1. 1 Manipulation of Artificial Gauge Fields for Ultra-cold Atoms for Ultra-cold Atoms Shi-Liang Zhu ( Shi-Liang Zhu ( 朱 诗 亮 )slzhu@scnu.edu.cn Laboratory of Quantum Information Technology and School of Physics South China Normal University, Guangzhou, China Collaborators: L.M.Duan (Michigan Univ); Z.D.Wang （ Univ.Hong Kong ） B.G.Wang, L.Sheng, D.Y.Xiong (Nanjing Univ.) C.Wu(UC) S.C.Zhang(Stanford Univ.) Students: L.B.Shao （ Nanjing Univ ） ; D.W.Zhang (SCNU) H.Fu (Michigan Univ.) “Condensed matter physics of cold atoms” (Sep 21-Nov.6, 2009) KITP (Beijing, Sep.24,2009)

2. 2 Outlines 1 Background Quantum Simulation with ultra-cold atoms 2 Geometric phase and artificial gauge fields in ultra-cold atoms 3Applications: Atomic SHE, Atomic QHE, Dirac-like equation

3. 3 1 Background: Quantum Simulation with Cold atoms Simulation of a quantum system with a classical computer is very hard 1 Simulate a quantum system by a quantum computer 2 Simulate a quantum system by a quantum simulator Quantum simulator with ultrocold atoms

4. 4 Atoms at optical lattices You can control almost all aspects of the periodic structure and the interactions between the atoms M. Greiner et al., Nature (2002) D.Jaksch et al (PRL 1998) Bose-Hubbard Hamiltonian Time of flight measurement

5. 5 Simulation of Condensed Matter Physics with ultrocold atoms One of the key topics in condensed matter physics is to study the response of electrons to an electromagnetic field V B e I E

6. 6 Quantum Hall effects B + + + + + - - - - - - - - 1980 1982 J However, atoms are electrically neutral and then a real electromagnetic field does not work Atomic QHE ?

7. 7 Three typical methods: Effective magnetic fields G. Juzeliunas PRL (2004) S.L.Zhu et al., PRL (2006) 1)Rotating N.K.Wilkin et al PRL (1998) 2) Optical Lattice set-up D.Jaksch and P.Zoller NJP(2003) 3) Light-induced geometric phase Laser

8. 8 How to realize the QHE with cold atoms Main Challenges (a)Realization: Strong uniform magnetic fields; (b) Detection: Transport measurement is not workable Atomic QHE Our work: Realization: Haldane’s QHE without Landau level Detection: establish a relation between Chern number and density profile L.B.Shao et al., Phys.Rev.Lett. (2008)

9. 9 2 Geometric phase and Artificial gauge fields in ultra-cold atoms

10. 10 Introduction: Geometric phase (Berry phase) Transport a closed path in parameter space: The initial state is one of non-degenerate energy eigenstates The final state differs from the initial one only by a phase factor Where Dynamic phase Berry phase M. V. Berry (1984) Geometric Phase---Depends on the geometry of the trajectory in parameter space, not on rate of passage --Non-integrable phase Berry considered a Hamiltonian which depends on a set of parameters

11. 11 Geometric phase: adiabatic Berry phase Berry curvature: Many applications in physics: it turns out to provide the fundamental structures that govern the physical universe (2) Nonintegrable phase factor---Related to Gauge potential and gauge field an artificial electromagnetic field for a neutral atom i) C.N.Yang, PRL (1974) ii) Concept of Nonintegrable phase factors and global formulation of gauge fields T.T.Wu and C. N. Yang, PRD (1975) (1) Geometric Quantum computation [a recent review paper: E.Sjoqvist, Physics 1, 35 (2008)]

12. 12 Geometric phase and Artificial gauge fields in ultra-cold atoms Wilczek and Zee, PRL 52, 211 (1984) C.P.Sun and M.L.Ge,PRD (1990) Ruseckas et al., PRL 95, 010404 (2005) N internal states The wave function: One diagonalizes to get a set of N dressed states with eigenvalues The full quantum state where

13. 13 obeys the Schrödinger equation where Abelian gauge potential U(1) : if the off-diagonal terms can be neglected Non-Abelian gauge potential : at least some off-diagonal terms can not be neglected

14. 14 Example: Gauge field for a Lambda-level configuration Three-level  type Atoms S. L. Zhu et al, Phys. Rev. Lett. 97,240401 (2006) Wilczek and Zee, PRL 52, 211 (1984) C.P.Sun and M.L.Ge,PRD (1990) Ruseckas et al., PRL 95, 010404 (2005)

15. 15 Gauge field induced by laser-atom interactions The vector potential The scalar potential Where  obey the Schrodinger eq. with the effective Hamiltonian given by F.Wilczek and A.Zee PRL 52,2111(1984)

16. 16 3 Application of the artificial gauge fields

17. 17 Application I: Spin Hall Effects B B x yz B + + + + + + + _ _ _ _ Charge Hall Effect Spin Hall Effect S. L. Zhu et al, Phys. Rev. Lett. 97,240401 (2006)

18. 18 SHE: Spin-dependent trajectories S. L. Zhu et al, Phys. Rev. Lett. 97,240401 (2006) Electronic field

19. 19 Experiments at NIST Y.J.Lin,R.L.Compton,A.R.Perry. W.D.Philips, J.V.Porto,and I.B.Spielman, PRL 102, 130401 (2009) Energy-momentum dispersion curves A group at NIST The experimental data are in agreement with the calculations predicted by a single-particle picture based on geometric phase.

20. 20 Application II: A periodic magnetic field can be used to realize the Haldane’s QHE without Landau levels A periodic magnetic field

21. 21 Application II: A periodic magnetic field can be used to realize the Haldane’s QHE without Landau levels F.D.M.Haldane PRL(1988) (nonzero Chern number) L.B.Shao,S.L.Zhu*,L.Sheng,D.Y.Xing, and Z.D.Wang, PRL 101, 246810 (2008)

22. 22 (1) The different site-energies of sublattices A and B can be controlled by the phase of laser beam  Realization of Haldane’s QHE (Different on-site energies)

23. 23 Realization of Haldane’s QHE

24. 24 With the Fourier transformation Spinor The Chern number: D.H.Lee,G.M.Zhang,T.Xiang PRL(2007) Haldane PRL

25. 25 Detection ? B=0 Streda JPA R. O. Umucalilar et al PRL (2008)

26. 26

27. 27 Application III: relativistic Dirac-Like equation S.L.Zhu,D.W.Zhang and Z.D.Wang, PRL 102, 210403 (2009).

28. 28 Realization of relativistic Dirac equation with cold atoms In the k space, x G. Juzeliunas et al, PRA (2008); S.L.Zhu,D.W.Zhang and Z.D.Wang, PRL 102, 210403 (2009).

29. 29 If and in one-dimensional case The effective mass is or Tripod-level configuration of For Rubidium 87 x

30. 30 Relativistic behaviors (1) Zitterbewegung (ZB) effect (2) Klein tunneling (Klein 1929) E V E<V Totally reflection (Classic) Quantum tunneling (non-relativistic QM) Klein tunneling (relativistic QM) T Transmission coefficient T Vaishnav and Clark, PRL(2008).

31. 31 Anderson localization in disordered 1D chains Scaling theory monotonic nonsingular function All states are localized for arbitrary weak random disorders For non-relativistic particles:

32. 32 a localized state for a massive particle However, for a massless particle break down the famous conclusion that the particles are always localized for any weak disorder in 1D disordered systems. S.L.Zhu,D.W.Zhang and Z.D.Wang, PRL 102, 210403 (2009). for a massless particle, all states are delocalized

33. 33 The chiral symmetry The chiral operator The chirality is conserved for a massless particles. Note that

34. 34 must be zero for a massless particle

35. 35 Detection of Anderson Localization Nonrelativistic case: non-interacting Bose–Einstein condensate Billy et al., Nature 453, 891 (2008) BEC of Rubidium 87 Relativistic case: three more laser beams G. Roati et al., Nature (London) 453, 895 (2008).

36. 36 Conclusions 1. Create artificial gauge fields for ultra-cold atoms 2. reviewed several applications, such as atomic QHE, atomic SHE and relativistic Dirac-like equation 1 Spin Hall effects for cold atoms in a light-induced gauge potential S. L. Zhu, H. Fu, C. J. Wu, S. C. Zhang, and L. M. Duan, Phys. Rev. Lett 97,240401 (2006) 2 Simulation and Detection of Dirac fermions with cold atoms in an optical lattice S. L. Zhu, B. G. Wang, and L. M. Duan, Phys. Rev. Lett. 98, 260402 (2007) 3 Realizing and detecting the quantum Hall effect without Landau levels by using ultracold atoms L.B.Shao, S.L.Zhu*,L.Sheng,D.Y.Xing, and Z.D.Wang, Phys. Rev. Lett 101, 246810 (2008) 4 Delocalization of relativistic Dirac particles in disordered one-dimensional systems and its implementation with cold atoms S.L.Zhu,D.W.Zhang and Z.D.Wang, Phys. Rev. Lett 102, 210403 (2009). References:

37. 37 Thank you for your attention 谢 谢 ！

38. 38 Typical examples …… Three-level Λ type Four-level tripod type N+1-level atoms The Hamiltonian admits dark states and it implies a gauge field.

39. 39 A general result Suppose the first atomic states are degenerate, and these levels are well separated from the remaining, where and are the truncated matrices and The vector potential is related to an effective “magnetic” field as Experiments: A type of laser-induced gauge potential has been experimentally realized Y.J.Lin et al., PRL(2009), A group at NIST