Quasiclassical analysis of Bloch oscillations in non-Hermitian tight-binding lattices

While Hermiticity lies at the heart of quantum mechanics, recent experimental advances in controlling dissipation have brought about unprecedented flexibility in engineering non-Hermitian Hamiltonians in open classical and quantum systems. Examples include parity-time-symmetric optical systems with gain and loss, dissipative Bose-Einstein condensates, exciton-polariton systems and biological networks. A particular interest centers on the topological properties of non-Hermitian systems, which exhibit unique phases with no Hermitian counterparts. However, no systematic understanding in analogy with the periodic table of topological insulators and superconductors has been achieved. In this paper, we develop a coherent framework of topological phases of non-Hermitian systems. After elucidating the physical meaning and the mathematical definition of non-Hermitian topological phases, we start with one-dimensional lattices, which exhibit topological phases with no Hermitian counterparts and are found to be characterized by an integer topological winding number even with no symmetry constraint, reminiscent of the quantum Hall insulator in Hermitian systems. A system with a nonzero winding number, which is experimentally measurable from the wave-packet dynamics, is shown to be robust against disorder, a phenomenon observed in the Hatano-Nelson model with asymmetric hopping amplitudes. We also unveil a novel bulk-edge correspondence that features an infinite number of (quasi-)edge modes. We then apply the K-theory to systematically classify all the non-Hermitian topological phases in the Altland-Zirnbauer (AZ) classes in all dimensions. The obtained periodic table unifies time-reversal and particle-hole symmetries, leading to highly nontrivial predictions such as the absence of non-Hermitian topological phases in two dimensions. We provide concrete examples for all the nontrivial non-Hermitian AZ classes in zero and one dimensions. In particular, we identify a Z2 topological index for arbitrary quantum channels (CPTP maps). Our work lays the cornerstone for a unified understanding of the role of topology in non-Hermitian systems. *

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“…We mention that a similar semi-classical analysis on nonunitary wave-packet dynamics is made in Ref. [172].…”

Section: Appendix E: Derivation Of the Semiclassical Equation Of Motimentioning

confidence: 90%

Topological Phases of Non-Hermitian Systems

Gong¹,

Ashida²,

Kawabata³

et al. 2018

Phys. Rev. X

1,411

1,447

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“…We illustrate such an intriguing behavior by considering a paradigmatic model of non-Hermitian transport in tight-binding lattices with asymmetric hopping, namely the Hatano-Nelson model [56][57][58]. This model shows a rich physics and has seen a renewed interest very recently [59][60][61][62][63][64][65][66][67][68][69][70][71][72][73][74][75][76][77][78].…”

Section: Introductionmentioning

confidence: 99%

Loschmidt Echo and Fidelity Decay Near an Exceptional Point

Longhi

2019

Annalen der Physik

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Non-Hermitian classical and open quantum systems near an exceptional point (EP) are known to undergo strong deviations in their dynamical behavior under small perturbations or slow cycling of parameters as compared to Hermitian systems. Such a strong sensitivity is at the heart of many interesting phenomena and applications, such as the asymmetric breakdown of the adiabatic theorem, enhanced sensing, non-Hermitian dynamical quantum phase transitions, and photonic catastrophe. Like for Hermitian systems, the sensitivity to perturbations on the dynamical evolution can be captured by Loschmidt echo and fidelity after imperfect time reversal or quench dynamics. Here, a rather counterintuitive phenomenon in certain non-Hermitian systems near an EP is disclosed, namely the deceleration (rather than acceleration) of the fidelity decay and improved Loschmidt echo as compared to their Hermitian counterparts, despite large (non-perturbative) deformation of the energy spectrum introduced by the perturbations. This behavior is illustrated by considering the fidelity decay and Loschmidt echo for the single-particle hopping dynamics on a tight-binding lattice under an imaginary gauge field.

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“…We mention that a similar semiclassical analysis on nonunitary wave-packet dynamics is made in Ref. [172].…”

Section: Appendix D: Long-lived Quasieigenstates and Their Absence Inmentioning

confidence: 78%

Non-Hermitian Topological Systems

Bandres

Segev

2018

Physics

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While Hermiticity lies at the heart of quantum mechanics, recent experimental advances in controlling dissipation have brought about unprecedented flexibility in engineering non-Hermitian Hamiltonians in open classical and quantum systems. Examples include parity-time-symmetric optical systems with gain and loss, dissipative Bose-Einstein condensates, exciton-polariton systems, and biological networks. A particular interest centers on the topological properties of non-Hermitian systems, which exhibit unique phases with no Hermitian counterparts. However, no systematic understanding in analogy with the periodic table of topological insulators and superconductors has been achieved. In this paper, we develop a coherent framework of topological phases of non-Hermitian systems. After elucidating the physical meaning and the mathematical definition of non-Hermitian topological phases, we start with one-dimensional lattices, which exhibit topological phases with no Hermitian counterparts and are found to be characterized by an integer topological winding number even with no symmetry constraint, reminiscent of the quantum-Hall insulator in Hermitian systems. A system with a nonzero winding number, which is experimentally measurable from the wave-packet dynamics, is shown to be robust against disorder, a phenomenon observed in the Hatano-Nelson model with asymmetric hopping amplitudes. We also unveil a novel bulk-edge correspondence that features an infinite number of (quasi)edge modes. We then apply the K theory to systematically classify all the non-Hermitian topological phases in the Altland-Zirnbauer (AZ) classes in all dimensions. The obtained periodic table unifies time-reversal and particle-hole symmetries, leading to highly nontrivial predictions such as the absence of non-Hermitian topological phases in two dimensions. We provide concrete examples for all the nontrivial non-Hermitian AZ classes in zero and one dimensions. In particular, we identify a Z 2 topological index for arbitrary quantum channels (completely positive tracepreserving maps). Our work lays the cornerstone for a unified understanding of the role of topology in non-Hermitian systems.

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Quasiclassical analysis of Bloch oscillations in non-Hermitian tight-binding lattices

Cited by 15 publications

References 35 publications

Topological Phases of Non-Hermitian Systems

Topological Phases of Non-Hermitian Systems

Loschmidt Echo and Fidelity Decay Near an Exceptional Point

Non-Hermitian Topological Systems

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