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

 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)

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

1.1. BEC of ideal gas
7Li 6Li

1.2. BEC in dilute gas

1.3. BEC near Feshbach resonance

1.4. BEC in optical lattices

1.5. Fermionic condensation

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

2. Non-Abelian Jesephson effect
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)

U(1) × U(1)

U(1) diagonal two goldstone modes
 one gapless mode (goldstone mode) and one gapped mode (pseudo goldstone mode)
SO(N), U(1) × SO(N)…
Multiple pseudo goldstone modes

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

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

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

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

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

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

Antiferromagnetic phase m=0

m= ± 2

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

Cyclic phase m= ± 1 m=0, ± 2

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


Rb-87, F=2

AFM: c
2
<0, c
1
-c
2
/20>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


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

3. Jesephson effect of photons
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)

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

Fig. 2 Excitations of a polariton condensate

Fig. 3 Chemical potential-current relation in polariton condensates

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

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

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)

A
B
A
B
B
A

Fig. 1 The scaling function

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).

5. Half vortex
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)

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

( i
 
) h
 
 t

1 h
2
2 m
 
V tr
  m
  
L z
 c
0
  c
2
(


1
 
0
  m 1
)

W

]


1
 c
2 0
2
_
  m 1
( i
 
) h
 
 t
0 [ h
2
2 m
 
V tr
   
L z
 c
0

 c
2
(

1
 

1
)]

0

2 c
  
2 1
_

1 0
V tr
   m


2
2
 
1
   x
2  
1
   y
2




2
 
250 Hz
W

   m
 m


2
2
 x 2  y 2


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.

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