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.cn

Supported by NSFC, MOST, CAS

Collaborators

• An-Chun Ji

• Zhi-Bing Li (Zhongshan Univ)

• Ran Qi

• Qing Sun

• Xin-Cheng Xie (Peking Univ)

• Xiao-Lu Yu

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

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

No Josephson effect

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

L L L

di Edt R R R

di Edt Li

L Le

RiR Re

LS RS

L R

E0

Nambu-Goldstone modes

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.

E0

(c) a low-energy spin-wave excitation.

Ee>E0

2008 Nobel Physics Prize

Spontaneous symmetry breaking

Josephson effect

Direct Current Josephson effect: EL=ER=J.

Alternating Current Josephson effect:

EL E+V, ER E−V.

L L L R

R R R L

di E Jdtdi E Jdt

SQUIDs (superconducting quantum interference devices)

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)

Laser cooling and Bose-Einstein condensation

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.

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).

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.

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.

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)

Non-Abelin case

Abelin

Non-Abelin

operator

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.

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

Non-Abelian Josephson effect (S=2 BEC)

Parameters:

c0=(4a2+3a4)/7, c1=(a4-a2)/7, c2=(7a0-10a2+3a4)/7,

T21012 ,,,,

ff

aaa

2

2

21

222

1

2

0 52

1 cfccF

For S=2 Spinor BEC

Free energy

Wave function for BEC

Spin operator

Density operator

Parameters for some atoms

Three ground states of S=2 BEC

★ Ferromagnetic phase

★ Antiferromagnetic phase

★ Cyclic phase

Ferromagnetic phase

U(1)XU(1)Nambu-Goldstone modesAbelian Josephson effect

01C 02021 CC

00001ien 10 4CCn

Antiferromagnetic phase

U(1)XSO(3)Pseudo Nambu-Goldstone modesNon-Abelian Josephson effect

02 C 02021 CC

Four of them correspond to the symmetry 31 SOU

Cyclic phase

U(1)XSO(3)Pseudo Nambu-Goldstone modesNon-Abelian Josephson effect

01 C 02 C

20

20

2

202

iiii eeeen

0Cn 022 2

Cyclic

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

Abelian Josephson effect

Ferromagnetic Ferromagnetic

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

Non-Abelian Josephson effect

CyclicAnti-ferromagnetic

Non-Abelian Josephson effect

*aL

coupleaL

V

dt

di

*aR

coupleaR

V

dt

di

LRRLRLcouple JFFV

222

1

2

0 52

1 cfccF

Pseudo Goldstone modes for antiferromagnetic phase

m=0

m=±1

m=±2

5

22

2

20

ncJJ

220

1

52

2JJ

ccn

221

2

202

2JJ

ccn

221 5

2 JJc

cn

Pseudo Goldstone modes for cyclic phase

m=±1

m=0,±2

nJcJ 12

1 22

nJcJ 021

2,0 2

nJcJ 122

2,0 22

Experimental proposal

Experimental data:Rb-87, F=2AFM: c2<0, c1-c2/20>0Cyclic: c1>0, c2>0c1:0-10nK, c2:0-0.2nK, c0:150nKfluctuation time scale-10mspseudo Goldstone modes:1-10nk

Suggested steps for experiment:1. Initiate a density oscillation2. Detect time dependence of atom numbers in different spin component3. Measure density oscillation in each of spin components4. Non-Abelian Josephson effect

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

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: ●SandwichAnd others

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.

1 21,2

ˆ ˆ ˆ ˆ .Di

i

H H H c

, , , ,ˆ ˆˆ ˆˆ ˆi i j i j i i i j i jg b a a b

, , , ,1

ˆ ˆˆ ˆ ˆ ˆ ˆ2

aND Ai C i i i j i j i j i j

j

H b b a a

Ψi is the single mode annihilation operator of the photons in each cavity; ai;j and bi;j are fermion operators, which are associated with the lower and upper levels of each atom; K is the intercavity tunneling amplitude, ωC and ω A are the cavity and atom resonance frequencies, gi is the modulated local atom-field coupling rate.

Fig. Top Excitations of a polariton condensate. Fig. Bottom Chemical potential-current relation in polariton condensates.

δ=(N1-N2)/N,

Φ=θ1-θ2

2. BEC in optical lattices2. BEC in optical lattices

Quantum phase transition

Superfluid － Mott insulator

Insulator + disorder = Bose glassInsulator + weak disorder = Anderson glass

Berezinskii–Kosterlitz–Thouless transation

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

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

Strong correlated system

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

Z.W. Xie, W.M. Liu,Superfluid–Mott insulator transition of dipolar bosons in an optical lattice,Phys. Rev. A70, 045602 (2004)

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)

Honeycomb Lattice

i i

iiiiji

ji ccnnUcctH

,,

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)

Fig. 1 evolution of density of states Fig 2 double occupancy as function of interaction U for various temperature

Fig 3 Fermi surface for several interaction U=1t, 3t ,4.5t

A

B A

A B

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)

Fig. 1 The scaling function

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

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)

Fig.1 evolution of density of states

Fig 2 phase diagram of cold atoms in triangular optical lattice

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