  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 1/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.
