Impact of the dark path on quantum dot single photon emitters in small cavities

Incoherent pumping in quantum dots (QDs) can create a biexciton state through two paths: via the formation of bright or dark exciton states. The latter, dark-pumping, path is shown to enhance the probability of two-photon simultaneous emission, and hence increase $g^{(2)}(0)$ by a factor $\propto 1/\gamma_S$, due to the slow spin relaxation rate $\gamma_S$ in QDs. The existence of the dark path is shown to impose a limitation on the single photon (SP) emission process, especially in nanocavities which exhibit a large exciton-cavity coupling and a Purcell enhancement for fast quantum telecommunications.