Thermodynamics and dynamics of atomic self-organization in an optical cavity

Pattern formation of atoms in high-finesse optical resonators results from the mechanical forces of light associated with superradiant scattering into the cavity mode. It occurs when the laser intensity exceeds a threshold value such that the pumping processes counteract the losses. We consider atoms driven by a laser and coupling with a mode of a standing-wave cavity and describe their dynamics with a Fokker-Planck equation, in which the atomic motion is semiclassical but the cavity field is a full quantum variable. The asymptotic state of the atoms is a thermal state, whose temperature is solely controlled by the detuning between the laser and the cavity frequency and by the cavity loss rate. From this result we derive the free energy and show that in the thermodynamic limit self-organization is a second-order phase transition. The order parameter is the field inside the resonator to which one can associate a magnetization in analogy to ferromagnetism, the control field is the laser intensity, but the steady state is intrinsically out of equilibrium. In the symmetry-broken phase, quantum noise induces jumps of the spatial density between two ordered patterns: We characterize the statistical properties of this temporal behavior at steady state and show that the thermodynamic properties of the system can be extracted by detecting the light at the cavity output. The results of our analysis are in full agreement with previous studies; we extend them by deriving a self-consistent theory which is valid also when the cavity field is in the shot-noise limit and elucidate the nature of the self-organization transition.