Adiabatic elimination for ensembles of emitters in cavities with dissipative couplings

We consider an ensemble of cavity-coupled two-level emitters interacting via full (coherent and dissipative) dipole-dipole interactions. Using an adiabatic elimination procedure we derive effective equations of motion for a subsystem consisting of the cavity and a single emitter. Those equations can be used for schemes to enhance the cavity-coupling of single emitters as shown in [S. Schütz, J. Schachenmayer, D. Hagenmüller, 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 joint dissipative decay processes in the subsystem affect cavity-coupling properties of the single emitter and cavity transmission spectra.