  In this work we numerically investigate the fate of the
Bardeen-Cooper-Schrieffer (BCS) pairing in the presence of quenched phase under
Peierls substitution using time-dependent real space and momentum space
Bogoliubov-de Gennes equation methods and Anderson pseudospin representation
method. This kind of phase imprint can be realized by modulating electric field
in ultracold atoms and illumining of THz optical pump pulse in solids with
conventional and unconventional superconductors. In the case of weak phase
imprint, the BCS pairing is stable; while in the strong phase imprint,
instability towards finite-momentum pairing is allowed, in which the real space
and momentum space methods yield different results. In the pulsed gauge
potential, we find that this instability will not happen even with much
stronger vector potential. We also show that the uniform and staggered gauge
potentials yield different behaviors. While the staggered potential induces
transition from the BCS pairing to over-damped phase, the uniform gauge may
enhance the pairing and will not induce to the over-damped phase. These result
may shade light on the realization of finite momentum pairing, such as
Fulde-Ferrell-Larkin-Ovchinnikov phase with dynamical modulation.
