2004/05/06 by Martijn van Noort, van Noort, Martijn, Mason A. Porter +5
Mathematics · Physics and Astronomy · #03.75.Lm #03.75.Nt #05.30.Jp #05.45.Ac #37N20 #70H99 #Atomic Physics (physics.atom-ph) #Chaotic Dynamics (nlin.CD) #Cold Atom Physics and Bose-Einstein Condensates #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #MSC: 37J40 #PACS: 05.45.-a #Quantum, superfluid, helium dynamics #Soft Condensed Matter (cond-mat.soft) #Strong Light-Matter Interactions #cond-mat.soft #math.DS #msc:03.75.Lm #msc:03.75.Nt #msc:05.30.Jp #msc:05.45.-a #msc:05.45.Ac #msc:37J40 #msc:37N20 #msc:70H99 #nlin.CD #physics.atom-ph
paper · pdf · doi:10.48550/arxiv.math/0405112
29 pages, 6 figures (several with multiple parts; higher-quality versions of some of them available at http://www.its.caltech.edu/~mason/papers), to appear very soon in Journal of Nonlinear Science
openalex publication_date 2004/05/06 · arxiv created 2006/09/18 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We employ KAM theory to rigorously investigate quasiperiodic dynamics in cigar-shaped Bose-Einstein condensates (BEC) in periodic lattices and superlattices. Toward this end, we apply a coherent structure ansatz to the Gross-Pitaevskii equation to obtain a parametrically forced Duffing equation describing the spatial dynamics of the condensate. For shallow-well, intermediate-well, and deep-well potentials, we find KAM tori and Aubry-Mather sets to prove that one obtains mostly quasiperiodic dynamics for condensate wave functions of sufficiently large amplitude, where the minimal amplitude depends on the experimentally adjustable BEC parameters. We show that this threshold scales with the square root of the inverse of the two-body scattering length, whereas the rotation number of tori above this threshold is proportional to the amplitude. As a consequence, one obtains the same dynamical picture for lattices of all depths, as an increase in depth essentially only affects scaling in phase space. Our approach is applicable to periodic superlattices with an arbitrary number of rationally dependent wave numbers.