Stochastic Heisenberg Limit: Optimal Estimation of a Fluctuating Phase

The ultimate limits to estimating a fluctuating phase imposed on an optical beam can be found using the recently derived continuous quantum Cramer-Rao bound. For Gaussian stationary statistics, and a phase spectrum scaling asymptotically as omega(-p) with p > 1, the minimum mean-square error in any (single-time) phase estimate scales as N-2(p-1)/(p+1), where N is the photon flux. This gives the usual Heisenberg limit for a constant phase (as the limit p -> infinity) and provides a stochastic Heisenberg limit for fluctuating phases. For p = 2 (Brownian motion), this limit can be attained by phase tracking.