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We theoretically analyze superradiant emission of light from a cold atomicgas, when mechanical effects of photon-atom interactions are considered. Theatoms are confined within a standing-wave resonator and an atomic metastabledipolar transition couples to a cavity mode. The atomic dipole is incoherentlypumped in the parameter regime that would correspond to stationarysuperradiance in absence of inhomogeneous broadening. Starting from the masterequation for cavity field and atomic degrees of freedom we derive a mean-fieldmodel that allows us to determine a threshold temperature, above which thermalfluctuations suppress superradiant emission. We then analyze the dynamics ofsuperradiant emission when the motion is described by a mean-field model. Inthe semiclassical regime and below the threshold temperature we observe thatthe emitted light can be either coherent or chaotic, depending on theincoherent pump rate. We then analyze superradiant emission from an ideal Bosegas at zero temperature when the superradiant decay rate $\Lambda$ is of theorder of the recoil frequency $\omega_R$. We show that the quantized exchangeof mechanical energy between the atoms and the field gives rise to a threshold,$\Lambda_c$, below which superradiant emission is damped down to zero. When$\Lambda>\Lambda_c$ superradiant emission is accompanied by the formation ofmatter-wave gratings diffracting the emitted photons. The stability of thesegratings depends on the incoherent pump rate $w$ with respect to a secondthreshold value $w_c$. For $w>w_c$ the gratings are stable and the systemachieves stationary superradiance. Below this second threshold the coupleddynamics becomes chaotic. We characterize the dynamics across these twothresholds and show that the three phases we predict (incoherent, coherent,chaotic) can be revealed via the coherence properties of the light at thecavity output.

Superradiant optomechanical phases of cold atomic gases in optical resonators