Fractional diffusion of cold atoms in optical lattices

Fractional calculus is an old branch of mathematics which deals with fractional order derivatives. Recently the Davidson's group (Weizmann) has recorded the spatial diffusion of cold atoms in optical lattices, fitting the results to the solution of a fractional diffusion equation. Within the semi classical theory of Sisyphus cooling we derive this fractional equation and discuss its meaning and its limitations [1,2]. An asymptotically weak friction force, induced by the laser field, is responsible for the large deviations from normal transport theory (and from Boltzmann-Gibbs equilibrium concepts [3]) at least below a critical value of the depth of the optical lattice.

1. E. Barkai, E. Aghion, and D. Kessler From the area under the Bessel excursion to anomalous diffusion of cold atoms Physical Review X 4, 021036 (2014)
2. D. A. Kessler, and E. Barkai Theory of fractional-Levy kinetics for cold atoms diffusing in optical lattices Phys. Rev. Lett. 108, 230602 (2012).
3. A. Dechant, D. A. Kessler and E. Barkai Deviations from Boltzmann-Gibbs equilibrium in confined optical lattices arXiv:1412.5402 [cond-mat.stat-mech] (2014).