  Quantum material systems upon applying ultrashort laser pulses provide a rich
platform to access excited material phases and their transformations that are
not entirely like their equilibrium counterparts. The addressability and
potential controls of metastable or long-trapped out-of-equilibrium phases have
motivated interests both for the purposes of understanding the nonequilibrium
physics and advancing the quantum technologies. Thus far, the dynamical
spectroscopic probes eminently focus on microscopic electronic and phonon
responses. For characterizing the long-range dynamics, such as order parameter
fields and fluctuation effects, the ultrafast scattering probes offer direct
sensitivity. Bridging the connections between the microscopic dynamics and
macroscopic responses is central toward establishing the nonequilibrium physics
behind the light-induced phases. Here, we present a path toward such
understanding by cross-examining the structure factors associated with
different dynamical states obtained from ultrafast electrons scattering,
imaging, and modeling. We give the basic theoretical framework on describing
the non-equilibrium scattering problems and briefly describe how such framework
relates to the out-of-equilibrium phenomena. We give effective models outlining
the emergences of nonthermal critical points, hidden phases, and
non-equilibrium relaxational responses from vacuum-suspended rare-earth
tritellurides, tantalum disulfides thin films, and vanadium dioxide
nanocrystalline materials upon light excitations.
