  We derive a non-perturbative solution to the Floquet-nonequilibrium Green
function (Floquet-NEGF) describing open quantum systems periodically driven by
an external field of arbitrary strength of frequency. By adopting the
reduced-zone scheme, we obtain expressions rendering conserved charge currents
for any given maximum number of photons, distinguishable from other existed
Floquet-NEGF-based expressions where, less feasible, infinite number of photons
needed to be taken into account to ensure the conservation. To justify our
derived formalism and to investigate spin-charge conversions by spin-orbit
coupling (SOC), we consider the spin-driven setups as reciprocal to the
electric-driven setups in S. Souma et. al., Phys. Rev. B 70, 195346 (2004) and
Phys. Rev. Lett. 94, 106602 (2005). In our setups, pure spin currents are
driven by the magnetization dynamics of a precessing ferromagnetic (FM) island
and then are pumped into the adjacent two- or four-terminal mesoscopic
Aharonov-Casher (AC) ring of Rashba SOC where spin-charge conversions take
place. Our spin-driven results show reciprocal features that excellently agree
with the findings in the electric-driven setups mentioned above. We propose two
types of symmetry operations, under which the AC ring Hamiltonian is invariant,
to argue the relations of the pumped/converted currents in the leads within the
same or between different pumping configurations. The symmetry arguments are
independent of the ring width and the number of open channels in the leads,
terminals, and precessing FM islands, In particular, net pure in-plane spin
currents and pure spin currents can be generated in the leads for certain
setups of two terminals and two precessing FM islands with the current
magnitude and polarization direction tunable by the pumping configuration, gate
voltage covering the two-terminal AC ring in between the FM islands.
