We investigate the out-of-equilibrium dynamics of ultracold atoms trapped inan optical lattice and loaded into an optical resonator that is driventransversely. We derive an effective quantum master equation for weakatom-light coupling that can be brought into Lindblad form both in the bad andgood cavity limits. In the so-called bad cavity regime, we find that the steadystate is always that of infinite temperature, but that the relaxation dynamicscan be highly non-trivial. For small hopping, the interplay between dissipationand strong interactions generally leads to anomalous diffusion in the space ofatomic configurations. However, for a fine-tuned ratio of cavity-mediated andon-site interactions, we discover a limit featuring normal diffusion. Incontrast, for large hopping and vanishing on-site interactions, the system canbe described by a linear rate equation leading to an exponential approach ofthe infinite-temperature steady state. Finally, in the good cavity regime, weshow that for vanishing on-site interactions, the system allows for opticalpumping between momentum mode pairs enabling cavity cooling.
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