  We theoretically investigate a spin-orbit coupled $s$-wave superfluid Fermi
gas, to examine the time evolution of the system, after an $s$-wave pairing
interaction is replaced by a $p$-wave one at $t=0$. In our recent paper, we
proposed that this manipulation may realize a $p$-wave superfluid Fermi gas,
because the $p$-wave pair amplitude that is induced in the $s$-wave superfluid
state by a parity-broken antisymmetric spin-orbit interaction gives a
non-vanishing $p$-wave superfluid order parameter, immediately after the
$p$-wave interaction is turned on. In this paper, using a time-dependent
Bogoliubov-de Gennes theory, we assess this idea under various conditions with
respect to the $s$-wave and $p$-wave interaction strengths, as well as the
spin-orbit coupling strength. From these, we clarify that the momentum
distribution of Fermi atoms in the initial $s$-wave state ($t<0$) is a key to
produce a large $p$-wave superfluid order parameter. Since the realization of a
$p$-wave superfluid state is one of the most exciting and difficult challenges
in cold Fermi gas physics, our results may provide a possible way to accomplish
this.
