  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.
