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Civil War Book Review

Two-Mode Squeezed Vacuum: Phase Estimation and Parity Detection

Petr M. Anisimov, William N. Plick, Gretchen M. Raterman, Hwang Lee, Jonathan P. Dowling, G. S. Agarwal

Published Jan 1, 2011• 0 citations
Publisher

Abstract

We present a parity-measurement-based phase estimation protocol with two-mode squeezed vacuum states; effects of loss and excess noise in squeezed vacuum are discussed; and a parity detection scheme without number-resolving detectors is proposed. © 2010 Optical Society of America.
DOI
10.1364/qels.2011.qfd6

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•Jan 1, 2011•0 citations
A recently proposed phase-estimation protocol that is based on measuring the parity of a two-mode squeezed-vacuum state at the output of a Mach-Zehnder interferometer shows that Cramer-Rao bound sensitivity can be obtained [P. M. Anisimov, et al., Phys. Rev. Lett. {\bf104}, 103602 (2010)]. This sensitivity, however, is expected in the case of an infinite number of parity measurements made on an infinite number of photons. Here we consider the case of a finite number of parity measurements and a finite number of photons, implemented with photon-number-resolving detectors. We use Bayesian analysis to characterize the sensitivity of the phase estimation in this scheme. We have found that our phase estimation becomes biased near 0 or π/2\pi/2π/2 phase values. Yet there is an in-between region where the bias becomes negligible. In this region, our phase estimation scheme saturates the Cramer-Rao bound and beats the shot-noise limit.
A proposed phase-estimation protocol based on measuring the parity of a two-mode squeezed-vacuum state at the output of a Mach-Zehnder interferometer shows that the Cram\'er-Rao sensitivity is sub-Heisenberg [P. M. Anisimov et al., Phys. Rev. Lett. 104, 103602 (2010)]. However, these measurements are problematic, making it unclear if this sensitivity can be obtained with a finite number of measurements. This sensitivity is only for a phase near zero, and in this region there is a problem with ambiguity because measurements cannot distinguish the sign of the phase. Here, we consider a finite number of parity measurements and show that an adaptive technique gives a highly accurate phase estimate regardless of the phase. We show that the Heisenberg limit is reachable, where the number of trials needed for mean photon number n‾=1\overline{n}=1n=1 is approximately 100. We show that the Cram\'er-Rao sensitivity can be achieved approximately, and the estimation is unambiguous in the interval ($\ensuremath{-}\ensuremath{\pi}/2,\ensuremath{\pi}/2$).
We investigate the sensitivity and resolution of phase estimation in a Mach–Zehnder interferometer with photon-added two-mode squeezed vacuum states (PA-TMSVS) with a photon-number parity measurement scheme. Our results indicate that, when the phase to be estimated is small, the PA-TMSVS offers an improved phase resolution over both the two-mode squeezed vacuum and the photon-subtracted two-mode squeezed vacuum for a given initial squeezing level. In addition, we study the behavior of the quantum Fisher information of the PA-TMSVS, and show that the optimum phase uncertainty predicted by the corresponding Cramer–Rao bound can be achieved with a parity measurement.