  We discuss a numerical method to study electron transport in mesoscopic
devices out of equilibrium. The method is based on the solution of operator
equations of motion, using efficient Chebyshev time propagation techniques. Its
peculiar feature is the propagation of operators backwards in time. In this way
the resource consumption scales linearly with the number of states used to
represent the system. This allows us to calculate the current for
non-interacting electrons in large one-, two- and three-dimensional lead-device
configurations with time-dependent voltages or potentials. We discuss the
technical aspects of the method and present results for an electron pump device
and a disordered system, where we find transient behaviour that exists for a
very long time and may be accessible to experiments.
