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We consider an ensemble of cavity-coupled two-level emitters interacting viafull (coherent and dissipative) dipole-dipole interactions. Using an adiabaticelimination procedure we derive effective equations of motion for a subsystemconsisting of the cavity and a single emitter. Those equations can be used forschemes to enhance the cavity-coupling of single emitters as shown in [S.Sch\"{u}tz, J. Schachenmayer, D. Hagenm\"{u}ller, G. K. Brennen, T. Volz, V.Sandoghdar, T. W. Ebbesen, C. Genes, and G. Pupillo, e-print arXiv:1904.08888].We analyze limitations of effective subsystem parameters, and study how jointdissipative decay processes in the subsystem affect cavity-coupling propertiesof the single emitter and cavity transmission spectra.

Adiabatic elimination for ensembles of emitters in cavities with dissipative couplings