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.
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