Quantum Optical Sensing: Single Mode, MultiMode, and Continuous Time

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

• Jeffrey H. Shapiro

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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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Phase-Sensing Interferometry with Classical Light

• Phase-conjugate Mach-Zehnder interferometer
• Homodyne measurement of
  

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Phase-Sensing Interferometry with Coherent States

• Phase-conjugate Mach-Zehnder interferometer
• Homodyne measurement of
  

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Phase-Sensing Interferometry with Squeezed States

• Phase-conjugate Mach-Zehnder interferometer
• Homodyne measurement of
  

Caves, PRD (1981) Bondurant Shapiro, PRD (1984)
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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)
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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)
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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)
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N00N-State Phase Measurement

• Phase-conjugate interferometer with
  measurement
• and N00N-state source
• Phase-measurement probability density function

Lee, Kok, Dowling JMO (2002)
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Phase Measurement with Guaranteed Precision

• Phase-conjugate interferometer with
  measurement
• and N00N-state sum
• Optimum phase-measurement probability density
  function

Shapiro, Phys Scripta (1993)
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Performance Comparison for ? 0 and N 50

• Phase-conjugate interferometry

• Two-mode measurement

• Only the coherent-state case degrades gracefully
  with loss!

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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

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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!

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Can Classical Light Do Even Better than 1/N3/2?

• 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)
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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)
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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