Mathematical Analysis and Numerical Simulation for Bose-Einstein Condensation

1. Mathematical Analysis and Numerical Simulation for Bose-Einstein Condensation Weizhu Bao Department of Mathematics & Center of Computational Science and Engineering National University of Singapore Email: bao@math.nus.edu.sg URL: http://www.math.nus.edu.sg/~bao

2. Collaborators • External • P.A. Markowich, Institute of Mathematics, University of Vienna, Austria • D. Jaksch, Department of Physics, Oxford University, UK • Q. Du, Department of Mathematics, Penn State University, USA • J. Shen, Department of Mathematics, Purdue University, USA • L. Pareschi, Department of Mathematics, University of Ferarra, Italy • I-Liang Chern, Department of Mathematics, National Taiwan University, Taiwan • C. Schmeiser & R.M. Weishaeupl, University of Vienna, Austria • W. Tang & L. Fu, Beijing Institute of Appl. Phys. & Comput. Math., China • Internal • Yanzhi Zhang, Hanquan Wang, Fong Ying Lim, Ming Huang Chai • Yunyi Ge, Fangfang Sun, etc.

3. Outline • Part I: Predication & Mathematical modeling • Theoretical predication • Physical experiments and results • Applications • Gross-Pitaevskii equation • Part II: Analysis & Computation for Ground states • Existence & uniqueness • Energy asymptotics & asymptotic approximation • Numerical methods • Numerical results

4. Outline • Part III: Analysis & Computation for Dynamics in BEC • Dynamical laws • Numerical methods • Vortex stability & interaction • Part IV: Rotating BEC & multi-component BEC • BEC in a rotational frame • Two-component BEC • Spinor BEC • BEC at finite temperature • Conclusions & Future challenges

5. Part I Predication & Mathematical modeling

6. Theoretical predication • Particles be divided into two big classes • Bosons: photons, phonons, etc • Integer spin • Like in same state & many can occupy one obit • Sociable & gregarious • Fermions: electrons, neutrons, protons etc • Half-integer spin & each occupies a single obit • Loners due to Pauli exclusion principle

7. Theoretical predication • For atoms, e.g. bosons • Get colder: • Behave more like waves & less like particles • Very cold: • Overlap with their neighbors • Extremely cold: • Most atoms behavior in the same way, i.e gregarious • quantum mechanical ground state, • `super-atom’ & new matter of wave & fifth state

8. Theoretical predication • S.N. Bose: Z. Phys. 26 (1924) • Study black body radiation: object very hot • Two photons be counted up as either identical or different • Bose statistics or Bose-Einstein statistics • A. Einstein: Sitz. Ber. Kgl. Preuss. Adad. Wiss. 22 (1924) • Apply the rules to atoms in cold temperatures • Obtain Bose-Einstein distribution in a gas

11. Experimental results • JILA (95’, Rb, 5,000): Science 269 (1995) • Anderson et al.,Science, 269 (1995), • 198: JILA Group; Rb • Davis et al.,Phys. Rev. Lett., 75 (1995), • 3969: MIT Group; Rb • Bradly et al., Phys. Rev. Lett., 75 (1995), • 1687, Rice Group; Li

12. Experimental results • Experimental implementation • JILA (95’): First experimental realization of BEC in a gas • NIST (98’): Improved experiments • MIT, ENS, Rice, • ETH, Oxford, • Peking U., … • 2001 Nobel prize in physics: • C. Wiemann: U. Colorado • E. Cornell:NIST • W. Ketterle: MIT ETH (02’, Rb, 300,000)

13. Experimental difficulties • Low temperatures absolutely zero (nK) • Low density in a gas

14. Experimental techniques • Laser cooling • Magnetic trapping • Evaporative Cooling ($100k—300k)

15. Possible applications • Quantized vortex for studying superfluidity • Test quantum mechanics theory • Bright atom laser: multi-component • Quantum computing • Atom tunneling in optical lattice trapping, ….. Square Vortex lattices in spinor BECs Vortex latticedynamics Giant vortices

16. Mathematical modeling • N-body problem • (3N+1)-dim linear Schroedinger equation • Mean field theory: • Gross-Pitaevskii equation (GPE): • (3+1)-dim nonlinear Schroedinger equation (NLSE) • Quantum kinetic theory • High temperature: QBME (3+3+1)-dim • Around critical temperature: QBME+GPE • Below critical temperature: GPE

17. Gross-Pitaevskii equation (GPE) • Physical assumptions • At zero temperature • N atoms at the same hyperfine species (Hartree ansatz) • The density of the trapped gas is small • Interatomic interaction is two-body elastic and in Fermi form

18. Second Quantization model • The second quantized Hamiltonian: • A gas of bosons are condensed into the same single-particle state • Interacting by binary collisions • Contained by an external trapping potential

19. Second quantization model • Crucial Bose commutation rules: • Atomic interactions are low-energy two-body s-wave collisions, i.e. essentially elastic & hard-sphere collisions • The second quantized Hamiltonian

20. Second quantization model • The Heisenberg equation for motion: • For a single-particle state with macroscopic occupation • Plugging, taking only the leading order term • neglecting the fluctuation terms (i.e., thermal and quantum depletion of the condensate) • Valid only when the condensate is weakly-interacting & low tempertures

21. Gross-Pitaevskii equation • The Schrodinger equation (Gross, Nuovo. Cimento., 61; Pitaevskii, JETP,61 ) • The Hamiltonian: • The interaction potential is taken as in Fermi form

22. Gross-Pitaevskii equation • The 3d Gross-Pitaevskii equation ( ) • V is a harmonic trap potential • Normalization condition

23. Gross-Pitaevskii equation • Scaling (w.l.o.g.) • Dimensionless variables • Dimensionless Gross-Pitaevskii equation • With

24. Gross-Pitaevskii equation • Typical parameters ( ) • Used in JILA • Used in MIT

25. Gross-Pitaevskii equation • Reduction to 2d (disk-shaped condensation) • Experimental setup • Assumption: No excitations along z-axis due to large energy • 2d Gross-Pitaevskii equation ( )

26. Numerical Verification

27. Numerical Results Bao, Y. Ge, P. Markowich & R. Weishaupl, 06’

28. Gross-Pitaevskii equation • General form of GPE ( ) with Normalization condition

29. Gross-Pitaevskii equation • Two kinds of interaction • Repulsive (defocusing) interaction • Attractive (focusing) interaction • Four typical interaction regimes: • Linear regime: one atom in the condensation • Weakly interacting condensation

30. Gross-Pitaevskii equation • Strongly repulsive interacting condensation • Strongly attractive interaction in 1D • Other potentials • Box potential • Double-well potential • Optical lattice potential • On a ring or torus

31. Gross-Pitaevskii equation • Conserved quantities • Normalization of the wave function • Energy • Chemical potential

32. Semiclassical scaling • When , re-scaling With • Leading asymptotics (Bao & Y. Zhang, Math. Mod. Meth. Appl. Sci., 05’)

33. Comparison of two scaling

34. Quantum Hydrodynamics • Set • Geometrical Optics:(Transport + Hamilton-Jacobi) • Quantum Hydrodynamics (QHD): (Euler +3rd dispersion)

35. Part II Analysis & Computation for Ground states

36. Stationary states • Stationary solutions of GPE • Nonlinear eigenvalue problem with a constraint • Relation between eigenvalue and eigenfunction

37. Stationary states • Equivalent statements: • Critical points of over the unit sphere • Eigenfunctions of the nonlinear eigenvalue problem • Steady states of the normalized gradient flow:(Bao & Q. Du, SIAM J. Sci. Compu., 03’) • Minimizer/saddle points over the unit sphere: • For linear case (Bao & Y. Zhang, Math. Mod. Meth. Appl. Sci., 05’) • Global minimizer vs saddle points • For nonlinear case • Global minimizer, local minimizer vs saddle points

38. Ground state • Ground state: • Existence and uniqueness of positive solution : • Lieb et. al., Phys. Rev. A, 00’ • Uniqueness up to a unit factor • Boundary layer width & matched asymptotic expansion • Bao, F. Lim & Y. Zhang, Bull. Institute of Math., Acad. Scinica , 07’

39. Excited & central vortex states • Excited states: • Central vortex states: • Central vortex line states in 3D: • Open question: (Bao & W. Tang, JCP, 03’; Bao, F. Lim & Y. Zhang, TTSP, 06’)

40. Approximate ground states • Three interacting regimes • No interaction, i.e. linear case • Weakly interacting regime • Strongly repulsive interacting regime • Three different potential • Box potential • Harmonic oscillator potential • BEC on a ring or torus

41. Energies revisited • Total energy: • Kinetic energy: • Potential energy: • Interaction energy: • Chemical potential

42. Box Potential in 1D • The potential: • The nonlinear eigenvalue problem • Case I: no interaction, i.e. • A complete set of orthonormal eigenfunctions

43. Box Potential in 1D • Ground state & its energy: • j-th-excited state & its energy • Case II: weakly interacting regime, i.e. • Ground state & its energy: • j-th-excited state & its energy

44. Box Potential in 1D • Case III: Strongly interacting regime, i.e. • Thomas-Fermi approximation, i.e. drop the diffusion term • Boundary condition is NOT satisfied, i.e. • Boundary layer near the boundary

45. Box Potential in 1D • Matched asymptotic approximation • Consider near x=0, rescale • We get • The inner solution • Matched asymptotic approximation for ground state

46. Box Potential in 1D • Approximate energy • Asymptotic ratios: • Width of the boundary layer:

47. Numerical observations:

48. Box Potential in 1D • Matched asymptotic approximation for excited states • Approximate chemical potential & energy

49. Fifth excited states

50. Energy & Chemical potential