  We analyze a similar scheme for producing light-mediated entanglement between
atomic ensembles, as first realized by Julsgaard, Kozhekin and Polzik [Nature
{\bf 413}, 400 (2001)]. In the standard approach to modeling the scheme, a
Holstein-Primakoff approximation is made, where the atomic ensembles are
treated as bosonic modes, and is only valid for short interaction times. In
this paper, we solve the time evolution without this approximation, which
extends the region of validity of the interaction time. For short entangling
times, we find this produces a state with similar characteristics as a two-mode
squeezed state, in agreement with standard predictions. For long entangling
times, the state evolves into a non-Gaussian form, and the two-mode squeezed
state characteristics start to diminish. This is attributed to more exotic
types of entangled states being generated. We characterize the states by
examining the Fock state probability distributions, Husimi $Q$ distributions,
and non-local entanglement between the ensembles. We compare and connect
several quantities obtained using the Holstein-Primakoff approach and our exact
time evolution methods.
