  We investigate the temperature dependence of photon coherence properties
through two photon interference (TPI) measurements from a single QD under
resonant excitation. We show that the loss of indistinguishability is only
related to the electron-phonon coupling without being affected by spectral
diffusion. Through these measurements, and a complementary microscopic theory,
we identify two independent separate decoherence processes each associated to
phonons. Below 10K, we find that the relaxation of the vibrational lattice is
the dominant contribution to the loss of TPI visibility. This process is
non-Markovian in nature, and corresponds to real phonon transitions resulting
in a broad phonon sideband in the QD emission spectra. Above 10K, virtual
phonon transitions to higher lying excited states in the QD become the dominant
dephasing mechanism, this leads to broadening of the zero phonon line, and a
corresponding rapid decay in the visibility. The microscopic theory we develop
provides analytic expressions for the dephasing rates for both virtual phonon
scattering and non-Markovian lattice relaxation.
