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

TL;DR

This paper develops a continuous-time quantum theory for optical phase measurement, designing phase-locked loops that achieve quantum-limited accuracy and exploring fundamental uncertainties in quantum estimation.

Contribution

It introduces a state-variable approach for quantum phase estimation and demonstrates how post-processing and delay can improve measurement accuracy beyond traditional limits.

Findings

Homodyne phase-locked loops can measure quantum-limited optical phase.

Post-processing with delay can enhance estimation accuracy.

Delayed estimation can seemingly surpass Heisenberg uncertainty bounds.

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.