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

TL;DR

This paper develops a fractional-Lévy kinetic theory for cold atoms in optical lattices, explaining observed superdiffusion and phase transitions in their dynamics based on a semiclassical Sisyphus cooling framework.

Contribution

It introduces a fractional diffusion equation model capturing the transition between normal and Lévy superdiffusion in cold atom systems.

Findings

Identification of three dynamical phases as potential depth varies

Derivation of a fractional diffusion equation for atomic cloud dynamics

Connection of superdiffusive behavior to Lévy walk characteristics

Abstract

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.