Optimal phase measurements in a lossy Mach-Zehnder interferometer with coherent

This work discusses the single-intensity and difference-intensity detection schemes for lossy Mach-Zehnder Interferometer (MZI) and gives the corresponding optimal conditions to achieve the best phase measurements. The interferometer is fed by a coherent state in one port and a vacuum state in the other. When the core parameters (reflectivities, phase difference) are optimal, the phase sensitivity of these two detection schemes can reach a generalized precision bound: the standard interferometric limit (SIL). In the experiment, we design an MZI with adjustable beam splitters to verify the phase sensitivity optimization compared to the conventional MZI with 50:50 beam splitters. The sensitivity improvements at loss rates from 0.4 to 0.998 are demonstrated based on difference-intensity detection, matching the theoretical results well. With a loss up to 0.998 in one arm, we achieve a sensitivity improvement of 2.5 dB by adjusting reflectivity, which equates to a 5.5 dB sensitivity improvement in single-intensity detection. Such optimal phase measurement methods provide practical solutions for the correct use of resources in lossy interferometers.