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

## TL;DR

This paper presents a theoretical framework to analyze how the dimensionality of the phonon bath affects the spectral efficiency of single-photon emitters in micro-cavities, revealing the role of phonon interactions in tuning emission properties.

## Contribution

The study introduces a simple formula linking spectral efficiency to free-space spectra and loss rates, accounting for phonon bath dimensionality effects in solid-state emitters.

## Key findings

- Spectral efficiency varies with phonon bath dimensionality.
- Asymmetric energy exchange enhances tuning range via phonon-assisted cavity feeding.
- The formula simplifies analysis of cavity-coupled emitter dynamics.

## Abstract

We develop a theoretical frame to investigate the single-photon emission spectral efficiency of a solid-state nano-emitter embedded in a high quality factor micro-cavity. This study encompasses the case of localized excitons embedded in a one, two or three-dimensional matrix. The populations evolutions are calculated based on a spin-boson model, using the non-interacting blip approximation (NIBA). We find that the single-photon spectral efficiency of the cavity-coupled emitter can be expressed by a simple formula, taking as inputs the free-space emission and absorption spectra of the emitter as well as the loss rates of the system. In other words, the information on the interaction between the exciton and the phonon bath, encoded in the free-space behavior of the emitter, is sufficient to obtain the dynamics of the system in the cavity. We compute numerically the spectral efficiency for several types of localized emitters and show distinct behaviors depending on the phonon bath dimensionality. In particular, a pronounced asymmetric energy exchange between the emitter and the cavity on the side-bands can yield a considerable extension of the tuning range of the sources through phonon-assisted cavity feeding.