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Simulation and Detection of Relativistic Effects with Ultra-Cold Atoms

Shi-Liang Zhu ( 朱诗亮 )

slzhu@scnu.edu.cn

School of Physics and Telecommunication Engineering, South China Normal University, Guangzhou, China

The 3rd International Workshop on Solid-State Quantum Computing & the Hong Kong Forum on Quantum Control

12 - 14 December, 2009

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Collaborators:

Lu-Ming Duan (Michigan Univ.)Z. D. Wang (HKU)Bai-Geng Wang (Nanjing Univ.)Dan-Wei Zhang (South China Normal Univ.)

References:

1) 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.102,210403 (2009).

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)

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Outline

• Introduction:

two typical relativistic effects: Klein tunneling and Zitterbewegung • Two approaches to realize Dirac Hamiltonian with tunable

parameters Honeycomb lattice and Non- Abelian gauge fields

• Observation of relativistic effects with ultra-cold atoms

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一、 Introduction: quantum Tunneling

V(x)1V

1V)( KEEa

T

1Va

Rectangular potential barrier

Transmission coefficient T

)(2

2

22

xVxd

d

mH

Ht

i

2 4 6 8 10

0.2

0.4

0.6

0.8

1

2 4 6 8 10

0.2

0.4

0.6

0.8

1

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一、 Introduction: Klein Paradox

E

V(x)

V

x0

Klein paradox (1929)

2

2

1

1

4 ( )( ) =

(1 ) ( )( )

s

s

R

V E m E mT

V E m E m

a finite limit for , then tends to a non-zero limitsV T

Dirac eq. in one dimension

0)()(

xmxVEx zx

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– Scattering off a square potential barrier

x

Klein tunneling

E

V(x)

0

V

a

V>E

Totally reflection (classical)

Quantum tunneling (non-relativistic QM)

Klein tunneling (relativistic QM)

Transmission

coefficient

0 a

1Klein tunneling

Quantum tunneling

Quantized energiesof antiparticle states

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2 2 316

8

-100

10 / ( / )

free electron: c~3.0 10

graphene: ~c/300 m=0

Ultracold atom ~10 c

e

F

mc m cE V cm

e mc e

v

v

，

Challenges in observation of klein tunneling

In the past eighty years, Klein tunneling has never been directly observed for elementary particles.

It is not feasible to create such a barrier for free electrons due to the enormous electric fields required.

E

Overcome: Masseless particles or particles with ultra-slow speed

Compton length

Rest energy

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M.I.Katsnelson et al., Nature Phys.2,620 (2006)A.F.Young and P. Kim, Nature Phys. Phys.(2009)N.Stander et al., PRL102,026807 (2009)

Klein paradox in Graphene

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Klein tunneling in graphene

Nature Phys. 2, 620 (2006).

Phys. Rev. B 74, 041403(R) (2006).

Experimental evidences: Theory:

Graphene hetero-junction:

Phys. Rev.Lett. 102, 026807 (2009).

Nature Phys. 2, 222 (2009)

disadvantages:

i) Disorder, hard to realize full ballistic transport

ii) Massive cases can’t be directly tested

iii) 2D system, hard to distinguish perfect from near-perfect transmission

The transmission probability crucially depends on the incident angle

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一、 Introduction: Zitterbewegung effect

10 20 30 40 50

-1

-0.5

0.5

1

10/ :Amplitude 12 mmc （ free electron ）

Newton Particles

Non-relativistic quantum particles

r

r

The trajectory of a free particle

Zitterbegwegung (trembling motion) Schrodinger (1930)

The order of the Compton wavelength

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Dirac-Like Equation with tunable parameters in Cold Atoms

)( 2 xVcmdx

dciHH

ti zeffeffxeffDD

1010/~/0.1~

light of speed effective the

mass effective the

cscmc

c

m

eff

eff

eff

Implementation of a Dirac-like equation by using ultra-cold atoms where

can be well controllable

effeff

effeff

cme

cmE

:bewegung Zitter ~ :elingKlein tunn32

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三、 Realization of Dirac equation with cold atoms

• honeycomb lattice

• NonAbelian gauge field

Interesting results: the parameters in the effective Hamiltonian are tunable masse less and massive Dirac particles

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Simulation and detection of Relativistic Dirac fermions in an optical honeycomb lattice

S. L. Zhu, B. G. Wang, and L. M. Duan, Phys. Rev. Lett.98,260402 (2007)

]2/)sincos([sin),( 2

3,2,1

jjLj

j yxkVyxV

0,3/2,3/ 321 where

)(sin 2 kxVx

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ji

jiij chbatH,

.).(

Single-component fermionic atoms in the honeycomb lattice

2211 amamA

),0(,)2/)(1,3( 21 aaaa

bAB

)0,3/(

)2/)(1,3/1(

)2/)(1,3/1(

3

2

1

ab

ab

ab

)3/(2 Lka

ttttt 321 ;

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Roughly one atom per unit cite and in the low-energy

),(),( 00yyxxyx qkqkkk

22222yyxxq qvqvE

yyyxxxD

Dt

pvpvH

Hi

0

212/,2/3),2(

204/1,2/3,0 2

fortavtavt

fortavtav

yx

yx

The Dirac Eq.

Massless:

Massive:

light of speed effective :,

mass effective :

yx vv

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Local density approximation

2/||)(,)( 220 rmrVrV

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

yxyx dkdkkkfS

n ),,(1

)(0

qE yxdkdkS

n0

1)(

220 3/8 aSwhere

The method of Detection (1) : Density profile

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The method of Detection (2) : The Bragg spectroscopy

Atomic transition rate ~~~ dynamic structure factor

quadratic

Linear

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三、 Dirac-like equation with Non-Abelian gauge field

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).

Ht

i

21 DD

20

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

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Tunneling with a Gaussian potential

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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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Two results:

(1) a localized state for a massive particle

(2) D S

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 particle.

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

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Observation of Zitterbewegung with cold atoms

J.Y.Vaishnav and C.W.Clark, PRL100,153002 (2008)

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Summary

)( 2 xVcmdx

dciHH

ti zeffeffxeffDD

1010/~/0.1~

light of speed effective the

mass effective the

cscmc

c

m

eff

eff

eff

where

can be well controllable

effeff

effeff

cme

cmE

:bewegung Zitter ~ :elingKlein tunn32

(1) Two approaches to realize Dirac Hamiltonian

(2)

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The end