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arXiv:2602.00700

Enhanced Phase Estimation via Photon-Added Two-Mode Squeezed States and Kerr Nonlinearity

Zekun Zhao, Qingqian Kang, Teng Zhao, Cunjin Liu, Liyun Hu

quant-phSubmitted Jan 31, 2026• Announced Feb 3, 2026• 0 citations
v1 · Jan 31, 2026v2 · Feb 19, 2026
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Abstract

Quantum metrology harnesses quantum resources to achieve measurement precision beyond classical limits. This work investigates a Mach-Zehnder interferometer incorporating a Kerr nonlinear phase shifter, using photon-added two-mode squeezed coherent states generated via four-wave mixing as input. This study demonstrates that increasing both the photon-addition order and input resource strength systematically enhances phase sensitivity, quantum Fisher information, and the quantum Cramér-Rao bound. The system not only surpasses the standard quantum limit but also approaches or exceeds the Heisenberg limit with linear phase shifts, while Kerr nonlinearity enables overcoming the super-Heisenberg limit. The proposed scheme exhibits enhanced robustness against photon loss, providing a promising approach for high-precision quantum metrology applications.

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Quantum metrology harnesses quantum resources to achieve measurement precision beyond classical limits. This work investigates a Mach-Zehnder interferometer incorporating a Kerr nonlinear phase shifter, using photon-added two-mode squeezed coherent states generated via four-wave mixing as input. This study demonstrates that increasing both the photon-addition order and input resource strength systematically enhances phase sensitivity, quantum Fisher information, and the quantum Cramér-Rao bound. The system not only surpasses the standard quantum limit but also approaches or exceeds the Heisenberg limit with linear phase shifts, while Kerr nonlinearity enables overcoming the super-Heisenberg limit. The proposed scheme exhibits enhanced robustness against photon loss, providing a promising approach for high-precision quantum metrology applications.
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.
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.
Abstract A scheme is proposed based on a Mach–Zehnder interferometer with high phase sensitivity, utilizing a two-mode squeezed coherent state, generated by four-wave mixing, as input. The phase sensitivity of this scheme easily surpasses the Heisenberg limit when intensity difference detection is applied. Under phase-matching conditions, the quantum Cramér–Rao bound significantly exceeds the Heisenberg limit. Additionally, the scheme exhibits robustness against photon loss. When compared with the modified SU(1,1) interferometer with two coherent state inputs, this approach demonstrates superior measurement sensitivity, evaluated through various detection methods and the quantum Cramér–Rao bound. This work holds potential applications in quantum metrology.
ABSTRACT We enhance phase sensitivity in an SU(1,1) interferometer by combining a Kerr nonlinear phase shifter with multi‐photon addition. Using coherent and vacuum states as inputs, we analyze phase sensitivity and the quantum Fisher information under ideal and lossy conditions. We show that both phase sensitivity and the quantum Cramér‐Rao bound (QCRB) improve gradually as the total mean photon number increases. For the standard SU(1,1) interferometer, phase sensitivity does not saturate the QCRB, even with multi‐photon addition. In contrast, for the Kerr‐based SU(1,1) interferometer, phase sensitivity nearly saturates the QCRB under ideal conditions and remains optimal at high transmissivity under loss. The Kerr‐based scheme surpasses the standard quantum limit, approaches the sub‐Heisenberg limit, and exhibits enhanced robustness against moderate losses. Our scheme is feasible with current technology and provides a promising route for high‐precision quantum metrology.
A nonlinear phase shift is introduced to a Mach-Zehnder interferometer (MZI), and we present a scheme for enhancing the phase sensitivity. In our scheme, one input port of a standard MZI is injected with a coherent state and the other input port is injected with one mode of a two-mode squeezed-vacuum state. The final interference output of the MZI is detected with the method of active correlation output readout. Based on the optimal splitting ratio of beam splitters, the phase sensitivity can beat the standard quantum limit and approach the quantum Cram\'er-Rao bound. The effects of photon loss on phase sensitivity are discussed. Our scheme can also provide some estimates for units of ${\ensuremath{\chi}}^{(3)}$, due to the relation between the nonlinear phase shift and the susceptibility ${\ensuremath{\chi}}^{(3)}$ of the Kerr medium.
We investigate phase estimation in an SU(1,1) interferometer employing mmm-photon-added even and odd coherent states as nonclassical input resources. The phase sensitivity is evaluated through intensity detection and the error propagation method, while the ultimate precision limit is determined from the quantum Cramér-Rao bound with the quantum Fisher information serving as the relevant metrological quantity. Our results demonstrate that photon addition significantly enhances the phase sensitivity, increases the quantum Fisher information, and reduces the quantum Cramér-Rao bound, leading to a clear improvement over the corresponding even and odd coherent states. Furthermore, the achievable sensitivity exceeds the standard quantum limit and gradually approaches the Heisenberg scaling with increasing photon-addition number. We also find that the mmm-photon-added even coherent states exhibit a modest advantage over their odd counterparts. As mmm increases, however, this distinction becomes progressively weaker, suggesting that photon addition diminishes the role of the initial parity of the coherent state in determining the interferometric performance.