  Recently, anomalous superdiffusion of ultra cold 87Rb atoms in an optical
lattice has been observed along with a fat-tailed, L\'evy type, spatial
distribution. The anomalous exponents were found to depend on the depth of the
optical potential. We find, within the framework of the semiclassical theory of
Sisyphus cooling, three distinct phases of the dynamics, as the optical
potential depth is lowered: normal diffusion; L\'evy diffusion; and x ~ t^3/2
scaling, the latter related to Obukhov's model (1959) of turbulence. The
process can be formulated as a L\'evy walk, with strong correlations between
the length and duration of the excursions. We derive a fractional diffusion
equation describing the atomic cloud, and the corresponding anomalous diffusion
coefficient.
