We show theoretically that the dynamics of cold atoms in the lowest energy band of a stationary optical lattice can be transformed and controlled by a second, weaker, periodic potential moving at a constant speed along the axis of the stationary lattice. The atom trajectories exhibit complex behavior, which depends sensitively on the amplitude and speed of the propagating lattice. When the speed and amplitude of the moving potential are low, the atoms are dragged through the static lattice and perform drifting orbits with frequencies an order of magnitude higher than that corresponding to the moving potential. Increasing either the speed or amplitude of the moving lattice induces Bloch-like oscillations within the energy band of the static lattice, which exhibit complex resonances at critical values of the system parameters. In some cases, a very small change in these parameters can reverse the atoms direction of motion. In order to understand these dynamics we present an analytical model, which describes the key features of the atom transport and also accurately predicts the positions of the resonant features in the atoms phase space. The abrupt controllable transitions between dynamical regimes, and the associated set of resonances, provide a mechanism for transporting atoms between precise locations in a lattice: as required for using cold atoms to simulate condensed matter or as a stepping stone to quantum information processing. The system also provides a direct quantum simulator of acoustic waves propagating through semiconductor nanostructures in sound analogs of the optical laser (SASER).
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