Quantum Kalman Filtering and the Heisenberg Limit in Atomic Magnetometry

The shot-noise detection limit in current high-precision magnetometry [I. Kominis, T. Kornack, J. Allred, and M. Romalis, Nature (London) 422, 596 (2003)] is a manifestation of quantum fluctuations that scale as $1/\sqrt{N}$ in an ensemble of $N$ atoms. Here, we develop a procedure that combines continuous measurement and quantum Kalman filtering [V. Belavkin, Rep. Math. Phys. 43, 405 (1999)] to surpass this conventional limit by exploiting conditional spin squeezing to achieve $1/N$ field sensitivity. Our analysis demonstrates the importance of optimal estimation for high bandwidth precision magnetometry at the Heisenberg limit and also identifies an approximate estimator based on linear regression.