What Is the Spectral Density of the Reservoir for a Lossy Quantized Cavity?

By considering a two-level system weakly coupled to a single lossy three-dimensional cavity mode, and using a quasinormal mode expansion for the photonic medium, we show that the frequency dependence of the quantum coupling interaction between the cavity and its photonic reservoir can not be simply constructed to be independent of the cavity contents. For the single quantum dipole case, we identify the correct form of the local spectral density, revealing a surprising gauge-dependent ∼ω^{±1} prefactor scaling, as well as a spatially dependent correction. We thus establish the correct quantum form for the cavity-reservoir interaction, which so far has only been worked out rigorously for simple 1D geometries, and show its significant impact on broadband strong coupling.