  We discuss phonon-induced non-Markovian and Markovian features in QD-based
optics. We cover lineshapes in linear absorption experiments, phonon-induced
incoherence in the Heitler regime, and memory correlations in two-photon
coherences. To quantitatively and qualitatively understand the underlying
physics, we present several theoretical models which model the non-Markovian
properties of the electron-phonon interaction accurately in different regimes.
Examples are the Heisenberg equation of motion approach, the polaron master
equation, and Liouville-propagator techniques in the independent boson limit
and beyond via the path-integral method. Phenomenological modeling
overestimates typically the dephasing due to the finite memory kernel of
phonons and we give instructive examples of phonon-mediated coherence such as
phonon-dressed anticrossings in Mollow physics, robust quantum state
preparation, cavity-feeding and the stabilization of the collapse and revival
phenomenon in the strong coupling limit.
