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 Cram\'er-Rao bound. For Gaussian stationary statistics, and a phase spectrum scaling asymptotically as ω^-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→∞) and provides a stochastic Heisenberg limit for fluctuating phases. For $p=2$ (Brownian motion), this limit can be attained by phase tracking.