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We argue that it is possible in principle to reduce the uncertainty of anatomic magnetometer by double-passing a far-detuned laser field through theatomic sample as it undergoes Larmor precession. Numerical simulations of thequantum Fisher information suggest that, despite the lack of explicitmulti-body coupling terms in the system's magnetic Hamiltonian, the parameterestimation uncertainty in such a physical setup scales better than theconventional Heisenberg uncertainty limit over a specified but arbitrary rangeof particle number N. Using the methods of quantum stochastic calculus andfiltering theory, we demonstrate numerically an explicit parameter estimator(called a quantum particle filter) whose observed scaling follows that of ourcalculated quantum Fisher information. Moreover, the quantum particle filterquantitatively surpasses the uncertainty limit calculated from the quantumCramer-Rao inequality based on a magnetic coupling Hamiltonian with onlysingle-body operators. We also show that a quantum Kalman filter isinsufficient to obtain super-Heisenberg scaling, and present evidence that suchscaling necessitates going beyond the manifold of Gaussian atomic states.

Magnetometry via a double-pass continuous quantum measurement of atomic spin