Two-body exceptional points in open dissipative systems

We study two-body non-Hermitian physics in the context of an open dissipative system depicted by the Lindblad master equation.Adopting a minimal lattice model of a handful of interacting fermions with single-particle dissipation,we show that the non-Hermitian effective Hamiltonian of the master equation gives rise to two-body scattering states with state-and interaction-dependent parity-time transition.The resulting two-body exceptional points can be extracted from the trace-preserving density-matrix dynamics of the same dissipative system with three atoms.Our results not only demonstrate the interplay of parity-time symmetry and interaction on the exact few-body level,but also serve as a minimal illustration on how key features of non-Hermitian few-body physics can be probed in an open dissipative many-body system.     

All-Optical Modulation of Microcavity Emission by Parity-Time Symmetry

Integration of microcavity-based coherent light sources and waveguides is of central importance for building optical networks and photonic circuits. However, such combined devices face severe challenges in distributing the generated signals to designed ends despite that the significant advances are accomplished in emission modulation and controlling. Herein, dynamically all-optical control of the outputs from the microcavity by employing parity-time (PT) symmetry is demonstrated. By exploiting the interplay between gain and loss in a waveguide-connected microsquare cavity, the mode field distributions can be strongly modified when the non-Hermitian system enters PT-symmetry breaking phase. Numerical calculations and the corresponding Husimi projections unambiguously show the PT-symmetry-induced mode localization and output redistributions in microsquare cavity. Notably, the intensity ratios of the lights collected by the two channels can be directly tuned over two orders of magnitude with the increase of pumping strength. These findings will be essential for understanding the fundamentals of non-Hermitian optics and advancing the application potentials of microcavity system in integrated photonics.     

Optical resonance in inhomogeneous parity-time symmetric systems

We show that inhomogeneous waveguides of slowly varied parity-time(PT) symmetry support localized optical resonances.The resonance is closely related to the formation of exceptional points separating exact and broken PT phases. Salient features of this kind of non-Hermitian resonance, including the formation of half-vortex flux and the discrete nature,are discussed. This investigation highlights the unprecedented uniqueness of field dynamics in non-Hermitian systems with many potential adaptive applications.     

Topology of a parity-time symmetric non-Hermitian rhombic lattice

We investigate the topological properties of a trimerized parity–time(PT)symmetric non-Hermitian rhombic lattice.Although the system is PT-symmetric,the topology is not inherited from the Hermitian lattice;in contrast,the topology can be altered by the non-Hermiticity and depends on the couplings between the sublattices.The bulk–boundary correspondence is valid and the Bloch bulk captures the band topology.Topological edge states present in the two band gaps and are predicted from the global Zak phase obtained through the Wilson loop approach.In addition,the anomalous edge states compactly localize within two diamond plaquettes at the boundaries when all bands are flat at the exceptional point of the lattice.Our findings reveal the topological properties of the??PT-symmetric non-Hermitian rhombic lattice and shed light on the investigation of multi-band non-Hermitian topological phases.     

PT symmetric Floquet topological phase in SSH model

We consider periodically modulated Su–Schrieffer–Heeger (SSH) model with gain and loss. This model, which can be realized with current technology in photonics using waveguides, allows us to study Floquet topological insulating phase. By using Floquet theory, we find the quasi-energy spectrum of this one dimensional PT symmetric topological insulator. We show that stable Floquet topological phase exists in our model provided that oscillation frequency is large and the non-Hermitian degree is below than a critical value.     

Complex energy plane and topological invariant in non-Hermitian systems

Non-Hermitian systems as theoretical models of open or dissipative systems exhibit rich novel physical properties and fundamental issues in condensed matter physics. We propose a generalized local–global correspondence between the pseudo-boundary states in the complex energy plane and topological invariants of quantum states. We find that the patterns of the pseudo-boundary states in the complex energy plane mapped to the Brillouin zone are topological invariants against the parameter deformation. We demonstrate this approach by the non-Hermitian Chern insulator model. We give the consistent topological phases obtained from the Chern number and vorticity. We also find some novel topological invariants embedded in the topological phases of the Chern insulator model, which enrich the phase diagram of the non-Hermitian Chern insulators model beyond that predicted by the Chern number and vorticity. We also propose a generalized vorticity and its flipping index to understand physics behind this novel local–global correspondence and discuss the relationships between the local–global correspondence and the Chern number as well as the transformation between the Brillouin zone and the complex energy plane. These novel approaches provide insights to how topological invariants may be obtained from local information as well as the global property of quantum states, which is expected to be applicable in more generic non-Hermitian systems.     

Resonance Trapping and Dynamical Phase Transitions

In the non-Hermitian quantum physics, resonance trapping occurs due to width bifurcation in the regime of overlapping resonances. It causes dynamical phase transitions in many-level quantum systems. In the present contribution, three different examples, observed experimentally, are considered. In any case, resonance trapping breaks the symmetry characteristic of the system at low level density due to the alignment of a few states with the scattering states of the environment.     

PT symmetry and spontaneous symmetry breaking in a microwave billiard

We demonstrate the presence of parity-time (PT) symmetry for the non-Hermitian two-state Hamiltonian of a dissipative microwave billiard in the vicinity of an exceptional point (EP). The shape of the billiard depends on two parameters. The Hamiltonian is determined from the measured resonance spectrum on a fine grid in the parameter plane. After applying a purely imaginary diagonal shift to the Hamiltonian, its eigenvalues are either real or complex conjugate on a curve, which passes through the EP. An appropriate basis choice reveals its PT symmetry. Spontaneous symmetry breaking occurs at the EP.     

Non-Hermitian Kitaev chain with complex periodic and quasiperiodic potentials

We study the topological properties of the one-dimensional non-Hermitian Kitaev model with complex either periodic or quasiperiodic potentials. We obtain the energy spectrum and the phase diagrams of the system by using the transfer matrix method as well as the topological invariant. The phase transition points are given analytically. The Majorana zero modes in the topological nontrivial regimes are obtained. Focusing on the quasiperiodic potential, we obtain the phase transition from the topological superconducting phase to the Anderson localization, which is accompanied with the Anderson localization–delocalization transition in this non-Hermitian system. We also find that the topological regime can be reduced by increasing the non-Hermiticity.     

Parity-time对称性对电注入半导体激光器的模式控制

非厄米的描述对于开放系统有重要意义,满足parity-time对称性的哈密顿量,其参数在一定范围内可以使能量具有实的本征值.本文通过模拟,研究了损耗大小以及结构对称性对条形波导中的parity-time对称性的影响,并通过实验发现了电注入条件下由parity-time对称破缺导致的脊条波导模式间隔加倍、模式数减半的现象.     

Observation of the exceptional point in superconducting qubit with dissipation controlled by parametric modulation

Open physical systems described by the non-Hermitian Hamiltonian with parity-time-reversal(PT) symmetry show peculiar phenomena, such as the presence of an exceptional point(EP) at which the PT symmetry is broken and two resonant modes of the Hamiltonian become degenerate. Near the EP, the system could be more sensitive to external perturbations and this may lead to enhanced sensing. In this paper, we present experimental results on the observation of PT symmetry broken transition and the EP using a tunable superconducting qubit. The quantum system of investigation is formed by the two levels of the qubit and the energy loss of the system to the environment is controlled by a method of parametric modulation of the qubit frequency. This method is simple with no requirements for additional elements or qubit device modifications. We believe it can be easily implemented on multi-qubit devices that would be suitable for further exploration of non-Hermitian physics in more complex and diverse systems.     

Shannon Entropy and Diffusion Coefficient in Parity-Time Symmetric Quantum Walks

Non-Hermitian topological edge states have many intriguing properties, however, to date, they have mainly been discussed in terms of bulk–boundary correspondence. Here, we propose using a bulk property of diffusion coefficients for probing the topological states and exploring their dynamics. The diffusion coefficient was found to show unique features with the topological phase transitions driven by parity–time (PT)-symmetric non-Hermitian discrete-time quantum walks as well as by Hermitian ones, despite the fact that artificial boundaries are not constructed by an inhomogeneous quantum walk. For a Hermitian system, a turning point and abrupt change appears in the diffusion coefficient when the system is approaching the topological phase transition, while it remains stable in the trivial topological state. For a non-Hermitian system, except for the feature associated with the topological transition, the diffusion coefficient in the PT-symmetric-broken phase demonstrates an abrupt change with a peak structure. In addition, the Shannon entropy of the quantum walk is found to exhibit a direct correlation with the diffusion coefficient. The numerical results presented herein may open up a new avenue for studying the topological state in non-Hermitian quantum walk systems.     

Dirac光子晶体

Dirac费米子作为粒子物理中的基本粒子之一,其理论在近年来蓬勃发展的拓扑电子理论领域中被广泛提及并用来刻画具有Dirac费米子性质的电子态.这种特殊的能态通常被称为Dirac点,在能谱上表现为两条不同能带之间的线性交叉点.由于Dirac点往往是发生拓扑相变的转变点,因而也被视为实现各种拓扑态的重要母态.作为可与拓扑电子体系类比的拓扑光子晶体因其独特的潜在应用价值也受到人们的广泛关注,实现包含Dirac点的光子能带已成为研究拓扑光子晶体的核心课题.本文基于电子的拓扑理论,简要地回顾了Dirac点在光子系统中的研究进展,特别介绍了如何在光子晶体中利用不同晶格对称性实现在高对称点/线上的Dirac点,以及由Dirac点衍生的Weyl点.     

Exact Solutions of Klein--Gordon Equation with Scalar and Vector Rosen--Morse-Type Potentials

We obtain an exact analytical solution of the Klein Gordon equation for the equal vector and scalar Rosen Morse and Eckart potentials as well as the parity-time （PT） symmetric version of the these potentials by using the asymptotic iteration method. Although these PT symmetric potentials are non-Hermitian, the corresponding eigenvalues are real as a result of the PT symmetry.     

Topological Phase Transition with Larger Winding Number in the Non-Hermitian Dimerized Lattice

The topological phase transitions among normal insulator phase, two kinds of topological insulator phases, and topological semimetal phase are shown based on the non-Hermitian dimerized Su–Schrieffer–Heeger (SSH) model with the nonreciprocal intercell and long-range hopping. In contrast to the previous work, it is found that the topological insulator phase in the present SSH model can hold the larger non-Bloch winding number accompanied by exceptional winding of the generalized Brillouin zone around the gap-closing points. Compared with the usual topological insulator phase in non-Hermitian SSH model, the topological insulator with the larger winding number owns two pairs of zero energy modes with a distinct form of edge localization in the gap. The physical mechanism of the distinct edge localization for zero energy modes via a equivalent Hermitian version of the non-Hermitian SSH model is revealed. Additionally, the process of the phase transition is visualized among normal insulator phase, topological insulator phases, and topological semimetal phase in detail via the evolution of the gap-closing points on the plane of generalized Brillouin zone. This work further verifies the non-Bloch theory and enrich the investigation about the topologically nontrivial phase with the larger topological invariant in the non-Hermitian SSH model.     

Symmetry-Protected Scattering in Non-Hermitian Linear Systems

Symmetry plays fundamental role in physics and the nature of symmetry changes in non-Hermitian physics.Here the symmetry-protected scattering in non-Hermitian linear systems is investigated by employing the discrete symmetries that classify the random matrices.The even-parity symmetries impose strict constraints on the scattering coefficients:the time-reversal(C and K) symmetries protect the symmetric transmission or reflection;the pseudo-Hermiticity(Q symmetry) or the inversion(P) symmetry protects the symmetric transmission and reflection.For the inversion-combined time-reversal symmetries,the symmetric features on the transmission and reflection interchange.The odd-parity symmetries including the particle-hole symmetry,chiral symmetry,and sublattice symmetry cannot ensure the scattering to be symmetric.These guiding principles are valid for both Hermitian and non-Hermitian linear systems.Our findings provide fundamental insights into symmetry and scattering ranging from condensed matter physics to quantum physics and optics.     

Symmetry and Topology of the Topological Counterparts in Non-Hermitian Systems

The symmetries and topological properties of the topological counterparts in 1D non-Hermitian systems are investigated. It is found that, after applying the non-unitary similarity transformation, the non-unitary topological counterpart in real space exhibits completely different global symmetries except for the sublattice symmetry and reveals many brand new local symmetries. Due to the abundant symmetries of non-unitary topological counterparts, it is also found that the unique overlapping projections about the unit sphere vector representing the eigenstates appear in the nontrivial regions, and the triviality of the point-gap topology of non-unitary topological counterpart completely eliminate the intrinsic skin effect in non-Hermitian systems. It is also shown that the unitary topological counterpart never arises any changes for the original symmetries and topological structures even in real space. Unitary topological counterparts are further summarized about the two-band Bloch Hamiltonian, which can expand the definition of non-Bloch winding number. Furthermore, it is demonstrated theoretically that the Bloch Hamiltonian would still hold time-reversal symmetry, abnormal particle-hole symmetry, and sublattice symmetry even suffering from the non-unitary transformation. This work provides a new way to understand the roles of symmetry and topology in non-Hermitian systems from the perspective of topological counterparts.     

拓扑光学体系中合成维度方法的进展

光子学中的合成维度是近年来微纳光学和拓扑光子学的研究热点。通常意义上,一个光学系统的物理维度受限于其空间几何维度,因此极大地制约了光学系统所支持研究的物理现象。而研究者通过引入合成维度,可以突破几何维度对物理系统维度的制约,研究高维空间的物理问题。同时,合成维度的高度可控性和选择的丰富多样性,为简化系统设计、观察高维物理现象提供了很大的便利。本文介绍了光子学中合成维度的基本概念,回顾了近年来实现合成维度的各种设计方案,并初步探讨了其在基础物理研究和应用上的未来前景。     

Characterizing the Topological Properties of 1D Non-Hermitian Systems without the Berry–Zak Phase

A new method is proposed to predict the topological properties of 1D periodic structures in wave physics, including quantum mechanics. From Bloch waves, a unique complex valued function is constructed, exhibiting poles and zeros. The sequence of poles and zeros of this function is a topological invariant that can be linked to the Berry–Zak phase. Since the characterization of the topological properties is done in the complex plane, it can easily be extended to the case of non-Hermitian systems. The sequence of poles and zeros allows to predict topological phase transitions.     

Topological Anderson insulator in two-dimensional non-Hermitian systems

We study the disorder-induced phase transition in two-dimensional non-Hermitian systems. First, the applicability of the noncommutative geometric method(NGM) in non-Hermitian systems is examined. By calculating the Chern number of two different systems(a square sample and a cylindrical one), the numerical results calculated by NGM are compared with the analytical one, and the phase boundary obtained by NGM is found to be in good agreement with the theoretical prediction. Then, we use NGM to investigate the evolution of the Chern number in non-Hermitian samples with the disorder effect. For the square sample, the stability of the non-Hermitian Chern insulator under disorder is confirmed. Significantly,we obtain a nontrivial topological phase induced by disorder. This phase is understood as the topological Anderson insulator in non-Hermitian systems. Finally, the disordered phase transition in the cylindrical sample is also investigated. The clean non-Hermitian cylindrical sample has three phases, and such samples show more phase transitions by varying the disorder strength:(1) the normal insulator phase to the gapless phase,(2) the normal insulator phase to the topological Anderson insulator phase, and(3) the gapless phase to the topological Anderson insulator phase.