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

TL;DR

The paper addresses simultaneous estimation of two optical phases in a three-mode interferometer using a path-entangled N-particle W-type input and uniform optical-parametric amplification (OPA). ItDevelops the normally ordered characteristic-function formalism to derive all photon-number moments and the quantum Fisher information matrix, enabling analysis in both lossless and lossy scenarios. The main findings show that OPA substantially improves the quantum Cramér–Rao bound when applied to signal modes, with the enhancement linked to increased intra-mode photon-number correlations rather than inter-mode correlations; this advantage persists under moderate loss, though it saturates at high amplification due to dissipation. These results offer physical insight into OPA-enabled multiparameter metrology and provide practical guidelines for robust phase-estimation protocols in realistic noisy environments, including how to exploit path entanglement and local amplification for best performance.

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.

Figures (5)

• Figure 1: (Color online) Figure 1. Schematic diagram of an OPA-assisted tritter interferometer for simultaneous two-phase estimation. A three-mode W-type state is injected into the tritter, where modes $a_0$ and $b_0$ encode the unknown phases, while mode $c_0$ serves as the reference. After the tritter, each mode undergoes single-mode optical parametric amplification followed by phase accumulation on the signal modes. The output state is finally measured to infer both phases simultaneously.
• Figure 2: (Color online) The QCRB for simultaneous estimation of two phases as a function of the total photon number in a lossless interferometer. (a) Identical OPA gain applied to all three modes, with $r_a=r_b=r_c=0$, $0.25$, and $0.5$. (b) Mode-selective OPA configurations: $r_a=r_b=r_c=0$, $r_a=0.5,, r_b=r_c=0$, and $r_c=0.5,, r_a=r_b=0$.
• Figure 3: (Color online) Second-order photon-number correlations of the output state in a lossless interferometer. (a) Intra-mode second-order correlation function $g^{(2)}a$ versus the total photon number for identical OPA gain $r_a=r_b=r_c=0$, $0.25$, and $0.5$. (b) Inter-mode second-order correlation function $g^{(2)}{a,b}$ for the same gain configurations as in panel (a).
• Figure 4: (Color online) Comparison of phase-estimation performance between a three-mode W state and a separable Fock state in a lossless interferometer. The QCRB is shown as a function of the total photon number for both input states under identical interferometric and OPA conditions.
• Figure 5: (Color online) Effects of photon loss on simultaneous two-phase estimation. (a) QCRB as a function of the total photon number $N$ for different OPA gains and photon transmissivities, with $r_a=r_b=r_c=0,\,\eta=1$; $r_a=r_b=r_c=0,\,\eta=0.7$; $r_a=r_b=r_c=0.5,\,\eta=1$; and $r_a=r_b=r_c=0.5,\,\eta=0.7$. (b) QCRB as a function of the OPA gain $r$ for a fixed photon number $N=10$, shown for two different transmissivities $\eta=1$ and $\eta=0.6$.