  We investigate plasmon-emitter interactions in a nanoparticle-on-a-mirror
cavity. We consider two different sorts of emitters, those that sustain dipolar
transitions, and those hosting only quadrupolar, dipole-inactive, excitons. By
means of a fully analytical two-dimensional transformation optics approach, we
calculate the light-matter coupling strengths for the full plasmonic spectrum
supported by the nanocavity. We reveal the impact of finite-size effects in the
exciton charge distribution and describe the population dynamics in a
spontaneous emission configuration. Pushing our model beyond the quasi-static
approximation, we extract the plasmonic dipole moments, which enables us to
calculate the far-field scattering spectrum of the hybrid plasmon-emitter
system. Our findings, tested against fully numerical simulations, reveal the
similarities and differences between the strong coupling phenomenology for
bright and dark excitons in nanocavities.
