Continuous real-time tracking of a quantum phase below the standard quantum limit

Abstract

We propose a scheme for continuously measuring the evolving quantum phase of a collective spin composed of $N$ pseudospins. Quantum non-demolition measurements of a lossy cavity mode interacting with an atomic ensemble are used to directly probe the phase of the collective atomic spin without converting it into a population difference. Unlike traditional Ramsey measurement sequences, our scheme allows for real-time tracking of time-varying signals. As a bonus, spin-squeezed states develop naturally, providing real-time phase estimation significantly more precise than the standard quantum limit of $\Delta {\varphi }_{\text{SQL}}=1/\sqrt{N}$ radians.

6 + 6 pages, 3 + 2 figures, includes Supplemental Material; close to journal version