熱場の量子論とその応用 2018
光格子冷却原子系におけるDirac fermion量子シミュレータの構築法とそのトポロジカル物性
𝑌𝑜𝑠ℎ𝑖ℎ𝑖𝑡𝑜 𝐾𝑢𝑛𝑜1 ,𝐼𝑘𝑢𝑜 𝐼𝑐ℎ𝑖𝑛𝑜𝑠𝑒 2 ,𝑌𝑜𝑠ℎ𝑖𝑟𝑜 𝑇𝑎𝑘𝑎ℎ𝑎𝑠ℎ𝑖1

1𝐾𝑦𝑜𝑡𝑜 𝑈𝑛𝑖𝑣.

2𝑁𝑎𝑔𝑜𝑦𝑎 𝐼𝑛𝑠𝑡. 𝑜𝑓 𝑇𝑒𝑐ℎ.

Y. K, I. Ichinose, Y. Takahashi, Sci. Rep. 8, 10699 (2018)
8/31/20181

Summary
How to construct 1D topological insulating model 1D generalized Wilson-Dirac model
Experimental proposal

Generalize matrix:
Implement fourPeierls phases

I. Two parallel tilted optical lattice

Energy offset term

II. Laser assisted hopping scheme

Dirac gamma matricesSymmetry aspect
Only requirement for Piers phase

This condition preserves
Chiral symmetry.

8/31/2018

1D Wilson-Dirac type simulator
in topological BDI class.

Application
・Simulator of 1D topological insulator
・(1+1)-D lattice Gross-Neveu model2

Outline
1. Introduction
2. 1D generalized Wilson-Dirac model
3. General construction scheme

4. Symmetry aspects and non-trivial topological phase
5. Other related topics

8/31/20183

Ultracold atomic gas in an optical lattice & quantum simulator
・Parameter controllability
・Synthetic dimension, artificial gauge field・High-resolution measurement scheme= Ideal platform of quantum simulator
for various theoretical models
Recent experimental progress
長寿命励起状態を用いた人工磁場
cf.
内部自由度を用いた人工次元
M. Mancini, et al.,
＆人工磁場

Science 349, 6255 (2015).

F. Gerbier and J. Dalibard,
New J. Phys. 12, 033007(2010).

実験では光格子冷却原子系（フェルミ、ボソン）で原子の内部自由度を巧みに用いた
8/31/2018人工ゲージ場等が実装できる段階に技術が進歩してきている4

Ultracold atomic gas in an optical lattice & quantum simulator
Quantum Simulation:
・ 1D topological insulating models.
・ Simulate relativistic physics

Question：
How to realize 1D lattice Dirac fermion
in cold-atomic gas in an optical lattice ?

Dirac fermion

1次元トポロジカル絶縁体
格子上のDirac Fermion

物性物理学、格子ゲージ理論の理論モデルに広く現れるフェルミ粒子系トイモデルを冷却原子系上で深く理解する

・実験に先立ち理論から考え得る目標を提示する
・技術的困難をクリアーにする・その系でシミュレートされうる物理現象を予測する
8/31/20185

Outline
1. Introduction
2. 1D generalized Wilson-Dirac model
3. General construction scheme

4. Symmetry aspects and non-trivial topological phase
5. Other related topics

8/31/20186

1D generalized Wilson-Dirac model: spinorGamma matrix in 1D system

Generalized gamma matrix

Parameters
Hopping amplitudes
Peierls phases
Energy-offset8/31/20187

1D generalized Wilson-Dirac model: Ordinary form
The model is reduced to (1+1)-D ordinary Wilson-Dirac model= Standard model of 1D topological insulator
Set parameter

Non-trivial topological phase

Trivial insulating phase
8/31/20188

Outline
1. Introduction
2. 1D generalized Wilson-Dirac model
3. General construction scheme

4. Symmetry aspects and non-trivial topological phase
5. Other related topics

8/31/20189

1D generalized Wilson-Dirac model: Fermion atom picture

Energy-offset term

Hopping termswith Peierls phase
Origin: Dirac-gamma matrix
exchangeexchange
8/31/2018

How to implement
Dirac-gamma matrices
in optical lattice system ?
10

General construction scheme“Combine two experimental techniques”
I. Two parallel tilted optical lattice*O.Mandel, et.al, PRL 91,010407 (2003).

Energy offset term

・Spin dependent lattice (vector light shift)+ Zeeman splitting
・Lattice tilt: Gravity, Electric field gradientII. Laser assisted hopping scheme

Dirac gamma matrices
Particle hops with Peierls phase

Apply four kinds oflaser-assisted hopping
8/31/201811Implement “four kinds of laser-assisted hopping”
・Lattice tilt
・Choice suitable Energy levels
・Use both polarized lasers

Create exchange hopping
offset energy
tilt energy

j-1

8/31/2018

jj+1

12

Feasible theoretical proposal: Concrete example 171 Yb
・Making use of polalizationof lasers in the laser-assisted hopping scheme

・Large energy splittinginexcited state manifold.

・Select suitable polalizability for

Pseudo-spin picture

excited states.

Experimental candidate
Two hyperfine state inUp spin
Down spin
8/31/201813

Schematic figure of four kinds of laser-assisted hopping

8/31/201814

Outline
1. Introduction
2. 1D generalized Wilson-Dirac model
3. General construction scheme

4. Symmetry aspects and non-trivial topological phase
5. Other related topics

8/31/201815

Symmetry classification*S. Ryu, et.al.,
New J. Phys.12, 065010 (2010).
For the 1D model
Quantum simulator of topological model:⇒ Target: BDI class:・Chiral・Time reversal
・Charge conjugation (particle-hole)

Four Piers phase parameters

Change physical properties
・symmetries of the Hamiltonian,・energy spectrum,・ground state
8/31/2018“How to design 1D GWDM to be BDI class”
16Symmetry aspect

ChiralSymmetric about E=0This condition preserves
Chiral symmetry.

TR and CC
Pauli matrices rotatedaround the z-spin axis are given asWe find8/31/2018

We need not to implement
the ordinary 1D Wilson-Dirac fermion
to simulate topological properties.
17

Phase structure of non-trivial topological phase
Same phase structurewith the ones of
the “topological Haldane model”.

*F. D. M. Haldane,
PRL 61, 2015 (1988).

Ordinary Wilson–Dirac model

・Bulk-Topological properties
can be characterized byNon-trivial

Non-trivial

Trivial8/31/201818

Outline
1. Introduction
2. 1D generalized Wilson-Dirac model
3. General construction scheme

4. Symmetry aspects and non-trivial topological phase
5. Other related topics

8/31/201819

Quantum simulator of (1+1)-D lattice Gross-Neveu model

Possibility of atomic quantum simulator for high-energy physics

Finite s-wave scattering(1+1)-D Gross-Neveu model (Wilson fermion)*D. J. Gross and A. Neveu,
PRD 10, 3235 (1974).

There exists non-trivial phase transition
8/31/201820

Summary
How to construct 1D topological insulating model 1D generalized Wilson-Dirac model
Experimental proposal

Generalize matrix:Implement fourPeierls phases

I. Two parallel tilted optical lattice

Energy offset term

II. Laser assisted hopping scheme

Dirac gamma matrices

Symmetry aspect
Only requirement for Piers phase

This condition preserves
Chiral symmetry.

8/31/2018

1D Wilson-Dirac type simulator
in topological BDI class.

Application・Simulator of 1D topological insulator
・(1+1)-D lattice Gross-Neveu model21