  We show how the decoherence that occurs in an entangling atomic spin-light
interface can be simply modeled as the dynamics of a bosonic mode. Although one
seeks to control the collective spin of the atomic system in the
permutationally invariant (symmetric) subspace, diffuse scattering and optical
pumping are local, making an exact description of the many-body state
intractable. To overcome this issue we develop a generalized Holstein-Primakoff
approximation for collective states which is valid when decoherence is uniform
across a large atomic ensemble. In different applications the dynamics is
conveniently treated as a Wigner function evolving according to a thermalizing
diffusion equation, or by a Fokker-Planck equation for a bosonic mode decaying
in a zero temperature reservoir. We use our formalism to study the combined
effect of Hamiltonian evolution, local and collective decoherence, and
measurement backaction in preparing nonclassical spin states for application in
quantum metrology.
