From the Area under the Bessel Excursion to Anomalous Diffusion of Cold Atoms

Abstract

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 þ 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 ψðχ; τÞ 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 Pðx; tÞ 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.

Large Fluctuations for Spatial Diffusion of Cold Atoms

“…, for large τ [17,24]. The slow decaying power-law tail of this function, means that the duration of the last step might be as long as the sum of all the prior ones and it cannot be neglected.…”

“…between zero crossings), is approximated by a Bessel excursion in velocity space [24,25] (see Fig. 2).…”

“…Clearly the statistics of χ and the zero crossing times, τ , determines the random position of the particle, x(t). Since the τ s are independent and identically distributed, the zero crossings form a renewal process [24,33], which allows us to analyze the problem analytically.…”

Observing Power-Law Dynamics of Position-Velocity Correlation in Anomalous Diffusion

“…An intriguing system to look at in this context is that of anomalous dynamics for which the mean square displacement (MSD) scales as x 2 ∼ t 2α , with α = 1/2. This type of dynamics, found in a wide variety of systems in nature ranging from dynamics of "bubbles" in denaturing DNA molecules [1], through fluctuations in the stockmarket [2] to models describing brief awakenings in the course of a night's sleep [3], is generally non-universal and system-dependent [4][5][6].A uniquely interesting model system for the study of anomalous diffusion is that of cold atoms diffusing in a dissipative 1D lattice, closely related to Lévy walks and motion in logarithmic potentials, displaying such phenomena as the breakdown of ergodicity and of equipartition, memory effects and slow relaxation to equilibrium [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. The major advantage of such a system is the high degree of control it enables over the physical parameters governing the dynamics.…”

“…To test the generality of our scaling argument we numerically study the dynamics of the position-velocity correlations within the framework of two distinct models featuring anomalous diffusion. The first describes semiclassical atomic motion in a 1D Sisyphus lattice [19], using the Langevin phase-space equationṡThe white noise term ξ(t) is Gaussian and has zero mean. The initial conditions are Gaussian, uncorrelated distributions of standard deviation σ = 1 in both velocity and position.…”