Numerical Method for Computing Ground States of Spin-1 Bose-Einstein Condensates

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 Collaborators: Weizhu Bao (National University of Singapore) 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 ETH (02’,87Rb) • Hyperfine spin F = I + S • 2F+1 hyperfine components: mF = -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

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

5. Gross-Pitaevskii Equation • GPE describes BEC at T <<Tc • 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 • Boundary eigenvalue method • Runge-Kutta space-marching(Edward & Burnett, PRA, 95’) (Adhikari, Phys. Lett. A, 00’) • 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’) • 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 • Replace in the time-dependent GPE (imaginary time method) and form gradient flow with discrete normalization (GFDN) in each time interval • BEFD – implicit, unconditionally stable, energy diminishing, second order accuracy in space • 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=(U0, U1, …, UM)T; U satisfying U0 = UM = 0 • The sine transform coefficients

11. Backward Euler Sine-pseudospectral Method (BESP) • A stabilization parameter a 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 • a 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: mF = -1, 0 ,1 • Coupled Gross-Pitaevskii equations (CGPE) • βn -- spin-independent mean-field interaction βs -- 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 • Gradient flow • 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 mF = +1 • Explicit • Computationally efficient

23. 87Rb in 1D harmonic potential • Repulsive and ferromagnetic interaction (βn >0 , βs < 0) • Initial condition

24. 87Rb in 1D harmonic potential • Repulsive and ferromagnetic interaction (βn >0 , βs < 0)

25. 23Na in 1D harmonic potential • Repulsive and antiferromagnetic interaction (βn >0 , βs > 0) • Initial condition

26. 87Rb in 3D optical lattice

27. 23Na in 3D optical lattice

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

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

30. Spin-1 BEC in 1D Harmonic Potential 23Na (N=104) 87Rb (N=104) • E, µ are independent of M • λ = 0 for all M (You et. al., PRA, 02’)

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