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We look at the time dependent fluctuations of the electrical charge in anopen 1D quantum system represented by a quantum dot experiencing random lateralmotion. In essentially non-adiabatic settings we study both diffusive andballistic (Levy) regimes of the barrier motion where the electric current aswell as the net pumped electric charge experience random fluctuations over thestatic background. We show that in the large-time limit, $t \to \infty$, thewavefunction is naturally separated into the Berry-phase component (resultingfrom the singular part of the wave amplitude in the co-moving frame) and thenon-adiabatic correction (arising from fast oscillating, slow decaying tails ofthe same amplitude). Based on this separation we report two key results:Firstly, the disorder averaged wave function and current are asymptoticallymainly determined by the same Berry phase contribution that applies in the caseof adiabatic motion. Secondly, after a short transition period the pumpedelectric charge exhibits fluctuations that grow much faster than predicted bythe adiabatic theory. We also derive the exact expressions for the averagepropagator (in the co-moving basis representation) for the diffusive andballistic types of motion considered.

Non-adiabatic quantum pumping by a randomly moving quantum dot