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

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

Bose-Einstein condensates GP equation FFEM Conclusion and future work

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)

Bose-Einstein condensates GP equation FFEM Conclusion and future work

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)

Bose-Einstein condensates GP equation FFEM Conclusion and future work

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

Bose-Einstein condensates GP equation FFEM Conclusion and future work

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

Bose-Einstein condensates GP equation FFEM Conclusion and future work

Vortices in fluids and superfluids

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)

Bose-Einstein condensates GP equation FFEM Conclusion and future work

Vortices in fluids and superfluids

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)

Bose-Einstein condensates GP equation FFEM Conclusion and future work

Vortices in fluids and superfluids

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)

Bose-Einstein condensates GP equation FFEM Conclusion and future work

Vortices in fluids and superfluids

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)

Bose-Einstein condensates GP equation FFEM Conclusion and future work

Vortices in fluids and superfluids

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

Bose-Einstein condensates GP equation FFEM Conclusion and future work

Vortices in fluids and superfluids

Identification of a vortex (2)

optical lattice

giant vortex

Bose-Einstein condensates GP equation FFEM Conclusion and future work

Vortices in fluids and superfluids

Creating vortices in BEC

Rotation

Wake of moving objects

Phase imprint

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

Bose-Einstein condensates GP equation FFEM Conclusion and future work

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

Bose-Einstein condensates GP equation FFEM Conclusion and future work

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

Bose-Einstein condensates GP equation FFEM Conclusion and future work

Gross-Pitaevskii equation

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.

Bose-Einstein condensates GP equation FFEM Conclusion and future work

Gross-Pitaevskii equation

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

Bose-Einstein condensates GP equation FFEM Conclusion and future work

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)

Bose-Einstein condensates GP equation FFEM Conclusion and future work

Previous numerical simulations

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

Bose-Einstein condensates GP equation FFEM Conclusion and future work

Previous numerical simulations

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

Bose-Einstein condensates GP equation FFEM Conclusion and future work

Previous numerical simulations

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

Bose-Einstein condensates GP equation FFEM Conclusion and future work

Previous numerical simulations

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

Bose-Einstein condensates GP equation FFEM Conclusion and future work

Previous numerical simulations

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.

Bose-Einstein condensates GP equation FFEM Conclusion and future work

Previous numerical simulations

Beyond the physical experiment (1)

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

Bose-Einstein condensates GP equation FFEM Conclusion and future work

Previous numerical simulations

Beyond the physical experiment (2)

moment cinetique

vu de haut

coupe z=0

Bose-Einstein condensates GP equation FFEM Conclusion and future work

Previous numerical simulations

Suggesting new configurations

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

3D simulation

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

Bose-Einstein condensates GP equation FFEM Conclusion and future work

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

Bose-Einstein condensates GP equation FFEM Conclusion and future work

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

]

Bose-Einstein condensates GP equation FFEM Conclusion and future work

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.

Bose-Einstein condensates GP equation FFEM Conclusion and future work

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).

Bose-Einstein condensates GP equation FFEM Conclusion and future work

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)

Bose-Einstein condensates GP equation FFEM Conclusion and future work

Computing physical cases: Abrikosov lattice

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

Bose-Einstein condensates GP equation FFEM Conclusion and future work

Computing physical cases: giant vortex

Quartic trapping potential: Vtrap = 12 r2 + 1

4 r4, g = 1000.

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

Bose-Einstein condensates GP equation FFEM Conclusion and future work

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.