Quantum theory of optical temporal phase and instantaneous frequency. II. Continuous time limit and state-variable approach to phase-locked loop design

Abstract

We consider the continuous-time version of our recently proposed quantum theory of optical temporal phase and instantaneous frequency [Tsang, Shapiro, and Lloyd, Phys. Rev. A 78, 053820 (2008)]. Using a state-variable approach to estimation, we design homodyne phase-locked loops that can measure the temporal phase with quantum-limited accuracy. We show that post-processing can further improve the estimation performance, if delay is allowed in the estimation. We also investigate the fundamental uncertainties in the simultaneous estimation of harmonic-oscillator position and momentum via continuous optical phase measurements from the classical estimation theory perspective. In the case of delayed estimation, we find that the inferred uncertainty product can drop below that allowed by the Heisenberg uncertainty relation. Although this result seems counter-intuitive, we argue that it does not violate any basic principle of quantum mechanics.

11 pages, 6 figures, v2: accepted by PRA

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• Adaptive homodyne measurement of optical phase
• Adaptive Quantum Measurements of a Continuously Varying Phase
• State determination in continuous measurement
• Master Equation for Retrodiction of Quantum Communication Signals
• Adaptive phase measurements for narrowband squeezed beams
• Quantum theory of optical temporal phase and instantaneous frequency
• Adaptive phase estimation is more accurate than non-adaptive phase estimation for continuous beams of light

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