Vortices in Bose-Einstein condensates. Ionut Danaila

Transcription

1 Vortices in Bose-Einstein condensates 3D numerical simulations Ionut Danaila Laboratoire Jacques Louis Lions Université Pierre et Marie Curie (Paris 6) danaila October 16, 2008

2 Outline 1 Introduction Experimental Bose-Einstein condensate Vortices in fluids and superfluids 2 Numerical method Gross-Pitaevskii energy Imaginary time evolution : 3D code 3 3D structure of vortices Vortices in non rotating condensates Rotating condensate: harmonic potential Rotating condensate: quartic potential Rotating condensate: optical lattice Conclusion and future work

3 Outline 1 Introduction Experimental Bose-Einstein condensate Vortices in fluids and superfluids 2 Numerical method Gross-Pitaevskii energy Imaginary time evolution : 3D code 3 3D structure of vortices Vortices in non rotating condensates Rotating condensate: harmonic potential Rotating condensate: quartic potential Rotating condensate: optical lattice Conclusion and future work

4 Experimental BEC Bose-Einstein condensate (1) New state of the matter: super-atom Properties: superfluid, super-conductor. Predicted in 1924 S. Bose A. Einstein

5 Experimental BEC Bose-Einstein condensate (1) New state of the matter: super-atom Properties: superfluid, super-conductor. Predicted in 1924 S. Bose A. Einstein Created in 1995 Nobel Prize 2001 C. E. Wieman (Univ. Colorado) E. A. Cornell (Univ. Colorado) W. Ketterle (MIT, Cambridge)

6 Experimental BEC Bose-Einstein condensate (2) Experiment of Wieman and Cornell (1995) 1000 atoms of Rubidium (Rb) magnetic trap cooling by lasers + radio-frequency T 20nK size 100µm, t 1s

7 Experimental BEC Bose-Einstein condensate (2) 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

8 Vortices in fluids and superfluids Vortices in classical fluids easy physical intuition (velocity - pressure) (controversial) mathematical description Example of a classical vortex: the vortex ring flow injection through an orifice in a quiescent surrounding (Gharib et al., 1998)

9 Vortices in fluids and superfluids Identification of the vortex Passive scalar (smoke) (

10 Vortices in fluids and superfluids Identification of the vortex Passive scalar (smoke)

11 Vortices in fluids and superfluids Identification of the vortex Passive scalar / vorticity / pressure

12 Vortices in fluids and superfluids Identification of the vortex Passive scalar / vorticity / pressure U velocity field, p pressure field

13 Vortices in fluids and superfluids Identification of the vortex Passive scalar / vorticity / pressure U velocity field, p pressure field ω = U, Γ = ω nda vortex = iso-surface of ω (max) or of p (min)

14 Vortices in fluids and superfluids Vortices in superfluids difficult physical intuition (flow without viscosity) simple mathematical description (wave function) Bose-Einstein condensate (LKB, ENS Paris) (JILA, University of Colorado)

15 Vortices in fluids and superfluids Identification of a 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

16 Vortices in fluids and superfluids Identification of a vortex (2) optical lattice giant vortex

17 Vortices in fluids and superfluids Vortex in a Bose-Einstein condensate Wake of moving objects

18 Vortices in fluids and superfluids Vortex in a Bose-Einstein condensate Wake of moving objects Phase imprint

19 Vortices in fluids and superfluids Vortex in a Bose-Einstein condensate Rotation Wake of moving objects Phase imprint

20 Outline 1 Introduction Experimental Bose-Einstein condensate Vortices in fluids and superfluids 2 Numerical method Gross-Pitaevskii energy Imaginary time evolution : 3D code 3 3D structure of vortices Vortices in non rotating condensates Rotating condensate: harmonic potential Rotating condensate: quartic potential Rotating condensate: optical lattice Conclusion and future work

21 Gross-Pitaevskii energy Gross-Pitaevskii theory (1) 3D Gross-Pitaevskii energy 2 E(ψ) = D 2m ψ 2 + Ω (iψ, ψ x) + V }{{}}{{} trap ψ 2 + Ng 3D ψ 4 }{{}}{{} kinetic rotation trap interactions scaling : [A. Aftalion, T. Rivière, Phys. Rev. A, 2001.] 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 ū ( y x u x y u ) H(u) = 1 2 u ε 2 V trap(r) u ε 2 u 4

22 Gross-Pitaevskii energy Gross-Pitaevskii theory (1) 3D Gross-Pitaevskii energy 2 E(ψ) = D 2m ψ 2 + Ω (iψ, ψ x) + V }{{}}{{} trap ψ 2 + Ng 3D ψ 4 }{{}}{{} kinetic rotation trap interactions scaling : [A. Aftalion, T. Rivière, Phys. Rev. A, 2001.] 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 ū ( y x u x y u ) H(u) = 1 2 u ε 2 V trap(r) u ε 2 u 4

23 Gross-Pitaevskii energy Gross-Pitaevskii theory (2) Lagrange equation equilibrium states :: critical points min E(u) u t u + i(ω r). u = u 2ε 2 (V trap u 2 ) + µ ε u constraint : D u2 = 1

24 Gross-Pitaevskii energy Gross-Pitaevskii theory (2) Lagrange equation equilibrium states :: critical points min E(u) u t u + i(ω r). u = u 2ε 2 (V trap u 2 ) + µ ε u constraint : D u2 = 1

25 Gross-Pitaevskii energy Gross-Pitaevskii theory (2) Lagrange equation equilibrium states :: critical points min E(u) u t u + i(ω r). u = u 2ε 2 (V trap u 2 ) + µ ε u constraint : D u2 = 1

26 Imaginary time evolution : 3D code Imaginary time evolution 3D numerical code 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 )

27 Imaginary time evolution : 3D code 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

28 Imaginary time evolution : 3D code Imaginary time evolution parameters of the simulation V trap, Ω initial condition: ansatz for the vortex / field for Ω = 0 convergence: δe/e 10 6

29 Outline 1 Introduction Experimental Bose-Einstein condensate Vortices in fluids and superfluids 2 Numerical method Gross-Pitaevskii energy Imaginary time evolution : 3D code 3 3D structure of vortices Vortices in non rotating condensates Rotating condensate: harmonic potential Rotating condensate: quartic potential Rotating condensate: optical lattice Conclusion and future work

30 Vortices in non rotating condensates Vortices in non rotating condensates L.-C. Crasovan, V. M. Pérez-García, I. Danaila, D. Mihalache and L. Torner, Phys Rev A, series of Hermite polynomials 3D simulation ψ = j c j e ie j t 3 k=1 H jk (λ 1/2 k x k )e λ k x 2 k /2 φ (x, y, z) = H 2 (x)h 0 (y)h 0 (z) + ih 0 (x)h 2 (y)h 0 (z)

31 Rotating condensate: harmonic potential 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]).

32 Rotating condensate: harmonic potential Harmonic potential: V trap = x 2 + α 2 y 2 + β 2 z 2 P. Rosenbusch, V. Bretin, J. Dalibard, Phys. Rev. Lett A. Aftalion, I. Danaila, Phys. Rev. A, U vortex S vortex 3D U-vortex

33 Rotating condensate: harmonic potential The U vortex(ω 0.42, global minimum) Validation of theoretical results A. Aftalion, T. Rivière, Phys Rev A, E γ = γ ρ TF dl Ω ρ 2 TF ln ε dz γ 1 no vortex for small Ω 2 β > 1 min= straight vortex 3 β 1 min= vortex en U 4 γ (x, z) or γ (y, z) 5 Ω, β large ; min= straight vortex

34 Rotating condensate: harmonic potential The U vortex bifurcation diagram

35 Rotating condensate: harmonic potential The S vortex (Ω 0, local minimum) energy diagram

36 Rotating condensate: harmonic potential Multiple vortices

37 Rotating condensate: quartic potential Fast rotating condensate harmonic potential: singularity when 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 + Gaussian potential: remove singularity 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

38 Rotating condensate: quartic potential Quartic 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

39 Rotating condensate: quartic potential Quartic-harmonic potential (α = 1.1) (1) top view angular momentum 2D cut (z=0)

40 Rotating condensate: quartic potential Quartic-harmonic potential (α = 1.1) (2) top view angular momentum 2D cut (z=0)

41 Rotating condensate: quartic potential Quartic-harmonic potential (α = 1.1) (3) top view angular momentum 2D cut (z=0)

42 Rotating condensate: quartic potential Quartic-harmonic potential (α = 1.1) (4) top view angular momentum 2D cut (z=0)

43 Rotating condensate: quartic potential Quartic 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

44 Rotating condensate: quartic potential Quartic-harmonic potential (α = 1.2) new transition: giant vortex (giant + array of vortices)

45 Rotating condensate: quartic potential Quartic 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

46 Rotating condensate: quartic potential Quartic+harmonic potential (1) 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.

47 Rotating condensate: quartic potential Quartic+harmonic potential (2)

48 Rotating condensate: quartic potential Quartic+harmonic potential (3) μ [nk] 10 0 Quantitative information condensate characteristics R, µ lattice characteristics, r v, b v R [μm] Ω/2π Ω/2π ρ z TF(r) ρ z v(r) = Ae 1 2 (r r 0) 2 /r v 2 ξ(r) = [8πa s ρ z TF(r)] 1/2

49 Rotating condensate: quartic potential Quartic+harmonic potential (4) I. Danaila, Phys. Rev. A, Good quantitative agreement D. E. Sheehy and L. Radzihovsky, Phys. Rev. A, 2004.

50 Rotating condensate: optical lattice 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

51 Introduction Numerical method 3D structure of vortices Rotating condensate: optical lattice Rotating condensate in an optical lattice Ω = 0.87 U = 0.1 U = 0.5 U = 0.7

52 Conclusion Conclusion Conclusion and future work Papers rich variety of vortex configurations remarkably good (qualitative/quantitative) agreement with experiments physical exploration of the results for the rotating condensate in optical lattices Phys. Rev A, 72, (2005) (with L.C. Crasovan, V.M. Perez-Garcia, D. Mihalache, L. Torner) Phys. Rev A, 70, (2004) (with A. Aftalion) Phys. Rev A, 69, (2004) Phys. Rev A, 68, (2003)