Numerical methods for computing vortex states in rotating Bose-Einstein condensates. Ionut Danaila

Transcription

1 Numerical methods for computing vortex states in rotating Bose-Einstein condensates Ionut Danaila Laboratoire de mathématiques Raphaël Salem Université de Rouen Conference Non-linear optical and atomic systems, Lille, January 22, 2013

2 Outline 1 Vortices in Bose-Einstein condensates 2 Mathematical description and numerical simulation The Gross-Pitaevskii equation 3 Imaginary-time propagation of the wave function Simulation of BEC experiments 4 Direct minimization of the energy functional Descent methods using Sobolev gradients FreeFem++ implementation 2D results 5 Conclusion

3 Outline 1 Vortices in Bose-Einstein condensates 2 Mathematical description and numerical simulation The Gross-Pitaevskii equation 3 Imaginary-time propagation of the wave function Simulation of BEC experiments 4 Direct minimization of the energy functional Descent methods using Sobolev gradients FreeFem++ implementation 2D results 5 Conclusion

4 Bose-Einstein condensate Experiment of Wieman and Cornell (1995) 1000 atoms of Rubidium (Rb) magnetic trap cooling by lasers + radio-frequency T 20nK size 100µm, t 1s

5 Bose-Einstein condensate Experiment of Wieman and Cornell (1995) 1000 atoms of Rubidium (Rb) magnetic trap cooling by lasers + radio-frequency T 20nK size 100µm, t 1s explosion in experimental and theoretical activity(wikipedia) Experiments in Lab. Kastler Brossel, ENS Paris

6 Vortices in fluids and superfluids classical fluids easy intuition (velocity - pressure) solid rotation

7 Vortices in fluids and superfluids classical fluids easy intuition (velocity - pressure) complicated math description solid rotation

8 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

9 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) (JILA, Colorado)

10 Identification of a quantized vortex (1) Macroscopic description ψ wave function ψ = ρ(r)e iθ(r) vortex :: ρ = 0 + rotation velocity field v(r) = h m θ quantified circulation Γ = v(s)ds = n h m

11 Identification of a quantized vortex (2) phase portraits optical lattice giant vortex

12 Creating vortices in BEC Wake of moving objects Q. Du, Penn State

13 Creating vortices in BEC Wake of moving objects Q. Du, Penn State Phase imprint L.-C. Crasovan, V. M. Pérez-García, I. Danaila, D. Mihalache, L. Torner, PRA, 2004.

14 Creating vortices in BEC Rotation Wake of moving objects Q. Du, Penn State Phase imprint L.-C. Crasovan, V. M. Pérez-García, I. Danaila, D. Mihalache, L. Torner, PRA, 2004.

15 Rotating Bose-Einstein condensate Experiments in Lab Kastler Brossel, ENS Paris Cold Atoms Group of J. Dalibard Condensate of Rb made of atoms ; T = 90nK Thomas Fermi regime: Na s /a h 500 (a s =5 [nm]) << (ξ=0.3 [µm]) << (a h =1 [µm]) << (R=3 [µm]).

16 Outline 1 Vortices in Bose-Einstein condensates 2 Mathematical description and numerical simulation The Gross-Pitaevskii equation 3 Imaginary-time propagation of the wave function Simulation of BEC experiments 4 Direct minimization of the energy functional Descent methods using Sobolev gradients FreeFem++ implementation 2D results 5 Conclusion

17 The Gross-Pitaevskii equation The Gross-Pitaevskii theory (1) 3D Gross-Pitaevskii energy 2 E(ψ) = D 2m ψ 2 + N }{{} 2 g 3D ψ 4 + V trap ψ 2 + Ω (iψ, ψ x) }{{}}{{}}{{} kinetic interactions trap rotation scaling: [Aftalion Rivière (2001), Tsubota et al (2002), Fetter et al (2005)] r = x/r, u(r) = R 3/2 ψ(x), R = d/ ε d = ( /mω ) 1/2, ε = (d/8πna s ) 2/5, Ω = Ω/(εω ). Dimensionless energy E(u) = H(u) ΩL z (u), L z (u) = i u ( A t ) u, A = (y, x, 0) t H(u) = 1 2 u 2 + Ṽtrap(r) u 2 + g 2 u 4

18 The Gross-Pitaevskii equation The Gross-Pitaevskii theory (1) 3D Gross-Pitaevskii energy 2 E(ψ) = D 2m ψ 2 + N }{{} 2 g 3D ψ 4 + V trap ψ 2 + Ω (iψ, ψ x) }{{}}{{}}{{} kinetic interactions trap rotation scaling: [Aftalion Rivière (2001), Tsubota et al (2002), Fetter et al (2005)] r = x/r, u(r) = R 3/2 ψ(x), R = d/ ε d = ( /mω ) 1/2, ε = (d/8πna s ) 2/5, Ω = Ω/(εω ). Dimensionless energy E(u) = H(u) ΩL z (u), L z (u) = i u ( A t ) u, A = (y, x, 0) t H(u) = 1 2 u 2 + Ṽtrap(r) u 2 + g 2 u 4

19 The Gross-Pitaevskii equation Gross-Pitaevski theory (2) D R 3 et u = 0 on D 1 E(u) = 2 u 2 + V trap (r) u 2 + g 2 u 4 Ωi D under the unitary norm constraint u 2 = 1 (meta-)stable states :: local minima of the energy min E(u) Numerical methods D D u ( A t ) u Direct minimization of the energy Sobolev gradients. Imaginary time propagation.

20 The Gross-Pitaevskii equation Evolution of the numerical wave function parameters of the simulation V trap, Ω initial condition: ansatz for the vortex / field for Ω = 0 convergence: δe/e 10 6

21 Outline 1 Vortices in Bose-Einstein condensates 2 Mathematical description and numerical simulation The Gross-Pitaevskii equation 3 Imaginary-time propagation of the wave function Simulation of BEC experiments 4 Direct minimization of the energy functional Descent methods using Sobolev gradients FreeFem++ implementation 2D results 5 Conclusion

22 (3D) Imaginary time propagation E(u) = 1 2 u 2 + V trap (r) u 2 + g 2 u 4 Ωi u ( A t ) u Euler-Lagrange eq/ stationary Gross-Pitaevskii eq u t u + i(ω r). u = u 2ε 2 (V trap u 2 ) + µ ε u constraint : D u2 = 1 normalized gradient flow (Bao and Du, 2004) u t = 1 E(u) = 1 2 u 2 L 2E(u)

23 (3D) Imaginary time propagation E(u) = 1 2 u 2 + V trap (r) u 2 + g 2 u 4 Ωi u ( A t ) u Euler-Lagrange eq/ stationary Gross-Pitaevskii eq u t u + i(ω r). u = u 2ε 2 (V trap u 2 ) + µ ε u constraint : D u2 = 1 normalized gradient flow (Bao and Du, 2004) u t = 1 E(u) = 1 2 u 2 L 2E(u)

24 (3D) Imaginary time propagation E(u) = 1 2 u 2 + V trap (r) u 2 + g 2 u 4 Ωi u ( A t ) u Euler-Lagrange eq/ stationary Gross-Pitaevskii eq u t u + i(ω r). u = u 2ε 2 (V trap u 2 ) + µ ε u constraint : D u2 = 1 normalized gradient flow (Bao and Du, 2004) u t = 1 E(u) = 1 2 u 2 L 2E(u)

25 Finite difference 3D code 3D numerical code :: BETI solves :: u t = H(u) + 2 u, u C combined Runge Kutta + Crank-Nicolson scheme u l+1 u l δt ADI factorization = a l H l + b l H l 1 + c l 2 ( ul+1 + u l 2 (I c l δt 2 ) = (I c l δt 2 x )(I c l δt 2 y )(I c l δt 2 z ) projection after 3 steps of R-K u = u D u 2 )

26 Spatial discretization compact schemes (Padé) of order u i 1 + u i u i+1 = 14 u i+1 u i u i+2 u i 2, 9 2h 9 4h i+1 = 12 u i+1 2u i + u i 1 11 h u i 1 +u i u u i+2 2u i + u i 2 4h 2 boundary conditions : u = 0 computational domain D {ρ TF = ρ 0 V trap = 0}, D ρ TF = 1 grid

27 Simulation of BEC experiments Simulation of experiments (harmonic potential) P. Rosenbusch, V. Bretin, J. Dalibard, Phys. Rev. Lett A. Aftalion, I. Danaila, Phys. Rev. A, U vortex S vortex 3D U-vortex

28 Simulation of BEC experiments The S vortex (Ω 0, local minimum) energy diagram

29 Simulation of BEC experiments Fast rotating condensate towards the giant vorex [Kasamatsu, Tsubota and Ueda, 2002] harmonic potential : singularity for Ω = (ω (0) ) V h (r, z) = 1 2 m(ω(0) )2 r mω2 zz 2 V eff (r) = V h (r) 1 2 mω2 r 2 harmonic potential + Gaussian potential V (r, z) = V (r, z) = V h (r, z) + U 0 e 2r 2 /w 2 [ 1 2 m(ω(0) )2 2U ] 0 w 2 r 2 + 2U 0 w 4 r mω2 zz 2

30 Simulation of BEC experiments Potential : V trap = (1 α)r 2 + k 4 r 4 + β 2 z 2 A. Aftalion, I. Danaila, Phys. Rev. A, V eff (r) = V trap (r) ε 2 Ω 2 r 2 ε = 0.02, k/α = α < 1 weak attractive case 2 1 < α < 1 + β 1/4 k 5/8 / π weak repulsive case 3 α > 1 + β 1/4 k 5/8 / π strong repulsive case

31 Simulation of BEC experiments Suggestion for new configurations Quartic-harmonic potential: A. Aftalion, I. Danaila, PRA, top view angular momentum 2D cut (z=0)

32 Simulation of BEC experiments Quartic-harmonic potential top view angular momentum 2D cut (z=0)

33 Simulation of BEC experiments Simulation of real experiments 3D simulation of the experimental configuration (10 7 grid points). V. Bretin, S. Stock, Y. Seurin, J. Dalibard, Phys. Rev. Lett I. Danaila, Phys. Rev. A, 2005.

34 Simulation of BEC experiments Quartic+harmonic potential (1)

35 Simulation of BEC experiments Quartic+harmonic potential (2) I. Danaila, Phys. Rev. A, Good quantitative agreement D. E. Sheehy and L. Radzihovsky, Phys. Rev. A, 2004.

36 Simulation of BEC experiments Optical lattice potential: V trap = r 2 + U sin 2 (πz/d) Non rotating BEC in optical lattices Z. Handzibababic, S. Stock, B. Battelier, V. Bretin, J. Dalibard, Phys. Rev. Lett D simulation

37 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

38 Outline 1 Vortices in Bose-Einstein condensates 2 Mathematical description and numerical simulation The Gross-Pitaevskii equation 3 Imaginary-time propagation of the wave function Simulation of BEC experiments 4 Direct minimization of the energy functional Descent methods using Sobolev gradients FreeFem++ implementation 2D results 5 Conclusion

39 Sobolev Direct minimization of the GP energy search critical points E(u) Normalized gradient flow u t = E(u) 1 2 L 2E(u) = 2 u 2 V trapu g u 2 u + iωa t u Sobolev gradients: J. W. Neuberger, Springer, 1997/2010 L 2 (D, C) :: u, v L 2 = u, v H 1 (D, C) :: u, v H = D D u, v + u, v García-Ripoll and Pérez-García, SISC and PRA, 2001

40 Sobolev Sobolev gradient method/ preconditionners Classical descent method (L 2 gradient) u = t L 2φ(u) = u k+1 = u k α L 2φ(u k ) similar to Richardson steepest descent method! Sobolev gradient descent method u t = H φ(u), P H φ(u k ) = L 2φ(u k ) u k+1 = u k αp 1 L 2φ(u k ) similar to preconditionned Richardson method!

41 Sobolev New descent method (1) (I. Danaila and P. Kazemi, SIAM J. Sci Computing, 2010) E(u) = D 1 2 u + iωat u 2 + (V trap Ω2 r 2 2 ) u 2 + g 2 u 4 New gradient u, v HA = D u, v + A u, A v, A = + iωa t H A (D, C) = H 1 (D, C) L 2 (D, C) < HA E, v > HA =< L 2E, v > L 2, v H 1 (D, C)

42 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 = {L 2, H 1, H A P u,x G = G B v X from < X E, v > X =< L 2E, v > L 2 R v X, v X = β (u)v = R u, v L 2 from R u, P u,x G L 2 = 0 [ ] R u, G L 2 B = R u, v X L 2

43 FreeFem++ implementation Implementation of the new method FreeFem++ ( 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.

44 FreeFem++ implementation FreeFem++ implementation compute the gradient for X = H 1 [ ] G h + Gh = RHS = u h + 2h V trap u + g u 2 u iωa t u D D compute the gradient X = H A [ ] 1 + Ω 2 (y 2 + x 2 ) Gh + G h 2iΩ(A t G)h = RHS D projection time advancement [ ] R u, G L 2 P u,x G = G B v X, B = R u, v X L 2 u n+1 = u n δt P u,x G(u n ).

45 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 = H 1 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;

46 2D results Academic test cases (manufactured solutions) New Sobolev method H A more efficient than H 1 for Ω ; CPU gain : 40% to 300 % New projection method for the unitary norm faster convergence that with normalization methods.

47 2D results Mesh adaptivity with FreeFem++ (1) (I. Danaila, F. Hecht, J. Computational Physics, 2010.) Mesh refinement by metrics control χ = u or χ = [u r, u i ] ; P1 finite elements+ adaptivity high order (6th order FD) V trap = 1 2 r r 4, Ω = 2 g = adapt U M=200 adapt [Ur, Ui] M=200 no-adapt M=200 no-adapt M=400 6th order FD 12.4 E(u) iterations

48 2D results Mesh adaptivity with FreeFem++ (2) (I. Danaila, F. Hecht, J. Computational Physics, 2010.) Good refinement strategy χ = [u r, u i ] ; V trap = 1 2 r r 4, Ω = 2 Ω = 2.5. a) 4 3 ε =10-3 b) 4 3 ε = y 2 1 y 2 1 E(u) c) x 4 ε =10-3 d) x 4 ε = y 3 2 y iterations x x

49 2D results Mesh isotropy

50 2D results Computing physical cases: Abrikosov lattice Harmonic trapping potential: V trap = 1 2 r 2, Ω = 0.95.

51 2D results Computing physical cases: giant vortex Quartic trapping potential: V trap = 1 2 r r 4, g = 1000.

52 Outline 1 Vortices in Bose-Einstein condensates 2 Mathematical description and numerical simulation The Gross-Pitaevskii equation 3 Imaginary-time propagation of the wave function Simulation of BEC experiments 4 Direct minimization of the energy functional Descent methods using Sobolev gradients FreeFem++ implementation 2D results 5 Conclusion

53 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) information to experiments, and suggest new configurations. Future work: ANR project BECASIM ( ) 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 available free and performant codes!