Describing Local Decoherence of Spin Ensembles using a Fokker-Planck Equation in a Bosonic Mode

Many protocols seek to prepare nonclassical states of spin ensembles through an atom-light interface. When uniformly coupled, and evolving under unitary dynamics, this is simply described by the collective spin in the symmetric subspace. However, optical pumping, in which spins flip their orientation and scatter a photon into the environment, breaks this symmetry as the scattered photon will carry information about the location of the spin into the environment. This results in local decoherence (LD), which is typically inefficient to describe for large ensembles of atoms. In this work we develop a formalism that represents LD for a large number of spins using a Wigner function on a bosonic mode by making use of the Holstein-Primakoff approximation. The dynamics of the bosonic mode can be described using a Fokker-Planck equation at zero temperature. We use this formalism to study the combined effect of Hamiltonian evolution, local and collective decoherence, and measurement backaction for preparing nonclassical spin states for quantum metrology.