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PhLAM. LKB. LPT. Dominique Delande Nicolas Cherroet. Matthias Lopez Benoît Vermersch Radu Chicireanu J.F. Clément Véronique Zehnlé Pascal Szriftgiser JCG. Gabriel Lemarié. Équipe Chaos Quantique. - PowerPoint PPT Presentation
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Dynamiques chaotiques : classique x quantique, discret x continu
Le rotateur frapp: Rapport avec la localisation dAnderson et ralisation exprimentale
quipe Chaos Quantique
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Groupe de travail NLSE-CEMPI
10/9/2012
Matthias Lopez
Benot Vermersch
Radu Chicireanu
J.F. Clment
Vronique Zehnl
Pascal Szriftgiser
JCG
Dominique Delande
Nicolas Cherroet
Gabriel Lemari
PhLAM
LKB
LPT
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Le modle dAnderson
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Liaisons fortes (tight-binding)
Alatoire:
Modle dAnderson
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Quantum dynamics in (perfect) lattices
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Perfect crystal: Delocalized Bloch waves diffusive dynamics
Conducteur
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Ordered crystal
Cliquer sur la figure pour voir lanimation
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Quantum dynamics in disordered lattices
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Disordered crystal
Insulator
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Disordered crystal: Localized states (3D: mobility edge)
Cliquer sur la figure pour voir lanimation
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Simple picture of Anderson dynamics
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Nombre de sites visits ~
Localisation
Diffusion
Temps de tunneling
Temps de sjour
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Impact of the Anderson model
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5300 citations
Increase of computer power
End of citing life
Cold-atom experiments
One-parameter scaling
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Impact of the Anderson model
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S. Redner arXiv:physics/0407137
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Consequences and limitations
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1D : Exponential localization of the eigenfunctions
Suppression of the diffusion Insulator
3D Mobility edge Metal-insulator transition
One-particle model No particle interactions
Zero-temperature
Oversimplified description of a crystal lattice
Limitations of the Anderson model
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Consequences of the Anderson model
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Transition dAnderson pour les nuls
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Transition dAnderson pour les nuls
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L
Insulator
Insulator
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La transition dAnderson pour les nuls
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L
L
L
Conductor
Insulator
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La transition dAnderson pour les nuls
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2D
4D
5D
Insulator
Conductor
1D
3D
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Experiments in condensed-matter and ultracold atoms
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Condensed matter
Decoherence (ill-defined quantum phases)
No access to the wave function
Electron-electron coulombian interactions
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Ultracold atoms
Control of decoherence
Access to probability distributions (and even the full wavefunction)
Control of interactions (Feschbach resonance)
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Experiments with ultracold atoms
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3D: S. S. Kondov et al., Three-Dimensional Anderson Localization of Ultracold Fermionic Matter, Science 334, 66 (2011)
1D: J. Billy et al., Direct observation of Anderson localization of matter-waves in a controlled disorder, Nature 453, 891 (2008)
3D : F. Jendrzejewski et al., Three-dimensional localization of ultracold atoms in an optical disordered potential, Nature Physics 8, 398 (2012)
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Le rotateur frapp
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Mouvement libre
J
q
q
J+DJ
Frappe (kick)
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Le rotateur frapp dpli
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Mouvement libre
p
Frappe (kick)
p+Dp
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Comment faire cela avec des atomes froids?
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Potentiel optique
Cliquer sur la figure pour voir lanimation
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Comment faire cela avec des atomes froids?
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Acousto-optical modulator
Cold-atom cloud
Mirror
F. L. Moore et al., Atom optics realization of the quantum d-kicked rotator, Phys. Rev. Lett. 75, 4598 (1995)
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Problme
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Limite la dure de la manip quelques ms
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Solution
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Ce nest pas un rotateur frapp
(kicked accelerator)
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Mesurer la vitesse des atomes
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Mesure directe de la norme de Sobolev 2,1
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Le rotateur frapp simule le modle dAnderson 1D
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S. Fishman et al., Chaos, quantum recurrences, and Anderson localization, PRL 49, 509 (1982)
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Anderson
Kicked rotor
Time periodicity: Floquet analysis
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Le rotateur frapp simule le modle dAnderson 1D
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S. Fishman et al., Chaos, quantum recurrences, and Anderson localization, PRL 49, 509 (1982)
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Each Floquet state is a realization of the fixed disorder ~ W = cte
Pseudo disorder
Random
Eq. (1)
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Comment simuler le modle dAnderson 3D ?
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24
g
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Rotateur frapp quasi-priodique
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G. Casati et al., Anderson transition in a one-dimensional system with three incommensurate frequencies, PRL 62, 345 (1989)
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irrational
H3F NOT periodic: NO Floquet states
NO Fishman-Grempel-Prangue equivalence
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Rotateur frapp quasi-priodique
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Good news: H3D is periodic in time : Floquet analysis
Apply Fishman-Grempel-Prangue trick all over again
H3D is equivalent to a 3D Anderson model
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H3D et H3F sont-ils quivalents ?
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The underlying unit of nature: different systems described by the same equations
Feynman Lectures in Physics, vol.2 ch. 12
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La transition dAnderson (enfin!)
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Localized
Critical
Diffusive
e
K
4
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0.1
0.8
Metal
Insulator
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Caractrise
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Caractrise
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Fonction donde critique
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Fonction donde critique
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Universalit
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La suite
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Utiliser un condensat de Bose-Einstein
Atomes individuels Onde de matire collective
Mcanique quantique non linaire !
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-2
-1
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0
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-2
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0
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r
ln
L
d
R
d
ln
ln
=
a
na
x
=