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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
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
Introduction GP eq and numerics Imaginary-time propagation Sobolev gradients 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
Introduction GP eq and numerics Imaginary-time propagation Sobolev gradients 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
Introduction GP eq and numerics Imaginary-time propagation Sobolev gradients Conclusion
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)
Introduction GP eq and numerics Imaginary-time propagation Sobolev gradients Conclusion
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)
Introduction GP eq and numerics Imaginary-time propagation Sobolev gradients Conclusion
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)
Introduction GP eq and numerics Imaginary-time propagation Sobolev gradients Conclusion
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)
Introduction GP eq and numerics Imaginary-time propagation Sobolev gradients Conclusion
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
Introduction GP eq and numerics Imaginary-time propagation Sobolev gradients Conclusion
Identification of a quantized vortex (2)
• phase portraits
optical lattice
giant vortex
Introduction GP eq and numerics Imaginary-time propagation Sobolev gradients Conclusion
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.
Introduction GP eq and numerics Imaginary-time propagation Sobolev gradients Conclusion
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.
Introduction GP eq and numerics Imaginary-time propagation Sobolev gradients Conclusion
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.
Introduction GP eq and numerics Imaginary-time propagation Sobolev gradients Conclusion
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]).
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
Introduction GP eq and numerics Imaginary-time propagation Sobolev gradients 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
Introduction GP eq and numerics Imaginary-time propagation Sobolev gradients 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
Introduction GP eq and numerics Imaginary-time propagation Sobolev gradients Conclusion
The 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.
Introduction GP eq and numerics Imaginary-time propagation Sobolev gradients Conclusion
The 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
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
Introduction GP eq and numerics Imaginary-time propagation Sobolev gradients 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)
Introduction GP eq and numerics Imaginary-time propagation Sobolev gradients 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)
Introduction GP eq and numerics Imaginary-time propagation Sobolev gradients 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)
Introduction GP eq and numerics Imaginary-time propagation Sobolev gradients Conclusion
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
Introduction GP eq and numerics Imaginary-time propagation Sobolev gradients Conclusion
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
Introduction GP eq and numerics Imaginary-time propagation Sobolev gradients Conclusion
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
Introduction GP eq and numerics Imaginary-time propagation Sobolev gradients Conclusion
Simulation of BEC experiments
The S vortex (Ω ≥ 0, local minimum)
energy diagram
Introduction GP eq and numerics Imaginary-time propagation Sobolev gradients Conclusion
Simulation of BEC experiments
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
Introduction GP eq and numerics Imaginary-time propagation Sobolev gradients Conclusion
Simulation of BEC experiments
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
Introduction GP eq and numerics Imaginary-time propagation Sobolev gradients Conclusion
Simulation of BEC experiments
Suggestion for new configurationsQuartic-harmonic potential: A. Aftalion, I. Danaila, PRA, 2004.
angular momentum
top view
2D cut (z=0)
Introduction GP eq and numerics Imaginary-time propagation Sobolev gradients Conclusion
Simulation of BEC experiments
Quartic-harmonic potential
angular momentum
top view
2D cut (z=0)
Introduction GP eq and numerics Imaginary-time propagation Sobolev gradients Conclusion
Simulation of BEC experiments
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.
Introduction GP eq and numerics Imaginary-time propagation Sobolev gradients Conclusion
Simulation of BEC experiments
Quartic+harmonic potential (1)
Introduction GP eq and numerics Imaginary-time propagation Sobolev gradients Conclusion
Simulation of BEC experiments
Quartic+harmonic potential (2)
I. Danaila, Phys. Rev. A, 2005. Good quantitative agreementD. E. Sheehy and L. Radzihovsky, Phys. Rev. A, 2004.
Introduction GP eq and numerics Imaginary-time propagation Sobolev gradients Conclusion
Simulation of BEC experiments
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
Introduction GP eq and numerics Imaginary-time propagation Sobolev gradients Conclusion
Simulation of BEC experiments
Rotating condensate in an optical lattice
Ω = 0.87 U = 0.1 U = 0.5 U = 0.7
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
Introduction GP eq and numerics Imaginary-time propagation Sobolev gradients 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
Introduction GP eq and numerics Imaginary-time propagation Sobolev gradients Conclusion
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!
Introduction GP eq and numerics Imaginary-time propagation Sobolev gradients Conclusion
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)
Introduction GP eq and numerics Imaginary-time propagation Sobolev gradients Conclusion
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
]
Introduction GP eq and numerics Imaginary-time propagation Sobolev gradients Conclusion
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.
Introduction GP eq and numerics Imaginary-time propagation Sobolev gradients Conclusion
FreeFem++ implementation
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).
Introduction GP eq and numerics Imaginary-time propagation Sobolev gradients Conclusion
FreeFem++ implementation
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;
Introduction GP eq and numerics Imaginary-time propagation Sobolev gradients Conclusion
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.
Introduction GP eq and numerics Imaginary-time propagation Sobolev gradients Conclusion
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
Introduction GP eq and numerics Imaginary-time propagation Sobolev gradients Conclusion
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)
Introduction GP eq and numerics Imaginary-time propagation Sobolev gradients Conclusion
2D results
Mesh isotropy
Introduction GP eq and numerics Imaginary-time propagation Sobolev gradients Conclusion
2D results
Computing physical cases: Abrikosov lattice
Harmonic trapping potential: Vtrap = 12 r2, Ω = 0.95.
Introduction GP eq and numerics Imaginary-time propagation Sobolev gradients Conclusion
2D results
Computing physical cases: giant vortex
Quartic trapping potential: Vtrap = 12 r2 + 1
4 r4, g = 1000.
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
Introduction GP eq and numerics Imaginary-time propagation Sobolev gradients 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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