  We theoretically analyze superradiant emission of light from an ultracold gas
of bosonic atoms confined in a bad cavity. A metastable dipolar transition of
the atoms couples to the cavity field and is incoherently pumped, the
mechanical effects of cavity-atom interactions tend to order the atoms in the
periodic cavity potential. By means of a mean-field model we determine the
conditions on the cavity parameters and pump rate that lead to the buildup of a
stable macroscopic dipole emitting coherent light. We show that this occurs
when the superradiant decay rate and the pump rate exceed threshold values of
the order of the photon recoil energy. Above these thresholds superradiant
emission is accompanied by the formation of stable matter-wave gratings that
diffract the emitted photons. Outside of this regime, instead, the
optomechanical coupling can give rise to dephasing or chaos, for which the
emitted light is respectively incoherent or chaotic. These behaviors exhibit
the features of a dynamical phase transitions and emerge from the interplay
between global optomechanical interactions, quantum fluctuations, and noise.
