  We investigate the non-equilibrium behavior of BCS superconductors subjected
to slow ramps of their internal interaction strength. We identify three
dynamical regimes as a function of ramp duration. For short ramp times, these
systems become non-superconducting; however, fermions with opposite momenta
remain paired albeit with reduced amplitudes, and the associated pair amplitude
distribution is non-thermal. In this first regime, the disappearance of
superconductivity is due to the loss of phase coherence between pairs. By
contrast, for intermediate ramp times, superconductivity survives but the
magnitude of the order parameter is reduced and presents long-lived
oscillations. Finally, for long ramp times, phase coherence is almost fully
retained during the slow interaction quench, and the steady-state is
characterized by a thermal-like pair amplitude distribution. Using this
approach, one can therefore dynamically tune the coherence between pairs in
order to control the magnitude of the superconducting order parameter and even
engineer a non-equilibrium state made of pre-formed pairs.
