1. Quantum simulation for frustrated many body interaction models
Zheng-Wei Zhou(周正威）
Key Lab of Quantum Information , CAS, USTC
In collaboration with:
Univ. of Sci. & Tech. of China
X.-F. Zhou (周祥发)
Z.-X. Chen (陈志心)
X.-X. Zhou (周幸祥)
M.-H. Chen (陈默涵)
L.-X. He (何力新)
G.-C. Guo (郭光灿)
Fudan Univ.
Y. Chen (陈焱)
H. Ma (马涵）
KITPC, June 2, 2011

2. 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

3. 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)

5. Why quantum simulation is important?
Answer 2: simulate and build new virtual quantum materials.
Kitaev’s models
topological quantum computing

6. Physical Realizations for quantum simulation
Iulia Buluta and Franco Nori, Science 326,108

7. 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

8. Theoretical treatment of strong frustrated systems is very difficult.
One dimension
Three dimension
F.D.M. Haldane, PRB 25, 4925 (1982)
Zhao J, et. al., Phys. Rev. Lett. 101, (2008)
Theoretical treatment of strong frustrated systems is very difficult.

11. The tunable interactions are realized in the measurement-induced fashion.

12. (arXiv: )

13. 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.

14. II. Simulation for 1D frustrated spin ½ models
Zhi-Xin Chen, Zheng-Wei Zhou, Xingxiang Zhou, et al., Phys. Rev. A 81, (2010)
Basic idea and difficulty
nonlocal modes
Fourth order effective Hamiltonian
Effective spin 1/2

15. 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!

16. The XXZ chain with next-nearest-neighbor interactions
Our Model
Key points π
The index j represent j-th cavity

17. The effective Hamiltonian reads
where

19. 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.

20. III. Simulation for 2D J_1,J_2 spin ½ model
J_2>0, frustrated spin model

21. 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)

22. Optical lattice +

23. t_2
t_1
V_2/V_1
Schrieffer-Wolf transformation

24. 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.

25. Theoretical prediction:?
Theoretical prediction:
0.38
0.6
Néel state
RVB state
Spin-striped state

26. 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, ® (2010).

27. 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

28. initially the atoms reside in state 0
Here,

29. Controllability:
In principle,
Crossover from unfrustrated BH model to frustrated BH model,
from frustrated BH model to frustrated spin model.

30. In the hard-core limit Frustrated XY-model
The ground state consists of two independent
√2 × √2 sublattices with antiferromagnetic
order.
The mean-field phase diagram
of the spin model (t_0=0.4)

31. Frustrated superfluidity
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_0=0.12
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.

32. Frustration in various lattice geometries:
kagomé lattice
honeycomb geometry
Optical sublattice with and polarization
J can be negative or positive.

33. Summary
Quantum simulation of one-dimensional,two-dimensional 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.

34. Thanks for your attention

35. 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, (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, ® (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.