Quantum simulation for frustrated many body interaction models

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Quantum simulation for frustrated many body
interaction models
Zheng-Wei Zhou(???) Key Lab of Quantum
Information , CAS, USTC In collaboration with
Univ. of Sci. Tech. of China X.-F. Zhou (???)
Z.-X. Chen (???) X.-X. Zhou (???) M.-H. Chen
(???) L.-X. He (???) G.-C. Guo (???)
Fudan Univ. Y. Chen (??) H. Ma (??)
Lanzhou Aug. 2, 2011
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Outline

• I. Some Backgrounds on Quantum Simulation
• II. Simulation for 1D frustrated spin ½ models
• III. Simulation for 2D J_1,J_2 spin ½ model
• IV. Simulation for 2D Bose-Hubbard model with
  frustrated tunneling
• Summary

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I. Backgrounds on Quantum Simulation
Nature isn't classical, and if you want to make
a simulation of Nature, you'd better make it
quantum mechanical, and it's a wonderful problem,
because it doesn't look so easy. (Richard
Feynman)
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Why quantum simulation is important?
Answer 2 simulate and build new virtual quantum
materials.
Kitaevs models
topological quantum computing
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Physical Realizations for quantum simulation
Iulia Buluta and Franco Nori, Science 326,108
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About frustration

• Frustration is a very important phenomenon in
  condensed matter systems. It is usually induced
  by the competing interaction or lattice geometry.

AF
AF
AF
AF
or
AF
AF
AF
AF
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One dimension
Three dimension
Zhao J, et. al., Phys. Rev. Lett. 101,167203
(2008)
F.D.M. Haldane, PRB 25, 4925 (1982)
Theoretical treatment of strong frustrated
systems is very difficult.
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The tunable interactions are realized in the
measurement-induced fashion.
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(arXiv1103.5944)
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The results demonstrate the realization of a
quantum simulator for classical magnetism in a
triangular lattice. One succeeded in observing
all the various magnetic phases and phase
transitions of first and second order as well as
frustration induced spontaneous symmetry breaking.
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II. Simulation for 1D frustrated spin ½ models
Zhi-Xin Chen, Zheng-Wei Zhou, Xingxiang Zhou, et
al., Phys. Rev. A 81, 022303 (2010)
Basic idea and difficulty
nonlocal modes
Fourth order effective Hamiltonian
Effective spin 1/2
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The spin chain with next-nearest-neighbor
interactions
Two-photon detuning
The interaction strength decay rapidly along with
the distance between different sites,
So, long-range interaction can be omitted!
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The XXZ chain with next-nearest-neighbor
interactions
Our Model
0 0 0
Key points
0 p 0
The index j represent j-th cavity
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The effective Hamiltonian reads
where
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Experimental Requirements
In this model, the effective decay rate is the
effective cavity field decay rate is Here,
is the linewidth of the upper level and
describes the cavity decay of photons.
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III. Simulation for 2D J_1,J_2 spin ½ model
Model
J_2gt0, frustrated spin model
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Cold Atoms Trapped in Optical Lattices to
Simulate condensed matter physics
D. Jaksch, C. Bruder, C.W. Gardiner, J.I. Cirac
and P. Zoller (1998)
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Basic idea
The physical origin of the confinement of cold
atoms with laser light is the dipole force
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Optical lattice

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t_2
t_1
V_2/V_1
Schrieffer-Wolf transformation
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• Detection of various exotic quantum phases

Possible quantum phases
K. Eckert, et al., Nature Physics, 4, 50 (2008)
The feature of the regime of RVB remains open.
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?
Theoretical prediction
0.38
0.6
Spin-striped state
RVB state
Néel state
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Theoretical prediction
0.8
1.02
decoupled Heisenberg spin chains
Néel state
valence bond crystal
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IV. Simulation for 2D Bose-Hubbard model with
frustrated tunneling
We wonder what will happen if frustration effects
beyond quantum spin models are induced. Here, we
propose a scheme to experimentally realize
frustrated tunneling of ultracold atoms in a
two-dimensional (2D) state-dependent optical
lattice.
Traditional Bose-Hubbard model
For typical optical trapping potential, J is
always positive, and next-nearest-neighbor
interaction is much smaller than the
nearest-neighbor tunnneling rate.
J
Xiang-Fa Zhou, Zhi-Xin Chen, Zheng-Wei Zhou, et
al., Phys. Rev. A 81, 021602 (2010).
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Frustrated tunneling basic idea
state-dependent trapping potential
Two sublattices are displaced so that the
potential minima of one sublattice overlaps with
the potential maxima of the other lattice. The
(red) dotted arrow indicates a lattice-induced
tunneling of atoms
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initially the atoms reside in state 0
Here,
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Controllability
In principle,
Crossover from unfrustrated BH model to
frustrated BH model, from frustrated BH model
to frustrated spin model.
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In the hard-core limit
Frustrated XY-model
The ground state consists of two independent v2
v2 sublattices with antiferromagnetic order.
The mean-field phase diagram of the spin model
(t_00.4)
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In the soft-core case, it is expected the
transition from a Mott insulator (MI) to a
superfluid (SF) occurs at finite hopping
amplitudes for integer filling.
SF superfluid. MI Mott insulator
t_00.12
Frustrated superfluidity
The phase diagram shows a strong asymmetry for
positive and negative J_2. Additionally, for a
finite t_0, there also exists a first-order
transition between the two SF states.
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Frustration in various lattice geometries
honeycomb geometry
kagomé lattice
Optical sublattice with and
polarization
J can be negative or positive.
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Summary

• Quantum simulation of one-dimensional,two-dimensio
  nal frustrated spin models in photon coupled
  cavities and optical lattices.
• Realization of frustrated tunneling of ultracold
  atoms in the optical lattice.This enables us to
  investigate the physics of frustration in both
  bosonic SF and spin systems.

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Thanks for your attention
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References
Quantum simulation of Heisenberg spin chains with
next-nearest-neighbor interactions in coupled
cavities, Zhi-Xin Chen, Zheng-Wei Zhou, Xingxiang
Zhou, Xiang-Fa Zhou, Guang-Can Guo, Phys. Rev. A
81, 022303 (2010)
Frustrated tunneling of ultracold atoms in a
state-dependent optical lattice, Xiang-Fa Zhou,
Zhi-Xin Chen, Zheng-Wei Zhou, Yong-Sheng Zhang,
Guang-Can Guo, Phys. Rev. A 81, 021602 (2010).
The J1-J2 frustrated spin models with ultracold
fermionic atoms in a square optical lattice,
Zhi-Xin Chen, Han Ma, Mo-Han Chen, Xiang-Fa Zhou,
Xingxiang Zhou, Lixin He, Guang-Can Guo, Yan
Chen, and Zheng-Wei Zhou, to be submitted.