manipulation of artificial gauge fields for ultra-cold atoms

1

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

Shi-Liang Zhu (Shi-Liang Zhu ( 朱诗亮 ) [email protected]@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)

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Outlines

1 Background Quantum Simulation with ultra-cold atoms2 Geometric phase and artificial gauge fields in ultra-cold atoms 3 Applications: Atomic SHE, Atomic QHE, Dirac-like equation

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 computer2 Simulate a quantum system by a quantum simulator

Quantum simulator with ultrocold atoms

4

ij i i

iiiiji nnU

nbbJH )1ˆ(ˆ2

ˆ

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

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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

Quantum Hall effects

B

+ + + + + + + + + +

- - - - - - - - - - - - - - - -

1980

1982

J

However, atoms are electrically neutral and However, atoms are electrically neutral and then a real electromagnetic field does not workthen a real electromagnetic field does not workAtomic QHE ?

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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 phaseLaser Laser

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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 levelDetection: establish a relation between Chern number and density profile

L.B.Shao et al., Phys.Rev.Lett. (2008)

2

1

e

h

I

VR

x

yxy

zB

yV

xI

9

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

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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

)}(),(),({)( 21 tRtRtRtR n

)0()( ))(( dnn CieT

T

ndn dttRE

i0

))((

dRRnRniCC Rn )()()(

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

)0()( RTR

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Geometric phase: adiabatic Berry phase

1221

iBerry curvature:

Many applications in physics: it turns out to provide the fundamental structures that govern the physical universe

)()( :connectionBerry RnRnA R

(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)

factor phase a contral toused becan that factors phase bleNonintegra Cie (1)

Geometric Quantum computation [a recent review paper: E.Sjoqvist, Physics 1, 35 (2008)]

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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)C.P.Sun and M.L.Ge,PRD (1990)Ruseckas et al., PRL 95, 010404 (2005)

2

int( )2

pH V r H

m

N internal states j ( 1, , )j N

The wave function:1

( ) N

jjr j i H

t

One diagonalizes to get a set of N dressed states with eigenvalues intH ( )n r

(n=1,2, ,N) n U j

The full quantum state where1

N

Tot j jj

1 2( , , , )T

N

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obeys the Schrödinger equation

21( )

2i i A V

t m

where

,

( )

nm n m

nm n n m n m

A i U U A i

V r I V

V V

Abelian gauge potential U(1) : if the off-diagonal terms can be neglectedNon-Abelian gauge potential : at least some off-diagonal terms can not be neglected

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Example: Gauge field for a Lambda-level configuration

3

1int

2

)()()(2 j

jrrHrVm

PH

2

00

00

*2

*1

2

1

intH

Three-level type Atoms

3

2

1

cossincossinsin

sincoscoscossin

0sincos

3

2

1

i

i

i

e

e

e

cos,sin 21 ie

0~/arctan 22

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)C.P.Sun and M.L.Ge,PRD (1990)Ruseckas et al., PRL 95, 010404 (2005)

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Gauge field induced by laser-atom interactions

j jj rrr )()()(

)(~~

2

1~ 2rVAi

mH

UUiA ~

The vector potential

The scalar potential UrUVIrV )()(~

Where obey the Schrodinger eq. with the effective Hamiltonian given by

F.Wilczek and A.Zee PRL 52,2111(1984)

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3 Application of the artificial gauge fields

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Application I: Spin Hall Effects

2cos1sin

2sin1cos

2

1

i

i

e

e

H

HH eff 0

0

)(2

1 2 rVAim

H

iA

)2sin(

sin 2

AB

AABB

x

yzB

+ + + + + + +_ _ _ _ _ _ _ _ Charge Hall EffectSpin Hall Effect


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SHE: Spin-dependent trajectories


Electronic field

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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 calculationspredicted by a single-particle picture based on geometric phase.

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zII

yII

exkkB

exkkA

)'2sin('2

)'(sin'22

2

0 1 2 3

B

B

X (/k)

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

A periodic magnetic field

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

jljjjj

i

jl jjjjjjl

cHbbaaet

bbaaMcHbtaH

jl

,

,

..'

..

Insulator Chern insulator Normal

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)

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(1) The different site-energies of sublattices A and Bcan be controlled by the phase of laser beam

Realization of Haldane’s QHE (Different on-site energies)

3/2 60/39

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Realization of Haldane’s QHE

jljjjj

i

jl jjjjjjl

cHbbaaet

bbaaMcHbtaH

jl

,

,

..'

..

0 1 2 3

B

B

X (/k)

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k

k

b

a

With the Fourier transformation

Spinor

The Chern number: D.H.Lee,G.M.Zhang,T.Xiang PRL(2007)

)sgn()sgn(2

1 mmC

sin'33 tMm

Haldane PRL

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Detection ?

B=0 0B

sin'33 tMm

00

)sgn()sgn(2

1

eB

CeB

mm

Txy B ,|/

Streda JPAR. O. Umucalilar et al PRL (2008)

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Application III: relativistic Dirac-Like equation

S.L.Zhu,D.W.Zhang and Z.D.Wang, PRL 102, 210403 (2009).

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Realization of relativistic Dirac equation with cold atoms

2

int

3

int1

2

( 0 . .)

H La

jj

pH V V H

m

H j h c

1

2

3

sin 2

sin 2

cos

ikx

ikx

iky

e

e

e

2 2 2

1 2 2

In the k space, ( )( , ) i k r tkr t e

cos' ''2

1' 2

LkkVkimt

i

xG. Juzeliunas et al, PRA (2008); S.L.Zhu,D.W.Zhang and Z.D.Wang, PRL 102, 210403 (2009).

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22( )

2kH k k Vm

If and in one-dimensional case

2 24

( ) ( )

cos sin2

k x x z z H L

za a

H c p V x V x

k kc

m m

The effective mass is 2 2tan sin2

amm

87Rb

1Fm 0Fm 1Fm

23 25 ( 0) P F

21 25 ( 1) S F

or 23 25 ( 2, =0) FP F m

Tripod-level configuration

of 87Rb

'kk

For Rubidium 87

ml

scmv

kp

k

a

a

a

1

/5

'

x

ml

ml

x

1

0

10

10

30

Relativistic behaviors

(1) Zitterbewegung (ZB) effect

10 20 30 40 50

-1

-0.5

0.5

1

atom) (Ultracold 10~

electron) (free 10/ :Amplitude6

12

m

mmc

(2) Klein tunneling (Klein 1929)

c10~ atom Ultracold

c/300~ :graphene

103.0~c :electron free

/10)/(

10-0

8

16322

v

v

cmVe

cm

mce

mcE

F

e

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).

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Anderson localization in disordered 1D chains

Scaling theory

ln

ln

d g

d L monotonic nonsingular function

All states are localized for arbitrary weak random disorders

[ , ]nV

For non-relativistic particles:

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a localized state for a massive particle

However, for a massless particle

1

1 a delocalized state

N

nn

D

Npb p a

g

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

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The chiral symmetry

The chiral operator 5 in 1Dx

5

5 5 2 ( )

c

c D x x

dH H i c mc V x

dx

The chirality is conserved for a massless particles.

Note that 5 1

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then ( )

for ( ) the outgoing wave function

i ipx px

ipx

in

x A e B e

x A e

( )i i

px px

out x A e B e

B must be zero for a massless particle

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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).

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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 i

mplementation with cold atoms S.L.Zhu,D.W.Zhang and Z.D.Wang, Phys. Rev. Lett 102, 210403 (2009).

References:

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Thank you for your attention

谢谢！

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Typical examples

e

1g 2g

1 2

e

1g 2g 3g

1 2 3 ……

e

1g 2g 1Ng Ng

1 2 1N N

Three-level Λ type Four-level tripod type N+1-level atoms

2

int

int1

( )2

. .N

k kk

pH V r H

m

H e g h c

The Hamiltonian admits dark states and it implies

a gauge field.

1N iD

( 1)U N

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A general result

Suppose the first atomic states are degenerate, and these levels are well separated from the remaining ,

q

N q

1

2

( , , )

1( )

2

q

i i A Vt m

where and are the truncated matrices and V

,1

1

2

N

n m nl lml q

A Am

The vector potential is related to an effective “magnetic” field asA

B��

iB A A A ��

Experiments: A type of laser-induced gauge potential has been experimentally realized Y.J.Lin et al., PRL(2009), A group at NIST

manipulation of artificial gauge fields for ultra-cold atoms

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