Stochastic Heisenberg limit: Optimal estimation of a fluctuating phase

TL;DR

This paper derives the fundamental quantum limits for estimating fluctuating optical phases, revealing a stochastic Heisenberg limit that depends on the phase spectrum's scaling, with practical attainment for Brownian motion.

Contribution

It introduces a stochastic Heisenberg limit for fluctuating phase estimation based on the quantum Cramer-Rao bound, extending the understanding of quantum limits to dynamic phases.

Findings

Minimum mean-square error scales as N^{-2(p-1)/(p+1)} for phase spectrum ~ 1/omega^p

Heisenberg limit for constant phase as p→ infinity

Attainability of the limit for p=2 (Brownian motion) through phase tracking

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/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.