Kohn-Sham approach to quantum electrodynamical density-functional theory: Exact time-dependent effective potentials in real space

Abstract

The density-functional approach to quantum electrodynamics extends traditional density-functional theory and opens the possibility to describe electron-photon interactions in terms of effective Kohn-Sham potentials. In this work, we numerically construct the exact electron-photon Kohn-Sham potentials for a prototype system that consists of a trapped electron coupled to a quantized electromagnetic mode in an optical high-Q cavity. Although the effective current that acts on the photons is known explicitly, the exact effective potential that describes the forces exerted by the photons on the electrons is obtained from a fixed-point inversion scheme. This procedure allows us to uncover important beyond-mean-field features of the effective potential that mark the breakdown of classical light-matter interactions. We observe peak and step structures in the effective potentials, which can be attributed solely to the quantum nature of light; i.e., they are real-space signatures of the photons. Our findings show how the ubiquitous dipole interaction with a classical electromagnetic field has to be modified in real space to take the quantum nature of the electromagnetic field fully into account.

Keywords: photon matter correlations; quantum electrodynamical density functional theory; quantum electrodynamics; strong light matter interaction; time-dependent density functional theory.

Fig. 1.

The figure schematically illustrates a 2D optical cavity containing one atom, with a single electron. The coupling of the electron to the cavity mode at resonance frequency ${\omega }_{\alpha }$ and with electron–photon coupling strength ${\lambda }_{\alpha }$ modifies the dynamics of the electron density $n\left(\mathbf{r}t\right)$, which moves in the external potential $v_{\text{ext}}\left(\mathbf{r}t\right)$.

Fig. S1.

Correlated electron–photon propagation with nonfactorizable initial state and driven by external laser pulse: Initially the system populated the combined electron–photon ground state. A shows the applied external laser pulse, B shows the exact (black) and classical (red) dipole moment. C shows the projections of the time-dependent wave function on the first three electronic eigenstates. D shows the evolution of the Mandel parameter Q(t) and purity γ(t) for the exact (black) and classical approximation (red). E, G, and I show the v_Mxc(r,t) potential for different times; F, H, and J show the electron density n(r,t) at the corresponding times. The applied laser field v(r,t) = E(t)e_α·r was added to Hamiltonian in Eq. 1 with E(t) = E₀exp(-(t-t₀)²/σ²)sin(ωt) and polarization e_α=(1,1). We choose t₀ = 0.29 ps, σ= 0.058 ps, ħω = 1.41 meV, and E₀ = 0.23 meV/nm.

Fig. 2.

A shows the ground-state density for a weak-coupling case with ${\lambda }_{\alpha }=1.68\hspace{0.17em}\cdot \hspace{0.17em}{10}^{-3}$ meV^1/2/nm and in B we illustrate a strong-coupling case with ${\lambda }_{\alpha }=0.134$ meV^1/2/nm. The corresponding ground-state Mxc potential for ${\lambda }_{\alpha }=1.68\hspace{0.17em}\cdot \hspace{0.17em}{10}^{-3}$ meV^1/2/nm is displayed in C and for ${\lambda }_{\alpha }=0.134$ meV^1/2/nm displayed in D. In E and F, cuts (blue ${\lambda }_{\alpha }=1.68\hspace{0.17em}\cdot \hspace{0.17em}{10}^{-3}$ meV^1/2/nm and red ${\lambda }_{\alpha }=0.134$ meV^1/2/nm) through $v_{\text{Mxc}}$ along the diagonal (C) /antidiagonal (D) are shown. The white arrow in A indicates the polarization direction of the field mode.

Fig. 3.

Coherent state as initial state for the photon mode: In A we display the dipole of the exact (black) and mean-field (red) time evolution. In B we contrast the exact Mandel parameter $Q\left(t\right)$ (see main text for definition) (solid black) and $\gamma \left(t\right)$ (dotted black) with the corresponding mean-field values (red). C, F, I, and L show the corresponding Mxc potentials at different times ($t=\mathrm{0,3.67},\hspace{0.17em}4.53,\hspace{0.17em}\text{and}\hspace{0.17em}7.29\hspace{0.17em}\text{ps}$); D, G, J, and M show the corresponding diagonal cuts of the Mxc potentials; and in E, H, K, and N we present the corresponding densities. The Inset in B shows Q(t) between t = 3 and 5 ps. The negative Q(t) in the exact solution indicates nonclassical behavior in the photon mode. Movie S2 shows the full time evolution from 0 to 23 ps.

Fig. 4.

(Color online) Superposition of Fock number states as initial state for the photon mode: In A we display the dipole of the exact (black) and mean-field (red) time evolution. In B we contrast the exact $Q\left(t\right)$ (solid black) and $\gamma \left(t\right)$ (dotted black) with the corresponding mean-field values (red). In C, F, I, and L, we show the corresponding Mxc potentials at different times ($t=\mathrm{0,1.56},\hspace{0.17em}13.92,\hspace{0.17em}\text{and}\hspace{0.17em}37.83\hspace{0.17em}\text{ps}$); in D, G, J, and M, we show the corresponding diagonal cuts of the Mxc potentials; and in E, H, K, and N, we present the corresponding densities. Movie S3 shows the full time evolution from 0 to 50 ps.