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.
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