Entangled collective-spin states of atomic ensembles under nonuniform atom-light interaction

We consider the optical generation and characterization of entanglement in atomic ensembles under nonuniform interaction between the ensemble and an optical mode. We show that for a wide range of parameters a system of nonuniformly coupled atomic spins can be described as an ensemble of uniformly coupled spins with a reduced effective atom-light coupling and a reduced effective atom number, with a reduction factor of order unity given by the ensemble-mode geometry. This description is valid even for complex entangled states with arbitrary phase-space distribution functions as long as the average total spin remains large, and the detection does not resolve single spins. Furthermore, we derive an analytic formula for determining the observable entanglement in the case, of relevance in practice, where the ensemble-mode coupling differs between state generation and measurement.