  We study equilibrium and nonequilibrium properties of electron-phonon systems
described by the Hubbard-Holstein model using the dynamical mean-field theory.
In equilibrium, we benchmark the results for impurity solvers based on the
one-crossing approximation and slave-rotor approximation against
non-perturbative numerical renormalization group reference data. We also
examine how well the low energy properties of the electron-boson coupled
systems can be reproduced by an effective static electron-electron interaction.
The one-crossing and slave-rotor approximations are then used to simulate
insulator-to-metal transitions induced by a sudden switch-on of the
electron-phonon interaction. The slave-rotor results suggest the existence of a
critical electron-phonon coupling above which the system is transiently trapped
in a non-thermal metallic state with coherent quasiparticles. The same quench
protocol in the one-crossing approximation results in a bad metallic state.
