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We study the problem of estimating a time dependent magnetic field bycontinuous optical probing of an atomic ensemble. The magnetic field is assumedto follow a stochastic Ornstein-Uhlenbeck process and it induces Larmorprecession of the atomic ground state spin, which is read out by the Faradaypolarization rotation of a laser field probe. The interactions and themeasurement scheme are compatible with a hybrid quantum-classical Gaussiandescription of the unknown magnetic field, and the atomic and field variables.This casts the joint conditional quantum dynamics and classical parameterestimation problem in the form of update formulas for the first and secondmoments of the classical and quantum degrees of freedom. Our hybridquantum-classical theory is equivalent with the classical theory of Kalmanfiltering and with the quantum theory of Gaussian states. By reference to theclassical theory of smoothing and with the quantum theory of past quantumstates, we show how optical probing after time $t$ improves our estimate of thevalue of the magnetic field at time $t$, and we present numerical simulationsthat analyze and explain the improvement over the conventional filteringapproach.

Estimating a fluctuating magnetic field with a continuously monitored atomic ensemble