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Journal of Physics B Atomic Molecular and Optical Physics

Optimization of the phase sensitivity through photon-subtraction inside the Mach-Zehnder interferometer

Qisi Zhou, Qingqian Kang, Teng Zhao, Xin Su, Cunjin Liu, Liyun Hu

Published Apr 22, 2026• 0 citations
Publisher

Abstract

Abstract We propose a photon subtraction scheme within the Mach–Zehnder interferometer using a coherent state and a squeezed vacuum state as inputs. By systematically comparing intensity detection and homodyne detection methods, we demonstrate that strategically applied photon subtraction ( a -mode subtraction) significantly enhances the phase sensitivity of all detection methods. Under this configuration, homodyne detection achieves superior phase sensitivity and loss robustness compared to intensity detection. Remarkably, under 30% photon loss, the system surpasses the Heisenberg limit, enabled by the synergy among the squeezed vacuum state, non-Gaussian operations, and detection optimization. Furthermore, we quantify performance through the quantum Fisher information and the quantum Cramér-Rao bound analyses under both ideal and lossy conditions.

DOI
10.1088/1361-6455/ae637a

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Abstract We propose a photon subtraction scheme within the Mach–Zehnder interferometer using a coherent state and a squeezed vacuum state as inputs. By systematically comparing intensity detection and homodyne detection methods, we demonstrate that strategically applied photon subtraction ( a -mode subtraction) significantly enhances the phase sensitivity of all detection methods. Under this configuration, homodyne detection achieves superior phase sensitivity and loss robustness compared to intensity detection. Remarkably, under 30% photon loss, the system surpasses the Heisenberg limit, enabled by the synergy among the squeezed vacuum state, non-Gaussian operations, and detection optimization. Furthermore, we quantify performance through the quantum Fisher information and the quantum Cramér-Rao bound analyses under both ideal and lossy conditions.
Based on the conventional Mach-Zehnder interferometer, we propose a metrological scheme to improve phase sensitivity. In this scheme, we use a coherent state and a squeezed vacuum state as input states, employ multi-photon-subtraction operations and make intensity-detection or homodyne-detection. We study phase sensitivity, quantum Fisher information and quantum Cramér-Rao bound under both ideal and lossy conditions. The results indicate that choosing an appropriate detection method and photon subtraction scheme can significantly enhance the phase sensitivity and robustness against photon losses. Even under lossy conditions, the multi-photon subtraction schemes can surpass the standard quantum limit. Notably, the homodyne detection method can even break through the Heisenberg limit. Moreover, increasing the number of photon-subtracted can enhance both phase sensitivity and quantum Fisher information. This research highlights the significant value of this scheme in quantum precision measurement.
ABSTRACT Photon addition operations applied to squeezed states have been shown to significantly enhance phase sensitivity. In this work, we extend this approach by applying photon addition not only to coherent states but also within a Mach–Zehnder interferometer, using coherent and squeezed vacuum states as input. Both intensity‐difference and homodyne detection are used to evaluate photon addition schemes, and their phase sensitivities are compared under ideal and lossy conditions. We also analyze the quantum Fisher information of these two schemes. Results show that both schemes improve phase sensitivity, quantum Fisher information, and robustness against losses. In particular, photon addition within the interferometer performs better. Homodyne detection outperforms intensity difference detection under photon losses. Notably, the two schemes exhibit distinct parameter dependencies, rendering them suitable for different operational scenarios. When the squeezing parameter is small, photon addition employed at the coherent input with intensity difference detection can approach the Heisenberg limit in ideal conditions and can exceed the standard quantum limit in high‐loss conditions. The proposed scheme represents a valuable method for quantum precision measurements. PACS : 03.67.‐a, 05.30.‐d, 42.50,Dv, 03.65.Wj
Abstract To improve the phase sensitivity, multi-photon subtraction schemes (multi-PSS) within the SU(1,1) interferometer are proposed. The input states are the coherent state and the vacuum state, and the detection method is homodyne detection. The effects of multi-photon subtraction on phase sensitivity, quantum Fisher information (QFI), and quantum Cramér-Rao bound (QCRB) are analyzed under both ideal and photon losses situations. It is shown that the internal subtraction operation can improve the phase sensitivity, which becomes better performance by increasing subtraction number. It can also efficiently improve the robustness of the SU(1,1) interferometer against internal photon losses. By comparing separatively arbitrary photon subtraction on the two-mode inside SU(1,1) interferometer, the performance differences under different conditions are analyzed, including the asymmetric properties of non-Gaussian operations on the phase precision and the QFI. Our proposed scheme represents a valuable method for achieving quantum precision measurements.
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 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.
We theoretically study the phase estimation based on a Mach-Zehnder interferometer (MZI) with a two-mode squeezed coherent state. By maximizing the quantum Fisher information, we find that the quantum Cramér-Rao bounds (QCRB) can reach sub-Heisenberg limit under the phase-matched condition. The optimal phase sensitivity can reach the sub-shot noise limit (SNL) and approach the QCRB by employing the intensity difference detection. Meanwhile, compared with the MZI fed with a coherent plus a single-mode squeezed vacuum state, this scheme can have better performance by adjusting the squeezing parameter and the mean photon number. With the same parameter, our scheme shows more sensitive phase measurement than the SU(1,1) interferometer with a coherent plus a vacuum state. We also show that the phase sensitivity of our proposal can still reach the SNL when the loss of the photon is 36%. This scheme can provide potential applications in optical sensors.