  Using a minimum model consisting of a magnetic quantum dot and an electron
lead, we investigate spin pumping by its precessing magnetization. Focusing on
the "adiabaticity", which is quantified using a comparison between the
frequency of precession and the relaxation rate of the relevant system, we
investigate the role of nonadiabaticity in spin pumping by obtaining the
dependence of the spin current generated on the frequency of precession using
full counting statistics. This evaluation shows that the steady-state
population of the quantum dot remains unchanged by the precession owing to the
rotational symmetry about the axis of precession. This implies that in the
adiabatic limit the spin current is absent and that spin pumping is entirely a
nonadiabatic effect. We also find that the nonadiabatic spin current depends
linearly on the frequency in the low-frequency regime and exhibits an
oscillation in the high-frequency regime. The oscillation points to an
enhancement of spin pumping by tuning the frequency of precession.
