非エルミート系の物理：
量子ウォークによるアプローチ
Non-Hermitian Physics:
An approach of quantum walks
北海道大学 工学研究院
小布施 秀明統計物理学懇談会 2022年3月10日

Importance of (discrete-time) quantum walks
Quantum walk defines a fundamental equation
for discrete-time quantum dynamics.
Quantum walk:
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Quantum counterpart of classical random walks
Faster propagations of probability
Useful for quantum computations and information (Grover search).
Interdisciplinary science; physics, math, quantum info, …
Various experimental setup (photon, cold atom, ion trap, …)••

Dirac equation in continuum limit
Chalker-Coddington network model
frequently used in the study of the Anderson localization

Importance of quantum walks for non-Hermitian physics
Quantum walk is an ideal platform
for experiments of non-Hermitian systems.
Photonic quantum walks with entangled photons•••

Quantitative control of loss of photons
Quantumness is guaranteed by the entanglement
Various physical phenomena can be reproduced due tohigh controllability of system parameters (quantum simulator).

Table of contents

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1. Non-Hermitian systems
2. Quantum walks

3. Non-unitary quantum walks for Non-Hermitian physics:
topological edge states, skin effects, Anderson transition in 1D[1] K. Mochizuki, D. Kim, and H. Obuse, Phys. Rev. A 93, 062116 (2016).
[2] L. Xiao, X. Zhan, Z.H. Bian, K. K. Wang, X. Zhang, X.P. Wang, J.Li, K. Mochizuki, D. Kim, N. Kawakami,Y. Wi,H. Obuse, B. Sanders, P. Xue, Nature Phys. 13, 1117 (2017).
[3] L. Xiao, X. Qin, K. Wang, Z. Bian, X. Zhan, H. Obuse, B.Sanders, W. Yi, P. Xue, Phys. Rev. A 98, 063847 (2018).
[4] K. Mochizuki, D. Kim, N. Kawakami, and H. Obuse, Phys. Rev. A, 102, 062202 (2020).
[5] M. Kawasaki, K. Mochizuki, N. Kawakami, and H. Obuse, Prog. Theor. Exp. Phys. 2020, 12A105 (2020).
[6] N. Hatano and H. Obuse, Annals of Physics 435, 168615 (2021).
[7] T. Bessho, K. Mochizuki, H. Obuse, and M. Sato, Phys. Rev. B (accepted).
[8] R. Okamoto, N. Kawakami, and H. Obuse (in preparation).
解説• 「量子ウォークのトポロジカル相と光の振幅制御への応用」 小布施、望月、金、川上、日本物理学会誌 74, 780 (2019)• 「アンダーソン局在の臨界現象：最近の実験・理論の新展開」 小布施、日本物理学会誌 70, 14 (2015)

Acknowledgements■Theory :• Makio Kawasaki (Hokkaido Univ.)• Dakyeong Kim (Hokkaido Univ.)• Ken Mochizuki (Hokkaido Univ.  Tohoku Univ.)
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Takumi Bessho (Kyoto Univ.)
Naomichi Hatano (U. Tokyo)
Norio Kawakami (Kyoto Univ.)
Masatoshi Sato (Kyoto Univ.)
Manami Yamagishi (U. Tokyo)■ Experiment :• Ryo Okamoto (Kyoto Univ.)• Peng Xue group (BCSRC, China)

1.Non-Hermitian Systems

Non-Hermitian physics and open systems

Open quantumopensystems
Phenomenologicalsystems(gainasymmetrichopping)(eg.
gainand/or& loss orloss,asymetrichopping)

Breakdown of conservation of probabilities
There are various phenomena which are peculiar to non-Hermitian systems.

Anderson transition in 1D non-Hermitian systems
N. Hatano and D. R. Nelson, PRL 77, 570 (1996)
Hatano-Nelson model : asymmetric hopping with onsite disorder in 1DSpectrum on a complex plain

Real 𝜺𝜺: localized states

Complex 𝜺𝜺: extended states

In 1D non-Hermitian system the Anderson transition occurs,
while it never happens in Hermitian systems.

PT symmetric non-Hermitian quantum mechanics
C.M. Bender and S. Boettcher, PRL 80, 5243 (1998)

Exceptional point :the Hamiltonian is non-diagonalizable at a parameter space

Importance of symmetry in non-Hermitian systems.

Non-Hermitian topological phases
K. Esaki, M. Sato, K. Hasebe, M. Kohmoto, PRB 84, 205128 (2011)
Z. Gong, Y. Ashida, K. Kawabata, K. Takasan, S. Higashikawa, M. Ueda, PRX 8, 031079 (2018)
K. Kawabata, K. Shiozaki, M. Ueda, M. Sato, PRX 9, 041015 (2019)• 38 symmetry classesRamification of symmetries

Time-reversal sym. (TRS):• Two kinds of energy gaps
Line gap

Point gap

Particle-hole sym. (PHS):
Chiral & sublattice symm.:

Hatano-Nelson model
Skin effectNon-Hermitian systems possess richer topological phenomena.

Experiment of non-Hermitian systems : classical optics

No experiment of non-Hermitian quantum systems
No experimental verification of PT symmetricopen quantum systems in 2014.
• Difficulty of controlling non-Hermitian effects(gain and/or loss, asymmetric hopping)• How to guarantee quantumness• Evidencing resultImportance of quantum walks for non-Hermitian physics
Quantum walk is an ideal platform
for experiments of non-Hermitian systems.
Photonic quantum walks with entangled photons•••

Quantitative control of loss of photons
Quantumness is guaranteed by the entanglement
Various physical phenomena can be reproduced due tohigh controllability of system parameters (quantum simulator).

2.Quantum Walks

Quantum walk (QW) in 1D
1.Basis:positioninternal states::Time-evolution operators
3. Time-evolution operator
A time evolution operator is defined by combining the basic operators.
E.g.)Single step QW:
2 step QW:Continuum limit for position and time [Strauch PRA (2006)]
Dirac equation

Time-evolution
4. Time evolution“Floquet system”, “Floquet topological phase”
c.f. Bessho, Mochizuki, Obuse, Sato, PRB(accepted), arXiv:2112.03167

c.f. 2D Grover walk for quantum search

Ambainis, Kempe, Rivosh (2005)

Effective Hamiltonian & Dispersion relation

Unitary operator:

Non-unitary operator:

Symmetry of quantum walks
Kitagawa, Rudner, Berg, Demler, PRA (2010)
Asboth and Obuse, PRB (2013)

TRS:PHS:
Chiral sym.:Topological phases of 1D quantum walks with chiral2 step QWs:

Topological phases of 1D quantum walks with chiral2 step QWs:

3.Non-unitary Quantum walks
for Non-Hermitian Physic

Introduction of non-unitary operators
K. Mochizuki, D. Kim, H. Obuse, PRA (2016)

Gain & loss operator:

truncateNon-unitary quantum walks
Non-unitary time evolution operator:
A time evolution operator is defined by combining the basic operators.
E.g.)Single step QW:
2 step QW:
3 step QW:Time evolution:Non-Hermitian topological phases
K. Esaki, M. Sato, K. Hasebe, M. Kohmoto, PRB 84, 205128 (2011)
Z. Gong, Y. Ashida, K. Kawabata, K. Takasan, S. Higashikawa, M. Ueda, PRX 8, 031079 (2018)
K. Kawabata, K. Shiozaki, M. Ueda, M. Sato, PRX 9, 041015 (2019)• 38 symmetry classes
Ramification of symmetries

Time-reversal sym. (TRS):• Two kinds of energy gaps
Line gap

Point gap

Particle-hole sym. (PHS):
Chiral & sublattice sym.:

Hatano-Nelson model
Skin effectNon-Hermitian systems possess richer topological phenomena.

Topological phases for real line gaps
2step non-unitary quantum walk

Z topological phases forreal line gaps in 1D

K. Mochizuki, D. Kim, and H. Obuse, Phys. Rev. A 93, 062116 (2016).
K. Mochizuki, D. Kim, N. Kawakami, and H. Obuse,Phys. Rev. A, 102, 062202 (2020).

Topological phases for real line gaps : experiment

Observation of edge states for a real line gaps

Breakdown of bulk-edge correspondence
M. Kawasaki, K. Mochizuki, N. Kawakami, and H. Obuse,Prog. Theor. Exp. Phys. 2020, 12A105 (2020).

3step non-unitary quantum walkwith chiral symmetry breaking term
When

Breakdown of bulk-edge correspondence
M. Kawasaki, K. Mochizuki, N. Kawakami, and H. Obuse,
Prog. Theor. Exp. Phys. 2020, 12A105 (2020).

Cf. K. Sone, Y. Ashida, T. Sagawa,Nat. Commun. 11, 5745 (2020).

Skin effects in non-Hermitian systems
Non-Hermitian system with the asymmetric hopping
Hatano-Nelson model (w/o disorder) N. Hatano and D. R. Nelson, PRL 77, 570 (1996)

Periodic BC(PBC)
Open BC(OBC)
Point gap  #topo 1

OBC
S. Yao et al., PRL (2018)
Z. Gong et al., PRX (2018)
K. Yokomizo et al., PRL (2019)
N. Okuma et al., PRL (2020)

Skin effect :All eigenstates localize at the edge.

Unique topological phenomena in the non-Hermitian systems

Anderson transition 1D by non-unitary QWHatano-Nelson model : asymmetric hopping with onsite disorder in 1D

N. Hatano and D. R. Nelson, PRL 77, 570 (1996)

Spectrum on a complex plain

Real 𝜺𝜺: localized states

Complex 𝜺𝜺: extended states

The Anderson transition in 1D non-Hermitian system occurs.

c.f. Kawabata and Ryu, PRL 126, 166801 (2021)
Luo, Xiao, Kawabata, Ohtsuki, Shindou, arXiv:2105.02514

Model & Results
N. Hatano and H. Obuse, Annals of Physics (2021).

:

No symmetry  class A (the same with the original Hatano-Nelson model)
Spectrum:No symmetry & No band gap due toLocalization length κ:periodicity  All states simultaneously delocalize at.

Summary
Topological edge states:•We have demonstrated the non-Hermitian quantum systems byusing the photonic quantum walk first time.

Skin effects:•We have shown the ballistic behaviors in the system exhibiting
the skin effects by using the quantum walk, which can be realized
by using the photonic quantum walk with fine control of loss.

Anderson transition in 1D non-unitary QW:• All states simultaneously delocalize due to no symmetry andband gap originating fromperiodicity of quasi-energy.

The quantum walk is an ideal platform to study
non-Hermitian quantum physics.

国際会議 Localisation 2022
https://2022.localisation.cloud
不規則性に伴う局在現象に関する国際会議(ハイブリッド形式)
LT29(8/18～24, 札幌)のサテライト
期間：2022年8月25～30日
場所：北海道大学 鈴木章ホール, Zoom
4月から口頭発表・ポスター発表の募集を開始予定
Co-chairs: Stefan Kettemann, Hideaki Obuse, Keith Slevin
Organizers: Tomi Ohtsuki, Dragana Popovic, Satoshi Tanda, Kousuke Yakubo