  In the adiabatic and weak-modulation quantum pump, net electron flow is
driven from one reservoir to the other by absorbing or emitting an energy
quantum $\hbar \omega $ from or to the reservoirs. In our approach, high-order
dependence of the scattering matrix on the time is considered. Non-sinusoidal
behavior of strong pumping is revealed. The relation between the pumped current
and the ac driving amplitude varies from power of 2, 1 to 1/2 when stronger
modulation is exerted. Open experimental observation can be interpreted by
multi-energy-quantum-related processes.
