Effect of pure dephasing on the Jaynes-Cummings nonlinearities

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.