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Quantum Optical Sensing: Single Mode, Multi-Mode, and Continuous Time

2. Quantum Optical Sensing. Single-mode optical interferometrysemiclassical theory: shot-noise limited performancequantum theory: coherent-state versus squeezed-state operationQuantum phase measurementSusskind-Glogower positive operator-valued measurement two-mode phase measurement: N00N-st

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Quantum Optical Sensing: Single Mode, Multi-Mode, and Continuous Time

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    1. October 1, 2007 Quantum Optical Sensing: Single Mode, Multi-Mode, and Continuous Time Jeffrey H. Shapiro

    2. 2 Quantum Optical Sensing Single-mode optical interferometry semiclassical theory: shot-noise limited performance quantum theory: coherent-state versus squeezed-state operation Quantum phase measurement Susskind-Glogower positive operator-valued measurement two-mode phase measurement: N00N-state performance two-mode phase measurement with guaranteed precision Continuous-time optical sensing semiclassical theory: shot-noise limited broadband performance quantum theory: what are the ultimate limits? Conclusions

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    6. 6 Single-Mode Number and Phase Wave Functions Single-mode field with annihilation operator Number kets and phase kets Number-ket and phase-ket state representations Fourier transform relation

    7. 7 Susskind-Glogower Phase Measurement Susskind-Glogower (SG) phase operator SG positive operator-valued measurement (POVM) SG-POVM probability density function

    8. 8 Two-Mode Phase Measurement Signal and conjugate modes: A pair of commuting observables: When conjugate mode is in its vacuum state, measurement yields outcome with the SG-POVM probability density BUT other behavior is possible when signal and conjugate are entangled

    9. 9 N00N-State Phase Measurement Phase-conjugate interferometer with measurement and N00N-state source Phase-measurement probability density function

    10. 10 Phase Measurement with Guaranteed Precision Phase-conjugate interferometer with measurement and N00N-state sum Optimum phase-measurement probability density function

    11. 11 Performance Comparison for ? = 0 and N = 50 Phase-conjugate interferometry Two-mode measurement

    12. 12 Continuous-Time Coherent-State Vibration Sensing Multi-bounce interrogation of vibrating mirror Coherent-state source and heterodyne detection receiver gives instantaneous frequency swing Work in the wideband frequency modulation (WBFM) regime

    13. 13 Continuous-Time Coherent-State Vibration Sensing Above-threshold WBFM reception requires Above-threshold WBFM rms velocity error is beating behavior seen earlier for nonclassical light is the average number of detected signal photons in the vibration-signature bandwidth Because classical light is used, loss degradation is graceful!

    14. 14 Can Classical Light Do Even Better than 1/N3/2? Pulse-frequency modulation analog communication transmitted as a coherent state and received by heterodyning Cramr-Rao bound on rms error in estimate is Cramr-Rao-bound performance prevails when With exponential bandwidth expansion, goes down exponentially with increasing

    15. 15 Towards the Ultimate Quantum Limit The Fourier duality between the number kets and phase kets for a single-mode field suggests that we seek a similar duality for continuous time For unity quantum efficiency continuous-time direct detection the measurement eigenkets are known: produces a photocount waveform on with counts at (and only at) A suitable Fourier transform of this state may guide us to the ultimate quantum measurement for instantaneous frequency

    16. 16 Conclusions Single-mode interferometric phase measurements standard quantum limit achieved by coherent states Heisenberg limit achieved by squeezed states Two-mode phase measurements Heisenberg limit achieved by N00N states guaranteed precision at Heisenberg limit achieved by N00N sum The BAD news highly squeezed states and high-order N00N states hard to generate nonclassical-state phase sensors do not degrade gracefully with loss The GOOD news continuous-time, coherent-state, wideband systems may offer superior performance and are robust to loss effects theorists still have some fundamental quantum limits to determine

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