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arXiv:2601.05595

Squeezing-Enhanced Two-Phase Estimation with N-Particle W-type States

Huan Zhang, Guofu Yin, Ying Xia, Xiuxing Zhang, Shoukang Chang, Wei Ye

quant-phSubmitted Jan 9, 2026• Announced Jan 12, 2026• 0 citations• Revised Mar 19, 2026 (v2)
v1 · Jan 9, 2026v2 · Mar 19, 2026
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Abstract

We investigate the simultaneous estimation of two optical phases in a three-mode interferometer assisted by optical parametric amplification (OPA). By employing the normally ordered characteristic-function formalism, we analytically obtain all photon-number moments of the output quantum state, enabling an explicit evaluation of the quantum Fisher information matrix for multiparameter phase estimation. In the lossless scenario, we show that uniformly applied OPA significantly enhances the attainable precision beyond that of an unamplified interferometer. By analyzing the second-order correlation functions, we demonstrate that this enhancement originates from the amplification of intra-mode photon-number correlations, rather than from inter-mode correlations. We further extend our analysis to realistic interferometers with photon loss using a purification-based variational approach. Although loss degrades the achievable precision, the OPA-assisted scheme retains a clear advantage for moderate loss, indicating a degree of robustness against dissipation. Our results clarify the physical mechanism underlying OPA-enhanced multiparameter quantum metrology and provide guidelines for optimizing phase estimation protocols in realistic noisy environments.

Comments
10 pages,5 figures

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We investigate the simultaneous estimation of two optical phases in a three-mode interferometer assisted by optical parametric amplification (OPA). By employing the normally ordered characteristic-function formalism, we analytically obtain all photon-number moments of the output quantum state, enabling an explicit evaluation of the quantum Fisher information matrix for multiparameter phase estimation. In the lossless scenario, we show that uniformly applied OPA significantly enhances the attainable precision beyond that of an unamplified interferometer. By analyzing the second-order correlation functions, we demonstrate that this enhancement originates from the amplification of intra-mode photon-number correlations, rather than from inter-mode correlations. We further extend our analysis to realistic interferometers with photon loss using a purification-based variational approach. Although loss degrades the achievable precision, the OPA-assisted scheme retains a clear advantage for moderate loss, indicating a degree of robustness against dissipation. Our results clarify the physical mechanism underlying OPA-enhanced multiparameter quantum metrology and provide guidelines for optimizing phase estimation protocols in realistic noisy environments.
ABSTRACT Simultaneous estimation of two optical phases in a three‐mode interferometer assisted by optical parametric amplification (OPA) is investigated. The normally ordered characteristic function formalism is employed to derive analytical expressions for all photon number moments of the output state, allowing explicit evaluation of the quantum Fisher information matrix for multiparameter estimation. In the lossless regime, uniformly applied OPA significantly improves the attainable precision compared with an unamplified interferometer. As the OPA gain increases, the quantum Cramér–Rao bound gradually approaches Heisenberg scaling, resulting from the combined effects of increased photon resources and squeezing‐induced modification of photon number statistics. Examination of second‐order correlation functions indicates that this enhancement originates from amplified intra mode photon number correlations, rather than inter mode correlations. The analysis is extended to lossy interferometers using a purification‐based variational approach. Although photon loss reduces the achievable precision, the OPA assisted scheme preserves a clear advantage in the moderate loss regime, demonstrating robustness against dissipation. These findings clarify the physical origin of OPA enhanced performance and provide guidance for optimizing multiparameter phase estimation protocols in realistic noisy settings.
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Interferometry with quantum light is known to provide enhanced precision for estimating a single phase. However, depending on the parameters involved, the quantum limit for the simultaneous estimation of multiple parameters may not attainable, leading to trade-offs in the attainable precisions. Here we study the simultaneous estimation of two parameters related to optical interferometry: phase and loss, using a fixed number of photons. We derive a trade-off in the estimation of these two parameters which shows that, in contrast to single-parameter estimation, it is impossible to design a strategy saturating the quantum Cramer-Rao bound for loss and phase estimation in a single setup simultaneously. We design optimal quantum states with a fixed number of photons achieving the best possible simultaneous precisions. Our results reveal general features about concurrently estimating Hamiltonian and dissipative parameters, and has implications for sophisticated sensing scenarios such as quantum imaging.
The hybrid interferometer, integrating an optical parametric amplifier and a beam splitter, exhibits the potential to outperform the SU(1,1) interferometer. However, photon loss remains a key constraint for practical implementation. To address this challenge, we propose a quantum metrology scheme that utilizes multi-photon subtraction at the output and replaces the standard 50:50 beam splitter with a variable beam splitter—thereby enhancing robustness to photon loss. We adopt a coherent state (injected into mode a or b ) and a vacuum state as inputs to the interferometer, while performing homodyne detection. Our results demonstrate that the selection of input modes exerts a significant influence on phase estimation, and the optimization of the beam splitter’s transmittance is essential to maximize phase sensitivity under lossy conditions. Furthermore, photon subtraction notably enhances phase sensitivity, quantum Fisher information, and noise robustness. Notably, our scheme attains sensitivities exceeding the Heisenberg limit even in the presence of 20% photon loss.
Multiparameter quantum estimation in open optomechanical systems is fundamentally constrained by dissipation, thermal fluctuations, and measurement incompatibility. In this work, we investigate a coupled cavity optomechanical platform in which two mechanical modes interact with driven optical cavities, two mode squeezed vacuum, intracavity degenerate parametric amplification, and coherent optical feedback. Using the continuous variable Gaussian state formalism, we derive the linearized quantum Langevin dynamics and steady state covariance matrix and evaluate the quantum Fisher information matrices associated with simultaneous estimation of the optomechanical coupling strength and cavity dissipation rate. We characterize the precision bounds using the symmetric and right logarithmic derivative formalisms and employ BMI=max⁡{BS,BR}\mathcal{B}_{\rm MI}=\max\{\mathcal{B}_S,\mathcal{B}_R\}BMI​=max{BS​,BR​} as a comparative figure of merit within the SLD/RLD framework. We find that parametric amplification can substantially reduce the most informative bound, BMI\mathcal{B}_{\rm MI}BMI​, thereby enhancing multiparameter estimation precision over a broad range of operating conditions. Coherent feedback can further reduce BMI\mathcal{B}_{\rm MI}BMI​ and improve the estimation precision, particularly in the strong feedback regime. Moreover, the simultaneous presence of coherent feedback, parametric amplification, and squeezed light injection provides a combined mechanism for controlling quantum fluctuations and parameter dependent correlations, leading to enhanced and more robust multiparameter estimation. We further analyze the dynamical and steady state regimes to identify the parameter regions in which these quantum resources provide the largest metrological gain. Our results establish coupled cavity optomechanics as a flexible platform for quantum-noise engineering in multiparameter sensing.
Quantum metrology exploits quantum correlations to make precise measurements with limited particle numbers. By utilizing inter- and intramode correlations in an optical interferometer, we find a state that combines entanglement and squeezing to give a sevenfold enhancement in the quantum Fisher information (QFI)---a metric related to the precision---over the shot-noise limit, for low photon numbers. Motivated by practicality we then look at the squeezed cat state, which has recently been made experimentally, and shows further precision gains over the shot-noise limit and a threefold improvement in the QFI over the optimal Gaussian state. We present a conceptually simple measurement scheme that saturates the QFI, and we demonstrate a robustness to loss for small photon numbers. The squeezed cat state can therefore give a significant precision enhancement in optical quantum metrology in practical and realistic conditions.