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arXiv:2111.01375 · Physical Review A

Supersensitivity of Kerr phase estimation with two-mode squeezed vacuum states

Yun-Feng Guo, Wei Zhong, Lan Zhou, Yu‐Bo Sheng

Published Mar 17, 2022• 18 citations
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Abstract

We analytically investigate the sensitivity of Kerr nonlinear phase estimation in a Mach-Zehnder interferometer with two-mode squeezed vacuum states. We find that such a metrological scheme could access a sensitivity scaling over the Boixo et al.'s generalized sensitivity limit [S. Boixo et al., Phys. Rev. Lett. 98, 090401 (2007)], which is saturable with celebrated NOON states. We also show that parity detection is a quasioptimal measurement which can nearly saturate the quantum Cram\'er-Rao bound in the aforementioned situation. Moreover, we further clarify the supersensitive performance observed in the above scheme is due to the restriction of Boixo et al.'s generalized sensitivity limit (BGSL) to probe states with fixed photon numbers. To conquer this problem, we generalize the BGSL into the case with probe states of a fluctuating number of photons, to which our scheme belongs.

DOI
10.1103/physreva.105.032609

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We analytically investigate the sensitivity of Kerr nonlinear phase estimation in a Mach-Zehnder interferometer with two-mode squeezed vacuum states. We find that such a metrological scheme could access a sensitivity scaling over the Boixo et al.'s generalized sensitivity limit [S. Boixo et al., Phys. Rev. Lett. 98, 090401 (2007)], which is saturable with celebrated NOON states. We also show that parity detection is a quasioptimal measurement which can nearly saturate the quantum Cram'er-Rao bound in the aforementioned situation. Moreover, we further clarify the supersensitive performance observed in the above scheme is due to the restriction of Boixo et al.'s generalized sensitivity limit (BGSL) to probe states with fixed photon numbers. To conquer this problem, we generalize the BGSL into the case with probe states of a fluctuating number of photons, to which our scheme belongs.
We analytically investigate the sensitivity of Kerr nonlinear phase estimation in a Mach-Zehnder interferometer with two-mode squeezed vacuum states. We find that such a metrological scheme could access a sensitivity scaling over the Boixo et al.'s generalized sensitivity limit [S. Boixo et al., Phys. Rev. Lett. 98, 090401 (2007)], which is saturable with celebrated NOON states. We also show that parity detection is a quasi-optimal measurement which can nearly saturate the quantum Cramer-Rao bound in the aforementioned situation.
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.
We investigate the sensitivity of phase estimation in a Mach–Zehnder interferometer with photon-subtracted two-mode squeezed vacuum states. Our results show that, for given initial squeezing parameter, both symmetric and asymmetric photon subtractions can further improve the quantum Cramér–Rao bound (i.e., the ultimate phase sensitivity), especially for single-mode photon subtraction. On the other hand, the quantum Cramér–Rao bound can be reached by parity detection for symmetric photon-subtracted two-mode squeezed vacuum states at particular values of the phase shift, but it is not valid for asymmetric photon-subtracted two-mode squeezed vacuum states. In addition, compared with the two-mode squeezed vacuum state, the phase sensitivity via parity detection with asymmetric photon-subtracted two-mode squeezed vacuum states will be getting worse. Thus, parity detection may not always be the optimal detection scheme for nonclassical states of light when they are considered as the interferometer states.
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.