1. Numerical Method for Computing Ground States of Spin-1 Bose-Einstein Condensates Fong Yin Lim Department of Mathematics and Center for Computational Science & Engineering National University of Singapore Email: fongyin.lim@nus.edu.sg fongyin.lim@nus.edu.sg Collaborators: Weizhu Bao (National University of Singapore) I-Liang Chern (National Taiwan University) I-Liang Chern (National Taiwan University)

2. Outline Introduction The Gross-Pitaevskii equation Numerical method for single component BEC ground state Gradient flow with discrete normalization Backward Euler sine-pseudospectral method Spin-1 BEC and the coupled Gross-Pitaevskii equations Numerical method for spin-1 condensate ground state Conclusions

3. Single Component BEC Hyperfine spin F = I + S 2F+1 hyperfine components: m F = -F, -F+1, …., F-1, F Experience different potentials under external magnetic field Earlier BEC experiments: Cooling of magnetically trapped atomic vapor to nanokelvins temperature  single component BEC ETH (02’, 87 Rb)

4. Spinor BEC Optical trap provides equal confinement for all hyperfine components m F -independent multicomponent BEC Hamburg (03’, 87 Rb, F=1)Hamburg (03’, 87 Rb, F=2)

5. Gross-Pitaevskii Equation GPE describes BEC at T <<T c Non-condensate fraction can be neglected Interaction between particles is treated by mean field approximation where interparticle mean-field interaction Number density

6. Dimensionless GPE Non-dimensionalization of GPE Conservation of total number of particles Conservation of energy Time-independent GPE

7. BEC Ground State BEC Ground State Boundary eigenvalue method Runge-Kutta space-marching (Edward & Burnett, PRA, 95’) (Adhikari, Phys. Lett. A, 00’) (Adhikari, Phys. Lett. A, 00’) Variational method Variational method Direct minimization of energy functional with FEM approach (Bao & Tang, JCP, 02’) Nonlinear algebraic eigenvalue problem approach Gauss-Seidel type iteration (Chang et. al., JCP, 05’) Continuation method (Chang et. al., JCP, 05’) (Chien et. al., SIAM J. Sci. Comput., 07’) (Chien et. al., SIAM J. Sci. Comput., 07’) Imaginary time method Explicit imaginary time algorithm via Visscher scheme (Chiofalo et. al., PRE, 00’) Backward Euler finite difference (BEFD) and time-splitting sine-pseudospectral method (TSSP) (Bao & Du, SIAM J. Sci. Comput., 03’) -

8. Imaginary Time Method Imaginary Time Method Replace in the time-dependent GPE (imaginary time method) and form gradient flow with discrete normalization (GFDN) in each time interval BEFD – second order accuracy in space BEFD – implicit, unconditionally stable, energy diminishing, second order accuracy in space TSSP – explicit, conditionally stable, TSSP – explicit, conditionally stable, spectral accuracy in space -

9. Discretization Scheme The problem is truncated into bounded domain with zero boundary conditions Backward Euler sine-pseudospectral method (BESP) ( Bao, Chern & Lim, JCP, 06’) Backward Euler scheme is applied, except the non- linear interaction term Sine-pseudospectral method for discretization in space Consider 1D gradient flow in

10. Backward Euler Sine-pseudospectral Method (BESP) At every time step, a linear system is solved iteratively Differential operator of the second order spatial derivative of vector U=(U 0, U 1, …, U M ) T ; U satisfying U 0 = U M = 0 The sine transform coefficients

11. Backward Euler Sine-pseudospectral Method (BESP) A stabilization parameter  is introduced to ensure the convergence of the numerical scheme Discretized gradient flow in position space Discretized gradient flow in phase space

12. Backward Euler Sine-pseudospectral Method (BESP) Stabilization parameter  guarantees the convergence of the iterative method and gives the optimal convergence rate BESP is spectrally accurate in space and is unconditionally stable, thereby allows larger mesh size and larger time-step to be used

13. BEC in 1D Potentials

14. BEC in 1D Optical Lattice Comparison of spatial accuracy of BESP and backward-Euler finite difference (BEFD)

15. BEC in 3D Optical Lattice Multiscale structures due to the oscillatory nature of trapping potential High spatial accuracy is required, especially for 3D problems

16. Spin-1 BEC 3 hyperfine components: m F = -1, 0,1 Coupled Gross-Pitaevskii equations (CGPE) β n -- spin-independent mean-field interaction β s -- spin-exchange interaction Number density

17. Spin-1 BEC Conservation of total number of particles Conservation of energy Conservation of total magnetization

18. Spin-1 BEC Time-independent CGPE Chemical potentials Lagrange multipliers, µ and λ, are introduced to the free energy to satisfy the constraints N and M

19. Spin-1 BEC Ground State Imaginary time propagation of CGPE with initial complex Gaussian profiles with constant speed Continuous normalized gradient flow (CNGF) -- -- -- N and M conserved, energy diminishing -- Involve and implicitly (Zhang, Yi & You, PRA, 02’) (Bao & Wang, SIAM J. Numer. Anal., 07’)

20. Normalization Conditions Numerical approach with GFDN by introducing third normalization condition Time-splitting scheme to CNGF in 1. Gradient flow 2. Normalization/ Projection

21. Normalization Conditions Normalization step Third normalization condition Normalization constants

22. Discretization Scheme Backward-forward Euler sine-pseudospectral method (BFSP) Backward Euler scheme for the Laplacian; forward Euler scheme for other terms Sine-pseudospectral method for discretization in space 1D gradient flow for m F = +1 Explicit Computationally efficient

23. 87 Rb in 1D harmonic potential Repulsive and ferromagnetic interaction (β n >0, β s 0, β s < 0) Initial condition

24. 87 Rb in 1D harmonic potential Repulsive and ferromagnetic interaction (β n >0, β s 0, β s < 0)

25. 23 Na in 1D harmonic potential Repulsive and antiferromagnetic interaction (β n >0, β s > 0) Initial condition

26. 87 Rb in 3D optical lattice

27. 23 Na in 3D optical lattice

28. Relative Populations Relative populations of each component Same diagrams are obtained for all kind of trapping potential in the absence of magnetic field β s < 0 ( 87 Rb) β s < 0 ( 87 Rb) β s > 0 ( 23 Na) β s > 0 ( 23 Na)

29. Chemical Potentials Weighted error Minimize e with respect to µ and λ

30. Spin-1 BEC in 1D Harmonic Potential 87 Rb (N=10 4 ) 87 Rb (N=10 4 ) E, µ are independent of M E, µ are independent of M λ = 0 for all M λ = 0 for all M (You et. al., PRA, 02’) (You et. al., PRA, 02’) 23 Na (N=10 4 ) 23 Na (N=10 4 )

31. Spin-1 BEC in Magnetic Field Spin-1 BEC subject to external magnetic field Straightforward to include magnetic field in the numerical scheme Stability??

32. Conclusions Spectrally accurate and unconditionally stable method for single component BEC ground state computation Extension of normalized gradient flow and sine- pseudospectal method to spin-1 condensate Introduction of the third normalization condition in addition to the existing conservation of N and M Future works: -- Extension of the method to spinor condensates with higher spin degrees of freedom --Finite temperature effect