  Strong laser pulses can control superconductivity, inducing non-equilibrium
transient pairing by leveraging strong-light matter interaction. Here we
demonstrate theoretically that equilibrium ground-state phonon-mediated
superconductive pairing can be affected through the vacuum fluctuating
electromagnetic field in a cavity. Using the recently developed ab initio
quantum electrodynamical density-functional theory approximation, we
specifically investigate the phonon-mediated superconductive behavior of
MgB$_2$ under different cavity setups and find that in the strong light-matter
coupling regime its superconducting transition temperature can be, in
principles, enhanced by $\approx 73\%$ ($\approx 40\%$) in an in-plane
(out-of-plane) polarized cavity. However, in a realistic cavity, we expect the
T$_{\rm{c}}$ of MgB$_2$ can increase, at most, by $5$ K via photon vacuum
fluctuations. The results highlight that strong light-matter coupling in
extended systems can profoundly alter material properties in a non-perturbative
way by modifying their electronic structure and phononic dispersion at the same
time. Our findings indicate a pathway to the experimental realization of
light-controlled superconductivity in solid-state materials at equilibrium via
cavity-material engineering.
