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

Abstract

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 electronphoton 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. [7,8]. The description of realistic coupled matterphoton systems requires combining electronic structure methods from materials science with quantum optical models. Recently, a time-dependent density-functional theory (TDDFT) for QED has been developed [9][10][11][12], allowing for such a first-principles treatment. However, any application of this theory requires approximations to the electron-photon exchange-correlation (xc) functional, which are currently not available.In this Letter we construct the first approximation to the xc functional of QED-(TD)DFT. To achieve this goal, we extend the widely used optimized effective potential (OEP) approach in electronic structure methods [13][14][15][16][17][18] to the photon-mediated electron-electron coupling. The new functional is tested from low to high coupling regime in the Rabi model [19][20][21], through comparison with the exact and classical solutions. We also address the functional dependence on the initial many-body state, assumed to be either a fully interacting or a factorizable state. In both cases, the electron-photon OEP for the model performs well, providing a promising path to the ab initio description of strongly coupled matter-photon systems.Consider a system with an arbitrarily large number N of electrons at coordinates fr i g N i¼1 , e.g., an atom, an ion, or a molecule, interacting with M quantized electromagnetic modes of a microcavity with frequencies ω α . We denote bŷ H 0 ¼T þV ee þV ext the Hamiltonian of the electronic system with kinetic energyT, Coulomb interactionV ee , and (time-dependent) external potentialV ext ¼ P N i¼1 v ext ðr i tÞ, due to the nuclei and any classical field applied to the system. In the dipole approximation [22] the length-gauge Hamiltonian [9,23,24] of the total electron-photon system can be represented as follows:where the second term corresponds to the energy 1=8π R ðB 2 þÊ 2 Þdr of the transverse radiation field. The magnetic fieldB α ¼ ffiffiffiffiffi ffi 4π pp α in the α mode is proportional to the photon canonical momentump α , while the electric fieldÊ α ¼ ffiffiffiffiffi ffi 4π p ðω αqα − λ αR Þ is related to the canonical coordinateq α . The latter is defined via the displacemen...

Cavity Born–Oppenheimer Approximation for Correlated Electron–Nuclear-Photon Systems

“…in ref (14) for the specific case of electrons and ions, and here we extend it to the photon case. In general, the correlated electron–nuclear-photon Hamiltonian 1,18,25,30,31 can be written as follows [Throughout this work, we assume SI units, unless stated otherwise. ]consisting of the electronic Hamiltonian Ĥ e with n e electrons of mass m e the nuclear Hamiltonian Ĥ n with n n nuclei each with possibly different individual masses m i and charges Z i where T̂ n and Ŵ n are the nuclear kinetic energy and nuclear interaction, respectively.…”

“…The electron–nuclear interaction Hamiltonian Ĥ en is given byand the cavity photon Hamiltonian Ĥ p with n p quantized photon modes of frequency ω α takes the formThe displacement field operators consist of the usual photon creation and annihilation operators and [ q̂ α , p̂ α′ ] = i ℏδ α,α′ . Furthermore, the q̂ α are directly proportional to the electric displacement field operator of the α-th photon mode 30,31 at the charge-center of the system by the connection D̂ α = ϵ 0 ω α λ α q̂ α and the p̂ α are proportional to the magnetic field. In eq 5, the sum runs from 1 to 2 n p , to correctly account for the two possible polarization directions of the electromagnetic field.…”

Exact functionals for correlated electron–photon systems

“…When increasing the correlation, i.e. increasing the coupling strength l a | |, the accuracy of the mean-field or the exchange-only OEP [19] decreases. To improve and construct approximations that can treat strong-coupling situations more accurately we need a better understanding of the electron-photon contributions in the strong-coupling limit.…”

“…the potential an electron encounters due to its coupling to the electromagnetic field. For the electron-photon contributions first approximations for the xc potential along the lines of the optimized effective potential (OEP) approximation have been already demonstrated to be practical [17,19]. If, however, common approximations for the electron-electron many-body effects are used, then clearly QEDFT will face the same challenges as standard DFT when systems with strong electron-electron correlations are considered.…”

Time-dependent many-body treatment of electron-boson dynamics: Application to plasmon-accompanied photoemission

“…II B). The formalism is kept general and is thus applicable to related problems such as pseudo-particles 34 , electronphonon [35][36][37][38][39] , electron-vibron 40,41 or electron-photon 42 , or plasmonic nanojunctions 43,44 . In this study we go beyond the frozen boson scheme as often employed for electron-phonon coupling 45,46 and treat density oscillations in the system quan-tum mechanically.…”