Downloaded from orbit.dtu.dk on: Dec 20, 2017 On quantum amplifiers, quantum discrimination and adaptive phase estimation Usuga Castaneda, Mario A.; Andersen, Ulrik Lund Publication date: 2012 Document Version Publisher's PDF, also known as Version of record Link back to DTU Orbit Citation (APA): Castaneda, M. A. U., & Andersen, U. L. (2012). On quantum amplifiers, quantum discrimination and adaptive phase estimation. 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Usuga Physics Department Technical University of Denmark A thesis submitted for the degree of Doctor of Philosophy February 2012 To Angela, Diana, Cristina, Mom and Dad for their understanding and continuous support. ii iii Acknowledgements I want to thank the members of the QIV group at the Max Planck institute who made me feel welcome during my stay in Erlangen and specially to Professor Gerd Leuchs for allowing this fruitful experience to happen. Likewise, I thank the members of QUIN group at DTU who have accompanied me during the last 2 years. I am specially grateful to my supervisor Ulrik L. Andersen for the freedom that he gave me in pursuing my research but also for his advice that kept me in the right track, and for his disposition and interest. I also would like to thank my closest collaborators, Christian Mueller, Mikael Lassen, Lars Madsen, Bo Melholt Nielsen, Christoffer Wittmann, Christoph Mar- quardt, Radim Filip and Petr Marek. During this time, I enjoyed and appreciated discussions about physics (and/or life in general) with Mikael Lassen, Alex Huck, Metin Sabuncu, Josef Fuerst, Jonas Schou Neergaard-Nielsen, Amine Laghaout, Ulrich Busk Hoff, Niels Israelsen Kristiansen, Mathieu Manceau, Adriano Berni and Miroslav Jezek (among others). And specially the long discussions during and after the experiments I had with Christian Mueller who had a remarkable participation in two of the experiments here presented. I finally want to thank my family and friends who always supported me and made this walk easier to follow. iv Contents 1 Introduction 1 1.1 Overview .................................. 1 1.2 Thesis structure .............................. 1 2 Basic Quantum Optics 3 2.1 EM Quantization ............................. 4 2.1.1 Heisenberg uncertainty ...................... 4 2.2 Quadrature operators ........................... 5 2.3 Homodyne detection ........................... 7 2.4 Density Operator (generalized preparation) ............... 10 2.4.1 The Density Matrix ........................ 10 2.5 Wigner, P and Q functions ....................... 12 2.6 Gaussian states .............................. 14 2.6.1 Vacuum state ........................... 14 2.6.2 Coherent state .......................... 15 2.6.3 Squeezed state .......................... 16 2.6.4 Thermal state ........................... 17 2.7 POVMs (generalized measurements) .................. 17 2.8 Phase probability distribution ...................... 20 2.8.1 Holevo variance .......................... 20 3 Quantum Amplifiers 23 3.1 Introduction ................................ 23 3.2 On the Impossibility of perfect Amplification .............. 25 3.2.1 No-cloning argument ....................... 25 3.2.2 Phase Insensitive Linear Amplifier ................ 27 3.2.3 Probabilistic Noiseless Amplifiers ................ 29 3.3 Noiseless Concentration of Phase Information ............. 33 v CONTENTS 3.3.1 Introduction ............................ 33 3.3.2 Theory ............................... 35 3.3.3 Experimental implementation .................. 41 3.3.4 Results ............................... 43 3.3.5 Conclusion ............................. 51 4 Adaptive Single Shot Phase Estimation 53 4.1 Introduction ................................ 53 4.1.1 Measurement vs Estimation ................... 59 4.1.2 Phase estimation classification .................. 60 4.2 Semiclassical theory ............................ 61 4.3 Quantum theory ............................. 64 4.4 Feedback .................................. 65 4.5 Simulations ................................ 66 4.6 Experiment ................................ 69 4.6.1 Setup ............................... 69 4.6.2 Experimental Results ....................... 76 4.6.3 Conclusions ............................ 81 5 Quantum State Discrimination 83 5.1 Introduction ................................ 83 5.2 Binary decision problem for coherent states ............... 85 5.2.1 Detection Schemes and Helstrom bound ............ 86 5.3 Discrimination of a QPSK alphabet via a hybrid receiver ....... 90 5.3.1 Introduction ............................ 90 5.3.2 Description and Theory ..................... 91 5.3.3 Experiment ............................ 97 5.3.4 Experimental Results ....................... 99 5.3.5 Conclusion. ............................ 102 6 Conclusions 103 vi List of Figures 2.1 Field quadratures ............................. 5 2.2 Homodyne detection ........................... 7 2.3 HD time traces and Photon statistics .................. 9 2.4 Familiy of Gaussian states. ........................ 18 2.5 Phase variance problem .......................... 21 3.1 Overlap of coherent states ........................ 24 3.2 Perfect cloner. ............................... 26 3.3 Linear amplifier. .............................. 27 3.4 Example of a linear amplifier setup. ................... 28 3.5 Probabilistic amplifier. .......................... 29 3.6 Quantum scissors approach to quantum amplification. ........ 30 3.7 Probabilistic Amplifiers. ......................... 32 3.8 Effective gain comparison. ........................ 32 3.9 Sketch of procedure and states. ..................... 34 3.10 Phase space diagram ........................... 35 3.11 Theoretical plots ............................. 38 3.12 Normalized Holevo phase variance. ................... 40 3.13 thermal coherent state .......................... 42 3.14 Density matrices and populations .................... 45 3.15 Reconstructed Wigner functions ..................... 46 3.16 Phase probability distribution ...................... 47 3.17 Theoretical vs experimental phase variance .............. 48 3.18 Amplifier’s gain for different thresholds M. ............... 49 3.19 Success Probability for the different states considered. ...... 50 Psuc 3.20 Arbitrary coherent state noiseless amplification ............ 51 4.1 Phase Metrology ............................. 54 4.2 Simultaneous measurement of conjugate quadratures ......... 56 vii LIST OF FIGURES 4.3 Phase estimation scheme ......................... 59 4.4 Single shot homodyne measurement. .................. 63 4.5 Simulated feedback ............................ 66 4.6 Effect of feedback gain . ......................... 67 4.7 Summary of simulations ......................... 68 4.8 AdaptiveSetup .............................. 70 4.9 Time traces for Locking and Scanning the phase of LO ........ 72 4.10 Main components of the FPGA ..................... 74 4.11 Time scales ................................ 76 4.12 Time traces ................................ 77 4.13 Estimates Analysis ............................ 78 4.14 Phase Estimates at different phases ................... 80 4.15 variance of estimates with different state amplitudes .......... 81 5.1 Binary state comunication ........................ 85 5.2 Detection schemes and alphabets in phase space ............ 88 5.3 Error probabilities for the different detector schemes vs α 2 ..... 89 | | 5.4 Schematic of the hybrid discrimination scheme. ............ 92 5.5 Hybrid receivers in phase space. ..................... 93 5.6 Sketch illustrating the two elementary displacements. ......... 95 5.7 Optimal displacement parameters .................... 96 5.8 Optiml trasmittance ........................... 97 5.9 Setup for discrimination of the QPSK coherent states. ......... 98 5.10 Experimental results for the error rates of the hybrid receivers .... 100 5.11 Error rate comparison .......................... 102 1 FPGAspecs ................................ 113 2 FPGAfilter ................................ 114 3 Digitalboard ................................ 115 4 Analogboard ................................ 115 5 switch ................................... 119 6 Variable gain fast amplifier ........................ 119 7 PID Controller .............................. 120 viii List of Tables 2.1 Normalization of Quadrature Operators. ................ 6 2.2 Most common operators in quantum optics. .............. 22 3.1 Experimental Parameters. ........................ 43 3.2 Norm, purity and Fidelity for reconstructed density matrices ..... 44 4.1 Phase estimation classification. ..................... 61 1 WGM characteristics ........................... 118 ix List of Symbols and Notation Aˆ Operator