Recently nanomechanical devices composed of a long stationary inner carbon nanotube and a shorter, slowly-rotating outer tube have been fabricated. In this Letter, we study the possibility of using such devices as adiabatic quantum pumps. Using the Brouwer formula, we employ a Greens function technique to determine the pumped charge from one end of the inner tube to the other, driven by the rotation of a chiral outer nanotube. We show that there is virtually no pumping if the chiral angle of the two nanotubes is the same, but for optimal chiralities the pumped charge can be a significant fraction of a theoretical upper bound.
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