# Phase sensitivity for an unbalanced interferometer without input phase-matching restrictions

## TL;DR

This paper explores how unbalanced interferometers with variable input phase mismatch and beam splitter transmission can optimize phase sensitivity, revealing scenarios where unbalanced configurations outperform balanced ones.

## Contribution

It introduces a method to optimize Fisher information in unbalanced interferometers by varying input phase mismatch and beam splitter parameters, extending beyond phase-matched conditions.

## Key findings

- Maximum Fisher information achieved with unbalanced inputs for double coherent states.
- Existence of a threshold phase mismatch for coherent plus squeezed vacuum inputs.
- Optimal squeezing angles are in anti-phase for squeezed-coherent plus squeezed vacuum inputs.

## Abstract

The Cram\'er-Rao bound and the quantum Fisher information have been tools used extensively for interferometric phase sensitivity. Most scenarios considering a Mach-Zehnder interferometer with two input sources focused on the phase-matched case, when the Fisher information is maximal. Under this constraint, the best sensitivity is achieved for a balanced (50/50) input beam splitter. In this paper, we take a different approach: we allow the beam splitter transmission coefficient as well as the input phase mismatch to be variable parameters. We then search for a pair of these parameters that maximizes the Fisher information. We find that for the double coherent input the maximum Fisher information can always be reached in the unbalanced case for a carefully chosen input phase mismatch. For the coherent plus squeezed vacuum case we find that under certain circumstances a threshold phase mismatch exists, beyond which the optimum Fisher information is found for the degenerate case. For the squeezed-coherent plus squeezed vacuum case we find that the optimum actually occurs when the squeezing angles of the two inputs are in anti-phase.