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 φ. In the zero-frequency limit, our theory recovers the well-known adiabatic result that the pumped current I(φ)~sin φ. At a finite frequency, it predicts I(φ=0)≠0 while I(φ=π)~0, consistent with recent experimental data. We discuss a possible mechanism behind the nonzero pumped current at φ=0 from photon-assisted processes.