  We present a quantum simulation method that follows the dynamics of
out-of-equilibrium many-body systems of electrons and oscillators in real time.
Its cost is linear in the number of oscillators and it can probe timescales
from attoseconds to hundreds of picoseconds. Contrary to Ehrenfest dynamics, it
can thermalize starting from a variety of initial conditions, including
electronic population inversion. While an electronic temperature can be defined
in terms of a non-equilibrium entropy, a Fermi-Dirac distribution in general
emerges only after thermalization. The time evolution of the populations is
rationalized in terms of a kinetic model.
