  The BCS model of an isolated superconductor initially prepared in a
nonequilibrium state, predicts the existence of interesting dynamical phenomena
in the time-dependent order parameter as decaying oscillations, persistent
oscillations and overdamped dynamics. To make contact with real systems remains
an open challenge as one needs to introduce dissipation due to the environment
in a self-consistent computation. Here, we reach this goal with the use of the
Keldysh formalism to treat the effect of a thermal bath. We show that, contrary
to the dissipationless case, all dynamical phases reach the equilibrium order
parameter in a characteristic time that depends on the coupling with the bath.
Remarkably, as time evolves, the overdamped phase shows a fast crossover where
the superconducting order parameter recovers to reach a state with a
well-developed long range order that tends towards equilibrium with the damped
Higgs mode oscillations. Our results provide a benchmark for the description of
the dynamics of real out-of-equilibrium superconductors relevant for quantum
technological applications.
