Quantum simulation for frustrated many body interaction models

Lanzhou Aug. 2, 2011

Zheng-Wei Zhou(周正威） Key Lab of Quantum Information , CAS, USTC

In collaboration with:

Univ. of Sci. & Tech. of ChinaX.-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）

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

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)

Why quantum simulation is important?

Answer 2: simulate and build new virtual quantum materials.

Kitaev’s models

topological quantum computing

Physical Realizations for quantum simulation

Iulia Buluta and Franco Nori, Science 326,108

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

About frustration…

AF

AF AF

orAFAFAF

AFAF

One dimension

F.D.M. Haldane, PRB 25, 4925 (1982) Zhao J, et. al., Phys. Rev. Lett. 101,167203 (2008)

Three dimension

Theoretical treatment of strong frustrated systems is very difficult.

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

(arXiv:1103.5944)

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.

Basic idea and difficulty

Effective spin 1/2

Fourth order effective Hamiltonian

nonlocal modes

II. Simulation for 1D frustrated spin ½ models Zhi-Xin Chen, Zheng-Wei Zhou, Xingxiang Zhou, et al., Phys. Rev. A 81, 022303 (2010)

The spin chain with next-nearest-neighbor interactions

So, long-range interaction can be omitted!

The interaction strength decay rapidly along with the distance between different sites,

Two-photon detuning

The XXZ chain with next-nearest-neighbor interactions

2

1

3

4

21

1

bg

ag 12

i je

4i je 3

2

Our Model

Key points

0 π 0

0 0 0

The index j represent j-th cavity

The effective Hamiltonian reads

2 22 2 2 22 2 2 2

1 2 2 2 2 22 2 3 32 2 2 2

2 22 2 2 22 2 2 2

1 1 1 2 1 12 2 3 31 1 1 1

2 , 2 ,

2 , 2 .

a b a b

a b a b

a b a b

a b a b

J J J JJ A B J A B

J J J JA B A B

where

1 3 2 41 2 1 2

31 32 42 41

2 2 2 2

1 1 2 2 1 1 2 231 31 42 42

, , , ,

, , , .

a a b b

a a b ba a b b

g g g gA A B B

g g g g

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.

III. Simulation for 2D J_1,J_2 spin ½ model

Model

J_2>0, frustrated spin model

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)

Basic idea

The physical origin of the confinement of cold atoms with laser light is the dipole force:

Optical lattice

+

Schrieffer-Wolf transformation

V_2/V_1

t_2

t_1

K. Eckert, et al., Nature Physics, 4, 50 (2008)

The feature of the regime of RVB remains open.

Detection of various exotic quantum phases

Possible quantum phases:

0.38 0.6

Theoretical prediction:

Spin-striped stateRVB state

?

Néel state

0.8 1.02

Theoretical prediction:

Néel state

valence bond crystal

decoupled Heisenberg spin chains

IV. Simulation for 2D Bose-Hubbard model with frustrated tunneling

Xiang-Fa Zhou, Zhi-Xin Chen, Zheng-Wei Zhou, et al., Phys. Rev. A 81, 021602® (2010).

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:

J For typical optical trapping potential,J is always positive, and next-nearest-neighbor interaction is much smaller than the nearest-neighbor tunnneling rate.

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

Frustrated tunneling: basic idea

state-dependent trapping potential:

Here,

initially the atoms reside in state 0

Controllability:

In principle,

Crossover from unfrustrated BH model to frustrated BH model, from frustrated BH model to frustrated spin model.

In the hard-core limit

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)

Frustrated XY-model

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.

Frustrated superfluidity

SF = superfluid. MI = Mott insulator

t_0=0.12

The phase diagram shows a strong asymmetry for positive and negative J_2. Additionally, fora finite t_0, there also exists a first-order transition between the two SF states.

Frustration in various lattice geometries:

Optical sublattice with and polarization

honeycomb geometrykagomé lattice

J can be negative or positive.

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.

References：

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).

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)

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.