Quantum atom-light interfaces in the gaussian description for spin-1 systems

We extend the covariance-matrix description of atom--light quantum interfaces, originally developed for real and effective spin-1/2 atoms, to include "spin alignment" degrees of freedom. This allows accurate modeling of optically-probed spin-1 ensembles in arbitrary magnetic fields. We also include technical noise terms that are very common in experimental situations. These include magnetic field noise, variable atom number and the effect of magnetic field inhomogeneities. We demonstrate the validity of our extended model by comparing numerical simulations to a free--induction decay (FID) measurement of polarized $^{87}$Rb atoms in the $f = 1$ ground state. We qualitatively and quantitatively reproduce experimental results with all free parameters of the simulations fixed. The model can be easily extended to larger spin systems, and adapted to more complicated experimental situations.