I Danaila Bose Einstein

FreeFEm++ and quantized vortices infast-rotating Bose-Einstein condensates

Ionut Danaila

Laboratoire Jacques Louis LionsUniversite Pierre et Marie Curie (Paris 6)

http://www.ann.jussieu.fr/∼danaila

2nd Workshop on FreeFem++, Paris, September 2, 2010

Bose-Einstein condensates GP equation FFEM Conclusion and future work

Outline

1 Vortices in Bose-Einstein condensatesExperimental Bose-Einstein condensateVortices in fluids and superfluids

2 Mathematical description and numerical simulationGross-Pitaevskii equationPrevious numerical simulations

3 FreeFEm++ implementation: Sobolev gradients

4 Conclusion and future work


Outline






Experimental BEC

Bose-Einstein condensate (1)

New state of the matter: super-atomProperties: superfluid, super-conductor.

Predicted in 1924S. Bose A. Einstein

Created in 1995Nobel Prize 2001C. E. Wieman (Univ. Colorado)E. A. Cornell (Univ. Colorado)W. Ketterle (MIT, Cambridge)


Experimental BEC

Bose-Einstein condensate (1)

New state of the matter: super-atomProperties: superfluid, super-conductor.

Predicted in 1924S. Bose A. Einstein

Created in 1995Nobel Prize 2001C. E. Wieman (Univ. Colorado)E. A. Cornell (Univ. Colorado)W. Ketterle (MIT, Cambridge)


Experimental BEC

Bose-Einstein condensate (2)Experiment of Wieman and Cornell (1995)

1000 atoms of Rubidium (Rb)magnetic trapcooling by lasers + radio-frequencyT ∼ 20nKsize ∼ 100µm, t ∼ 1s

explosion in experimental and theoretical activity(Wikipedia)

Experiments in Lab. Kastler Brossel, ENS Paris


Experimental BEC

Bose-Einstein condensate (2)Experiment of Wieman and Cornell (1995)

1000 atoms of Rubidium (Rb)magnetic trapcooling by lasers + radio-frequencyT ∼ 20nKsize ∼ 100µm, t ∼ 1s

explosion in experimental and theoretical activity(Wikipedia)

Experiments in Lab. Kastler Brossel, ENS Paris


Vortices in fluids and superfluids


classical fluids• easy intuition (velocity - pressure)

• complicated math description

solid rotation

superfluids• difficult intuition(vanishing viscosity)• simple math description(wave function)

local rotation

(JILA, Colorado)




classical fluids• easy intuition (velocity - pressure)• complicated math description

solid rotation


local rotation

(JILA, Colorado)





solid rotation


local rotation

(JILA, Colorado)





solid rotation


local rotation

(JILA, Colorado)



Identification of a vortex (1)

Macroscopic descriptionψ wave function

ψ =√ρ(r)eiθ(r)

vortex :: ρ = 0 + rotationvelocity field

v(r) =hm∇θ

quantified circulation

Γ =

∫v(s)ds = n

hm



Identification of a vortex (2)

optical lattice

giant vortex



Creating vortices in BEC

Rotation

Wake of moving objects

Phase imprint


Outline






Gross-Pitaevskii equation

Gross-Pitaevski theory (1)3D Gross-Pitaevski energy

E(ψ) =

∫D

~2

2m|∇ψ|2︸︷︷︸

kinetic

+ ~Ω · (iψ,∇ψ × x)︸︷︷︸rotation

+ Vtrap|ψ|2︸︷︷︸trap

+ Ng3D|ψ|4︸︷︷︸interactions

scaling : [A. Aftalion, T. Riviere, Phys. Rev. A, 2001.]

r = x/R, u(r) = R3/2ψ(x), R = d/√ε

d = (~/mω⊥)1/2 , ε = (d/8πNas)2/5 , Ω = Ω/(εω⊥).

Dimensionless energy

E(u) = H(u)− ΩLz(u), Lz(u) = i∫

u(y∂xu − x∂yu

)H(u) =

∫12|∇u|2 +

12ε2 Vtrap(r)|u|2 +

14ε2 |u|

4



Gross-Pitaevski theory (1)3D Gross-Pitaevski energy

E(ψ) =

∫D

~2

2m|∇ψ|2︸︷︷︸

kinetic

+ ~Ω · (iψ,∇ψ × x)︸︷︷︸rotation

+ Vtrap|ψ|2︸︷︷︸trap

+ Ng3D|ψ|4︸︷︷︸interactions

scaling : [A. Aftalion, T. Riviere, Phys. Rev. A, 2001.]

r = x/R, u(r) = R3/2ψ(x), R = d/√ε

d = (~/mω⊥)1/2 , ε = (d/8πNas)2/5 , Ω = Ω/(εω⊥).

Dimensionless energy

E(u) = H(u)− ΩLz(u), Lz(u) = i∫

u(y∂xu − x∂yu

)H(u) =

∫12|∇u|2 +

12ε2 Vtrap(r)|u|2 +

14ε2 |u|

4



Gross-Pitaevski theory (2)

D ⊂ R3 et u = 0 on ∂D

E(u) =

∫D

12|∇u|2 + Vtrap(r)|u|2 +

g2|u|4 − Ωi

∫D

u∗(At∇

)u

under the unitary norm constraint∫D|u|2 = 1

(meta-)stable states :: local minima of theenergy min E(u)

Numerical methodsDirect minimization of the energy −→ Sobolev gradients.Imaginary time propagation.



Evolution of the numerical wave function

parameters of the simulation Vtrap, Ω

initial condition: ansatz for the vortex / field for Ω = 0convergence: |δE/E| ≤ 10−6


Previous numerical simulations

(3D) Imaginary time propagation

E(u) =

∫12|∇u|2 + Vtrap(r)|u|2 +

g2|u|4 − Ωi

∫u∗(At∇

)u

Euler-Lagrange eq/ stationary Gross-Pitaevskii eq

∂u∂t− 1

2∇2u − iΩ(At∇)u = −u(Vtrap + g|u|2)+µεu

constraint:∫D u2 = 1

normalized gradient flow (Bao and Du, 2004)

∂u∂t

= −12∂E(u)

∂u= −1

2∇L2E(u)



Finite difference 3D code3D numerical code :: BETI

solves :: ∂u∂t = H(u) +∇2u,u ∈ C

combined Runge Kutta + Crank-Nicolson schemeul+1 − ul

δt= alHl + blHl−1 + cl∇2

(ul+1 + ul

2

)ADI factorization

(I − clδt ∇2) = (I − clδt ∂2x )(I − clδt ∂2

y )(I − clδt ∂2z )

projection after 3 steps of R-K

u =u∫D |u|2



Spatial discretization

compact schemes (Pade) of order 613

u′

i−1 + u′

i +13

u′

i+1 =149

ui+1 − ui−1

2h+

19

ui+2 − ui−2

4h,

211

u′′

i−1+u′′

i +2

11u

′′

i+1 =1211

ui+1 − 2ui + ui−1

h2 +3

11ui+2 − 2ui + ui−2

4h2

boundary conditions : u = 0computational domain

D ⊃ ρTF = ρ0 − Vtrap = 0 ,∫

DρTF = 1

grid ≤ 240× 240× 240



Comparison with experimentsP. Rosenbusch, V. Bretin , J. Dalibard, Phys. Rev. Lett. 2002.

A. Aftalion, I. Danaila, Phys. Rev. A, 2003.U vortex S vortex 3D U-vortex



Comparison with mathematical theories

Validation of theoretical resultsA. Aftalion et al., Phys Rev A,2001, 2002.

Eγ =

∫γρTF dl − Ω

| ln ε|

∫γρ2

TF dz

1 no vortex for small Ω

2 β > 1 min= straight vortex3 β ≤ 1 min= U vortex4 γ ∈ (x , z) ou γ ∈ (y , z)

5 Ω, β large ; min = straightvortex



Simulation the real experiment

• 3D simulation(107 grid points).

V. Bretin, S. Stock, Y. Seurin, J. Dalibard, Phys. Rev. Lett. 2003.

I. Danaila, Phys. Rev. A, 2005.



Beyond the physical experiment (1)

I. Danaila, Phys. Rev. A, 2005.



Beyond the physical experiment (2)

moment cinetique

vu de haut

coupe z=0



Suggesting new configurations

Z. Handzibababic, S. Stock, B.Battelier, V. Bretin, J. Dalibard,Phys. Rev. Lett. 2004

3D simulation


Outline






Direct minimization of the energysearch critical points E(u)

Steepest descent method

∂u∂t

= −∇E(u)

−12∇L2E(u) =

∇2u2− Vtrapu − g|u|2u + iΩAt∇u

Sobolev gradients J. W. Neuberger, Springer 1997/2010

L2(D,C) :: 〈u, v〉L2 =

∫D〈u, v〉

H1(D,C) :: 〈u, v〉H =

∫D〈u, v〉+ 〈∇u,∇v〉

Garcıa-Ripoll and Perez-Garcıa, SISC and PRA, 2001


New descent method(I. Danaila and P. Kazemi, SIAM J. Sci Computing, 2010)

New gradient

〈u, v〉HA =

∫D〈u, v〉+ 〈∇Au,∇Av〉, ∇A = ∇+ iΩAt

HA(D,C) = H1(D,C) ⊂ L2(D,C)

New projection method for the constraint

projection on β′(u) = 0, with β(u) =∫D |u|

2

G = ∇X E(u), X =

L2,H1,HA

, 〈vX , v〉X = 〈u, v〉L2

Pu,XG = G − B vX , B =

[<〈u,G〉L2

<〈u, vX 〉L2

]


Implementation of the new method

2D implementationfinite difference (4th order) with Matlab,finite elements with FreeFem++ (www.freefem.org).

Appealing (new) features of FreeFem++easy to implement weak formulations,use combined P1, P2 and P4 elements,complex matrices available,mesh interpolation and adaptivity.


FreeFem++ implementation

• compute the gradient for X = H1∫D∇G∇h + Gh = RHS =

∫D∇u∇h + 2h

[Vtrapu + g|u|2u − iΩAt∇u

]• compute the gradient X = HA∫

D

[1 + Ω2(y2 + x2)

]Gh +∇G∇h − 2iΩ(At∇G)h = RHS

• projection

Pu,XG = G − B vX , B =

[<〈u,G〉L2

<〈u, vX 〉L2

]• time advancement

un+1 = un − δt Pu,XG(un).


Mesh adaptivity with FreeFem++(I. Danaila, F. Hecht, J. Computational Physics, 2010.)

Mesh refinement by metrics control χ = [ur ,ui ] ;P1 finite elements+ adaptivity ≡ high order (6th order FD)

Vtrap = 12 r2 + 1

4 r4,Ω = 2 → Ω = 2.5.

iterations

E(u

)

0 500 1000 15005

5.5

6

6.5

7

7.5

8

8.5

9

9.5

10

xy

0 1 2 3 40

1

2

3

4

ε = 10-3a)

x

y

0 1 2 3 40

1

2

3

4

ε = 10-5b)

x

y

0 1 2 3 40

1

2

3

4

ε = 10-3c)

x

y0 1 2 3 40

1

2

3

4

ε = 10-5d)


Computing physical cases: Abrikosov lattice

Harmonic trapping potential: Vtrap = 12 r2, Ω = 0.95.


Computing physical cases: giant vortex

Quartic trapping potential: Vtrap = 12 r2 + 1

4 r4, g = 1000.


Outline






Conclusion and future work

Simulations are needed for BEC!rich variety of configurationscomplementary information to experimentssuggest new configurations

Future work with FreFem++add time-step optimization in the steepest descend methodreal-time evolution of the condensate3D simulations.

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I Danaila Bose Einstein

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