  We describe, within the framework of quantum electrodynamics, an interaction
between a non-resonant hard x-ray pulse and an electronic system in the
presence of a temporally periodic laser field driving electron dynamics in this
system. We apply Floquet theory to describe the laser-driven electronic system,
and then obtain the scattering probability of an arbitrary nonresonant x-ray
pulse from such a system employing the density-matrix formalism. We show that
the scattering probability can be connected to the time-dependent electron
density of the driven electronic system only under certain conditions, in
particular, if the bandwidth of the probe x-ray pulse is sufficiently narrow to
spectroscopically resolve transitions to different final states. A special
focus is laid on application of the theory to laser-driven crystals in a
strongly nonperturbative regime. We show how the time-dependent electron
density of a crystal can be reconstructed from energy-resolved scattering
patterns. This is illustrated by a calculation of a diffraction signal from a
driven MgO crystal.
