  The Jaynes-Cummings (JC) model represents one of the simplest ways in which
single qubits can interact with single photon modes, leading to profound
quantum phenomena like superpositions of light and matter states. One system,
that can be described with the JC model, is a single quantum dot embedded in a
micropillar cavity. In this joint experimental and theoretical study we
investigate such a system using four-wave mixing (FWM) micro-spectroscopy.
Special emphasis is laid on the dependence of the FWM signals on the number of
photons injected into the microcavity. By comparing simulation and experiment,
which are in excellent agreement with each other, we infer that up to ~20
photons take part in the observed FWM dynamics. Thus we verify the validity of
the JC model for the system under consideration in this non-trivial regime. We
find that the inevitable coupling between the quantum dot exciton and
longitudinal acoustic phonons of the host lattice influences the real time FWM
dynamics and has to be taken into account for a sufficient description of the
quantum dot-microcavity system. Performing additional simulations in an
idealized dissipation-less regime, we observe that the FWM signal exhibits
quasi-periodic dynamics, analog to the collapse and revival phenomenon of the
JC model. In these simulations we also see that the FWM spectrum has a triplet
structure, if a large number of photons is injected into the cavity.
