  Time-resolved ultrafast x-ray scattering from photo-excited matter is an
emerging method to image ultrafast dynamics in matter with atomic-scale spatial
and temporal resolutions. For a correct and rigorous understanding of current
and upcoming imaging experiments, we present the theory of time-resolved x-ray
scattering from an incoherent electronic mixture using quantum electrodynamical
theory of light-matter interaction. We show that the total scattering signal is
an incoherent sum of the individual scattering signals arising from different
electronic states and therefore heterodyning of the individual signals is not
possible for an ensemble of gas-phase photo-excited molecules. We scrutinize
the information encoded in the total signal for the experimentally important
situation when pulse duration and coherence time of the x-ray pulse are short
in comparison to the timescale of the vibrational motion and long in comparison
to the timescale of the electronic motion, respectively. Finally, we show that
in the case of an electronically excited crystal the total scattering signal
imprints the interference of the individual scattering amplitudes associated
with different electronic states and heterodyning is possible.
