Optimized Effective Potential for Quantum Electrodynamical Time-Dependent Density Functional Theory

We propose a practical approximation to the exchange-correlation functional of (time-dependent) density functional theory for many-electron systems coupled to photons. The (time non-local) optimized effective potential (OEP) equation for the electron- photon system is derived. We test the new approximation in the Rabi model from weak to strong coupling regimes. It is shown that the OEP (i) improves the classical description, (ii) reproduces the quantitative behavior of the exact ground-state properties and (iii) accurately captures the dynamics entering the ultra-strong coupling regime. The present formalism opens the path to a first-principles description of correlated electron-photon systems, bridging the gap between electronic structure methods and quantum optics for real material applications.