  We study the time evolution of excitonic states after photo-excitation in the
one-dimensional spin-less extended Falicov-Kimball model. Several numerical
methods are employed and benchmarked against each other: time-dependent
mean-field simulations, the second-Born approximation (2BA) within the
Kadanoff-Baym formalism, the generalized Kadanoff-Baym Ansatz (GKBA)
implemented with the 2BA and the infinite time-evolving block decimation
(iTEBD) method. It is found that the GKBA gives the best agreement with iTEBD
and captures the relevant physics. We find that excitations to the
particle-hole continuum and resonant excitations of the equilibrium exciton
result in a qualitatively different dynamics. In the former case, the exciton
binding energy remains positive and the frequency of the corresponding coherent
oscillations is smaller than the band gap. On the other hand, resonant
excitations trigger a collective mode whose frequency is larger than the band
gap. We discuss the origin of these different behaviors by evaluating the
nonequilibrium susceptibility using the nonthermal distribution and a random
phase approximation. The peculiar mode with frequency larger than the band gap
is associated with a partial population inversion with a sharp energy cutoff.
We also discuss the effects of the cooling by a phonon bath. We demonstrate the
real-time development of coherence in the polarization, which indicates
excitonic condensation out of equilibrium.
