Effect of phonon-bath dimensionality on the spectral tuning of single-photon emitters in the Purcell regime

We develop a theoretical frame to investigate the spectral dependence of the brightness of a single-photon source made of a solid-state nanoemitter embedded in a high-quality factor microcavity. This study encompasses the cases of localized excitons embedded in a one-, two-, or three-dimensional matrix. The population evolution is calculated based on a spin-boson model, using the noninteracting blip approximation. We find that the spectral dependence of the single-photon source brightness (hereafter called spectral efficiency) can be expressed analytically through the free-space emission and absorption spectra of the emitter, the vacuum Rabi splitting, and the loss rates of the system. In other words, the free-space spectrum of the emitter encodes all the relevant information on the interaction between the exciton and the phonon bath to obtain the dynamics of the cavity-coupled system. We compute numerically the spectral efficiency for several types of localized emitters differing by the phonon bath dimensionality. In particular, in low-dimensional systems where this interaction is enhanced, a pronounced asymmetric energy exchange between the emitter and the cavity on the phonon sidebands yields a considerable extension of the tuning range of the source through phonon-assisted cavity feeding, possibly surpassing that of a purely resonant system.