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Optics Express

Effects of imperfect elements on resolution and sensitivity of quantum metrology using two-mode squeezed vacuum state

Jiandong Zhang, Zijing Zhang, Longzhu Cen, Miao Yu, Shuo Li, Feng Wang, Yuan Zhe Zhao

Published Oct 2, 2017• 13 citations
Publisher

Abstract

It has been demonstrated that using two-mode squeezed vacuum state for phase estimation can break the Heisenberg limit. Our results reveal that the two-mode squeezed vacuum state is also applied to the optical rotation angle measurement. In our scheme, the resolution and sensitivity of the optical rotation angle signal are the same as the case of phase estimation. For the parameter estimation, phase or rotation angle, we discuss the influences of several imperfect factors on the resolution and sensitivity. First, the effect that the upper limit of photon-number resolving has on the maximum amount of available quantum Fisher information has been analyzed. Then, we have also studied the impacts of both the transmission efficiency in the transmission process and the detection efficiency on the detection results. Finally, conditions where all of the above imperfect elements are taken into account at the same time have also been explored. Additionally, other imperfect factors such as squeezing efficiency and dark counts are briefly discussed.

DOI
10.1364/oe.25.024907

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It has been demonstrated that using two-mode squeezed vacuum state for phase estimation can break the Heisenberg limit. Our results reveal that the two-mode squeezed vacuum state is also applied to the optical rotation angle measurement. In our scheme, the resolution and sensitivity of the optical rotation angle signal are the same as the case of phase estimation. For the parameter estimation, phase or rotation angle, we discuss the influences of several imperfect factors on the resolution and sensitivity. First, the effect that the upper limit of photon-number resolving has on the maximum amount of available quantum Fisher information has been analyzed. Then, we have also studied the impacts of both the transmission efficiency in the transmission process and the detection efficiency on the detection results. Finally, conditions where all of the above imperfect elements are taken into account at the same time have also been explored. Additionally, other imperfect factors such as squeezing efficiency and dark counts are briefly discussed.
Quantum metrology is to estimate accurately the value of an unknown parameter with the assistance of the quantum effects, in order to break through the standard quantum limit, even reach the Heisenberg limit. In this work, we study the performance of a general photon-added two-mode squeezed vacuum state that is taken as a detection state of a Mach-Zehnder interferometer. Based on quantum Fisher information, within the constraint on the total mean photon number, symmetric and asymmetric photon addition cannot improve the ultimate phase sensitivity. However, for a given initial squeezing parameter, on this occasion, the symmetric and asymmetric photon addition can improve the ultimate phase sensitivity. Compared with the asymmetric photon-added two-mode squeezed vacuum state, the symmetric one can well improve the ultimate phase sensitivity. This may be because it is always better to implement the symmetric photon addition rather than the asymmetric one in order to increase the mean photon number of the resulting state. On the other hand, via parity detection, the symmetric and asymmetric photon-added two-mode squeezed vacuum state can indeed improve the phase sensitivity of a Mach-Zehnder interferometer for a given initial squeezing parameter. Based on the parity detection, within a constraint on the mean photon number, although the two-mode squeezed vacuum state can give the better phase sensitivity at the optimal phase shift (φ = 0), the phase sensitivity offered by the symmetric and asymmetric photon-added two-mode squeezed vacuum states are both more stable around φ = 0 than by the two-mode squeezed vacuum state. In addition, we show that for the symmetric photon-added two-mode squeezed vacuum state, parity detection is an optimal detection only when the optimal phase shift approaches to zero. When the phase shift slightly deviates from zero, the parity detection is not an optimal detection scheme. Finally, for all values of the phase shift, our results also clearly show that the parity detection is not an optimal detection scheme for the asymmetric photon-added two-mode squeezed vacuum state serving as an interferometer state.
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.
We study the sensitivity of phase estimation in a lossy Mach-Zehnder interferometer (MZI) using two general, and practical, resources generated by a laser and a nonlinear optical medium with passive optimal elements, which are readily available in the laboratory: One is a two-mode separable coherent and squeezed vacuum state at a beam splitter and the other is a two-mode squeezed vacuum state. In view of the ultimate precision given by quantum Fisher information, we show that the two-mode squeezed vacuum state can achieve a lower bound of estimation error than the coherent and squeezed vacuum state under a photon-loss channel. We further consider practical measurement schemes, homodyne detection and photon number resolving detection (PNRD), to characterize the accuracy of phase estimation in reality and find that the coherent and squeezed vacuum state largely achieves a lower bound than the two-mode squeezed vacuum in the lossy MZI while maintaining quantum enhancement over the shot-noise limit. By comparing homodyne detection and PNRD, we demonstrate that quadrature measurement with homodyne detection is more robust against photon loss than parity measurement with PNRD. We also show that double homodyne detection can provide a better tool for phase estimation than single homodyne detection against photon loss.
We theoretically investigate the quantum enhanced metrology using two-mode squeezed twin-Fock states and parity detection. Our results indicate that, for a given initial squeezing parameter, compared with the two-mode squeezed vacuum state, both phase sensitivity and resolution can be enhanced when the two-mode squeezed twin-Fock state is considered as an input state of a Mach–Zehnder interferometer. Within a constraint on the total photon number, although the two-mode squeezed vacuum state gives the better phase sensitivity when the phase shift φ to be estimated approaches to zero, the phase sensitivity offered by these non-Gaussian entangled Gaussian states is relatively stable with respect to the phase shift itself. When the phase shift slightly deviates from φ = 0, the phase sensitivity can be still enhanced by the two-mode squeezed twin-Fock state over a broad range of the total mean photon number where the phase uncertainty is still below the quantum standard noise limit. Finally, we numerically prove that the quantum Cramér–Rao bound can be approached with the parity detection.
While quantum metrology enables measurement precision beyond classical limits, its performance is often susceptible to experimental imperfections. Most prior studies have focused on imperfections in quantum states and operations. Here, we investigate the effect of coarse graining in quantum measurement through both theoretical analysis and experimental demonstration. Using an interferometer with a squeezed vacuum and a laser input, we analyze how coarse graining in homodyne detection affects the precision of phase estimation. We evaluate the Fisher information under various coarse-graining conditions and determine, in each case, an optimal estimation strategy that saturates the Cramér-Rao bound. Remarkably, even extremely coarse-grained measurement -- with only two bins -- enables phase estimation beyond the standard quantum limit and even achieves a precision that follows the Heisenberg scaling. We experimentally demonstrate quantum-enhanced phase estimation under coarse-grained homodyne detection. To determine an optimal estimation strategy, we employ the method of moments and present calibration procedures that enable its application to general experimental settings. Using only two bins, we observe a quantum enhancement of 1.2 dB compared to the classical method using the ideal measurement, improving towards 3.8 dB as the bin number increases. These results highlight a practical pathway to achieving quantum enhancement under the presence of severe experimental imperfections.