Theory of one-dimensional double-barrier quantum pump in two-frequency   signal regime
  A one-dimensional system with two $\delta$-like barriers or wells bi-chromaticaly oscillating at frequencies $\omega$ and $2\omega$ is considered. The alternating signal leads to the direct current across the structure (even in a symmetric system). The properties of this quantum pump are studied in a wide range of the system parameters. 
M. M. Mahmoodian, M. V. Entin