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

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/(4n) standard quantum limit with coherent states, do not apply. Here, by 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 κp-1/|ω|p for large ω, for some p>1. For coherent states with mean photon flux N, we give the quantum Cramér-Rao bound on the mean-square phase error as [psin(π/p)]-1(4N/κ)-(p-1)/p. Next, we consider whether the bound can be achieved by an adaptive homodyne measurement in the limit N/κ 1, which allows the photocurrent to be linearized. Applying the optimal filtering for the resultant linear Gaussian system, we find the same scaling with N, but with a prefactor larger by a factor of p. By contrast, if we employ optimal smoothing we can exactly obtain the quantum Cramér-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 N/κ is not large and find a more complicated picture.