We investigate decoherence in
the quantum kicked rotator (modeling cold atoms in a pulsed optical
field) subjected to noise with power-law tail waiting-time
distributions of variable exponent (LÚvy 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.