  We present a medium-dependent quantum optics approach to describe the
influence of electron-acoustic phonon coupling on the emission spectra of a
strongly coupled quantum-dot cavity system. Using a canonical Hamiltonian for
light quantization and a photon Green function formalism, phonons are included
to all orders through the dot polarizability function obtained within the
independent Boson model. We derive simple user-friendly analytical expressions
for the linear quantum light spectrum, including the influence from both
exciton and cavity-emission decay channels. In the regime of semiconductor
cavity-QED, we study cavity emission for various exciton-cavity detunings and
demonstrate rich spectral asymmetries as well as cavity-mode suppression and
enhancement effects. Our technique is nonperturbative, and non-Markovian, and
can be applied to study photon emission from a wide range of semiconductor
quantum dot structures, including waveguides and coupled cavity arrays. We
compare our theory directly to recent and apparently puzzling experimental data
for a single site-controlled quantum dot in a photonic crystal cavity and show
good agreement as a function of cavity-dot detuning and as a function of
temperature.
