  For a semiconductor quantum dot strongly coupled to a microcavity, we
theoretically investigate phonon-assisted transitions from the exciton to a
cavity photon, where the energy mismatch is compensated by phonon emission or
absorption. By means of a Schrieffer-Wolff transformation we derive an
effective Hamiltonian, which describes the combined effect of exciton-cavity
and exciton-phonon coupling, and compute the scattering rates within a Fermi
golden rule approach. The results of this approach are compared with those of a
recently reported description scheme based on the independent Boson model [U.
Hohenester et al., Phys. Rev. B 80, 201311(R) (2009)], and a numerical density
matrix approach. All description schemes are shown to give very similar
results. We present results for the spontaneous emission lifetime of a quantum
dot initially populated with a single exciton or biexciton, and for the
spectral properties of an optically driven dot-cavity system operating in the
strong coupling regime. Our results demonstrate that phonon-assisted feeding
plays a dominant role for strongly coupled dot-cavity systems, when the
detuning is of the order of a few millielectron volts.
