Noisy atomic magnetometry with Kalman filtering and measurement-based feedback

Sensing a magnetic field with an atomic magnetometer operated in real time presents significant challenges, primarily due to sensor non-linearity, the presence of noise, and the need for one-shot estimation. To address these challenges, we propose a comprehensive approach that integrates measurement, estimation and control strategies. Specifically, this involves implementing a quantum non-demolition measurement based on continuous light-probing of the atomic ensemble. The resulting photocurrent is then directed into an Extended Kalman Filter to produce instantaneous estimates of the system's dynamical parameters. These estimates, in turn, are utilised by a Linear Quadratic Regulator, whose output is applied back to the system through a feedback loop. This procedure automatically steers the atomic ensemble into a spin-squeezed state, yielding a quantum enhancement in precision. Furthermore, thanks to the feedback proposed, the atoms exhibit entanglement even when the measurement data is discarded. To prove that our approach constitutes the optimal strategy in realistic scenarios, we derive ultimate bounds on the estimation error applicable in the presence of both local and collective decoherence, and show that these are indeed attained. Additionally, we demonstrate that for large ensembles the EKF not only reliably predicts its own estimation error in real time, but also accurately estimates spin-squeezing at short timescales.