  The generation of photons in a microcavity coupled to a laser-driven quantum
dot interacting with longitudinal acoustic (LA) phonons is studied in the
regime of simultaneously strong driving and strong dot-cavity coupling. The
stationary cavity photon number is found to depend in a non-trivial way on the
detuning between the laser and the exciton transition in the dot. In
particular, the maximal efficiency of the cavity feeding is obtained for
detunings corresponding to transition energies between cavity-dressed states
with excitation numbers larger than one. Phonons significantly enhance the
cavity feeding at large detunings. In the strong-driving, strong-coupling
limit, the photon statistics is highly non-Poissonian. While without phonons a
double-peaked structure in the photon distribution is predicted, phonons make
the photon statistics thermal-like with very high effective temperatures $\sim
10^5$ K, even for low phonon temperatures $\sim 4$ K. These results were
obtained by numerical calculations where the driving, the dot-cavity coupling
and the dot-phonon interactions are taken into account without approximations.
This is achieved by a reformulation of an exact iterative path-integral scheme
which is applicable for a large class of quantum-dissipative systems and which
in our case reduces the numerical demands by 15 orders of magnitude.
