  We consider a nonadiabatic quantum pumping phenomena in a ballistic narrow
constriction. The pumping is induced by a potential that has both spatial and
temporal periodicity characterized by $K$ and $\Omega$. In the zero frequency
($\Omega=0$) limit, the transmission through narrow constriction exhibits
valley structures due to the opening up of energy gaps in the pumping region --
a consequence of the $K$ periodicity. These valley structures remain robust in
the regime of finite $\Omega$, while their energies of occurrence are shifted
by about $\hbar\Omega/2$. The direction of these energy shifts depend on the
directions of both the phase-velocity of the pumping potential and the
transmitting electrons. This frequency dependent feature of the valley
structures gives rise to both the asymmetry in the transmission coefficients
and the pumping current. An experimental setup is suggested for a possible
observation of our nonadiabatic quantum pumping findings.
