  Driving a conventional superconductor with an appropriately tuned classical
electromagnetic field can lead to an enhancement of superconductivity via a
redistribution of the quasiparticles into a more favorable non-equilibrium
distribution -- a phenomenon known as the Eliashberg effect. Here we
theoretically consider coupling a two-dimensional superconducting film to the
quantized electromagnetic modes of a microwave resonator cavity. As in the
classical Eliashberg case, we use a kinetic equation to study the effect of the
fluctuating, dynamical electromagnetic field on the Bogoliubov quasiparticles.
We find that when the photon and quasiparticle systems are out of thermal
equilibrium, a redistribution of quasiparticles into a more favorable
non-equilibrium steady-state occurs, thereby enhancing superconductivity in the
sample. We predict that by tailoring the cavity environment (e.g. the photon
occupation and spectral functions), enhancement can be observed in a variety of
parameter regimes, offering a large degree of tunability.
