1. Non-Abelian Josephson effect and fractionalized vortices Wu-Ming Liu （刘伍明） （ Institute of Physics, CAS ） Email: wmliu@aphy.iphy.ac.cnwmliu@aphy.iphy.ac.cn Supported by NSFC, MOST, CAS

2. Collaborators Jiang-Ping Hu (Purdue Univ) An-Chun Ji Zhi-Bing Li (Zhongshan Univ) Ran Qi Qing Sun Xin-Cheng Xie (Oklahoma State Univ) Xiao-Lu Yu Yan-Yang Zhang Fei Zhou (British Columbia Univ)

3. 1. Introduction 2. Non-Abelian Josephson effect 3. Josephson effect of photons 4. Localization 5. Fractionalized vortex 6. Outlook Outline

4. 1.1. BEC of ideal gas 7Li6Li 1. Introduction

5. 1.2. BEC in dilute gas

6. 1.3. BEC near Feshbach resonance

7. 1.4. BEC in optical lattices

8. 1.5. Fermionic condensation

9. 1.6. Molecule condensation? J.G. Danzl et al. Science 321, 1062 (2008)

10. R. Qi, X.L. Yu, Z.B. Li, W.M. Liu, Non-Abelian Josephson effect between two F=2 spinor Bose-Einstein condensates in double optical traps, Phys. Rev. Lett. 102, 185301 (2009) 2. Non-Abelian Jesephson effect

12. Abelian case: U(1) × U(1)  U(1) diagonal two goldstone modes  one gapless mode (goldstone mode) and one gapped mode (pseudo goldstone mode) Non-Abelian case: SO(N), U(1) SO(N)… SO(N), U(1) × SO(N)… Multiple Multiple pseudo goldstone modes

13. No Josephson effect U(1)XU(1) Nambu-Goldstone modes

14. Josephson effect Single mode: U(1)XU(1) Nambu-Goldstone modes Many modes: S=1, U(1)XS(2); S=2, U(1)XSO(3) Pseudo Nambu-Goldstone modes

15. Ground states of S=2 boson Ferromagnetic phase Antiferromagnetic phase Cyclic phase

16. Ferromagnetic phase U(1)XU(1) Nambu-Goldstone modes

17. Antiferromagnetic phase U(1)XSO(3) Pseudo Nambu-Goldstone modes

18. Cyclic phase U(1)XSO(3) Pseudo Nambu-Goldstone modes

20. Antiferromagnetic phase m=0

22. m=±2

23. Fig. 2 The frequencies of pseudo Goldstone modes as a function of coupling parameter J in the case of antiferromagnetic phase.

24. Cyclic phase m=±1 m=0,±2

25. Fig. 3 The frequencies of pseudo Goldstone modes as a function of coupling parameter J in the case of cyclic phase.

26. Experimental parameter  Rb-87, F=2  AFM: c 2 0  Cyclic: c 1 >0, c 2 >0  c 1 :0-10nK, c 2 :0-0.2nK, c 0 :150nK  fluctuation time scale-10ms  pseudo Goldstone modes:1-10nk

27. Experimental signatures  Initiate a density oscillation  Detect time dependence of atom numbers in different spin component ◆ Measure density oscillation in each of spin components  Non-Abelian Josephson effect

28. A.C. Ji, Q. Sun, X. C. Xie, W. M. Liu, Josephson effects of photons in two weakly-inked microcavities, Phys. Rev. Lett. 102, 023602 (2009) 3. Jesephson effect of photons

29. Fig. 1 Experimental setup and control of coupling along resonator axis

31. Fig. 2 Excitations of a polariton condensate

32. Fig. 3 Chemical potential-current relation in polariton condensates

33. 4. Localization J. Billy et al., Nature 453, 891 (2008).

34. G. Roati et al., Nature 453, 895 (2008)

35. Y.Y. Zhang, J.P. Hu, B.A. Bernevig, X.R. Wang, X.C. Xie, W.M. Liu, Localization and Kosterlitz-Thouless transition in disordered graphene, Phys. Rev. Lett. 102, 106401 (2009)

36. A B AA B B

37. Fig. 1 The scaling function

38. Fig. 2 Typical configurations of local currents In (red arrows) and potential V n (color contour) on two sides of K-T type MIT with N=56X32 sites, \xi=1:73a, n I =1% and E F =0:1t. (a) W=1:1t (delocalized); (b) W=2:9t (localized).

39. A.C. Ji, W.M. Liu, J.L. Song, F. Zhou, Dynamical creation of fractionalized vortices and vortex lattices, Phys. Rev. Lett. 101, 010402 (2008) 5. Half vortex

42. Dynamical creation of fractionalized vortices and vortex lattices Fig.1 Density and spin density of an individual half vortex Fig. 2 Interaction potentials between two half vortex

44. Fig. 3 Creation of a half-quantum vortex. The bottom panel shows that a single half vortex is formed at t=600 ms after magnetic trap has been adiabatically switched off.

45. (a) Creation of a triangular integer vortex lattice (b) A square half vortex lattice formation at t=1600 ms

46. 6. Outlook

47. Thanks!