  We study, by means of the time-dependent Gutzwiller approximation, the out of
equilibrium dynamics of a half-filled Hubbard-Holstein model of correlated
electrons interacting with local phonons. Inspired by pump-probe experiments,
where intense light pulses selectively induce optical excitations that trigger
a transient out-of-equilibrium dynamics, here we inject energy in the Hubbard
bands by a non-equilibrium population of empty and doubly-occupied sites. We
first consider the case of a global perturbation, acting over the whole sample,
and find evidence of a mean-field dynamical transition where the lattice gets
strongly distorted above a certain energy threshold, despite the weak strength
of the electron-phonon coupling by comparison with the Hubbard repulsion. Next,
we address a slab geometry for a correlated heterostructure and study the
relaxation dynamics across the system when the perturbation acts locally on the
first layer. While for weak deviations from equilibrium the excited surface is
able to relax by transferring its excess energy to the bulk, for large
deviations the excess energy stays instead concentrated into the surface layer.
This self-trapping occurs both in the absence as well as in the presence of
electron-phonon coupling. Phonons actually enforce the trapping by distorting
at the surface.
