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