  We present a quantum optics formalism to study intensity power broadening of
a semiconductor quantum dot interacting with an acoustic phonon bath and a high
$Q$ microcavity. Power broadening is investigated using a time-convolutionless
master equation in the polaron frame which allows for a nonperturbative
treatment of the interaction of the quantum dot with the phonon reservoir. We
calculate the full non-Lorentzian photoluminescence (PL) lineshapes and
numerically extract the intensity linewidths of the quantum dot exciton and the
cavity mode as a function of pump rate and temperature. For increasing field
strengths, multiphonon and multiphoton effects are found to be important, even
for phonon bath temperatures as low as 4 K. We show that the interaction of the
quantum dot with the phonon reservoir introduces pronounced features in the
power broadened PL lineshape, enabling one to observe clear signatures of
electron-phonon scattering. The PL lineshapes from cavity pumping and exciton
pumping are found to be distinctly different, primarily since the latter is
excited through the exciton-phonon reservoir. To help explain the underlying
physics of phonon scattering on the power broadened lineshape, an effective
phonon Lindblad master equation derived from the full time-convolutionless
master equation is introduced; we identify and calculate distinct Lindblad
scattering contributions from electron-phonon interactions, including effects
such as excitation-induced dephasing, incoherent exciton excitation and
exciton-cavity feeding. Our effective phonon master equation is shown to
reproduce the full intensity PL and the phonon-coupling effects very well,
suggesting that its general Lindblad form may find widespread use in
semiconductor cavity-QED.
