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arXiv:0902.3034 · Physical Review A

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

Mankei Tsang, Jeffrey H. Shapiro, Seth Lloyd

Published May 21, 2009• 52 citations
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Abstract

We consider the continuous-time version of our recently proposed quantum theory of optical temporal phase and instantaneous frequency [M. Tsang et al., 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 postprocessing 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 counterintuitive, we argue that it does not violate any basic principle of quantum mechanics.

DOI
10.1103/physreva.79.053843

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We consider the continuous-time version of our recently proposed quantum theory of optical temporal phase and instantaneous frequency [M. Tsang et al., 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 postprocessing 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 counterintuitive, we argue that it does not violate any basic principle of quantum mechanics.
We consider the continuous-time version of our recently proposed quantum theory of optical temporal phase and instantaneous frequency [M. Tsang et al., 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 postprocessing 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 counterintuitive, we argue that it does not violate any basic principle of quantum mechanics.
We propose a general quantum theory of optical phase and instantaneous frequency in the time domain for slowly varying optical signals. Guided by classical estimation theory, we design homodyne phase-locked loops that enable quantum-limited measurements of temporal phase and instantaneous frequency. Standard and Heisenberg quantum limits to such measurements are then derived. For optical sensing applications, we propose multipass and Fabry-P\'erot position and velocity sensors that take advantage of the signal-to-noise-ratio enhancement effect of wide-band angle modulation without requiring nonclassical light. We also generalize our theory to three spatial dimensions for nonrelativistic bosons and define a Hermitian fluid velocity operator, which provides a theoretical underpinning to the current-algebra approach of quantum hydrodynamics.
Adaptive homodyne estimation of a continuously evolving optical phase using time-symmetric quantum smoothing has been demonstrated experimentally to provide superior accuracy in the phase estimate compared to adaptive or non-adaptive estimation using filtering alone. Here, we illustrate how the mean-square error in the adaptive phase estimate may be further reduced below the standard quantum limit for the stochastic noise process considered by using a Rauch-Tung-Striebel smoother as the estimator, along with an optimal Kalman filter in the feedback loop. Further, the estimation using smoothing can be made robust to uncertainties in the underlying parameters of the noise process modulating the system phase to be estimated. This has been done using a robust fixed-interval smoother designed for uncertain systems satisfying a certain integral quadratic constraint.
Abstract Optical phase tracking is an important technique for use in high-precision measurement applications, including optical frequency metrology and ground- or space-based gravitational wave observation, and coherent optical communications. When measuring fast-varying real-time signals, the response time limitations of the measurement system’s phase-locked loop cause the best operating point to be mismatched, and the measurement then becomes nonlinear. To make these measurements possible, this work proposes a time delay loop that theoretically enables optimal homodyne detection. When the time delay loop is combined with an extended Kalman filter, the estimated measurement accuracy is improved by 2.4 dB when tracking a fast-varying random signal with a velocity of 107 rad/s. This phase estimation improvement also increases as the interference angle deviates further from the optimal measurement point. The proposed method shows potential for use in real-time sensing and measurement applications.