Diffusion of Cold Atoms in an Optical Lattice

We discuss the problem of diffusion of cold atoms in an atomic trap. In the semiclassical limit, this problem is equivalent to independent particles undergoing a weakly biased random walk in momentum space. The tails of the momentum distribution are determined by the 1/p fall-off of the bias, and have a power-law decay. This gives rise to anomalous behavior if the exponent of the power-law is sufficiently small in magnitude. Then, the equilibrium prediction is that the average kinetic energy is infinite, which is clearly unphysical. Instead, the system never reaches equilibrium, and the distribution is cut off at momenta of order sqrt(t), rendering all moments finite, but growing in time. We show how a harmonic oscillator with a randomly varying (positive) stiffness gives rise to the same phenomenon. Returning to the atomic trap, we consider the resulting distribution of the atomic positions, which is a cut-off Levy distribution. Finally, we discuss the comparison to experiment.