Stochastic Heisenberg limit: Optimal estimation of a fluctuating phase
Abstract
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/omegap 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.
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