  We theoretically analyze superradiant emission of light from a cold atomic
gas, when mechanical effects of photon-atom interactions are considered. The
atoms are confined within a standing-wave resonator and an atomic metastable
dipolar transition couples to a cavity mode. The atomic dipole is incoherently
pumped in the parameter regime that would correspond to stationary
superradiance in absence of inhomogeneous broadening. Starting from the master
equation for cavity field and atomic degrees of freedom we derive a mean-field
model that allows us to determine a threshold temperature, above which thermal
fluctuations suppress superradiant emission. We then analyze the dynamics of
superradiant emission when the motion is described by a mean-field model. In
the semiclassical regime and below the threshold temperature we observe that
the emitted light can be either coherent or chaotic, depending on the
incoherent pump rate. We then analyze superradiant emission from an ideal Bose
gas at zero temperature when the superradiant decay rate $\Lambda$ is of the
order of the recoil frequency $\omega_R$. We show that the quantized exchange
of 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 of
matter-wave gratings diffracting the emitted photons. The stability of these
gratings depends on the incoherent pump rate $w$ with respect to a second
threshold value $w_c$. For $w>w_c$ the gratings are stable and the system
achieves stationary superradiance. Below this second threshold the coupled
dynamics becomes chaotic. We characterize the dynamics across these two
thresholds and show that the three phases we predict (incoherent, coherent,
chaotic) can be revealed via the coherence properties of the light at the
cavity output.
