# Understanding Detailed Balance for an Electron-Radiation System Through Mixed Quantum-Classical Electrodynamics

## TL;DR

This paper examines how mixed quantum-classical methods can accurately simulate thermal equilibrium in electron-radiation systems, identifying the strengths and limitations of three semiclassical approaches.

## Contribution

It demonstrates that Ehrenfest+R dynamics can recover detailed balance and correct thermal equilibrium in quantum-electrodynamics simulations.

## Key findings

- Ehrenfest dynamics over-estimate excited state populations due to vacuum fluctuation failure.
- Maxwell-Bloch equations under-estimate populations due to double-counting self-interactions.
- Ehrenfest+R dynamics successfully recover detailed balance and thermal equilibrium.

## Abstract

We investigate detailed balance for a quantum system interacting with thermal radiation within mixed quantum-classical theory. For a two-level system coupled to classical radiation fields, three semiclassical methods are benchmarked: (1) Ehrenfest dynamics over-estimate the excited state population at equilibrium due to the failure of capturing vacuum fluctuations. (2) The coupled Maxwell-Bloch equations, which supplement Ehrenfest dynamics by damping at the full golden rule rate, under-estimate the excited state population due to double-counting of the self-interaction effect. (3) Ehrenfest+R dynamics recover detailed balance and the correct thermal equilibrium by enforcing the correct balance between the optical excitation and spontaneous emission of the quantum system. These results highlight the fact that, when properly designed, mixed quantum-classical electrodynamics can simulate thermal equilibrium in the field of nanoplasmonics.