Theory of Fractional Levy Kinetics for Cold Atoms Diffusing in Optical Lattices

Recently, anomalous superdiffusion of ultracold Rb-87 atoms in an optical lattice has been observed along with a fat-tailed, Levy 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; Levy diffusion; and x similar to t(3/2) scaling, the latter related to Obukhov's model (1959) of turbulence. The process can be formulated as a Levy 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.