Spectral densities of a dispersive dielectric sphere in the modified Langevin noise formalism

This paper deals with the spectral densities of a dispersive dielectric object in the framework of macroscopic quantum electrodynamics based on the modified Langevin noise formalism. In this formalism, the electromagnetic field in the presence of a dielectric object has two contributions, one taking into account the polarization current fluctuations of the object and the other taking into account the vacuum field fluctuations scattered by the object. The combined effect of these fields on the dynamics of a quantum emitter is described via two independent continuous bosonic reservoirs, a medium-assisted reservoir and a scattering-assisted reservoir, each characterized by its own spectral density and initial quantum state. For initial thermal states of the two reservoirs at different temperatures, the standard approach based on the knowledge of the dyadic Green function of the dielectric object at the quantum emitter position cannot be employed. We map the two reservoirs to a single equivalent reservoir with a temperature-dependent effective spectral density and initially in its vacuum state, focusing on the case of a homogeneous dielectric sphere. We derive analytical expressions for the medium-assisted, scattering-assisted, and effective spectral densities in this setting. We then study the dynamics of the quantum emitter for initial thermal states of the two reservoirs, adopting a non-perturbative approach.