  We study a generalized Dicke model, as recently realized in an atomic quantum
gas experiment, describing the collective interaction of N two-level atoms with
a single cavity mode. The model takes account of dissipation of the cavity
field, and includes a non-linear atom-photon coupling, not present in the
conventional Dicke model. We extend previous theoretical investigations of a
semiclassical model by including all quantum effects and considering finite
atom number N . Our results show good agreement between quantum expectation
values and the semiclassical model as N is increased, but also show exotic
behaviour for the corresponding quantum state as the non-linear atom-photon
coupling is varied.
