Non-adiabatic charge pump: an exact solution
  We derived a general and exact expression of current for quantum parametric charge pumps in the non-adiabatic regime at finite pumping frequency and finite driving amplitude. The non-perturbative theory predicts a remarkable plateau structure in the pumped current due to multi-photon assisted processes in a double-barrier quantum well pump involving only a {\it single} pumping potential. It also predicts a current reversal as the resonant level of the pump crosses the Fermi energy of the leads. 
Baigeng Wang, Jian Wang, and Hong Guo