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.

Figure 1

(a) Simulated single-shot atomic magnetometry photocurrent low pass filtered at $F_{\mathrm{c}}=2\pi \sqrt{J}/t_{\mathrm{t}\mathrm{o}\mathrm{t}}$. (b) Corresponding diffusion of the atomic Bloch vector as conditional squeezing is produced by continuous quantum nondemolition observation.

Figure 2

Comparison of the estimation errors for a quantum Kalman filter and a linear least squares magnetic field determination. The inset plot highlights the $1/J$ (Heisenberg limited) scaling of both procedures for $t\gg (JM{)}^{-1}$.