  Laser control of solids was so far mainly discussed in the context of strong
classical nonlinear light-matter coupling in a pump-probe framework. Here we
propose a quantum-electrodynamical setting to address the coupling of a
low-dimensional quantum material to quantized electromagnetic fields in quantum
cavities. Using a protoypical model system describing FeSe/SrTiO$_3$ with
electron-phonon long-range forward scattering, we study how the formation of
phonon polaritons at the 2D interface of the material modifies effective
couplings and superconducting properties in a Migdal-Eliashberg simulation. We
find that through highly polarizable dipolar phonons, large cavity-enhanced
electron-phonon couplings are possible but superconductivity is not enhanced
for the forward-scattering pairing mechanism due to the interplay between
coupling enhancement and mode softening. An analysis of critical temperature
dependencies on couplings and mode frequencies suggests that that
cavity-enhanced superconductivity is possible for more conventional short-range
pairing mechanisms. Our results demonstrate that quantum cavities enable the
engineering of fundamental couplings in solids paving the way to unprecedented
control of material properties.
