  Using a polaron master equation approach we investigate the resonance
fluorescence spectra from coherently driven quantum dots (QDs) coupled to an
acoustic phonon bath and a photonic crystal waveguide with a rich local density
of photon states (LDOS). Resonance fluorescence spectra from QDs in
semiconductor crystals are known to show strong signatures of electron-phonon
interactions, but when coupled to a structured photonic reservoir, the QD
emission properties are also determined by the frequency dependence of the LDOS
of the photon reservoir. Here, we investigate the simultaneous role of coupled
photon and phonon baths on the characteristic Mollow triplet spectra from a
strongly driven QD. As an example structured photonic reservoir, we first study
a photonic crystal coupled cavity waveguide, and find that photons and phonons
have counter-interacting effects near the upper mode-edge of the coupled-cavity
waveguide, thus establishing the importance of their separate roles in
determining the emission spectra. The general theory is developed for arbitrary
photonic reservoirs and is further applied to determine resonance the
fluorescence spectra from a realistic, disordered W1 photonic crystal waveguide
showing important photon-phonon interaction effects that are directly relevant
to emerging experiments and theoretical proposals.
