Adaptive estimation of a time-varying phase with coherent states: smoothing can give an unbounded improvement over filtering

Abstract

The problem of measuring a time-varying phase, even when the statistics of the variation is known, is considerably harder than that of measuring a constant phase. In particular, the usual bounds on accuracy - such as the $1/(4\bar{n})$ standard quantum limit with coherent states - do not apply. Here, restricting to coherent states, we are able to analytically obtain the achievable accuracy - the equivalent of the standard quantum limit - for a wide class of phase variation. In particular, we consider the case where the phase has Gaussian statistics and a power-law spectrum equal to ${\kappa }^{p-1}/|\omega {|}^p$ for large $\omega$, for some $p>1$. For coherent states with mean photon flux $\mathcal{N}$, we give the Quantum Cramér-Rao Bound on the mean-square phase error as $[p\sin (\pi /p){]}^{-1}(4\mathcal{N}/\kappa {)}^{-(p-1)/p}$. Next, we consider whether the bound can be achieved by an adaptive homodyne measurement, in the limit $\mathcal{N}/\kappa \gg 1$ which allows the photocurrent to be linearized. Applying the optimal filtering for the resultant linear Gaussian system, we find the same scaling with $\mathcal{N}$, but with a prefactor larger by a factor of $p$. By contrast, if we employ optimal smoothing we can exactly obtain the Quantum Cram{é}r-Rao Bound. That is, contrary to previously considered ($p=2$) cases of phase estimation, here the improvement offered by smoothing over filtering is not limited to a factor of 2 but rather can be unbounded by a factor of $p$. We also study numerically the performance of these estimators for an adaptive measurement in the limit where $\mathcal{N}/\kappa$ is not large, and find a more complicated picture.

12 pages, 3 figures

References in corpus (9)

• General optimality of the Heisenberg limit for quantum metrology
• Quantum-enhanced optical phase tracking
• The quantum Bell-Ziv-Zakai bounds and Heisenberg limits for waveform estimation
• Does nonlinear metrology offer improved resolution? Answers from quantum information theory
• Quantum theory of optical temporal phase and instantaneous frequency. II. Continuous time limit and state-variable approach to phase-locked loop design
• Optimal Heisenberg-style bounds for the average performance of arbitrary phase estimates
• Adaptive phase measurements for narrowband squeezed beams
• Adaptive estimation of a time-varying phase with a power-law spectrum via continuous squeezed states
• Signal detection without finite-energy limits to quantum resolution

Cited by in corpus (5)

• Unifying theory of quantum state estimation using past and future information
• Quantum State Smoothing for Linear Gaussian Systems
• Experimental quantum-enhanced response function estimation
• Binary Discrimination in Quantum Systems via Hypothesis Testing
• Quantum Cramer-Rao Precision Limit of Noisy Continuous Sensing