Optical estimation of unitary Gaussian processes without phase reference using Fock states

TL;DR

This paper demonstrates that Fock states enable efficient, phase-reference-free estimation of displacement and squeezing in Gaussian processes, outperforming Gaussian states especially at low energies and allowing simultaneous parameter estimation.

Contribution

It introduces a method using photon number states for phase-insensitive estimation of Gaussian processes, surpassing Gaussian states in accuracy and enabling joint parameter estimation.

Findings

Fock states outperform Gaussian states in displacement estimation at all energies.

Estimation errors can saturate the Cramér-Rao bound with realistic losses.

Simultaneous estimation of displacement and squeezing is feasible with Fock states.

Abstract

Since a general Gaussian process is phase-sensitive, a stable phase reference is required to take advantage of this feature. When the reference is missing, either due to the volatile nature of the measured sample or the measurement's technical limitations, the resulting process appears as random in phase. Under this condition, we consider two single-mode Gaussian processes, displacement and squeezing. We show that these two can be efficiently estimated using photon number states and photon number resolving detectors. For separate estimation of displacement and squeezing, the practical estimation errors for hundreds of probes' ensembles can saturate the Cram\'{e}r-Rao bound even for arbitrary small values of the estimated parameters and under realistic losses. The estimation of displacement with Fock states always outperforms estimation using Gaussian states with equivalent energy and optimal measurement. For estimation of squeezing, Fock states outperform Gaussian methods, but only when their energy is large enough. Finally, we show that Fock states can also be used to estimate the displacement and the squeezing simultaneously.