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

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

3 Phase-Sensing Interferometry with Classical Light  Phase-conjugate Mach-Zehnder interferometer:  Homodyne measurement of :

4 Phase-Sensing Interferometry with Coherent States  Phase-conjugate Mach-Zehnder interferometer:  Homodyne measurement of :

5 Phase-Sensing Interferometry with Squeezed States  Phase-conjugate Mach-Zehnder interferometer:  Homodyne measurement of Caves, PRD (1981); Bondurant & Shapiro, PRD (1984)

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 Shapiro & Shepard PRA (1991)

7 Susskind-Glogower Phase Measurement  Susskind-Glogower (SG) phase operator  SG positive operator-valued measurement (POVM)  SG-POVM probability density function Susskind & Glogower, Physics (1964) Shapiro & Shepard PRA (1991)

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 Shapiro & Shepard PRA (1991)

9 N00N-State Phase Measurement  Phase-conjugate interferometer with measurement and N00N-state source  Phase-measurement probability density function Lee, Kok, & Dowling JMO (2002)

10 Phase Measurement with Guaranteed Precision  Phase-conjugate interferometer with measurement and N00N-state sum  Optimum phase-measurement probability density function Shapiro, Phys Scripta (1993)

11 Performance Comparison for  = 0 and N = 50  Phase-conjugate interferometry  Two-mode measurement  Only the coherent-state case degrades gracefully with loss!

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 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 Can Classical Light Do Even Better than 1/N 3/2 ?  Pulse-frequency modulation analog communication transmitted as a coherent state and received by heterodyning  Cramér-Rao bound on rms error in estimate is  Cramér-Rao-bound performance prevails when  With exponential bandwidth expansion, goes down exponentially with increasing Yuen, Quantum Squeezing (2004)

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 Shapiro, Quantum Semiclass Opt. (1998)

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