611 research outputs found
Quasiperiodic Dynamics in Bose-Einstein Condensates in Periodic Lattices and Superlattices
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.Comment: 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 Scienc
The semiclassical tool in mesoscopic physics
Semiclassical methods are extremely valuable in the study of transport and
thermodynamical properties of ballistic microstructures. By expressing the
conductance in terms of classical trajectories, we demonstrate that quantum
interference phenomena depend on the underlying classical dynamics of
non-interacting electrons. In particular, we are able to calculate the
characteristic length of the ballistic conductance fluctuations and the weak
localization peak in the case of chaotic dynamics. Integrable cavities are not
governed by single scales, but their non-generic behavior can also be obtained
from semiclassical expansions (over isolated trajectories or families of
trajectories, depending on the system). The magnetic response of a
microstructure is enhanced with respect to the bulk (Landau) susceptibility,
and the semiclassical approach shows that this enhancement is the largest for
integrable geometries, due to the existence of families of periodic orbits. We
show how the semiclassical tool can be adapted to describe weak residual
disorder, as well as the effects of electron-electron interactions. The
interaction contribution to the magnetic susceptibility also depends on the
nature of the classical dynamics of non-interacting electrons, and is
parametrically larger in the case of integrable systems.Comment: Latex, Cimento-varenna style, 82 pages, 21 postscript figures;
lectures given in the CXLIII Course "New Directions in Quantum Chaos" on the
International School of Physics "Enrico Fermi"; Varenna, Italy, July 1999; to
be published in Proceeding
H = x p with interaction and the Riemann zeros
Starting from a quantized version of the classical Hamiltonian H = x p, we
add a non local interaction which depends on two potentials. The model is
solved exactly in terms of a Jost like function which is analytic in the
complex upper half plane. This function vanishes, either on the real axis,
corresponding to bound states, or below it, corresponding to resonances. We
find potentials for which the resonances converge asymptotically toward the
average position of the Riemann zeros. These potentials realize, at the quantum
level, the semiclassical regularization of H = x p proposed by Berry and
Keating. Furthermore, a linear superposition of them, obtained by the action of
integer dilations, yields a Jost function whose real part vanishes at the
Riemann zeros and whose imaginary part resembles the one of the zeta function.
Our results suggest the existence of a quantum mechanical model where the
Riemann zeros would make a point like spectrum embbeded in the continuum. The
associated spectral interpretation would resolve the emission/absortion debate
between Berry-Keating and Connes. Finally, we indicate how our results can be
extended to the Dirichlet L-functions constructed with real characters.Comment: 23 pages, 12 figures, minor correction
Functionals of the Brownian motion, localization and metric graphs
We review several results related to the problem of a quantum particle in a
random environment.
In an introductory part, we recall how several functionals of the Brownian
motion arise in the study of electronic transport in weakly disordered metals
(weak localization).
Two aspects of the physics of the one-dimensional strong localization are
reviewed : some properties of the scattering by a random potential (time delay
distribution) and a study of the spectrum of a random potential on a bounded
domain (the extreme value statistics of the eigenvalues).
Then we mention several results concerning the diffusion on graphs, and more
generally the spectral properties of the Schr\"odinger operator on graphs. The
interest of spectral determinants as generating functions characterizing the
diffusion on graphs is illustrated.
Finally, we consider a two-dimensional model of a charged particle coupled to
the random magnetic field due to magnetic vortices. We recall the connection
between spectral properties of this model and winding functionals of the planar
Brownian motion.Comment: Review article. 50 pages, 21 eps figures. Version 2: section 5.5 and
conclusion added. Several references adde
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