Strong coupling of quantum emitters with confined electromagnetic modes of nanophotonic structures may be used to change optical, chemical, and transport properties of materials, with significant theoretical effort invested toward a better understanding of this phenomenon. However, a full theoretical description of both matter and light is an extremely challenging task. Typical theoretical approaches simplify the description of the photonic environment by describing it as a single mode or few modes. While this approximation is accurate in some cases, it breaks down strongly in complex environments, such as within plasmonic nanocavities, and the electromagnetic environment must be fully taken into account. This requires the quantum description of a continuum of bosonic modes, a problem that is computationally hard. We here investigate a compromise where the quantum character of light is taken into account at modest computational cost. To do so, we focus on a quantum emitter that interacts with an arbitrary photonic spectral density and employ the cumulant, or cluster, expansion method to the Heisenberg equations of motion up to first, second, and third order. We benchmark the method by comparing it with exact solutions for specific situations and show that it can accurately represent dynamics for many parameter ranges.
In principle, this is not the ground state of the full system, as we include counter-rotating terms in the light-matter coupling, and thus ultrastrong-coupling effects such as ground-state modifications.24 For the cases we treat below, the error due to this approximation is negligible.