# Quantum enhancement of sensitivity achieved by photon-number-resolving detection in the dark port of a two-path interferometer operating at high intensities

## TL;DR

This paper demonstrates that photon-number-resolving detection in a high-intensity two-path interferometer can achieve quantum-limited phase sensitivity, with robustness to photon losses enhanced by squeezing, bridging multiphoton interference and noise suppression.

## Contribution

It introduces a method to reach the quantum Cramér-Rao bound using photon-number-resolving detection of a small photon number in a high-intensity interferometer with squeezed vacuum.

## Key findings

- Photon-number-resolving detection achieves quantum-limited sensitivity.
- Robustness to photon losses increases with field displacement and squeezing.
- Transition from low to high loss robustness occurs at field shifts proportional to the cube of the squeezing factor.

## Abstract

It is shown that the maximal phase sensitivity of a two-path interferometer with high-intensity coherent light and squeezed vacuum in the input ports can be achieved by photon-number-resolving detection of only a small number of photons in a dark output port. It is then possible to achieve the quantum Cram\'{e}r-Rao bound of the two-path interferometer using only the field displacement dependence of the photon number statistics in the single mode output of the dark port represented by a field-displaced squeezed vacuum state. We find that, at small field displacements, it is not sufficient to use the average photon number as the estimator, indicating that an optimal phase estimation depends critically on measurements of the precise photon number. We therefore analyze the effect of detection efficiency on the Fisher information and show that there is a transition from low robustness against photon losses associated with quantum interference effects at low field displacements to high robustness against photon losses at high field displacements. The transition between the two regimes occurs at field shifts proportional to the third power of the squeezing factor, indicating that squeezing greatly enhances the phase interval in which quantum effects are relevant in optimal phase estimations using photon resolving detectors. The case under study could thus be understood as a `missing link' between genuine multiphoton interference and the straightforward suppression of noise usually associated with squeezed light.