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 　　　　 　shilzhu@yahoo.com.cn 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

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

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 97,240401 (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, 260402 (2007) 王伯根 (南京大学) 段路明(Univ. Mich.)

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

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

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

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

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

Real space or momentum distributions of the atoms

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

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

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, 126603 (2004); Y. K. Kato et al Science 2004; Y. K. Kato et al Science

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

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

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

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.

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

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

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

Light-atom interaction Three-level L-type Atoms

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)

Pseudo-spin in a gauge field B B

The configurations of the Raman laser beams: I G. Juzeliunas et al., Phys. Rev. A 73, 025602 (2006)

The configurations of the Raman laser beams: II

Spin currents g

Spin-dependent trajectories

Spin-dependent trajectories

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

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

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

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.

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)

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

Simulation and detection of Dirac Fermions

Single-component fermionic atoms in this hexagonal lattice

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

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

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

The Bragg spectroscopy Lighe-atom interaction Hamiltonian:

Massless Dirac fermions: Non-relativistic case:

The Bragg spectroscopy

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

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 10674049 Thank you for your attention