  We study the effect of pure dephasing on the strong-coupling between a
quantum dot and the single mode of a microcavity in the nonlinear regime. We
show that the photoluminescence spectrum of the system has a robust tendency to
display triplet structures, instead of the expected Jaynes-Cummings pairs of
doublets at the incommensurate frequencies $\pm(\sqrt{n}\pm\sqrt{n-1})$ for
integer $n$. We show that current experimental works may already manifest
signatures of single photon nonlinearities.
