From the Area under the Bessel Excursion to Anomalous Diffusion of Cold Atoms - E. Barkai; E. Aghion; D. A. Kessler (Physical Review X, 2014)
Lévy flights are random walks in which the probability distribution of the step sizes is fat-tailed. Lévy spatial diffusion has been observed for a collection of ultracold ^{87}Rb atoms and single ^{24}Mg^{+} ions in an optical lattice, a system which allows for a unique degree of control of the dynamics. Using the semiclassical theory of Sisyphus cooling, we formulate the problem as a coupled Lévy walk, with strong correlations between the length χ and duration τ of the excursions. Interestingly, the problem is related to the area under the Bessel and Brownian excursions. These are overdamped Langevin motions that start and end at the origin, constrained to remain positive, in the presence or absence of an external logarithmic potential, respectively. In the limit of a weak potential, i.e., shallow optical lattices, the celebrated Airy distribution describing the areal distribution of the Brownian excursion is found as a limiting case. Three distinct phases of the dynamics are investigated: normal diffusion, Lévy diffusion and, below a certain critical depth of the optical potential, x∼t^{3/2} scaling, which is related to Richardson’s diffusion from the field of turbulence. The main focus of the paper is the analytical calculation of the joint probability density function ψ(χ,τ) from a newly developed theory of the area under the Bessel excursion. The latter describes the spatiotemporal correlations in the problem and is the microscopic input needed to characterize the spatial diffusion of the atomic cloud. A modified Montroll-Weiss equation for the Fourier-Laplace transform of the density P(x,t) is obtained, which depends on the statistics of velocity excursions and meanders. The meander, a random walk in velocity space, which starts at the origin and does not cross it, describes the last jump event χ in the sequence. In the anomalous phases, the statistics of meanders and excursions are essential for the calculation of the mean-square displacement, indicating that our correction to the Montroll-Weiss equation is crucial and pointing to the sensitivity of the transport on a single jump event. Our work provides general relations between the statistics of velocity excursions and meanders on the one hand and the diffusivity on the other, both for normal and anomalous processes.