Simultaneous tracking of spin angle and amplitude beyond classical limits

Measurement of spin precession is central to extreme sensing in physics,1,2 geophysics,3 chemistry,4 nanotechnology5 and neuroscience,6 and underlies powerful magnetic resonance spectroscopies.7 Because there is no spin-angle operator, any measurement of spin precession is necessarily indirect, e.g., inferred from 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. Here we show how to reduce this disturbance below [Formula: see text] the classical limit for N spins, by directing the QMBA almost entirely into an unmeasured spin component. This generates a planar squeezed state8 which, because spins obey non-Heisenberg uncertainty relations,9,10 allows simultaneous precise knowledge of spin angle and amplitude. We use high-dynamic-range optical quantum non-demolition measurements11–13 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.14 This method for the first time surpasses classical limits in non-commuting observables, and enables orders-of-magnitude sensitivity boosts for state-of-the-art sensing15–18 and spectroscopy.19,20