  We investigate decoherence in the quantum kicked rotator (modelling cold
atoms in a pulsed optical field) subjected to noise with power-law tail
waiting-time distributions of variable exponent (Levy noise). We demonstrate
the existence of a regime of nonexponential decoherence where the notion of a
decoherence rate is ill-defined. In this regime, dynamical localization is
never fully destroyed, indicating that the dynamics of the quantum system never
reaches the classical limit. We show that this leads to quantum subdiffusion of
the momentum, which should be observable in an experiment.
