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 ${\cal N}$, we give the Quantum Cram\'er-Rao Bound on the mean-square phase error as $[p \sin (\pi/p)]^{-1}(4{\cal N}/\kappa)^{-(p-1)/p}$. Next, we consider whether the bound can be achieved by an adaptive homodyne measurement, in the limit ${\cal N}/\kappa \gg 1$ which allows the photocurrent to be linearized. Applying the optimal filtering for the resultant linear Gaussian system, we find the same scaling with ${\cal N}$, but with a prefactor larger by a factor of $p$. By contrast, if we employ optimal smoothing we can exactly obtain the Quantum Cram{\'e}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 ${\cal N}/\kappa$ is not large, and find a more complicated picture.