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Physical Review A

Quantum metrology for simultaneously estimating the linear and nonlinear phase shifts

Jing Cheng

Published Dec 29, 2014• 24 citations
Publisher

Abstract

Quantum metrology allows the enhanced precision on estimating phase shifts in an optical interferometer to reach the $\ensuremath{\propto}1/N$ scaling of precision ($N$ is the mean photon number of the light field). But when $N$ 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 $1/{N}^{1/2}$ ($1/{N}^{3/2}$) 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 $1/N$ scaling for the linear phase and $1/{N}^{2}$ scaling for the nonlinear phase.

DOI
10.1103/physreva.90.063838

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Quantum metrology allows the enhanced precision on estimating phase shifts in an optical interferometer to reach the \ensuremath∝1/N\ensuremath{\propto}1/N\ensuremath∝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.
Abstract The precision of quantum measurements can be effectively improved by using both photon-added non-Gaussian operations and Kerr nonlinear phase shift. Here, we employ a coherent state mixed with a photon-added squeezed vacuum state as the input to a Mach–Zehnder interferometer with parity detection, thereby achieving a significant enhancement in phase measurement precision. Our research focuses on the phase sensitivity of linear phase shift under both ideal conditions and photon loss, as well as quantum Fisher information (QFI). The results demonstrate that employing the photon addition operations can markedly enhance phase sensitivity and QFI, and under optimal conditions, the measurement precision can approach the Heisenberg limit for the linear phase shift case. In addition, we delve deeper into the scenario of replacing the linear phase shift with a Kerr nonlinear one and systematically analyze the QFI under both ideal and photon loss conditions. By comparison, it is evident that employing both the photon addition operations and the Kerr nonlinear phase shift can further significantly enhance phase measurement precision while effectively improving the system’s robustness against photon loss. These findings are instrumental in facilitating the development and practical application of quantum metrology.
We propose a high-precision phase estimation scheme in a hybrid interferometer by synergistically combining a Kerr nonlinear phase shifter and multi-photon subtraction operations. Using a coherent state and a vacuum state as input resources, we systematically evaluate the phase sensitivity via homodyne detection and analyze the quantum Fisher information as well as the quantum Cramér-Rao bound under both ideal and lossy conditions. Our results show that the joint integration of Kerr nonlinearity and multi-photon subtraction yields remarkable advantages over either technique used alone. The proposed scheme enables the phase sensitivity to surpass the standard quantum limit, exceed the conventional Heisenberg scaling (1/N1/N1/N), and approach the super-Heisenberg scaling (1/N21/N^{2}1/N2)-a direct consequence of Kerr nonlinearity. More precisely, the super-Heisenberg scaling $\propto $ 1/N21/N^{2}1/N2 is the ultimate precision limit permitted by the k=2k=2k=2 Kerr nonlinearity and does not violate the fundamental Heisenberg limit for linear phase accumulation. Even under moderate internal photon loss, the system maintains high precision and exhibits enhanced robustness to decoherence. The Kerr nonlinearity introduces an intensity-dependent phase shift proportional to the squared photon number, while multi-photon subtraction tailors non-Gaussian states to strengthen phase information extraction. Compared with existing schemes based on hybrid interferometers or SU(1,1) interferometers, our architecture achieves superior precision and stronger loss resilience. All components are experimentally accessible with current quantum optical technologies. This work provides a promising route for practical high-precision quantum metrology and quantum sensing.
We investigate the phase enhancement of quantum states subject to nonlinear phase shifts. The optimal phase estimation of even entangled coherent states (ECSs) is shown to be better than that of NOON states with the same average particle number $\ensuremath{\langle}n\ensuremath{\rangle}$ and nonlinearity exponent kkk. We investigate the creation of an approximate entangled coherent state (AECS) from a photon-subtracted squeezed vacuum with current optical technology methods and show that a pure AECS is even better than an even ECS for large $\ensuremath{\langle}n\ensuremath{\rangle}$. Finally, we examine the simple, but physically relevant, cases of loss in the nonlinear interferometer for a fixed average photon number $\ensuremath{\langle}n\ensuremath{\rangle}$.
Quantum metrology employs quantum resources to achieve measurement precision beyond classical limits. This work investigates a Mach--Zehnder interferometer incorporating a Kerr nonlinear phase shifter, with photon-added two-mode squeezed coherent states generated via four-wave mixing as input. We demonstrate that increasing both the photon-addition order and the input resource strength systematically enhances phase sensitivity, quantum Fisher information, and the corresponding quantum Cramér--Rao bound. The proposed system not only surpasses the standard quantum limit but also approaches or exceeds the Heisenberg limit for linear phase shifts, while Kerr nonlinearity enables surpassing the super-Heisenberg limit. Furthermore, the scheme exhibits enhanced robustness against photon loss, providing a promising pathway toward practical high-precision quantum metrology applications.
Abstract We propose a generalized form of entangled coherent states (ECS) and apply them in a multi-arm optical interferometer to estimate multiple phase shifts. We obtain the quantum Cramér–Rao bounds for both the linear and nonlinear parametrization protocols. Through analysis utilizing simultaneous estimation we find that this generalized form of ECS gives better precision than the generalized NOON states (2013 Phys. Rev. Lett. 111 070403 ). Moreover, when compared with the independent estimation both the linear and nonlinear protocols have the same advantage in the relation to the number of parameters.
We present an improved phase estimation scheme employing entangled SU(1,1) coherent states in comparison to NOON and entangled coherent states under perfect and lossy conditions for a fixed mean photon number. The study is also devoted to the phase enhancement of the quantum states resulting from a generalized nonlinearity of the phase shifts, both without and with losses. Furthermore, we show that these states give the smallest variance in the phase parameter for a large number of photons in a different order of nonlinearity. Finally, the phase sensitivity of this interferometric setting with parity detection is discussed.