  We study the problem of estimating a time dependent magnetic field by
continuous optical probing of an atomic ensemble. The magnetic field is assumed
to follow a stochastic Ornstein-Uhlenbeck process and it induces Larmor
precession of the atomic ground state spin, which is read out by the Faraday
polarization rotation of a laser field probe. The interactions and the
measurement scheme are compatible with a hybrid quantum-classical Gaussian
description of the unknown magnetic field, and the atomic and field variables.
This casts the joint conditional quantum dynamics and classical parameter
estimation problem in the form of update formulas for the first and second
moments of the classical and quantum degrees of freedom. Our hybrid
quantum-classical theory is equivalent with the classical theory of Kalman
filtering and with the quantum theory of Gaussian states. By reference to the
classical theory of smoothing and with the quantum theory of past quantum
states, we show how optical probing after time $t$ improves our estimate of the
value of the magnetic field at time $t$, and we present numerical simulations
that analyze and explain the improvement over the conventional filtering
approach.
