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 Cramér-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.
Department of Physics and Astronomy, Macquarie University, Sydney, New South Wales 2109, Australia.
Dominic W
DW
Berry
Michael J W
MJ
Hall
Howard M
HM
Wiseman