Estimation of a classical parameter with Gaussian probes: Magnetometry with collective atomic spins

We present a theory for the estimation of a classical magnetic field by an atomic sample with a Gaussian distribution of collective spin components. By incorporating the magnetic field and the probing laser field as quantum variables with Gaussian distributions on equal footing with the atoms, we obtain a very versatile description which is readily adapted to include probing with squeezed light, dissipation, and loss and additional measurement capabilities on the atomic system.