Continuous monitoring of a quantum state beyond classical limits*

Continuous monitoring of a quantum system is essential to high-sensitivity measurement of time-varying quantities. Precession of a spin ensemble may be monitored by measuring spin projectors $F_\alpha$ at different times. Since these projectors do not commute, quantum measurement back-action (QMBA) necessarily enters the measurement record, introducing errors and limiting sensitivity. Here we show how to reduce this disturbance below $\delta F_\alpha \sim \sqrt{ N}$, the classical limit for $ N$ spins, by directing the QMBA almost entirely into an unmeasured spin component. This generates a {\bf planar squeezed state} which allows simultaneous precise knowledge of spin angle and amplitude. We demonstrate continuous monitoring of a precessing spin ensemble 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. 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.