Parametric pumping at finite frequency

We report a first-principles theory for analyzing the parametric electron pump at finite frequency. The pump is controlled by two pumping parameters with phase difference phi. In the zero-frequency limit. our theory recovers the well-known adiabatic result that the pumped current I(phi) similar to sin phi. At a finite frequency, it predicts I(phi = 0) not equal 0 while I(phi = pi) approximate to 0, consistent with recent experimental data. We discuss a possible mechanism behind the nonzero pumped current at phi = 0 from photon-assisted processes.