thermal Point Sources . . . . . . . . . . . . . . . . . . . . . 131 v