  We describe coupled electron-phonon systems semiclassically - Ehrenfest
dynamics for the phonons and quantum mechanics for the electrons - using a
classical Monte Carlo approach that determines the nonequilibrium response to a
large pump field. The semiclassical approach is quite accurate, because the
phonons are excited to average energies much higher than the phonon frequency,
eliminating the need for a quantum description. The numerical efficiency of
this method allows us to perform a self-consistent time evolution out to very
long times (tens of picoseconds) enabling us to model pump-probe experiments of
a charge density wave (CDW) material. Our system is a half-filled,
one-dimensional (1D) Holstein chain that exhibits CDW ordering due to a Peierls
transition. The chain is subjected to a time-dependent electromagnetic pump
field that excites it out of equilibrium, and then a second probe pulse is
applied after a time delay. By evolving the system to long times, we capture
the complete process of lattice excitation and subsequent relaxation to a new
equilibrium, due to an exchange of energy between the electrons and the
lattice, leading to lattice relaxation at finite temperatures. We employ an
indirect (impulsive) driving mechanism of the lattice by the pump pulse due to
the driving of the electrons by the pump field. We identify two driving
regimes, where the pump can either cause small perturbations or completely
invert the initial CDW order. Our work successfully describes the ringing of
the amplitude mode in CDW systems that has long been seen in experiment, but
never successfully explained by microscopic theory.
