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

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