Abstract

Continuous monitoring or tracking of a quantum system is essential to high- sensitivity measurement of time-varying quantities. We estimate spin precession of a magnetically-sensitive atomic ensemble by measuring spin projectors F_α at different times. Such projectors do not commute, and thus quantum measurement back-action (QMBA) necessarily enters the spin measurement record, introducing errors and limiting sensitivity. We show how to reduce this disturbance below $\delta F_{\alpha }~\sqrt{N}$, the classical limit for N spins, by directing the QMBA almost entirely into an unmeasured spin component. This generates a planar squeezed state [1,2] which allows simultaneous precise knowledge of spin angle and amplitude. We use high-dynamic-range optical quantum non-demolition measurements applied to a precessing magnetic spin ensemble, to demonstrate spin tracking with steady-state angular sensitivity 2.9 dB beyond the standard quantum limit, simultaneous with amplitude sensitivity 7.0 dB beyond Poisson statistics, surpassing classical limits in non-commuting observables for the first time [3]. Our method is close to practical application in high-performance atomic sensors, such as magnetometers and clocks, and is compatible with multi-pass and cavity build-up methods.

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