# Real-time solutions of coupled Ehrenfest-Maxwell-Pauli-Kohn-Sham equations: fundamentals, implementation, and nano-optical applications

## TL;DR

This paper develops a density-functional approach for simulating coupled photon-electron-nuclei systems in non-relativistic quantum electrodynamics, enabling realistic modeling of light-matter interactions in nano-optics and related fields.

## Contribution

It introduces a Kohn-Sham formalism for coupled light-matter systems and implements it in the Octopus code using a Schrödinger form of Maxwell's equations.

## Key findings

- Formalism establishes a one-to-one correspondence between external fields and internal variables.
- Implementation demonstrates feasibility of real-time simulations of coupled systems.
- Applicable to various nano-optical and light-matter interaction scenarios.

## Abstract

We present the theoretical foundations and the implementation details of a density-functional approach for coupled photons, electrons, and effective nuclei in non-relativistic quantum electrodynamics. Starting point of the formalism is a generalization of the Pauli-Fierz field theory for which we establish a one-to-one correspondence between external fields and internal variables. Based on this correspondence, we introduce a Kohn-Sham construction which provides a computationally feasible approach for ab-initio light-matter interactions. In the mean-field limit for the effective nuclei the formalism reduces to coupled Ehrenfest-Maxwell-Pauli-Kohn-Sham equations. We present an implementation of the approach in the real-space real-time code Octopus. For the implementation we use the Riemann-Silberstein formulation of classical electrodynamics and rewrite Maxwell's equations in Schr\"odinger form. This allows us to use existing time-evolution algorithms developed for quantum-mechanical systems also for Maxwell's equations. Overall, our approach is ideally suited for applications in nano-optics, nano-plasmonics, (photo) electrocatalysis, light-matter coupling in 2D materials, cases where laser pulses carry orbital angular momentum, or light-tailored chemical reactions in optical cavities to name but a few.