  We study electron and positron pair production from the vacuum by a strong
slow electric field superimposed by a weak fast electric field pointing in an
arbitrary direction as a perturbation (the dynamically assisted Schwinger
mechanism). An analytical formula for the production number is derived on the
basis of the perturbation theory in the Furry picture. The formula is found to
be in good agreement with non-perturbative results obtained by numerically
solving the Dirac equation if the perturbation is sufficiently weak and/or is
not very slow. We also find analytically/numerically that the Schwinger
mechanism becomes spin-dependent if the perturbation has a transverse component
with respect to the strong electric field. The number difference between spin
up and down particles is strongly suppressed by an exponential of the critical
field strength if the frequency of the perturbation is small, while it is only
weakly suppressed by powers of the critical field strength if the frequency is
large enough. We also find that the spin-imbalance exhibits non-trivial
oscillating behaviors in terms of the frequency of the perturbation, the
azimuthal angle, and the momentum of produced particles.
