Fractional diffusion of cold atoms in optical lattices

Duration: 58 mins 42 secs

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Description:	Eli Barkai; Bar-Ilan University 7 February 2022 – 18:00 to 19:00


Created:	2022-02-11 16:18
Collection:	Fractional differential equations
Publisher:	Isaac Newton Institute
Copyright:	Eli Barkai
Language:	eng (English)


Abstract:	Fractional calculus is an old branch of mathematics which deals with fractional order derivatives, e.g., d1=2=dt1=2. Davidson’s group (Weizmann) has recorded the spatial diffusion of cold atoms in optical lattices, fitting the results to the solution of a fractional diffusion equation @βP(x; t) @tβ = KµrµP(x; t): Within the semi classical theory of Sisyphus cooling we derive this fractional equation and discuss its meaning and its limitations [1,2]. An asymptotically weak friction force, induced by the laser field, is responsible for the large deviations from normal transport theory (and from Boltzmann-Gibbs equilibrium concepts [3]) at least below a critical value of the depth of the optical lattice. 1. D. A. Kessler, and E. Barkai Theory of fractional-L´evy kinetics for cold atoms diffusing in optical lattices Phys. Rev. Lett. 108, 230602 (2012). 2. E. Barkai, E. Aghion, and D. Kessler From the area under the Bessel excursion to anomalous diffusion of cold atoms Physical Review X 4, 021036 (2014) 3. A. Dechant, D. A. Kessler and E. Barkai Deviations from Boltzmann-Gibbs equilibrium in confined optical lattices Phys. Rev. Lett. 115, 173006 (2015).

Abstract:

Fractional calculus is an old branch of mathematics which deals with fractional
order derivatives, e.g., d1=2=dt1=2. Davidson’s group (Weizmann) has recorded the spatial
diffusion of cold atoms in optical lattices, fitting the results to the solution of a fractional
diffusion equation
@βP(x; t)
@tβ = KµrµP(x; t):
Within the semi classical theory of Sisyphus cooling we derive this fractional equation
and discuss its meaning and its limitations [1,2]. An asymptotically weak friction force,
induced by the laser field, is responsible for the large deviations from normal transport
theory (and from Boltzmann-Gibbs equilibrium concepts [3]) at least below a critical value
of the depth of the optical lattice.
1. D. A. Kessler, and E. Barkai Theory of fractional-L´evy kinetics for cold atoms diffusing in optical lattices Phys. Rev. Lett. 108, 230602 (2012).
2. E. Barkai, E. Aghion, and D. Kessler From the area under the Bessel excursion to
anomalous diffusion of cold atoms Physical Review X 4, 021036 (2014)
3. A. Dechant, D. A. Kessler and E. Barkai Deviations from Boltzmann-Gibbs equilibrium in confined optical lattices Phys. Rev. Lett. 115, 173006 (2015).

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