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We study the dynamics of cold atoms subjected to {\em pairs} of closelytime-spaced $\delta$-kicks from standing waves of light. The classical phasespace of this system is partitioned into momentum cells separated by trappingregions. In a certain range of parameters it is shown that the classical motionis well described by a process of anomalous diffusion. We investigate in detailthe impact of the underlying classical anomalous diffusion on the quantumdynamics with special emphasis on the phenomenon of dynamical localization.Based on the study of the quantum density of probability, its second moment andthe return probability we identify a region of weak dynamical localizationwhere the quantum diffusion is still anomalous but the diffusion rate is slowerthan in the classical case. Moreover we examine how other relevant time scalessuch as the quantum-classical breaking time or the one related to the beginningof full dynamical localization are modified by the classical anomalousdiffusion. Finally we discuss the relevance of our results for theunderstanding of the role of classical cantori in quantum mechanics.Comment: 9 pages, 3 figure

EFFECT OF CLASSICAL TRAPPING IN QUANTUM CHAOS: AN ANOMALOUS DIFFUSION APPROACH