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arXiv:1510.07381 · Physical Review A

Variational limits for phase precision in linear quantum optical metrology

Yang Gao, Ru-Min Wang

Published Jan 7, 2016• 5 citations
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Abstract

We apply the variational method to obtain the universal and analytical lower bounds for parameter precision in some noisy systems. We first derive a lower bound for phase precision in lossy optical interferometry at nonzero temperature. Then we consider the effect of both amplitude damping and phase diffusion on phase-shift precision. Finally, we extend the constant phase estimation to the case of continuously fluctuating phase estimation and find that due to photon losses the corresponding mean square error transits from the stochastic Heisenberg limit to the stochastic standard quantum limit as the total photon flux increases.

DOI
10.1103/physreva.93.013809

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We apply the variational method to obtain the universal and analytical lower bounds for parameter precision in some noisy systems. We first derive a lower bound for phase precision in lossy optical interferometry at nonzero temperature. Then we consider the effect of both amplitude damping and phase diffusion on phase-shift precision. Finally, we extend the constant phase estimation to the case of continuously fluctuating phase estimation and find that due to photon losses the corresponding mean square error transits from the stochastic Heisenberg limit to the stochastic standard quantum limit as the total photon flux increases.
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We consider the problem of estimating an unknown but constant carrier phase modulation $\ensuremath{\theta}$ using a general, possibly entangled, nnn-mode optical probe through nnn independent and identical uses of a lossy bosonic channel with additive thermal noise. We find an upper bound to the quantum Fisher information (QFI) of estimating $\ensuremath{\theta}$ as a function of nnn, the mean and variance of the total number of photons NS{N}_{\mathrm{S}}NS​ in the nnn-mode probe, the transmissivity $\ensuremath{\eta}$, and mean thermal photon number per mode n‾B{\overline{n}}_{\mathrm{B}}nB​ of the bosonic channel. Since the inverse of QFI provides a lower bound to the mean-square error (MSE) of an unbiased estimator $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\theta}}$ of $\ensuremath{\theta}$, our upper bound to the QFI provides a lower bound to the MSE. It already has found use in proving fundamental limits of covert sensing and could find other applications requiring bounding the fundamental limits of sensing an unknown parameter embedded in a correlated field.
Quantum metrology allows the enhanced precision on estimating phase shifts in an optical interferometer to reach the $\ensuremath{\propto}1/N$ scaling of precision (NNN is the mean photon number of the light field). But when NNN is very large, the nonlinear optical effects may be nonnegligible. Here we investigate theoretically the quantum uncertainty bounds for simultaneously detecting the linear and nonlinear phase shifts. Focusing our attentions only on the linear and the nonlinear Kerr phase shifts, we find the NOON states can not be used to measure the two phase parameters. Using the entangled coherent states, the estimation precision is limited by the 1/N1/21/{N}^{1/2}1/N1/2 (1/N3/21/{N}^{3/2}1/N3/2) scaling for the linear (nonlinear) phase. To increase the precisions and beat the standard quantum limit, we show that the entangled Schr\"odinger-cat states can be used to reach the 1/N1/N1/N scaling for the linear phase and 1/N21/{N}^{2}1/N2 scaling for the nonlinear phase.
We give a detailed discussion of optimal quantum states for optical two-mode interferometry in the presence of photon losses. We derive analytical formulae for the precision of phase estimation obtainable using quantum states of light with a definite photon number and prove that maximization of the precision is a convex optimization problem. The corresponding optimal precision, i.e., the lowest possible uncertainty, is shown to beat the standard quantum limit thus outperforming classical interferometry. Furthermore, we discuss more general inputs: states with indefinite photon number and states with photons distributed between distinguishable time bins. We prove that neither of these is helpful in improving phase estimation precision.