  We propose an experiment with a driven quantum gas coupled to a dissipative
optical cavity that realizes a novel kind of far-from-equilibrium phase
transition between spatial and temporal order. The control parameter of the
transition is the detuning between the drive frequency and the cavity
resonance. For negative detunings, the system features a spatially ordered
phase, while positive detunings lead to a phase with both spatial order and
persistent oscillations, which we call dissipative spatio-temporal lattice. We
give numerical and analytical evidence for this superradiant phase transition
and show that the spatio-temporal lattice originates from cavity dissipation.
In both regimes the atoms are subject to an accelerated transport, either via a
uniform acceleration or via abrupt transitions to higher momentum states. Our
work provides perspectives for temporal phases of matter that are not possible
at equilibrium.
