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The generation of photons in a microcavity coupled to a laser-driven quantumdot interacting with longitudinal acoustic (LA) phonons is studied in theregime of simultaneously strong driving and strong dot-cavity coupling. Thestationary cavity photon number is found to depend in a non-trivial way on thedetuning between the laser and the exciton transition in the dot. Inparticular, the maximal efficiency of the cavity feeding is obtained fordetunings corresponding to transition energies between cavity-dressed stateswith excitation numbers larger than one. Phonons significantly enhance thecavity feeding at large detunings. In the strong-driving, strong-couplinglimit, the photon statistics is highly non-Poissonian. While without phonons adouble-peaked structure in the photon distribution is predicted, phonons makethe photon statistics thermal-like with very high effective temperatures $\sim10^5$ K, even for low phonon temperatures $\sim 4$ K. These results wereobtained by numerical calculations where the driving, the dot-cavity couplingand the dot-phonon interactions are taken into account without approximations.This is achieved by a reformulation of an exact iterative path-integral schemewhich is applicable for a large class of quantum-dissipative systems and whichin our case reduces the numerical demands by 15 orders of magnitude.

Nonlinear cavity feeding and unconventional photon statistics in solid-state cavity QED revealed by many-level real-time path-integral calculations