Sub shot-noise interferometry from measurements of the one-body density

TL;DR

This paper derives the maximum-likelihood phase estimation uncertainty for interferometric protocols based on particle position measurements, demonstrating sub shot-noise sensitivity in certain configurations and analyzing practical implementation with current technology.

Contribution

It provides a general formula for phase estimation uncertainty in position-measurement-based interferometry and shows how to achieve sub shot-noise sensitivity with feasible experimental setups.

Findings

Achieves sub shot-noise phase uncertainty of Δθ∝N^{-2/3}

Sensitivity is independent of the phase value θ

Shot-noise limit can be surpassed with current technology

Abstract

We derive the asymptotic maximum-likelihood phase estimation uncertainty for any interferometric protocol where the positions of the probe particles are measured to infer the phase, but where correlations between the particles are not accessible. First, we apply our formula to the estimation of the phase acquired in the Mach-Zehnder interferometer and recover the well-know momentum formula for the phase sensitivity. Then, we apply our results to interferometers with two spatially separated modes, which could be implemented with a Bose-Einstein condensate trapped in a double-well potential. We show that in a simple protocol which estimates the phase from an interference pattern a sub shot-noise phase uncertainty of up to $\Delta\theta\propto N^{-2/3}$ can be achieved. One important property of this estimation protocol is that its sensitivity does not depend on the value of the phase $\theta$, contrary to the sensitivity given by the momentum formula for the Mach-Zehnder transformation. Finally, we study the experimental implementation of the above protocol in detail, by numerically simulating the full statistics as well as by considering the main sources of detection noise, and argue that the shot-noise limit could be surpassed with current technology.