Manipulation of Artificial Gauge Fields for Ultra-cold Atoms
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Manipulation of Artificial Gauge Fields for Ultra-cold Atoms

Manipulation of Artificial Gauge Fieldsfor Ultra-cold AtomsShi-LiangZhu ( 朱 诗 亮 ) of Quantum Information Technology and School of PhysicsSouth China Normal University, Guangzhou, ChinaCollaborators: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.)ICQFT’09 (Shanghai, July 18-22)1

Outlines1 BackgroundQuantum simulationCondensed matter physics: ( superconductivity or Super-fluid;Anderson Localization; Quantum Hall Effects …………. )Quantum Simulation with ultra-cold atoms2 Geometric phase and artificial gauge fields in ultra-cold atoms3 Applications:Atomic SHE,Atomic QHE,Anderson localization for relativistic particles2

1 Background:Quantum Simulation with Cold atomsSimulation of a quantum system with a classical computer is very har1 Simulate a quantum system by a quantum computer2 Simulate a quantum system by a quantum simulatorQuantum simulator with ultrocold atoms3

Atoms at optical lattices+UBose-Hubbard Hamiltonian H = − J ∑bibj+ ∑εinˆi+ ∑nˆi( nˆi−1)< ij >i 2 iD.Jaksch et al (PRL 1998) M. Greiner et al. , Nature (2002)Time of flight measurementYou can control almost all aspects of the periodic structure and theinteractions between the atoms4

Quantum Hall effectsB+ + + + + + + + + +----------------J19801982Atomic QHE ?However, atoms are electrically neutral andthen a real electromagnetic field does not work5

Effective magnetic fields1) RotatingN.K.Wilkin et al PRL (1998)2) Optical Lattice set-upD.Jaksch and P.Zoller NJP(2003)LaserLaser3) Light-induced geometric phaseG. Juzeliunas PRL (2004)S.L.Zhu et al., PRL (2006)6

Atomic QHEB zI xHow to realize the QHE with cold atomsMain Challenges(a) Realization: Strong uniform magnetic fields;Rxy=V yVIyx1 h=2ν e(b) Detection: Transport measurement is not workableOur work:Realization: Haldane’s QHE without Landau levelDetection: establish a relation between Chernnumber and density profileL.B.Shao et al., Phys.Rev.Lett. (2008)7

2 Geometric phase and Artificial gauge fieldsin ultra-cold atoms8

Introduction: Geometric phase (Berry phase)Berry considered a Hamiltonian which depends on a set of parametersrR 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 factorWhere• Dynamic phase• Berry phase(1 2ntψ(T)γ=dnγ ( C)nM. V. Berry (1984)i(γ (e=n C−) + γih∫dnT0)ψ(0)En( R(t))dt= i∫ n(R)∇ n(R)CRr⋅ dRrGeometric Phase---Depends on the geometry of the trajectory in parameterspace, not on rate of passage9

Geometric phase: adiabatic Berry phaseMany applications in physics: it turns out to provide the fundamentalstructures that govern the physical universeBerry connection :A =n(R)∇Rn(R)Berry curvature:⎡Ω = i⎢⎣∂ψ∂ψ∂λ∂λ−∂ψ∂ψ∂λ∂1 22λ1⎤⎥⎦Nonintegrable phase factor---Related to Gauge potential and gauge fieldi) C.N.Yang, PRL (1974)ii) Concept of Nonintegrable phase factors and global formulation of gauge fieldsT.T.Wu and C. N. Yang, PRD (1975)an artificial electromagnetic field for a neutral atom10

Example: Gauge field for a Lambda-level configurationS. L. Zhu et al, Phys. Rev. Lett. 97,240401 (2006)H =P2+ V ( r)+ HΦ(r)3= ∑ =intj2m⎛ 0 0 Ω1⎞⎜⎟H int= ⎜ 0 0 Ω2⎟⎜ * *Ω Ω 2∆⎟1 2⎝⎠1φ(r)jΩi= Ωsinθeϕ , Ω2= cosθ1ΩThree-level Λ−type AtomsWilczek and Zee, PRL 52, 211 (1984)C.P.Sun and M.L.Ge,PRD (1990)Ruseckas et al., PRL 95, 010404 (2005)⎛ χ ⎛1 ⎞ cosθ⎜ ⎟ ⎜⎜ χ2⎟ = ⎜sinθcosγe⎜ ⎟ ⎜⎝ χ3⎠ ⎝ sinθsin γeγ =arctaniϕiϕ− sinθe−iϕcosθcosγcosθsin γ[( ) ]2 2∆ + Ω − ∆ / ∆ ~ 00 ⎞⎛⎟⎜− sin γ ⎟⎜cosγ⎟⎜⎠⎝11123

Gauge field induced by laser-atominteractionsThe vector potentialΦ( r ) = ∑ Ψ ( r)χ ( r)jjWhere Ψ obey the Schrodinger eq. with the effective Hamiltonian given by~ 1 ~ 2H = h +2mj~( −i∇ − A) V ( r)~A = ihU∇U+The scalar potential~V ( r)= λI+ UV(r)U+F.Wilczek and A.Zee PRL 52,2111(1984)12

3 Application of the artificial gauge fields13

χB↑χ↓==χ1χzin Hall Effect2Application I: Spin Hall EffectsS. L. Zhu et al, Phys. Rev. Lett. 97,240401 (2006)= cosθ1B= sinθey− sinθeiϕH1−iϕ+ cos221 2σ σ+ABx↑σ= h2mAσ⎛ HH eff=⎜⎝ 0( −i∇ − A ) V ( r)= ihσχσ∇σχσσ↑0H= −A= −h sin 2 θ∇ϕ↓= ∇× A = −ηh sin(2θ) ∇θ× ∇ϕB+ + + + + + +_ _ _ _ _ _ _ _↓⎞⎟⎠Charge Hall Effect14

SHE: Spin-dependent trajectoriesElectronic fieldS. L. Zhu et al, Phys. Rev. Lett. 97,240401 (2006)15

Experiments at NISTLin et al.,PRL 102, 130401 (2009)A group at NISTEnergy-momentum dispersion curvesThe experimental data are in agreement with the calculationspredicted by a single-particle picture based on geometric phase.16

Application II: A periodic magnetic field can be used to realizethe Haldane’s QHE without Landau levelsA periodic magnetic fieldBBABIIσIIσ==2 r2ησhk'sin( k'x)ey2 r2ηhk'sin(2k'x)eσz0 1 2 3X (π /k)17

Application II: A periodic magnetic field can be used torealize the Haldane’s QHE without Landau levelsL.B.Shao,S.L.Zhu* ,L.Sheng,D.Y.Xing, and Z.D.Wang, PRL 101, 246810 (2008)=∑< l,j>+∑(+) (+ +ta b + H.c.+ M a a − b b )∑[ ( ) ]iϕjl + +t'e ajaj− bjbj+ H.c.l,jljjjjjjF.D.M.Haldane PRL(1988)Normal insulator ⇔ Chern Insulator(nonzero Chern numbe18

Realization of Haldane’s QHE(Different on-site energies)(1) The different site-energies of sublattices A and Bcan be controlled by the phase of laser beam χχ = 2π / 3χ = 39π / 6019

Realization of Haldane’s QHEH=++ +∑( talbj+ H.c.) + ∑ M ( ajaj− bjbj)< l,j>++ +( a a − b b )∑[ ]iϕt'ej j j j+ H.c.l,jjlj20

With the Fourier transformationSpinorψ =⎛a⎜⎝bkk⎞⎟⎠σ= ∂ρ / ∂ |µxyB ,TStreda JPAR. O. Umucalilar et al PRLhe Chern number:D.H.Lee,G.M.Zhang,T.Xiang PRL(2007)C1=−m2( sgn( m ) − sgn( ))m = M ± 3 3t'sinϕ±+Haldane PRL21

Detection ?B=0 B ≠ 0m = M ± 3 3t'sinϕ±1eB eBρ++ ρ−= ( sgn( m−)− sgn( m+))= C2φ φ0220

Application III: relativistic Dirac-Like equationS.L.Zhu,D.W.Zhang and Z.D.Wang, PRL 102, 210403 (2009).24

Realization of relativistic Dirac equation with cold atoms2pH = + VH+ VL+ H2minta3h∑j=1intH =− ( Ω 0 j + hc . .)j−ikxΩ =Ωsinθe21ikxΩ =Ωsinθe22Ω =Ωcosθe3−iky2 2 21 2 2Ω= Ω +Ω +Ωi ∂ ⎡ 1⎤h Ψ ' =∂ ⎢2m⎥⎣⎦=⊥2( − ih∇ + hk'σ ) + V Ψ'k'cosθvΨ ′( r, t)= Ψi( k⋅r−ωt)In the k space,G. Juzeliunas et al, PRA (2008);S.L.Zhu,D.W.Zhang and Z.D.Wang, PRL 102, 210403 (2009).kev vk L25x

h2mv22Hk= ( k + k′σ ⊥) + VIf k 1µ mor25 P32( F = 2, m F=0lρl≈100µ m1x≈10µmm = −1m = 0 m = 1FFF25 S12( F = 1)26

Relativistic behaviors(1) Zitterbewegung (ZB) effect10.5Amplitude:h / mc< 10−12m(free electron)-0.510 20 30 40 50~ 10−6m(Ultracold atom)-1E(2) Klein tunnelingE 10-1016c8V/ cm27

Anderson localization in disordered 1D chainsVn∈− [ δ , δ ]Scaling theoryβ =ddlnlngLmonotonic nonsingular functionFor non-relativistic particles:All states are localized for arbitrary weak random disorders28

Two results:) a localized state for a massive particle)ξDS≥ ξHowever, for a massless particleϕ =− Npb +gDN∑n=1p an≡1 a delocalized statefor a massless particle, all states are delocalizedbreak down the famous conclusion that the particles are always localizedfor any weak disorder in 1D disordered systems.S.L.Zhu,D.W.Zhang and Z.D.Wang, PRL 102, 210403 (2009).29

The chiral symmetryThe chiral operator5γ = σ xin 1Dψc=5γψ5 5 d 2Hc = γ HDγ =−ihcσx − mc σx+ V( x)dxThe chirality is conserved for a massless particles.Note that⎞γ Φ =±Φ Φ = ⎜ ⎟⎝±κ ⎠5 ⎛1± ± ±30

ipxhthen Ψ ( x)= AΦ e + BΦei pxi− pxhhfor Ψ ( x) = AΦe the outgoing wave functionin++ −pxout( x)A′ hΨ = Φ e B′e −++ Φ−iipxhB′ must be zero for a massless particle31

Detection of Anderson LocalizationNonrelativistic case: non-interacting Bose–Einstein condensateBilly et al., Nature 453, 891 (2008)BEC of Rubidium 87Relativistic case: three more laser beams32

Conclusions1. Create artificial gauge fields for ultra-cold atoms2. reviewed several applications, such as atomic QHE and atomic SHE andAnderson localization for relativistic Dirac particlesReferences:1 Spin Hall effects for cold atoms in a light-induced gauge potentialS. 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 latticeS. 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 ultracoldatomsL.B.Shao, S.L.Zhu* ,L.Sheng,D.Y.Xing, and Z.D.Wang, Phys. Rev. Lett 101, 246810 (20084 Delocalization of relativistic Dirac particles in disordered one-dimensional systems and itsimplementation with cold atomsS.L.Zhu,D.W.Zhang and Z.D.Wang, Phys. Rev. Lett 102, 210403 (2009).33

Thank you for your attention谢 谢 !34

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