Phase estimation via photon subtraction at the output of the hybrid interferometer

TL;DR

This work tackles phase estimation in a hybrid OPA-BS interferometer under realistic photon loss by introducing photon subtraction (PS) at the output and a tunable variable beam splitter (vBS). Using coherent-state inputs and homodyne detection, the authors show that PS markedly enhances phase sensitivity and quantum Fisher information (QFI), with a strong advantage for the input configuration where the coherent state enters mode $a$ (Scheme A). The study demonstrates that PS improves robustness to loss, that the optimal vBS transmittance shifts from 0.5 as loss increases, and that the scheme can surpass the Heisenberg limit under substantial loss (up to 20%), providing a practical route for non-Gaussian metrology in lossy environments. Collectively, the results highlight a synergistic effect between PS and vBS that enables high-precision phase estimation with manageable resource requirements and realistic imperfections.

Abstract

The hybrid interferometer integrating an optical parametric amplifier and a beam splitter has the potential to outperform the SU(1,1) interferometer. However, photon loss remains a critical limitation for practical implementation. To address this challenge, we propose a quantum metrology scheme utilizing multi-photon subtraction at the output and replacing the conventional 50:50 beam splitter with a variable beam splitter to enhance robustness against photon loss. We employ a coherent state and a vacuum state as inputs and perform homodyne detection. Our results show that the selection of input modes significantly affects phase estimation, and optimizing the beam splitter's transmittance is crucial for maximizing phase sensitivity in lossy conditions. Furthermore, photon subtraction markedly improves phase sensitivity, quantum Fisher information, and robustness against noise. Our scheme achieves sensitivities beyond the Heisenberg limit even under 20% photon loss.

Figures (13)

• Figure 1: Schematic diagram of the hybrid interferometer with PS under photon loss. Two input configurations are considered: Scheme A $\left( \left \vert \psi _{1}\right \rangle _{in}=\left \vert \alpha \right \rangle _{a}\left \vert 0\right \rangle _{b}\right)$ and Scheme B $\left( \left \vert \psi _{2}\right \rangle _{in}=\left \vert 0\right \rangle _{a}\left \vert \beta \right \rangle _{b}\right)$. OPA denotes an optical parametric amplifier, $\phi$ is the phase shift, vBS is a variable beam splitter, and $D_{a}$ is the homodyne detector. The operator $a^{m}$ corresponds to $m$-photons subtraction, and $a_{v}$ denotes vacuum mode.
• Figure 2: The contour plots of phase sensitivity as a function of parameters $\phi$ and $\tau$, fixed with $\alpha (\beta )=1$ and $g=1$, where $m$ denotes the PS order. Fig. 2(a)-(d) correspond respectively to $m=0$, $1$, $2$, and $3$ in Scheme A; Fig. 2(e)-(h) correspond to those in Scheme B.
• Figure 3: The phase sensitivity as a function of (a) phase $\phi$, with $\tau =0.5$, $\alpha \left( \beta \right) =1$, and $g=1$; (b) transmittance $\tau$, with $\phi =\pi /2$, $\alpha \left( \beta \right) =1$, and $g=1$. The solid and dashed lines represent Schemes A and B, respectively.
• Figure 4: The phase sensitivity of PS based on homodyne detection as a function of (a) $\alpha \left( \beta \right)$ with $\phi = \pi /2$, $\tau =0.5$, and $g=1$; (b) $g$ with $\phi = \pi /2$, $\tau =0.5$, and $\alpha \left( \beta \right) =1$. The solid and dashed lines represent Schemes A and B, respectively.
• Figure 5: The phase sensitivity of PS as a function of $T$, with $\phi = \pi /2$, $\alpha \left( \beta \right) =1$, and $g=1$. (a) Scheme A; (b) Scheme B. The solid and dashed line correspond to $\tau =0.5$, while the point set corresponds to $\tau =0.7$.