Understanding detailed balance for an electron-radiation system through mixed quantum-classical electrodynamics

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 overestimate 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, underestimate 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.