Integrable model in Bose-Einstein condensates

1. Integrable model in Bose-Einstein condensates Wu-Ming Liu (Institute of Physics, Chinese Academy of Sciences） http:// www.iphy.ac.cn Email: wmliu@aphy.iphy.ac.cn Phone: 86-10-82649249

2. Collaborators • Prof. S.T. Chui (Delaware Univ.) • Prof. I. Kats (ILL, France) • Prof. J.Q. Liang (Shanxi Univ.) • Prof. B. A. Malomed (Tel Aviv Univ.) • Prof. Q. Niu (Texas Univ.) • Prof. Y.Z. Wang (SIOM, CAS) • Prof. B. Wu (IOP, CAS) • Prof. W.P. Zhang (East China Normal Univ.) • Prof. W.M. Zheng (ITP, CAS)

3. Outline 1. Introduction 2. BEC tunneling－instanton 3. BEC interference－long time solution 4. BEC near Feshbach resonance– soliton 5. BEC in optical lattice – discrete soliton 6. Two component BEC－soliton inelastic collision 7. Spinor BEC－soliton 8. Conclusion

4. 1. Introduction 6Li 7Li

6. C. E. Wieman and E. A. Cornell, Science 269, 198 (1995).

7. 40 Lab. • Elements: Li, Na, K, H, Rb, He, Fermi gases • Y.Z. Wang, BEC in China, March 2002, Shanghai, China

8. 2. BEC tunneling- instanton W.M. Liu, W.B. Fan, W.M. Zheng, J.Q. Liang, S.T. Chui, Quantum tunneling of Bose-Einstein condensates in optical lattices under gravity, Phys. Rev. Lett. 88, 170408 (2002).

9. Fig. 1. The effective optical-plus-gravitational potential U/ER for parameters used in our experiment (ER = 2k2/2m is the photon recoil energy with k=2 / ). The horizontal oscillating curves illustrate de Broglie waves from the tunnel output of each well. In region A, the relative phases of the waves interfere constructively to form a pulse. Heavy lines illustrate the energies of the lowest bound states of harmonic oscillator potentials that match the shapes of the actual potentials near each local energy minimum. B.P. Anderson et al., Science 282, 1686 (1998).

10. Figure 1. (A) Combined potential of the optical lattice and the magnetic trap in the axial direction. The curvature of the magnetic potential is exaggerated by a factor of 100for clarity. (B) Absorption image of the BEC released from the combined trap. The expansion time was 26.5ms and the optical potential height was 5ER. .. F.S. Cataliotti, Science 293, 843 (2001).

11. Parameters: Wells: 30 or 200 Atoms number: 10³ /well Density: n₀=10¹³ cm⁻³ Hamiltonian Landau-Zener tunneling Wannier-Stark tunneling

12. Potential energy and Bloch bands Landau-Zener tunneling • Barrier between lattices is low • Localized level between lattices is coupling • Miniband • Adiabatic approximation • Tunneling between delocalized states in different Bloch bands

13. Tilted bands and WS ladders Wannier-Stark tunneling • An external field • Wavefunction of miniband is localization • Miniband is divided into discrete level • Wannier-Stark ladder • Tunneling between localized states in different individual wells—Wannier-Stark localized states

14. Bloch bands and WS Ladder WS Ladder

15. Resonances condition for discrete spectrum mean energy of \alpha band Actual energy spectrum for discrete spectrum Wannier-Stark energy spectrum I.W. Herbst et al., Commun. Math. Phys. 80, 23(1981) J. Agler et al., ibid 100, 161 (1985) J.-M. Combes et al., ibid 140, 291(1991)

16. Potential energy and energy bands

17. No crossing--condition Decay rate E: complex energy Transition amplitude

18. Transition amplitude Periodic instanton represents pseudo-condensed atom configuration responsible for tunneling under barrier at energy E

19. Euler-Lagrange equation Potential V(z)

20. Periodic instanton solution – solutions of classical Euler-Lagrange equations in Euclidean space-time with finite energy denote three roots of equation V(z)=E

21. All instanton contributions

22. Decay rate of metastable state is energy dependent frequency is frequency of small oscillations

23. Harmonic approximation Decay rate of nth low excited state Metastable ground state

24. Tunneling rate of Landau-Zener regime

25. Atoms:Yale experimental parameters Theory

26. Atoms:INFM (Istituto Nazionale di Fisica della Materia, Italy) Theory

27. At high temperature:Arrhenius law Temperature dependence

28. At intermediate temperature:Thermally assisted tunneling Crossover temperature At low temperature:Pure quantum tunneling

30. Experimental prediction Population Measure tunneling from lowest metastable state 1.Turn on a potential which has only one state in each well. 2. Accelerate potential in such a way that only band of states from these levels are swept along with potential, leaving all higher states behind (so they can be neglected). 3. Increase amplitude of potential, so that different wells become isolated from each other. 4. Tilt potential (by acceleration) to achieve Wannier-Stark regime described by present theory. 5. Observe how many atoms survive in time t.

31. Measure decays from excited states and at higher temperature 1. Starting with a thermal distribution of free atom states, turn on potential to some amplitude, so that eventually there are n bands lying in wells. 2. Accelerate potential so that n bands are taken along with wells, leaving atoms in higher bands behind. The acceleration must be such that occupation number of each of n bands is not changed during this process. 3. Same as (3) above. 4. Same as (4) above. 5. Same as (5) above.

32. 3. BEC interference–long time solution W.M. Liu, B. Wu, Q. Niu, Nonlinear effects in interference of Bose-Einstein condensates, Phys. Rev. Lett. 84, 2294 (2000).

33. W. Ketterle, Science 275, 637 (1997).

35. Experimental parameters: • Separation of two BEC ~ 40 μm • Fringe spacing ~ 15 μm • Expanding time ~ 40 ms • Demonstration: 1. laser-like 2. coherent 3. long-range correlation • Implication: 1. atomic laser 2. Josephson effect

36. Many-body Hamiltonian The mean field theory Gross-Pitaevskii equation

37. Parameters: • x is measured in unit of x0= 1μm • t in unit of mx0/ h, t= 120 • φ in unit of square root of n0 • G= 4πn0ax02= 5-10

38. Gross-Pitaevskii equation Long time solution

39. Theoretical explanation Fringe position Central fringe

40. Experimental prediction:1. Energy level 2. Many wave packets Ratio of level width to level spacing

42. 4. BEC near Feshbach resonance-soliton S. Inouye et al., Nature 392, 151 (1998).

43. Z. X. Liang, Z. D. Zhang, W. M. Liu, Dynamics of a bright soliton in Bose-Einstein condensates with time-dependent atomic scattering length in an expulsive parabolic potential, Phys. Rev. Lett. 74, 050402 (2005).

47. SupernovaS.L. Cornish et al., Phys. Rev. Lett. 85, 1795 (2000).

48. 5. BEC in optical lattice–discrete soliton L. Khaykovich et al., Science 296, 1290 (2002).

49. K.E. Strecker et al., Nature 417, 150 (2002).

50. K.E. Strecker et al., Nature 417, 150 (2002).