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

We propose an orbital exchange-correlation functional for applying time-dependent density functional theory to many-electron systems coupled to cavity photons. The time nonlocal equation for the electron-photon optimized effective potential (OEP) is derived. In the static limit our OEP energy functional reduces to the Lamb shift of the ground state energy. We test the new approximation in the Rabi model. It is shown that the OEP (i) reproduces quantitatively the exact ground-state energy from the weak to the deep strong coupling regime and (ii) accurately captures the dynamics entering the ultrastrong 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.

Figure 1

Comparison of the OEP (red), exact (black), and classical (green) (a) density difference $\mathrm{\Delta }n$ and (b) energy $E$ versus the coupling parameter $\lambda$ in a.u. Other parameters: $\omega =1$, $v_{\text{ext}}=0.2$, $T=0.7$.

Figure 2

Comparison of the (a) errors $\delta \mathrm{\Delta }n$ in the TDOEP (black) and classical (blue) density difference $\mathrm{\Delta }n$ and (b),(c) TDOEP (red), exact (black), and classical (green) effective potential $v_{\text{eff}}$ versus time $t$ in a.u. for the configurations: (a),(b) $v_{\text{ext}}=-0.2\text{sgn}(t)$, $\lambda =0.1$ and (c) $v_{\text{ext}}=0$, $\lambda =0.1\theta (t)$. Other parameters: $\omega =1$, $T=0.7$.