  Spin-polarised atomic ensembles probed by light based on the Faraday
interaction are a versatile platform for numerous applications in quantum
metrology and quantum information processing. Here we consider an ensemble of
Alkali atoms that are continuously optically pumped and probed. Due to the
collective scattering of photons at large optical depth, the steady state of
atoms does not correspond to an uncorrelated tensor-product state, as is
usually assumed. We introduce a self-consistent method to approximate the
steady state including the pair correlations, taking into account the
multilevel structure of atoms. We find and characterize regimes of Raman
lasing, akin to the model of a superradiant laser. We determine the spectrum of
the collectively scattered photons, which also characterises the coherence time
of the collective spin excitations on top of the stationary correlated
mean-field state, as relevant for applications in metrology and quantum
information.
