Climbing the Jaynes-Cummings ladder by photon counting

We present a new method to observe direct experimental evidence of Jaynes--Cummings nonlinearities in a strongly dissipative cavity quantum electrodynamics system, where large losses compete with the strong light-matter interaction. This is a highly topical problem, particularly for quantum dots in microcavities where transitions from higher rungs of the Jaynes--Cummings ladder remain to be evidenced explicitly. We compare coherent and incoherent excitations of the system and find that resonant excitation of the detuned emitter make it possible to unambiguously evidence few photon quantum nonlinearities in currently available experimental systems.