ionut danaila introduction bose-einstein condensates ... · v. bretin , j. dalibard phys. rev....

14/09/200414/09/2004 Ionut Danaila (www.ann.jussieu.fr/~danaila)Ionut Danaila (www.ann.jussieu.fr/~danaila) 11

Laboratoire Jacques-Louis LionsUniversité Pierre et Marie Curie (Paris 6)

Ionut DanailaIonut Danaila

Vortex configurations in Vortex configurations in rotatingrotating BoseBose--EinsteinEinstein

condensatescondensates


Introduction Bose-Einstein Condensates / Vortices

Outline

Numerical approach

Numerical results

Solving the 3D Gross-Pitaevskii equation

Comparison with experiments


BoseBose--EinsteinEinsteinCondensateCondensate

Predicted in 1924

Gas of bosons at very lowtemperature

Condensate= liquid in the gas= super-atom= same wave-function

SuperfluidsSuperconductors

Created in 1995C. E. Wieman (University of Colorado)E. A. Cornell (University of Colorado)W. Ketterle (MIT, Cambridge)

S. Bose A. Einstein


ENS (Ecole Normale Supérieure) experiments

BoseBose--EinsteinEinsteinCondensateCondensate


ENS experiments /Laboratoire Kastler BrosselVortex team // J. Dalibard

Rb rotating condensate ~ 500 000 atoms; T=90 nK

VorticesVortices in in BoseBose--EinsteinEinstein CondensatesCondensates


)()( rier θρ=Ψ

Macroscopic wave-function

Velocity field

θ∇=mhrv )(

Circulation

∫ ==Γmhndssv )(

VorticesVortices in in BoseBose--EinsteinEinstein CondensatesCondensates


NumericalNumerical approachapproach

),.(2

V2

)( 42trap

23 xiNg

mE D ×Ψ∇ΨΩ+Ψ+Ψ+Ψ∇=Ψ ∫∫∫ h

h

Gross-Pitaevskii energy

critical points

)(min 3 uE D

)V(2

).~(21 2*

trap22 uuuriu +−=∇×Ω+∇−

εuu)(εµ+t

u∂∂


Donu ∂=0

0),,( ≥⊃ zyxD TFρ

NumericalNumerical approachapproach

-Time integration: Runge-Kutta (3 steps)+ Crank-Nicolson

-Space discretization: finite differences6th order Padé (compact) scheme

2402402405127272××

××


P. Rosenbusch,V. Bretin ,J. DalibardPhys. Rev. Lett. 89 (2002)

Harmonic potential )(V 22222*trap zyx βα ++=

U vortex S vortex 3D-S vortex14/09/200414/09/2004 Ionut Danaila (www.ann.jussieu.fr/~danaila)Ionut Danaila (www.ann.jussieu.fr/~danaila) 1010

Ω/ωx =0.38 0.44 0.48

Ω/ωx =0.48

The S vortex

Branches canonly lie in the (x,z) or (y,z) planes


The U vortex


V. Bretin, S. Stock, Y. Seurin, J. Dalibard, PRL 2003

Harmonic+Quartic potential

2242*trap 4

)1(V zrkr βα ++−=


Intermediatequartic case(α=1.1)








Z. Handzibababic, S. Stock, B. Battelier,V. Bretin, J. Dalibard cond-mat, 2004

Harmonic+Optical lattice

)/(sinV 20

2*trap dzUr π+=

Ionut Danaila Ionut Danaila (www.ann.jussieu.fr/~danaila)(www.ann.jussieu.fr/~danaila) 181814/09/200414/09/2004

U=0.1 U=0.5 U=0.7Ω=0.87


ConclusionConclusion

Rich variety of vortex configurations

Harmonic potential -> single U, S vortex

Quartic + Harmonic potential -> giant vortex + single vortices

Harmonic + Optical lattice -> array of BECs

Related papers (with A. Aftalion)

Phys. Rev A, 68, 023603 (2003), cond-mat/0303416Phys. Rev A, 69, 033608 (2004), cond-mat/0309668

ionut danaila introduction bose-einstein condensates ... · v. bretin , j. dalibard phys. rev....

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