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

physical review. a, atomic, molecular, and optical physics/physical review, a, atomic, molecular, and optical physics

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