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We present a comprehensive theoretical study of the resonance fluorescencespectra of an exciton-driven quantum dot (QD) placed inside a high-$Q$semiconductor cavity and interacting with an acoustic phonon bath. We derive aquantum master equation (ME) in the polaron frame which includes exciton-phononand exciton-cavity coupling to all orders. This work details and extends thetheory used in a recent issue of {\em Physical Review Letters} (C. Roy and S.Hughes 2011: Phys. Rev. Lett. {\bf 106} 247403) to describe the QD Mollowtriplet in the regime of semiconductor cavity-QED. Here we introduce two MEforms, Nakajima-Zwanzig (NZ) and time-convolutionless (TC), both to secondorder in the system--phonon-reservoir perturbation. In the polaron frame, thesetwo ME forms are shown to yield equivalent population dynamics and fluorescencespectra for a continuous wave (cw) driving field. We also demonstrate that aMarkov approximation is valid for computing the incoherent scattering processesand we subsequently exploit the Markovian TC ME to explore the resonancefluorescence spectra of an exciton-driven QD. Both cavity-emitted andexciton-emitted spectra are studied and these are found to have qualitativelydifferent spectral features. Using a coherent driving field, the well knowncharacteristics of the atomic Mollow triplet are shown to be considerablymodified with electron--acoustic-phonon scattering and we highlight the keyeffects arising from both cavity coupling and electron-phonon coupling. Regimesof pronounced cavity feeding and anharmonic cavity-QED are exemplified, and wefind that the cavity coupling depends sensitively on the exciton-cavitydetuning and the temperature of the phonon bath. We show how the full width athalf maximum (linewidth) of the Mollow triplet sidebands varies as a functionof the square of the Rabi frequency of the cw pump.

Theory of the quantum-dot Mollow triplet in an exciton-driven semiconductor cavity