  We investigate the interplay between gap oscillations and damping in the
dynamics of superconductors taken out of equilibrium by strong optical pulses
with sub-gap Terahertz frequencies. A semi-phenomenological formalism is
developed to include the damping within the electronic subsystem that arises
from effects beyond BCS, such as interactions between Bogoliubov quasiparticles
and decay of the Higgs mode. Such processes are conveniently expressed as
$T_{1}$ and $T_{2}$ times in the standard pseudospin language for
superconductors. Comparing with data on NbN that we report here, we argue that
the superconducting dynamics in the picosecond time scale, after the pump is
turned off, is governed by the $T_{2}$ process.
