Quantum electrodynamical time-dependent density functional theory on a lattice

TL;DR

This paper develops a rigorous time-dependent density functional theory framework for lattice electrons strongly coupled to cavity photons, proving unique functional mappings and extending to many-electron systems with multiple photonic modes.

Contribution

It introduces a formalism for lattice electron-photon systems, proving the uniqueness of the wave function as a functional of density and photonic expectation, and generalizes to complex many-electron, multi-mode systems.

Findings

Proved the wave function is a unique functional of density and photonic coordinate for a single particle.

Extended the formalism to many interacting electrons on a lattice with multiple photonic modes.

Showed ground state evolved systems have locally v-representable densities.

Abstract

We present a rigorous formulation of the time-dependent density functional theory for interacting lattice electrons strongly coupled to cavity photons. We start with an example of one particle on a Hubbard dimer coupled to a single photonic mode, which is equivalent to the single mode spin-boson model or the quantum Rabi model. For this system we prove that the electron-photon wave function is a unique functional of the electronic density and the expectation value of the photonic coordinate, provided the initial state and the density satisfy a set of well defined conditions. Then we generalize the formalism to many interacting electrons on a lattice coupled to multiple photonic modes and prove the general mapping theorem. We also show that for a system evolving from the ground state of a lattice Hamiltonian any density with a continuous second time derivative is locally $v$-representable.