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We study charge pumping when a combination of static potentials andpotentials oscillating with a time period T is applied in a one-dimensionalsystem of non-interacting electrons. We consider both an infinite system usingthe Dirac equation in the continuum approximation, and a periodic ring with afinite number of sites using the tight-binding model. The infinite system istaken to be coupled to reservoirs on the two sides which are at the samechemical potential and temperature. We consider a model in which oscillatingpotentials help the electrons to access a transmission resonance produced bythe static potentials, and show that non-adiabatic pumping violates the simple\sin \phi rule which is obeyed by adiabatic two-site pumping. For the ring, wedo not introduce any reservoirs, and we present a method for calculating thecurrent averaged over an infinite time using the time evolution operator U(T)assuming a purely Hamiltonian evolution. We analytically show that the averagedcurrent is zero if the Hamiltonian is real and time reversal invariant.Numerical studies indicate another interesting result, namely, that theintegrated current is zero for any time-dependence of the potential if it isapplied to only one site. Finally we study the effects of pumping at two siteson a ring at resonant and non-resonant frequencies, and show that the pumpedcurrent has different dependences on the pumping amplitude in the two cases.

Nonadiabatic charge pumping by oscillating potentials in one dimension: results for infinite system and finite ring