Theory of fractional-L\'evy kinetics for cold atoms diffusing in optical lattices

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.