  We present a semi-analytical framework for studying interactions between
quantum emitters and general electromagnetic resonators. The method relies on
the Lippmann-Schwinger equation to calculate the complex resonance frequencies
of the coupled system based only on a single calculation for the
electromagnetic resonator without the quantum emitter and with no fitting
parameters. This is in stark contrast to standard approaches in the literature,
in which the properties of the coupled system are fitted from calculated
spectra. As an application example, we consider a recent dielectric cavity
design featuring deep subwavelength confinement of light. We find the expected
anti-crossing of the emitter and cavity resonance frequencies, and comparing to
independent reference calculations, we find an extraordinary quantitative
agreement with a relative error below one part in ten thousand. In order to
unambiguously connect with the Jaynes-Cummings model, we derive an explicit
expression relating the classical description of the emitter, as modeled by a
spherical inclusion with a Lorentzian material response, to the dipole moment
of the corresponding quantum optical model. The combined framework therefore
enables classical calculations to be used for evaluating the coupling strength
entering quantum optical theories in a transparent way.
