  We analyze the thermalization of a photoexcited charge carrier coupled to a
single branch of quantum phonons within the Holstein model. To this end, we
calculate the far-from-equilibrium time evolution of a pure many-body state and
compare it with predictions of the thermal Gibbs ensemble. We show that at
strong enough carrier excitation, the nonequilibrium system evolves towards a
thermal steady state. Our analysis is based on two classes of observables.
First, the occupations of fermionic momenta, which are the eigenvalues of the
one-particle density matrix, match in the steady state the values in the
corresponding Gibbs ensemble. This indicates thermalization of static fermionic
correlations on the entire lattice. Second, the dynamic current-current
correlations, including the time-resolved optical conductivity, also take the
form of their thermal counterparts. Remarkably, both static and dynamic
fermionic correlations thermalize with identical temperatures. Our results
suggest that the subsequent relaxation processes, observed in time-resolved
ultrafast spectroscopy, may be efficiently described by applying quasithermal
approaches, e.g., multi-temperature models.
