From the area under the Bessel excursion to anomalous diffusion of cold atoms

TL;DR

This paper develops a theoretical framework linking Bessel excursions to anomalous diffusion in cold atoms, revealing phase transitions and detailed statistical properties of atomic transport in optical lattices.

Contribution

It introduces a novel analytical approach to describe the area under Bessel excursions, connecting microscopic velocity correlations to macroscopic diffusion behavior.

Findings

Identifies three phases: normal diffusion, Levy diffusion, and superdiffusive scaling.

Derives the Airy distribution for the area under Brownian excursions in a weak potential.

Highlights the importance of velocity meanders in the anomalous diffusion process.

Abstract

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.