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Simulation and Detection of Relativistic Effects with Ultra-Cold Atoms
Shi-Liang Zhu ( 朱诗亮 )
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
5
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
6
– 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