1. Non-Abelian Josephson effect Wu-Ming Liu ( 刘伍明 ) (Institute of Physics, Chinese Academy of Sciences) ( 中国科学院物理所 ) Email: wmliu@iphy.ac.cnwmliu@iphy.ac.cn Supported by NSFC, MOST, CAS

2. Collaborators An-Chun Ji Zhi-Bing Li (Zhongshan Univ) Ran Qi Qing Sun Xin-Cheng Xie (Peking Univ) Xiao-Lu Yu

3. Outline 1. Cold atoms in double well 1.1. Josephson effect 1.2. Non-Abelian Josephson effect 1.3. Josephson effect for photons 2. Cold atoms in optical lattices

4. 1.1. Josephson effect ★ Two superconductors are brought into close together with a thin layer of insulator between them. ★ This interaction allows for tunneling of Cooper pairs between superconductors, across junction. B.D. Josephson, Phys. Lett. 1, 251 (1962) 1973 Nobel Physics Prize

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

6. E0E0 when ground state of a system does not share the full symmetry, spontaneous symmetry breaking occurs. A consequence of spontaneous symmetry breaking of a continuous symmetry like this one is that there are excitations whose energy goes to zero in the long wavelength limit. These are Nambu-Goldstone modes. (a)A ground state of the ferromagnet, with all spins aligned. (b) another ground state, with all spins rotated. E0E0 (c) a low-energy spin-wave excitation. E e >E 0 2008 Nobel Physics Prize

7. Spontaneous symmetry breaking

8. Josephson effect Direct Current Josephson effect: E L =E R =J. Alternating Current Josephson effect: E L  E+V, E R  E−V. SQUIDs (superconducting quantum interference devices)

9. Josephson effect Single mode: S=0, 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 Josephson effect corresponds to excitations of Nambu-Goldstone bosons. (Abelin) (Non-Abelin)

10. Laser cooling and Bose-Einstein condensation

11. B.P. Anderson, M.A. Kasevich, Science 282, 1686 (1998). Fig. Left Illustration of apparatus. Fig. Right (A) Absorption image of a BEC in a TOP trap. (B to E) Absorption images in optical lattice showing time development of pulse train; 3 ms (B), 5 ms (C), 7 ms (D), 10 ms (E). (F) The integrated absorption proble for (E), obtained by summing over horizontal cross-sections.

12. Fig. Left (A) Combined potential of optical lattice and magnetic trap in axial direction. (B) Absorption image of BEC released from combined trap. Fig. Right Frequency of atomic current in array of Josephson junctions as a function of interwell potential height. F.S. Cataliotti, S.Burger, C. Fort, P. Maddaloni, F. Minardi, A. Trombettoni, A. Smerzi, M. Inguscio, Science 293, 843 (2001).

13. S. Levy, E. Lahoud, I. Shomroni, J. Steinhauer, Nature 449, 579 (2007) Fig. Left Creating and imaging BEC Josephson junction. a, The application of high resolution potentials. b, In situ image of BEC Josephson junction. c, An enlargement of narrow tunnelling region of wavefunction. Fig. Right Time evolution of a BEC Josephson junction. a, Twelve in situ images of same BEC Josephson junction. b, The image integrated in z-direction. c, The phase evolution of BEC Josephson junction. Panels d and e show BEC in harmonic trap.

14. Fig. 2 Observation of a.c. and d.c. Josephson effects. a, The a.c. Josephson effect. The solid line shows ω/2π=Δμ/h. b, The decay of macroscopic quantum self- trapping. c, The Δμ–I relation and d.c. Josephson effect. d, Before imaging each point in c, is increased, to prevent plasma oscillations in potential.

15. 1.2. Non-Abelian Josephson effect A group for which the elements commute (i.e., AB=BA for all elements A and B) is called an Abelian group. Niels Henrik Abel (1802-1829) Abelian Josephson effect: Single mode: S=0, U(1) × U(1)  U(1) diagonal two Goldstone modes  one gapless mode (Goldstone mode)  another gapped mode (pseudo Goldstone mode)

16. Non-Abelin case

17. Abelin Non-Abelin operator

18. Many modes: S=1, U(1)XS(2); S=2, U(1)XSO(3) Pseudo Nambu-Goldstone modes Non-Abelian case: SO(N), U(1) × SO(N), … Multiple pseudo Goldstone modes Non-Abelian Josephson effect: the spontaneous breaking of non-Abelian gauged symmetries, or coexisting Abelian symmetries, if an interface arises.

19. R. Qi, X.L. Yu, Z.B. Li, W.M. Liu, Phys. Rev. Lett. 102, 185301 (2009) Non-Abelian Josephson effect (S=2 BEC)

20. Parameters: c0=(4a2+3a4)/7, c1=(a4-a2)/7, c2=(7a0-10a2+3a4)/7, For S=2 Spinor BEC Free energy Wave function for BEC Spin operator Density operator

21. Parameters for some atoms

22. Three ground states of S=2 BEC ★ Ferromagnetic phase ★ Antiferromagnetic phase ★ Cyclic phase

23. Ferromagnetic phase U(1)XU(1) Nambu-Goldstone modes Abelian Josephson effect

24. Antiferromagnetic phase U(1)XSO(3) Pseudo Nambu-Goldstone modes Non-Abelian Josephson effect Four of them correspond to the symmetry

25. Cyclic phase U(1)XSO(3) Pseudo Nambu-Goldstone modes Non-Abelian Josephson effect Cyclic

26. R. Qi, X.L. Yu, Z.B. Li, W.M. Liu, Phys. Rev. Lett. 102, 185301 (2009) Abelian Josephson effect Ferromagnetic

27. R. Qi, X.L. Yu, Z.B. Li, W.M. Liu, Phys. Rev. Lett. 102, 185301 (2009) Non-Abelian Josephson effect Cyclic Anti-ferromagnetic

28. Non-Abelian Josephson effect

29. Pseudo Goldstone modes for antiferromagnetic phase m=0 m=±1 m=±2

30. Pseudo Goldstone modes for cyclic phase m=±1 m=0,±2

31. Experimental proposal Experimental data:  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 Suggested steps for experiment: 1. Initiate a density oscillation 2. Detect time dependence of atom numbers in different spin component 3. Measure density oscillation in each of spin components 4. Non-Abelian Josephson effect

32. 1. Josephson effect corresponds to excitations of pseudo-Goldstone bosons. 2. Josephson effect allows for a generalization to non-abelian symmetries and the corresponding non-abelian Josephson effect. 3. Non-Abelian Jesophson effect: the spontaneous breaking of non-Abelian gauged symmetries, or coexisting Abelian symmetries, if an interface arises. 4. S=2 spinor BEC of Non-Abelian Jesophson case:  Anti-ferromagnetic system  Cyclic system Summary and Outlook

33. 5. The new non-Abelian systems: ● High density phases of QCD ● Two band superconductors, d-wave high Tc superconductors, p-wave heavy fermion ● A phase of liquid Helium-3 ● Nonlinear optics 6. The completed Non-Abelian system: ● Sandwich And others

34. A.C. Ji, Q. Sun, X.C. Xie, W.M. Liu, Phys. Rev. Lett. 102, 023602 (2009) 1.3. Josephson effect of photons FIG. 1 Experimental setup and control of coupling along resonator axis. (a) Two FFP cavities are linked. (b) The atoms are placed at a position x along the cavity axis and are loaded into optical lattice. (c) The loaded atoms show a strongly modulated coupling depending on local overlap between lattice and cavity mode.

35. Ψ i is the single mode annihilation operator of the photons in each cavity; a i;j and b i;j are fermion operators, which are associated with the lower and upper levels of each atom; K is the intercavity tunneling amplitude, ω C and ω A are the cavity and atom resonance frequencies, g i is the modulated local atom-field coupling rate.

36. Fig. Top Excitations of a polariton condensate. Fig. Bottom Chemical potential- current relation in polariton condensates. δ=(N 1 -N 2 )/N, Φ=θ 1 -θ 2

37. 2. BEC in optical lattices

38. Quantum phase transition Superfluid － Mott insulator Insulator + disorder = Bose glass Insulator + weak disorder = Anderson glass Berezinskii–Kosterlitz–Thouless transation

39. M. Greiner et al., Nature 415, 39 (2002)

40. Z. Hadzibabic et al., Nature 441, 1118 (2006)

41. Strong correlated system

42. J.J. Liang, J.Q. Liang, W.M. Liu, Quantum phase transition of condensed bosons in optical lattices, Phys. Rev. A68, 043605 (2003).

43. Z.W. Xie, W.M. Liu, Superfluid–Mott insulator transition of dipolar bosons in an optical lattice, Phys. Rev. A70, 045602 (200 4)

44. G.P. Zheng, J.Q. Liang, W.M. Liu, Phase diagram of two-species Bose-Einstein condensates in an optical lattice, Phys. Rev. A71, 053608 (2005)

45. Honeycomb Lattice W. Wu, Y. H. Chen, H. S. Tao, N. H. Tong, W.M. Liu, Interacting Dirac fermions on honeycomb lattice, Phys. Rev. B 82, 245102 (2010)

47. Fig. 1 evolution of density of statesFig 2 double occupancy as function of interaction U for various temperature Fig 3 Fermi surface for several interaction U=1t, 3t,4.5t

49. A B A AB B 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 honeycomb lattice, Phys. Rev. Lett. 102, 106401 (2009)

51. Fig. 1 The scaling function

52. 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).

53. Triangular optical lattice FIG. 1: Sketch of experimental setup to form triangular optical lattice Y. H. Chen, W. Wu, H. S. Tao, W.M. Liu, Cold atoms in a two-dimensional triangular optical lattice as an artificial frustrated system, Phys. Rev. A82, 043625 (2010)

55. Fig.1 evolution of density of states Fig 2 phase diagram of cold atoms in triangular optical lattice

56. Outlook

58. Thanks!