  We study a two-dimensional fermionic QFT used to model 1D strongly correlated
electrons in the presence of a time-dependent impurity that drives the system
out of equilibrium. In contrast to previous investigations, we consider a
dynamic barrier switched on at a finite time. We compute the total energy
density (TED) of the system and establish two well defined regimes in terms of
the relationship between the frequency of the time-dependent perturbation
$\Omega$ and the electron energy $\omega$. Finally, we derive a relaxation time
$t_{R}$ such that for times shorter than $t_{R}$ the finite-time switching
process is relevant.
