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/√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.