1. Simulation of Condensed Matter Physics with ultrocold atoms
朱诗亮 (Shi-Liang Zhu)
广州 华南师范大学物理与电信工程学院
　　　　
全国冷原子物理和量子信息青年学者学术讨论会
山西 太原
2007,7,2-6

2. Outlines Background Quantum simulation; Ultracold atoms
2 Spin-Hall effects
3 Relativistic Dirac fermions

3. 1 Spin-Hall effects for cold atoms in a light-induced gauge field
S. L. Zhu, H. Fu, C. J. Wu, S. C. Zhang, and L. M. Duan,
PRL 97, (2006).
傅浩 (FOCUS center and MCTP, University of Michigan
吴从军(C.J.Wu) (KITP, UC, Santa Barbara)
张首晟(S. C. Zhang) (Stanford University)
段路明(L. M. Duan) (FOCUS center and MCTP, University of Michigan)
2 Simulation and detection of Dirac fermions with cold atoms in an optical lattice
S. L. Zhu, B. Wang, and L. M. Duan, PRL98, (2007)
王伯根 (南京大学) 段路明(Univ. Mich.)

4. Simulate a quantum system with a classical computer is very hard
Simulate a quantum system by a quantum computer
accurately control every basic operation
2 Simulate a quantum system by a quantum simulator
Quantum simulator with ultrocold atoms

5. Quantum computation: a universal set of quantum gates
Two noncommutable single-qubit gates
One nontrivial two-qubit gate
If is a product state
is an entangled state
then U is nontrivial C U U T
A general U can be decomposed into single-qubit rotations
and a nontrivial two-qubit gate

6. 1 The parameters of a system are
well-controllable
2 measurable
Simulator
System

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

8. Cold atomic systems Quantum information Ultra cold astoms
Many-body system
(condensed matter physics)
Few-body system
Field theories

9. Real space or momentum distributions of the atoms

10. Spin Hall Effects 1 Gauge fields in atomic systems
2 Spin Hall effects and Atomic spin Hall effects
3 Detection
4 Conclusions

11. Hall effects B J + + + + + + + + + + - - - - - - - - - - - - - - - -
1980
1982

12. Spin Hall effect J J. E. Hirsch, PRL 83, 1834 (1999);
S. Murakami, N. Nagaosa, and S. C. Zhang, Science 301, 1348 (2003);
J. Sinova et al, PRL92, (2004);
Y. K. Kato et al Science 2004;
Y. K. Kato et al Science

13. Effective magnetic fields
G. Juzeliunas PRL (2004)
G. Juzeliunas et al PRA (2006)

14. Introduction: geometric phase
The Aharonov-Bohm effect
The adiabatic approximation
The Berry phase
The Aharonov-Anandan phase

15. The Aharonov-Bohm effect
Aharonov and Bohm 1959 q q
solenoid
The charge never “touches” field
The fringe shift is gauge invariant
The phase shift depends only on the path integral

16. The adiabatic approximation
H dependents on a set of parameters
Changed adiabatically
The adiabatic theorem:
If the system is initially in a non-degenerate energy eigenstate, it will remain in the corresponding eigenstate during the whole adiabatic evolution.

17. The Berry phase M. V. Berry (1984) A closed path in parameter space:
The initial state is one of energy eigenstates
The final state differs from the initial one by a phase factor
Where
Dynamic phase
Berry phase

18. The Berry phase
Depends on the geometry of the trajectory in parameter space, not on rate of passage
--- geometric phase
Depends on basis |n>
Gauge invariance is guaranteed by cyclic evolution
May be detected by an interference experiment
Berry curvature:
Many application in physics

19. An example: A spin ½ in a magnetic field
Parameter space
The Bloch sphere
Path by n(t)
Path by unit B(t)
Path by B(t)
Berry phase –1/2 solid angle
AA phase = 1/2 solid angle

20. Light-atom interaction
Three-level L-type Atoms

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

22. Pseudo-spin in a gauge fieldB B

23. The configurations of the Raman laser beams: I
G. Juzeliunas et al., Phys. Rev. A 73, (2006)

24. The configurations of the Raman laser beams: II

25. Spin currents g

26. Spin-dependent trajectories

27. Spin-dependent trajectories

28. Detection of the spin currents (spin separation)
A Raman transition with an effective Hamiltonian
After this Raman p-pulse, the initial different dressed spin states are mapped to different hyperfine levels, and the populations in different atomic hyperfine levels can be separately imaged

29. Quantum spin Hall effects
Satisfies the Schrodinger eq. with the Hamiltonian
where B
D.J.Thouless et al, PRL (1982) B

30. Atomic spin Hall effects
X. J. Liu et al (2006)
1 Interactions: Gross-Pitaevskii equations
2 Non-adiabatic effects

31. Conclusions
We propose an experimental scheme to observe spin Hall effects with cold Atoms in a light induced gauge potential. Through numerical simulation, we demonstrated that such a gauge field leads to observable spin Hall currents under realistic conditions. We also discuss the quantum spin Hall state in an optical lattice.

32. Simulation and detection of Dirac Fermions
Condensed Matter Systems:
Low velocity – non-relativistic
Governed by Schrodinger equ.
Relativistic
Dirac Equ.
Graphene: Single-layer carbon atoms
G.W.Semenoff, PRL53,2449(1984)

33. Kitaev Model and its realization with cold atoms
A. Kitaev, Ann Phys 321, 2 (2006)
L.M.Duan et al PRL 91,
090402(2003)

34. Simulation and detection of Dirac Fermions

36. Single-component fermionic atoms in this hexagonal lattice

38. Roughly one atom per lattice cite and in the low energy density
The Dirac Eq.

39. Real space or momentum distributions of the atoms time-of-flight imaging

40. Local density approximation
The local density profile n(r) is uniquely determined by n(m)

42. The Bragg spectroscopy
Lighe-atom interaction Hamiltonian:

43. Massless Dirac fermions:
Non-relativistic case:

44. The Bragg spectroscopy

45. Conclusions
We propose an experimental scheme to simulate and observe relativistic Dirac fermions with cold atoms
By controlling the lattice anisotropy, massive and massless fermions can be
observed and the quantum phase transition between them can be detected.
Atomic density profile and the Bragg spectroscopy are powerful tools

46. Thank you for your attention
S.L.Zhu acknowledges support by the State Key Program for Basic Research of China under No. (No. 2006CB921800), the NCET and NSFC under grant number
Thank you for your attention