Simulation and Detection of Relativistic Effects with Ultra-Cold Atoms

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Simulation and Detection of Relativistic Effects
with Ultra-Cold Atoms

• Shi-Liang Zhu
• (???)
• slzhu_at_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
• 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)

3
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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??Introduction quantum Tunneling
V(x)
Rectangular potential barrier
a
T
Transmission coefficient T
a
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??Introduction Klein Paradox
Klein paradox (1929)
Dirac eq. in one dimension
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Klein tunneling

• Scattering off a square potential barrier

Quantized energies of antiparticle states
VgtE

• Totally reflection (classical)
• Quantum tunneling (non-relativistic QM)
• Klein tunneling (relativistic QM)

x
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Challenges in observation of klein tunneling
In the past eighty years, Klein tunneling has
never been directly observed for elementary
particles.
E
Rest energy
Compton length
It is not feasible to create such a barrier for
free electrons due to the enormous electric
fields required.
Overcome Masseless particles or particles with
ultra-slow speed
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Klein paradox in Graphene
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)
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Klein tunneling in graphene

• Theory

• Experimental evidences

Graphene hetero-junction Phys. Rev.Lett.
102, 026807 (2009). Nature Phys. 2, 222
(2009)
Nature Phys. 2, 620 (2006).

• disadvantages

1. Disorder, hard to realize full ballistic
   transport
2. Massive cases cant be directly tested
3. 2D system, hard to distinguish perfect from
   near-perfect transmission

Phys. Rev. B 74, 041403(R) (2006).
The transmission probability crucially depends on
the incident angle
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??Introduction Zitterbewegung effect
The trajectory of a free particle
Newton Particles Non-relativistic quantum
particles
Zitterbegwegung (trembling motion)
Schrodinger (1930)
(free electron)
The order of the Compton wavelength
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Dirac-Like Equation with tunable parameters in
Cold Atoms
Implementation of a Dirac-like equation by using
ultra-cold atoms where
can
be well controllable
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??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)
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Single-component fermionic atoms in the honeycomb
lattice
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Roughly one atom per unit cite and in the
low-energy
Massless Massive
The Dirac Eq.
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The method of Detection (1) Density profile
Local density approximation
The local density profile n(r) is uniquely
determined by n(m)
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(No Transcript)
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The method of Detection (2) The Bragg
spectroscopy
Linear
quadratic
Atomic transition rate dynamic structure
factor
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??Dirac-like equation with Non-Abelian gauge field
x
In the k space,
G. Juzeliunas et al, PRA (2008)
S.L.Zhu,D.W.Zhang and Z.D.Wang, PRL 102, 210403
(2009).
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If and in one-dimensional case
For Rubidium 87
The effective mass is
Tripod-level configuration of
x
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Tunneling with a Gaussian potential
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Anderson localization in disordered 1D chains
Scaling theory
monotonic nonsingular function
For non-relativistic particles
All states are localized for arbitrary weak
random disorders
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Two results
(1) a localized state for a massive particle
(2)
However, for a massless particle
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 102, 210403
(2009).
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The chiral symmetry
The chiral operator
The chirality is conserved for a massless
particle.
Note that
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must be zero for a massless particle
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Detection of Anderson Localization
Nonrelativistic case non-interacting
BoseEinstein 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
(1) Two approaches to realize Dirac Hamiltonian
where
can be well controllable
(2)
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The end

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