Quantum kinetics of ultracold fermions coupled to an optical resonator

We study the far-from-equilibrium statistical mechanics of periodically driven fermionic atoms in a lossy optical resonator. We show that the interplay of the Fermi surface with cavity losses leads to sub-natural cavity linewidth narrowing, squeezed light, and out-of-equilibrium quantum statistics of the atoms. Adapting the Keldysh approach, we set-up and solve a quantum kinetic Boltzmann equation in a systematic $1/N$ expansion with $N$ the number of atoms. In the strict thermodynamic limit $N,V\rightarrow \infty$, $N/V=\text{const.}$ we find the atoms (fermions or bosons) remain immune against cavity-induced heating or cooling. At next-to-leading order in $1/N$, we find a "one-way thermalization" of the atoms determined by cavity decay. We argue that, in absence of an equilibrium fluctuation-dissipation relation, the long-time limit $\Delta t \rightarrow \infty$ does not commute with the thermodynamic limit $N\rightarrow \infty$, such that for the physically relevant case of large but finite $N$, the dynamics ultimately becomes strongly coupled, especially close to the superradiance phase transition.