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

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.