Manipulation of Artificial Gauge Fields for Ultra-cold Atoms

Transcription

1 Manipulation of Artificial Gauge Fields for Ultra-cold Atoms 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 (Naning Univ.) C.Wu(UC) S.C.Zhang(Stanford Univ.) Students: L.B.Shao(Naning Univ); D.W.Zhang (SCNU) H.Fu (Michigan Univ.) ICQFT 09 (Shanghai, July 18-) 1

2 Outlines 1 Background Quantum simulation Condensed matter physics: ( superconductivity or Super-fluid; Anderson Localization; Quantum Hall Effects. ) Quantum Simulation with ultra-cold atoms Geometric phase and artificial gauge fields in ultra-cold atoms 3 Applications: Atomic SHE, Atomic QHE, Anderson localization for relativistic particles

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

4 Atoms at optical lattices U Bose-Hubbard Hamiltonian H = J bi b ε inˆ i nˆ i ( nˆ i 1) < i > i i D.Jaksch et al (PRL 1998) M. Greiner et al., Nature (00) Time of flight measurement You can control almost all aspects of the periodic structure and the interactions between the atoms 4

5 Quantum Hall effects B J Atomic QHE? However, atoms are electrically neutral and then a real electromagnetic field does not work 5

6 Effective magnetic fields 1) Rotating N.K.Wilkin et al PRL (1998) ) Optical Lattice set-up D.Jaksch and P.Zoller NJP(003) Laser Laser 3) Light-induced geometric phase G. Juzeliunas PRL (004) S.L.Zhu et al., PRL (006) 6

7 Atomic QHE B z I x How to realize the QHE with cold atoms Main Challenges (a) Realization: Strong uniform magnetic fields; R xy = V y V I y x 1 h = ν e (b) Detection: Transport measurement is not workable 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. (008) 7

8 Geometric phase and Artificial gauge fields in ultra-cold atoms 8

9 Introduction: Geometric phase (Berry phase) Berry considered a Hamiltonian which depends on a set of parameters r R t) = { R ( t), R ( t), LR ( )} Transport a closed path in parameter space: R( T ) = R(0) 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 ( 1 n t ψ( T) γ = d n γ ( C) n M. V. Berry (1984) i( γ ( e = n C ) γ i h d n T 0 ) ψ(0) E n ( R( t)) dt = i n( R) n( R) C R r dr r Geometric Phase---Depends on the geometry of the traectory in parameter space, not on rate of passage 9

10 Geometric phase: adiabatic Berry phase Many applications in physics: it turns out to provide the fundamental structures that govern the physical universe Berry connection : A = n( R) R n( R) Berry curvature: Ω = i ψ ψ λ λ ψ ψ λ 1 λ1 Nonintegrable phase factor---related to Gauge potential and gauge field 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) an artificial electromagnetic field for a neutral atom 10

11 Example: Gauge field for a Lambda-level configuration S. L. Zhu et al, Phys. Rev. Lett. 97,40401 (006) H = P V ( r) H Φ( r) 3 = = int m 0 0 Ω1 H int = 0 0 Ω * * Ω Ω 1 1 φ( r) Ω i = Ωsinθe ϕ, Ω = cosθ 1 Ω Three-level Λ type Atoms Wilczek and Zee, PRL 5, 11 (1984) C.P.Sun and M.L.Ge,PRD (1990) Ruseckas et al., PRL 95, (005) χ 1 cosθ χ = sinθ cosγe χ3 sinθ sin γe γ = arctan iϕ iϕ sinθe iϕ cosθ cosγ cosθ sin γ [( ) ] Ω / ~ 0 0 sin γ cosγ

12 Gauge field induced by laser-atom interactions The vector potential Φ( r ) = Ψ ( r) χ ( r) Where Ψ obey the Schrodinger eq. with the effective Hamiltonian given by ~ 1 ~ H = h m ~ ( i A) V ( r) ~ A = ihu U The scalar potential ~ V ( r) = λi UV( r) U F.Wilczek and A.Zee PRL 5,111(1984) 1

13 3 Application of the artificial gauge fields 13

14 χ B χ = = χ 1 χ z in Hall Effect Application I: Spin Hall Effects S. L. Zhu et al, Phys. Rev. Lett. 97,40401 (006) = cosθ 1 B = sinθe y sinθe iϕ H 1 iϕ cos 1 σ σ A B x σ = h m A σ H H eff = 0 ( i A ) V ( r) = ih σ χ σ σ χ σ σ 0 H = A = h sin θ ϕ = A = η h sin(θ ) θ ϕ B Charge Hall Effect 14

15 SHE: Spin-dependent traectories Electronic field S. L. Zhu et al, Phys. Rev. Lett. 97,40401 (006) 15

16 Experiments at NIST Lin et al.,prl 10, (009) A group at NIST Energy-momentum dispersion curves The experimental data are in agreement with the calculations predicted by a single-particle picture based on geometric phase. 16

17 Application II: A periodic magnetic field can be used to realize the Haldane s QHE without Landau levels A periodic magnetic field B B A B II σ II σ = = r η σ hk'sin ( k' x) ey r η hk' sin(k' x) e σ z X (π /k) 17

18 Application II: A periodic magnetic field can be used to realize the Haldane s QHE without Landau levels L.B.Shao,S.L.Zhu*,L.Sheng,D.Y.Xing, and Z.D.Wang, PRL 101, (008) = < l, > ( ) ( ta b H. c. M a a b b ) [ ( ) ] iϕ l t' e a a b b H. c. l, l F.D.M.Haldane PRL(1988) Normal insulator Chern Insulator (nonzero Chern numbe 18

19 Realization of Haldane s QHE (Different on-site energies) (1) The different site-energies of sublattices A and B can be controlled by the phase of laser beam χ χ = π / 3 χ = 39π / 60 19

20 0 Realization of Haldane s QHE ( ) ( ) ( ) [ ] = > < l i l l H c b b a a e t b b a a M H c b ta H l,,.. '.. ϕ

21 With the Fourier transformation Spinor ψ = a b k k σ = ρ / µ xy B,T Streda JPA R. O. Umucalilar et al PRL he Chern number: D.H.Lee,G.M.Zhang,T.Xiang PRL(007) C 1 = m ( sgn( m ) sgn( )) m = M ± 3 3t'sinϕ ± Haldane PRL 1

22 Detection? B=0 B 0 m = M ± 3 3t'sinϕ ± 1 eb eb ρ ρ = ( sgn( m ) sgn( m )) = C φ φ 0 0

23 3

24 Application III: relativistic Dirac-Like equation S.L.Zhu,D.W.Zhang and Z.D.Wang, PRL 10, (009). 4

25 Realization of relativistic Dirac equation with cold atoms p H = VH VL H m int a 3 h = 1 int H = ( Ω 0 hc..) ikx Ω =Ωsinθe 1 ikx Ω =Ωsinθe Ω =Ωcosθe 3 iky 1 Ω= Ω Ω Ω i 1 h Ψ ' = m = ( ih hk' σ ) V Ψ' k' cosθ v Ψ ( r, t) = Ψ i( k r ωt) In the k space, G. Juzeliunas et al, PRA (008); S.L.Zhu,D.W.Zhang and Z.D.Wang, PRL 10, (009). k e v v k L 5 x

26 h m v H k = ( k k σ ) V If k << k' and in one-dimensional case H cσ p γσ V ( x) V ( x) c k x x z z H L hk h k = cos = sin m a 4 θ γ z θ ma For Rubidium 87 k p = h a << k' The effective mass is Tripod-level configuration of 87 Rb x m a m = tan θ sin θ 87 Rb v l a a 5 P3( F = 0) << 5cm / s >> 1µ m or 5 P3( F =, m F =0 l ρ l 10 0 µ m 1 x 10 µ m m = 1 m = 0 m = 1 F F F 5 S1( F = 1) 6

27 Relativistic behaviors (1) Zitterbewegung (ZB) effect Amplitude: h / mc < 10 1 m (free electron) ~ 10 6 m (Ultracold atom) -1 E () Klein tunneling E<V V T Transmission coefficient T E e mc e( h / mc) free electron : graphene : Ultracold atom Totally reflection (Classic) Quantum tunneling (non-relativistic QM) Klein tunneling (relativistic QM) = m c eh c ~ v F v 0 3 ~ c/300 ~ 10 > c 8 V / cm 7

28 Anderson localization in disordered 1D chains Vn [ δ, δ ] Scaling theory β = d d ln ln g L monotonic nonsingular function For non-relativistic particles: All states are localized for arbitrary weak random disorders 8

29 Two results: ) a localized state for a massive particle ) ξ D S ξ However, for a massless particle ϕ = Npb g D N n= 1 p a n 1 a delocalized state for a massless particle, all states are delocalized 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 10, (009). 9

30 The chiral symmetry The chiral operator 5 γ = σ x in 1D ψ c = 5 γψ 5 5 d Hc = γ HDγ = ihcσx mc σx V( x) dx The chirality is conserved for a massless particles. Note that γ Φ =±Φ Φ = ± κ 5 1 ± ± ± 30

31 i px h then Ψ ( x) = AΦ e BΦ e i px i px h h for Ψ ( x) = AΦ e the outgoing wave function in px out ( x) A h Ψ = Φ e B e Φ i i px h B must be zero for a massless particle 31

32 Detection of Anderson Localization Nonrelativistic case: non-interacting Bose Einstein condensate Billy et al., Nature 453, 891 (008) BEC of Rubidium 87 Relativistic case: three more laser beams 3

33 Conclusions 1. Create artificial gauge fields for ultra-cold atoms. reviewed several applications, such as atomic QHE and atomic SHE and Anderson localization for relativistic Dirac particles References: 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 97,40401 (006) 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, 6040 (007) 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, (008 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 10, (009). 33

34 Thank you for your attention 谢谢! 34