Large Fluctuations for Spatial Diffusion of Cold Atoms

Abstract

Large-deviations theory deals with tails of probability distributions and the rare events of random processes, for example spreading packets of particles. Mathematically, it concerns the exponential fall-of of the density of thin-tailed systems. Here we investigate the spatial density Pt(x) of laser cooled atoms, where at intermediate length scales the shape is fat-tailed. We focus on the rare events beyond this range, which dominate important statistical properties of the system. Through a novel friction mechanism induced by the laser fields, the density is explored with the recently proposed nonnormalized infinite-covariant density approach. The small and large fluctuations give rise to a bi-fractal nature of the spreading packet.PACS numbers: 05.40. Jc,46.65.+g In diffusion processes such as Brownian motion, the concentration of particles starting at the origin spreads out like a Gaussian, which is fully characterized by the mean squared-displacement. This is the result of the widely applicable Gaussian central limit theorem (CLT) [1]. Of no less importance is large-deviations theory [2], which deals with the rare fluctuations of processes such as simple coin tossing random walks (see e.g., [3]), extreme variations of the surface height in the KardarParisi-Zhang model [4] and the tails of the position distribution in single-file diffusion [5,6]. Mathematically, a prerequisite of the theory is that the cumulant generating function be "well behaved", i.e. smooth and differentiable. Large-deviations theory works when the decay of the probability of the observable of interest is exponential (see details in [2]). However many systems do not meet this requirement [2], for example Lévy fat-tailed processes [7][8][9], where the decay rate is a power-law. This is the case for a cloud of atoms undergoing Sisyphus laser-cooling [10], where both theoretically [11,12] and experimentally [13], it was shown that the central part of the spreading particle packet is described by the Lévy CLT [14]. The latter deals with the sum of independent identically-distributed random variables, but unlike the classical Gaussian CLT, here the summands' own distribution is heavy-tailed. As a result, Lévy's CLT yields an infinite mean squared-displacement for the sum [14], and consequently also the second cumulant. Largedeviations theory mainly deals with thin-tailed processes where extreme events are rare, but in Lévy processes these large fluctuations are dominant. To study the fluctuations in this system, we will show that the relevant tool is the asymptotic moment-generating function, which yields an infinite-covariant density (ICD) [12,15]. We will discuss the generality of this approach and its results below.In an experimental situation, diverging moments are unphysical. For example, although the experiment in [13] shows a nice fit of the particles' density to a symmetric Lévy distribution, clearly at finite times no particles traveling at finite velocities can ever be found infinitely far from their origin. The finiteness of all t...

Single-big-jump principle in physical modeling

“…The motion of the atoms in this framework has been studied in [21][22][23]. The dynamical evolution can be described in terms of a random walk where each step is defined by two subsequent events with v(t) = 0, as described in Fig.…”

“…However using a level crossing technique [21], we are able to reformulate the big jump principle also for continuous Langevin processes, thus extending its scope dramatically. For that aim we use a model of cold atoms diffusing in optical lattices [21][22][23].…”

Equivalence of position–position auto-correlations in the Slicer Map and the Lévy–Lorentz gas

“…For instance, after an initial transient all trajectories of the SM turn periodic. However, anomalous transport may be dominated by ballistic flights (Aghion et al, 2017). Indeed the SM features anomalous transport because the length of ballistic flights in the initial ensemble follows a power-law distribution.…”

Infinite ergodic theory meets Boltzmann statistics

“…It is important to note that logarithmic potentials, such as the example in Fig. 1b, become weakly-binding when the depth of the well is sufficiently shallow (namely, when V (x) ∼ V 0 log(|x|) at large |x|, and V 0 < k B T at a certain given temperature), and this class of potentials was studied in the context of infinite densities, in [2][3][4][5][6][7][8][9] (see also a related work in [10]). In addition, one may consider potentials whose structure keeps changing for any x ∈ (−∞, ∞) (be it periodic as in Fig.…”