i danaila bose einstein
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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.