Regular and Anomalous Quantum Diffusion in the Fibonacci Kicked Rotator

We study the dynamics of a quantum rotator kicked according to the almost-periodic Fibonacci sequence. A special numerical technique allows us to carry on this investigation for as many as $10^{12}$ kicks. It is shown that above a critical kick strength the excitation of the system is well described by regular diffusion, while below this border it becomes anomalous, and sub-diffusive. A law for the dependence of the exponent of anomalous sub-diffusion on the system parameters is established numerically. The analogy between these results and quantum diffusion in models of quasi-crystal and in the kicked Harper system is discussed.