Extended Data Fig. 5: Evaluation of the single-photon contribution via the phonon-assisted process to STPE.

(a) Schematics of the emission from the cQED system. Cavity photons are predominantly emitted from the top of the micropillar, while the direct QD emission can be probed from the side [Phys. Rev. B 93 115308 (2016)]. (b) Simulated emission spectra with and without accounting for QD-phonon interaction for the direct QD emission to the side (based on the first-order coherence for QD transition operators \(\langle {\sigma }_{Y}^{+}(t+\tau ){\sigma }_{Y}^{-}(t)\rangle \)). (c) Simulated emission spectra with and without accounting for QD-phonon interaction for the cavity emission to the top (based on the first-order coherence for cavity operators \(\langle {b}^{\dagger }(t+\tau )b(t)\rangle \), and consistent with what is measured in the experimental spectra). While the direct emission from the QD features sizable phonon side bands, the cavity effectively filters the emission, leading to a strong suppression of phonon sidebands, in agreement with literature [Nature Photon. 11, 521 (2017)]. (d) Simulated decay of an initially prepared biexciton state. Single-photon emission from the biexciton state occurs by a process involving only virtual occupations of the cavity. I.e. the slight mixing between state \(| XX,0\rangle \) and \(| Y,1\rangle \) enables emission via cavity losses towards the state \(| Y,0\rangle \) (see methods section on derivation of γ_XX). The photon energy corresponds to the difference between energies of the initial and final states, \(| XX,0\rangle \) and \(| Y,0\rangle \), respectively. This is opposed to a sequential process, which would involve significant occupations of the state \(| Y,1\rangle \), where the photon-emission to state \(| Y,0\rangle \) would contribute with a cavity photon energy resonant with the TPE transition. This is seen in panel (d) by the fact that the coherence between \(| XX,0\rangle \) and \(| Y,1\rangle \) are much larger than the occupations of \(| Y,1\rangle \). Furthermore, phonons do not significantly affect the dynamics of these observables. In particular, no significant phonon-assisted cavity feeding is found that would result in sizable real occupations of the state \(| Y,1\rangle \).