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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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

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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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r

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