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Center for Nonlinear Studies |
The semiclassical theory of atoms undergoing Sisyphus cooling maps onto the problem of particles in a heat bath with a nonlinear friction force that falls as 1/p for large momentum p. At long times, this leads to a power-law distribution of momenta, which is cut off at order sqrt(t). This power-law momentum distribution implies an anomalous spatial distribution of particles. This latter distribution is a microscopic realization of a Levy walk, where the power-law distributed jump lengths are correlated to the time duration of the jump. This correlation is given by the distribution of the area under a Bessel excursion, a Bessel process that describes first returns to the origin at a given time. The Bessel process itself corresponds to the random fluctuations of the radius of a random walk in arbitrary dimension. We use all this to derive the time-dependence of the distribution and mean-squared displacement of the particles, which exhibit transitions as the strength of the cooling is varied.
Host: Eli Ben-Naim