  We investigate non-equilibrium phase transitions for driven atomic ensembles,
interacting with a cavity mode, coupled to a Markovian dissipative bath. In the
thermodynamic limit and at low-frequencies, we show that the distribution
function of the photonic mode is thermal, with an effective temperature set by
the atom-photon interaction strength. This behavior characterizes the static
and dynamic critical exponents of the associated superradiance transition.
Motivated by these considerations, we develop a general Keldysh path integral
approach, that allows us to study physically relevant nonlinearities beyond the
idealized Dicke model. Using standard diagrammatic techniques, we take into
account the leading-order corrections due to the finite number of atoms N. For
finite N, the photon mode behaves as a damped, classical non-linear oscillator
at finite temperature. For the atoms, we propose a Dicke action that can be
solved for any N and correctly captures the atoms' depolarization due to
dissipative dephasing.
