Cavity engineering of Hubbard $U$ via phonon polaritons

Pump-probe experiments have suggested the possibility to control electronic correlations by driving infrared-active phonons with resonant midinfrared laser pulses. In this work we study two possible microscopic nonlinear electron-phonon interactions behind these observations, namely coupling of the squared lattice displacement either to the electronic density or to the double occupancy. We investigate whether photon-phonon coupling to quantized light in an optical cavity enables similar control over electronic correlations. We first show that inside a dark cavity electronic interactions increase, ruling out the possibility that $T_c$ in superconductors can be enhanced via effectively decreased electron-electron repulsion through nonlinear electron-phonon coupling in a cavity. We further find that upon driving the cavity, electronic interactions decrease. Two different regimes emerge: (i) a strong coupling regime where the phonons show a delayed response at a time proportional to the inverse coupling strength, and (ii) an ultra-strong coupling regime where the response is immediate when driving the phonon polaritons resonantly. We further identify a distinctive feature in the electronic spectral function when electrons couple to phonon polaritons involving an infrared-active phonon mode, namely the splitting of the shake-off band into three bands. This could potentially be observed by angle-resolved photoemission spectroscopy.