  Electron-phonon coupling in semiconductor quantum dots plays a significant
role in determining the optical properties of excited excitons, especially the
spectral nature of emitted photons. This paper presents a comprehensive theory
and analysis of emission spectra from artificial atoms or quantum dots coupled
to structured photon reservoirs and acoustic phonons, when excited with
incoherent pump fields. As specific examples of structured reservoirs, we chose
a Lorentzian cavity and a coupled cavity waveguide, which are of current
experimental interest. For the case of optical cavities, we directly compare
and contrast the spectra from three distinct theoretical approaches to treat
electron-phonon coupling, including a Markovian polaron master equation, a
non-Markovian phonon correlation expansion technique and a semiclassical linear
susceptibility approach, and we point out the limitations of these models. For
the cavity-QED polaron master equation, we give closed form analytical
solutions to the phonon-assisted scattering rates in the weak excitation
approximation, fully accounting for temperature, cavity-exciton detuning and
cavity dot coupling. We show explicitly why the semiclassical linear
susceptibility approach fails to correctly account for phonon-mediated cavity
feeding. For weakly coupled cavities, we calculate the optical spectra using a
more general reservoir approach and explain its differences from the above
approaches in the low Q limit of a Lorentzian cavity. We subsequently use this
general reservoir to calculate the emission spectra from quantum dots coupled
to slow-light photonic crystal waveguides, which demonstrate a number of
striking photon-phonon coupling effects. Our quantum theory can be applied to a
wide range of photonic structures including photonic molecules and
coupled-cavity waveguide systems.
