Lévy Motion and Cold Atoms
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Date
2020-09-24
Authors
Erickson, Wesley
Journal Title
Journal ISSN
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Publisher
University of Oregon
Abstract
Lévy processes are a universal model for characterizing the behavior of
extreme events and anomalously diffusive systems. They are also important in
modeling the transport of laser-cooled atoms. This dissertation contributes to the
understanding of Lévy processes themselves, and to their application in laser-cooled
atomic dynamics.
Lévy processes are models of systems that contain extreme events. However,
since any particular event could, upon closer inspection, be composed of multiple
smaller events, what makes an event “extreme”? To answer this question it is
useful to consider the history of the event, by studying Lévy processes conditioned
to have a fixed final state at some later time. We find that events that greatly
exceed a particular threshold are more likely to be composed of many small events
and a single large event, rather than a series of comparably sized events. Analysis
of the source of this threshold helps clarify the Gaussian limit of Lévy processes,
and serves as the foundation for a useful distinction between “short” and “long”
steps. These ideas also suggest possible experimental techniques that may be
employed for cold atoms.
Studies of anomalous diffusion in laser-cooled atoms have typically focused on
the expansion profiles of clouds of cold atoms. However, the results of experiments
and theoretical models have not been in close agreement, suggesting that existing
models are still incomplete. To address this, it is important to develop alternative
experimental approaches. One such approach is to use a single atom as a probe
of the diffusion, which allows for the collection of boundary crossing statistics,
such as the time between when an atom enters and leaves an imaging region.
Through simulations, we find that distributions of these statistics develop peaks
that correspond to atomic Lévy flights, in direct contrast to featureless power-law
distributions for atoms undergoing Brownian motion. We find that characterizing
these distributions gives information on the anomalous-diffusion exponent and
typical velocities. Furthermore, these distributions serve as the basis for a method
to directly detect Lévy flights at the level of a single atom.
Description
Keywords
anomalous diffusion, extreme events, Lévy flights, Lévy processes, probability bifurcation, Sisyphus cooling