Numerical implementation of the multiscale and averaging methods for quasi periodic systems

We consider the problem of numerically solving the Schr\"odinger equation with a potential that is quasi periodic in space and time. We introduce a numerical scheme based on a newly developed multi-time scale and averaging technique. We demonstrate that with this novel method we can solve efficiently and with rigorous control of the error such an equation for long times. A comparison with the standard split-step method shows substantial improvement in computation times, besides the controlled errors. We apply this method for a free particle driven by quasi-periodic potential with many frequencies. The new method makes it possible to evolve the Schrodinger equation for times much longer than was possible so far and to conclude that there are regimes where the energy growth stops in-spite of the driving.