Numerical methods for computing vortex states in...

Numerical methods for computing vortexstates in rotating Bose-Einstein

condensates

Ionut Danaila

Laboratoire de mathematiques Raphael SalemUniversite de Rouen

www.univ-rouen.fr/LMRS/Persopage/Danaila

Conference Non-linear optical and atomic systems,Lille, January 22, 2013

Introduction GP eq and numerics Imaginary-time propagation Sobolev gradients Conclusion

Outline

1 Vortices in Bose-Einstein condensates

2 Mathematical description and numerical simulationThe Gross-Pitaevskii equation

3 Imaginary-time propagation of the wave functionSimulation of BEC experiments

4 Direct minimization of the energy functionalDescent methods using Sobolev gradientsFreeFem++ implementation2D results

5 Conclusion


Outline





5 Conclusion


Bose-Einstein condensateExperiment 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 condensateExperiment 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 quantized 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 quantized vortex (2)

• phase portraits

optical lattice

giant vortex


Creating vortices in BEC

Rotation

Wake of moving objects Q. Du, Penn State

Phase imprint L.-C. Crasovan, V. M. Perez-Garcıa,I. Danaila, D. Mihalache, L. Torner, PRA, 2004.



Rotation





Rotation




Rotating Bose-Einstein condensate

Experiments in Lab Kastler Brossel, ENS ParisCold Atoms Group of J. Dalibard

Condensate of Rb made of ∼ 500 000 atoms ; T = 90nKThomas Fermi regime: Nas/ah ≈ 500(as=5 [nm]) << (ξ=0.3 [µm]) << (ah=1 [µm]) << (R=3 [µm]).


Outline





5 Conclusion


The Gross-Pitaevskii equation

The Gross-Pitaevskii theory (1)3D Gross-Pitaevskii energy

E(ψ) =

∫D

~2

2m|∇ψ|2︸︷︷︸

kinetic

+N2

g3D|ψ|4︸︷︷︸interactions

+ Vtrap|ψ|2︸︷︷︸trap

+ ~Ω · (iψ,∇ψ × x)︸︷︷︸rotation

scaling: [Aftalion–Riviere (2001), Tsubota et al (2002),Fetter et al (2005)]

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∗(At∇

)u, A = (y ,−x ,0)t

H(u) =

∫12|∇u|2 + Vtrap(r)|u|2 +

g2|u|4



The Gross-Pitaevskii theory (1)3D Gross-Pitaevskii energy

E(ψ) =

∫D

~2

2m|∇ψ|2︸︷︷︸

kinetic

+N2

g3D|ψ|4︸︷︷︸interactions

+ Vtrap|ψ|2︸︷︷︸trap

+ ~Ω · (iψ,∇ψ × x)︸︷︷︸rotation

scaling: [Aftalion–Riviere (2001), Tsubota et al (2002),Fetter et al (2005)]

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∗(At∇

)u, A = (y ,−x ,0)t

H(u) =

∫12|∇u|2 + Vtrap(r)|u|2 +

g2|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


Outline





5 Conclusion


(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 + i(Ω× r).∇u = − u

2ε2 (Vtrap − |u|2) + µεu

constraint :∫D u2 = 1

normalized gradient flow (Bao and Du, 2004)

∂u∂t

= −12∂E(u)

∂u= −1

2∇L2E(u)



E(u) =

∫12|∇u|2 + Vtrap(r)|u|2 +

g2|u|4 − Ωi

∫u∗(At∇

)u


∂u∂t− 1

2∇2u + i(Ω× r).∇u = − u




∂u∂t

= −12∂E(u)

∂u= −1

2∇L2E(u)



E(u) =

∫12|∇u|2 + Vtrap(r)|u|2 +

g2|u|4 − Ωi

∫u∗(At∇

)u


∂u∂t− 1

2∇2u + i(Ω× r).∇u = − u




∂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


Simulation of BEC experiments

Simulation of experiments (harmonic potential)P. Rosenbusch, V. Bretin , J. Dalibard, Phys. Rev. Lett. 2002.

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



The S vortex (Ω ≥ 0, local minimum)

energy diagram



Fast rotating condensate

• towards the giant vorex [Kasamatsu, Tsubota and Ueda, 2002]• harmonic potential : singularity for Ω = (ω

(0)⊥ )

Vh(r , z) =12

m(ω(0)⊥ )2r2 +

12

mω2z z2

V eff (r) = Vh(r)− 12

mΩ2r2

• harmonic potential + Gaussian potential

V (r , z) = Vh(r , z) + U0 e−2r2/w2

V (r , z) =

[12

m(ω(0)⊥ )2 − 2U0

w2

]r2 +

2U0

w4 r4 +12

mω2z z2



Potential : Vtrap = (1− α)r2 + k4r4 + β2z2

A. Aftalion, I. Danaila, Phys.Rev. A, 2004.V eff (r) = Vtrap(r)− ε2Ω2r2

ε = 0.02, k/α = 0.25

1 α < 1weak attractive case

2 1 < α < 1 + β1/4k5/8/√π

weak repulsive case3 α > 1 + β1/4k5/8/

√π

strong repulsive case



Suggestion for new configurationsQuartic-harmonic potential: A. Aftalion, I. Danaila, PRA, 2004.

angular momentum

top view

2D cut (z=0)



Quartic-harmonic potential

angular momentum

top view

2D cut (z=0)



Simulation of real experiments

• 3D simulation of the experimental configuration(107 grid points).

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

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



Quartic+harmonic potential (1)



Quartic+harmonic potential (2)

I. Danaila, Phys. Rev. A, 2005. Good quantitative agreementD. E. Sheehy and L. Radzihovsky, Phys. Rev. A, 2004.



Optical lattice potential: Vtrap = r2 + U sin2 (πz/d)

• Non rotating BEC in optical lattices

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

• 3D simulation



Rotating condensate in an optical lattice

Ω = 0.87 U = 0.1 U = 0.5 U = 0.7


Outline





5 Conclusion


Sobolev

Direct minimization of the GP energysearch critical points E(u)

Normalized gradient flow

∂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


Sobolev

Sobolev gradient method/ preconditionners

Classical descent method (L2 gradient)

∂u∂t

= −∇L2φ(u) =⇒ uk+1 = uk − α∇L2φ(uk )

similar to Richardson steepest descent method!

Sobolev gradient descent method

∂u∂t

= −∇Hφ(u), P · ∇Hφ(uk ) = ∇L2φ(uk )

uk+1 = uk − αP−1∇L2φ(uk )

similar to preconditionned Richardson method!


Sobolev

New descent method (1)(I. Danaila and P. Kazemi, SIAM J. Sci Computing, 2010)

E(u) =

∫D

12|∇u + iΩAtu|2 +

(Vtrap −

Ω2r2

2

)|u|2 +

g2|u|4

New gradient

〈u, v〉HA =

∫D〈u, v〉+ 〈∇Au,∇Av〉, ∇A = ∇+ iΩAt

HA(D,C) = H1(D,C) ⊂ L2(D,C)

< ∇HAE , v >HA=< ∇L2E , v >L2 , ∀v ∈ H1(D,C)


Sobolev

New descent method (2)(I. Danaila and P. Kazemi, SIAM J. Sci Computing, 2010)

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

Pu,XG = G − B vX

• from < ∇X E , v >X =< ∇L2E , v >L2

<〈vX , v〉X = β′(u)v = <〈u, v〉L2

• from <〈u,Pu,XG〉L2 = 0

B =

[<〈u,G〉L2

<〈u, vX 〉L2

]


FreeFem++ implementation

Implementation of the new method

FreeFem++ (www.freefem.org)Free Generic PDE solver using finite elements (2D and 3D)

powerful mesh generator,easy to implement weak formulations,use combined P1, P2 and P4 elements,complex matrices available,mesh interpolation and adaptivity.

You are welcome to participate in the:Workshop on FreeFem++ and Applications

Paris, December, 2013.




• 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).



FreeFem++ syntax

• create a mesh and a finite element space

border circle(t=0,2*pi)label=1;x=Rmax*cos(t);y=Rmax*sin(t);;

mesh Th=buildmesh(circle(nbseg));fespace Vh(Th,P1); fespace Vh4(Th,P4);

• compute the gradient for X = H1

Vh<complex> ug,v ;problem AGRAD(ug,v) =

int2d(Th)(ug*v + dx(ug)*dx(v)+dy(ug)*dy(v))- int2d(Th)(Ctrap*un*v)- int2d(Th)(CN*real(un*conj(un))*un*v)+ ...+ on(1,ug=0);

AGRAD;


2D results

Academic test cases (manufactured solutions)

New Sobolev method HA more efficient than H1 for Ω ;

→CPU gain : 40% to 300 %

New projection method for the unitary norm −→ fasterconvergence that with normalization methods.


2D results

Mesh adaptivity with FreeFem++ (1)(I. Danaila, F. Hecht, J. Computational Physics, 2010.)

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

Vtrap = 12 r2 + 1

4 r4, Ω = 2 g = 500

iterations

E(u

)

0 50 100 150 200 250 30011.8

12

12.2

12.4

12.6

12.8

adapt |U| M=200adapt [Ur, Ui] M=200no-adapt M=200no-adapt M=4006th order FD


2D results

Mesh adaptivity with FreeFem++ (2)(I. Danaila, F. Hecht, J. Computational Physics, 2010.)

Good refinement strategy χ = [ur ,ui ] ;

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)

xy

0 1 2 3 40

1

2

3

4

ε = 10-5d)


2D results

Mesh isotropy


2D results

Computing physical cases: Abrikosov lattice

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


2D results

Computing physical cases: giant vortex

Quartic trapping potential: Vtrap = 12 r2 + 1

4 r4, g = 1000.


Outline





5 Conclusion


Conclusion and future work

Advanced numerics are needed for BEC!Numerical Analysis→ new efficient methods,prove their capabilities on real (experimental) cases,bring complementary (qualitative/quantitative) informationto experiments, and suggest new configurations.

Future work: ANR project BECASIM (2013-2016)

3D methods for real and imaginary time GP,implementation using (HPC) parallel computing,huge simulations of physical configurations(turbulence in BEC)iteract with physics community and make availablefree and performant codes!

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