Quantizing open resonators with non-separable boundary conditions to solve the overcounting of modes problem

A widely used cavity–free-space interaction model represents the total field as a simple sum of cavity and free-space contributions, which can double-count electromagnetic modes in and near the cavity region. This is especially problematic when the cavity is not driven by one single mode, but by multiple modes and if strong loss to a reservoir is considered. In these cases, the bandwidth of the cavity modes and thus the overlap with the free space system increases, yielding (i) spurious double counting of the free space modes; (ii) loss of a well-defined discrete cavity sector in the bad-cavity/strong-leakage regime; (iii) invalid reservoir tracing and incorrect Lindblad structure when vacuum fluctuations are traced out using free-space modes.

To overcome this problem we model the interaction of a detector with an electric field, quantized in a basis considering the whole space relevant for the interaction, such that the hermiticity of the mode decomposition is preserved. Therefore, we construct analytic expressions of the modes of a cavity system, consisting of a smaller cavity centered inside a larger cavity. This is achieved via the Method of Moments, which follows solely from Maxwell's equations, and enables to connect both cavities through an opening of the interior cavity. We find that already for small openings connecting the two cavities, the eigenmodes of the smaller cavity are altered significantly. Due to our analytic expression of the field modes, the radius of the outer cavity can always be set large enough to account for no reflection on the walls of the outer cavity, bridging the gap to a cavity-free-space setup.