Levy flights are random walks in which the probability distribution of the
step sizes is fat-tailed. Levy spatial diffusion has been observed for a
collection of ultra-cold Rb atoms and single Mg+ ions in an optical lattice.
Using the semiclassical theory of Sisyphus cooling, we treat the problem as a
coupled Levy walk, with correlations between the length and duration of the
excursions. The problem is related to the area under Bessel excursions,
overdamped Langevin motions that start and end at the origin, constrained to
remain positive, in the presence of an external logarithmic potential. In the
limit of a weak potential, the Airy distribution describing the areal
distribution of the Brownian excursion is found. Three distinct phases of the
dynamics are studied: normal diffusion, Levy diffusion and, below a certain
critical depth of the optical potential, x~ t^{3/2} scaling. The focus of the
paper is the analytical calculation of the joint probability density function
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 (MW) equation for the density 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 in the sequence. In the anomalous
phases, the statistics of meanders and excursions are essential for the
calculation of the mean square displacement, showing that our correction to the
MW equation is crucial, and points to the sensitivity of the transport on a
single jump event. Our work provides relations between the statistics of
velocity excursions and meanders and that of the diffusivity.Comment: Supersedes arXiv: 1305.008
