We develop a theoretical frame to investigate the single-photon emissionspectral efficiency of a solid-state nano-emitter embedded in a high qualityfactor micro-cavity. This study encompasses the case of localized excitonsembedded in a one, two or three-dimensional matrix. The populations evolutionsare calculated based on a spin-boson model, using the non-interacting blipapproximation (NIBA). We find that the single-photon spectral efficiency of thecavity-coupled emitter can be expressed by a simple formula, taking as inputsthe free-space emission and absorption spectra of the emitter as well as theloss rates of the system. In other words, the information on the interactionbetween the exciton and the phonon bath, encoded in the free-space behavior ofthe emitter, is sufficient to obtain the dynamics of the system in the cavity.We compute numerically the spectral efficiency for several types of localizedemitters and show distinct behaviors depending on the phonon bathdimensionality. In particular, a pronounced asymmetric energy exchange betweenthe emitter and the cavity on the side-bands can yield a considerable extensionof the tuning range of the sources through phonon-assisted cavity feeding.
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