Dissipation-induced non-equilibrium phases with temporal and spatial order

Understanding spatial and temporal order in many-body systems is a key challenge, particularly in out-of-equilibrium settings. A major hurdle is developing controlled model systems to study these phases. We propose an experiment with a driven quantum gas coupled to a dissipative optical cavity, realizing a non-equilibrium phase diagram featuring both spatial and temporal order. The system's control parameter is the detuning between the drive frequency and cavity resonance. Negative detunings yield a spatially ordered phase, while positive detunings produce phases with both spatial order and persistent oscillations, forming dissipative spatio-temporal lattices. We also identify a phase where the dynamics dephase, leading to chaotic behavior. Numerical and analytical evidence supports these superradiant phases, showing that the spatio-temporal lattice originates from cavity dissipation. The atoms experience accelerated transport, either via uniform acceleration or abrupt momentum transitions. Our work provides insights into temporal phases of matter not possible at equilibrium.