non-Hermitian Lattices Research Articles

Angular excitation of exceptional points and pseudospetra of photonic lattices

Abstract Optical parity-time ( PT ) symmetry has triggered a plethora of theoretical and experimental studies, and as a result non-Hermitian photonics is one of the frontiers of optics today. From unidirectional invisibility and coherent perfect absorbers to wireless power transfer and ultrasensitive sensors, non-hermiticity led to numerous important applications that rely on gain-loss synthetic structures. In particular, one such structure is that of complex photonic lattices that may exhibit exceptional points (EPs) around which the corresponding sensitivity is increased by orders of magnitude. Here, we investigate a new multiparametric family of non-Hermitian lattices that respects PT -symmetry, and examine their spectral sensitivity in terms of pseudospectra, as well, the angular excitation of the underlying EPs.

Quantum walks of correlated photons in non-Hermitian photonic lattices

Quantum walks of correlated photons in non-Hermitian photonic lattices

Exact non-Hermitian mobility edges and robust flat bands in two-dimensional Lieb lattices with imaginary quasiperiodic potentials

The mobility edge (ME) is a critical energy delineates the boundary between extended and localized states within the energy spectrum, and it plays a crucial role in understanding the metal–insulator transition in disordered or quasiperiodic systems. While there have been extensive studies on MEs in one-dimensional non-Hermitian (NH) quasiperiodic lattices recently, the investigation of exact NH MEs in two-dimensional (2D) cases remains rare. In the present study, we introduce a 2D dissipative Lieb lattice (DLL) model with imaginary quasiperiodic potentials applied solely to the vertices of the Lieb lattice. By mapping this DLL model to the 2D NH Aubry–André–Harper model, we analytically derive the exact ME and find it associated with the absolute eigenenergies. We find that the eigenvalues of extended states are purely imaginary when the quasiperiodic potential is strong enough. Additionally, we demonstrate that the introduction of imaginary quasiperiodic potentials does not disrupt the flat bands inherent in the system. Finally, we propose a theoretical framework for realizing our model using the Lindblad master equation. Our results pave the way for further investigation of exact NH MEs and flat bands in 2D dissipative quasiperiodic systems.

Non-Hermitian dynamical topological winding in photonic mesh lattices.

Topological winding in non-Hermitian systems is generally associated to the Bloch band properties of lattice Hamiltonians. However, in certain non-Hermitian models, topological winding naturally arises from the dynamical evolution of the system and is related to a new form of geometric phase. Here we investigate dynamical topological winding in non-Hermitian photonic mesh lattices, where the mean survival time of an optical pulse circulating in coupled fiber loops is quantized and robust against Hamiltonian deformations. The suggested photonic model could provide an experimentally accessible platform for the observation of non-Hermitian dynamical topological windings.

Multiple skin transitions in two-band non-Hermitian systems with long-range nonreciprocal hopping

Non-Hermitian skin effect (NHSE) is a prominent feature in non-Hermitian physics, leading to novel topological properties and expanding the traditional energy band theories. In this paper, we investigate a two-band non-Hermitian system in which multiple skin transitions are induced by long-range nonreciprocal hopping. The spectral winding number under periodic boundary conditions reveals the localization directions of skin states. Further, we present the analytical solution of transition points by tracing the self-intersecting points on the complex plane. Interestingly, the current system exhibits the abundant NHSEs, including the normal, W-shaped, and bipolar localization properties, which the eigenstate distributions and the generalized Brillouin zone can clearly illustrate. We also provide a phase diagram to represent the skin transition properties of the system comprehensively. Further, we demonstrate that the multimer non-Hermitian lattices also present the anomalous skin effect and multiple transitions, which occur in the region of the bulk band touching, the same as the two-band lattice. Moreover, a feasible scheme is proposed to realize the current non-Hermitian system with long-range nonreciprocal hopping by a topoelectrical circuit. This work further supplies the content of skin transitions and may help us explore more plentiful localization features in the two-band non-Hermitian systems.

Selecting mode by the complex Berry phase in non-Hermitian waveguide lattices

Bloch oscillations (BOs) in a parity-time (PT)-symmetric Su–Schrieffer–Heeger (SSH) waveguide array are theoretically investigated. We show that the BOs are amplified or damped even for the systems to exhibit entirely real energy bands. The amplified and damped BOs stem from the complex Berry phase and closely relate to the topological properties of the lattice. For the topological nontrivial lattice, the amplification and attenuation of BOs are much more prominent than the trivial case and the output Bloch mode can be selected. Furthermore, we propose an experimental scheme and perform a numerical simulation based on a bent waveguide array. Our work uncovers the impact of the topological properties on the dynamics of the bulk Bloch modes and unveils a horizon in the study of non-Hermitian physics. The mode selection induced by the complex Berry phase may also find application in integrated photonic devices such as the mode filter.

Selection and enhancement of the frequency modes with Floquet exceptional points and chiral Zener tunneling

Mode selecting plays a vital role in the field of optoelectronics, such as optical communication, signal processing, on-chip light manipulation, mode conversion, and frequency synthesis. In this work, flexible selection and enhancement of the frequency modes in an unidirectional coupled Su–Schrieffer–Heeger (SSH) frequency lattice are obtained with Floquet exceptional points (EPs) and chiral Zener tunneling (ZT). The unidirectional coupled non-Hermitian SSH frequency lattices are synthesized by a double-ring system with complex dynamical modulations. Under an effective direct current (dc) force induced by the phase-mismatching of the modulations, the two Floquet bands of the non-Hermitian frequency lattices are degenerated and the Floquet EPs arise. Therefore, the unidirectional and irreversible frequency mode conversion takes place, which is the chiral ZT. Moreover, through perturbation analysis and numerical simulations, we prove that the frequency modes of the two-band system can be selected and enhanced by a multi-photon resonance dc force.

Occupation-Dependent Particle Separation in One-Dimensional Non-Hermitian Lattices.

We unveil an exotic phenomenon arising from the intricate interplay between non-Hermiticity and many-body physics, namely, an occupation-dependent particle separation for hardcore bosons in a one-dimensional lattice driven by unidirectional non-Hermitian pumping. Taking hardcore bosons as an example, we find that a pair of particles occupying the same unit cell exhibit an opposite non-Hermitian pumping direction to that of unpaired ones occupying different unit cells. By turning on an intracell interaction, many-body eigenstates split in their real energies, forming separable clusters in the complex energy plane with either left-, right-, or bipolar-types of non-Hermitian skin effect (NHSE). The dependency of skin accumulating directions on particle occupation is further justified with local sublattice correlation and entanglement entropy of many-body eigenstates. Dynamically, this occupation-dependent NHSE manifests as uni- or bidirectional pumping for many-body initial states, allowing for spatially separating paired and unpaired particles. Our results unveil the possibility of designing and exploring novel non-Hermitian phases originated from particle nonconservation in subsystems (e.g., orbitals, sublattices, or spin species) and their spatial configurations.

Delocalization-localization transitions in 1D non-Hermitian cross-stitch lattices

<sec>In this work, we investigate the influence of quasi-periodic modulation on the localization properties of one-dimensional non-Hermitian cross-stitch lattices with flat bands. The crystalline Hamiltonian for this non-Hermitian cross-stitch lattice is given by: </sec><sec><inline-formula><tex-math id="M232">\begin{document}$\hat{H}=\displaystyle\sum\limits_{n}\left[t(a_n^{\dagger} b_n + b_n^{\dagger}a_n ) + J{\mathrm{e}}^{h}\left(a_n^{\dagger}b_{n + 1} + a_n^{\dagger} a_{n + 1} + Ab_n^{\dagger}a_{n + 1} + Ab_n^{\dagger}b_{n + 1}\right) + J{\mathrm{e}}^{ - h} \left(Aa_{n + 1}^{\dagger}b_n + a_{n + 1}^{\dagger}a_n + b_{n + 1}^{\dagger}a_n + Ab_{n + 1}^{\dagger}b_n\right)\right] $\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M232.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M232.png"/></alternatives></inline-formula>with <inline-formula><tex-math id="M216">\begin{document}$A =\pm 1$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M216.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M216.png"/></alternatives></inline-formula>. When <i>A</i> = 1, the clean lattice supports two bands with dispersion relations <inline-formula><tex-math id="M217">\begin{document}$E_0=- t, $\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M217.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M217.png"/></alternatives></inline-formula><inline-formula><tex-math id="M217-1">\begin{document}$ E_1=4\cos (k - {\mathrm{i}}h) + t$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M217-1.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M217-1.png"/></alternatives></inline-formula>. The compact localized states (CLSs) within the flat band <i>E</i><sub>0</sub> are localized in one unit cell, indicating that the system is characterized by the <i>U</i> = 1 class. Conversely, for <i>A</i> = –1, there are two flat bands in the system: <inline-formula><tex-math id="M218">\begin{document}$E_{\pm}=\pm\sqrt{t^2 + 4}$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M218.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M218.png"/></alternatives></inline-formula>. The CLSs within the flat bands are localized in two unit cells, indicating that the system is marked by the <i>U</i> = 2 class. After introducing quasi-periodic modulations <inline-formula><tex-math id="M219">\begin{document}$\varepsilon_n^{\beta}=\lambda_{\beta}\cos(2\pi\alpha n + \phi_{\beta})$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M219.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M219.png"/></alternatives></inline-formula> (<inline-formula><tex-math id="M220">\begin{document}$\beta=\{a,b\}$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M220.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M220.png"/></alternatives></inline-formula>), delocalization-localization transitions can be observed by numerically calculating the fractal dimension <i>D</i><sub>2</sub> and imaginary part of the energy spectrum <inline-formula><tex-math id="M221">\begin{document}$\ln{|{\rm{Im}}(E)|}$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M221.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M221.png"/></alternatives></inline-formula>. Our findings indicate that the symmetry of quasi-periodic modulations plays an important role in determining the localization properties of the system. For the case of <inline-formula><tex-math id="M222">\begin{document}$U=1$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M222.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M222.png"/></alternatives></inline-formula>, the symmetric quasi-periodic modulation leads to two independent spectra <inline-formula><tex-math id="M223">\begin{document}$\sigma_f$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M223.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M223.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M224">\begin{document}$\sigma_p$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M224.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M224.png"/></alternatives></inline-formula>. The <inline-formula><tex-math id="M229">\begin{document}$\sigma_f$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M229.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M229.png"/></alternatives></inline-formula> retains its compact properties, while the <inline-formula><tex-math id="M225">\begin{document}$\sigma_p$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M225.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M225.png"/></alternatives></inline-formula> owns an extended-localized transition at <inline-formula><tex-math id="M226">\begin{document}$\lambda_{{\mathrm{c}}1}=4M$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M226.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M226.png"/></alternatives></inline-formula> with <inline-formula><tex-math id="M230">\begin{document}$M=\max\{{\mathrm{e}}^{h},\;{\mathrm{e}}^{ - h}\}$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M230.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M230.png"/></alternatives></inline-formula>. However, in the case of antisymmetric modulation, the system exhibits an exact mobility edge <inline-formula><tex-math id="M227">\begin{document}$\lambda_{{\mathrm{c}}2}=2\sqrt{2|E - t|M}$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M227.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M227.png"/></alternatives></inline-formula>. For the <i>U</i> = 2 class, all the eigenstates remain localized under any symmetric quasi-periodic modulation. In the case of antisymmetric modulation, all states transition from multifractal to localized states as the modulation strength increases, with a critical point at <inline-formula><tex-math id="M228">\begin{document}$\lambda_{{\mathrm{c}}3}=4M$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M228.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240510_M228.png"/></alternatives></inline-formula>. This work expands the understanding of localization properties in non-Hermitian flat-band systems and provides a new perspective on delocalization-localization transitions.</sec>

Experimental Realization of Geometry-Dependent Skin Effect in a Reciprocal Two-Dimensional Lattice.

Recent studies of non-Hermitian periodic lattices unveiled the non-Hermitian skin effect (NHSE), in which the bulk modes under the periodic boundary conditions (PBC) become skin modes under open boundary conditions. The NHSE is a topological effect owing to the nontrivial spectral winding, and such spectral behaviors appear naturally in nonreciprocal systems. Hence prevailing approaches rely on nonreciprocity to achieve the NHSE. Here, we report the experimental realization of the geometry-dependent skin effect in a two-dimensional reciprocal system, in which the skin effect occurs only at boundaries whose macroscopic symmetry mismatches with the lattice symmetry. The role of spectral reciprocity and symmetry is revealed by connecting reflective channels at given boundaries with the spectral topology of the PBC spectrum. Our work highlights the vital role of reciprocity, symmetry, and macroscopic geometry on the NHSE in dimensionality larger than one and opens new routes for wave structuring using non-Hermitian effects.

Loss Difference Induced Localization in a Non-Hermitian Honeycomb Photonic Lattice.

Non-Hermitian systems with complex-valued energy spectra provide an extraordinary platform for manipulating unconventional dynamics of light. Here, we demonstrate the localization of light in an instantaneously reconfigurable non-Hermitian honeycomb photonic lattice that is established in a coherently prepared atomic system. One set of the sublattices is optically modulated to introduce the absorptive difference between neighboring lattice sites, where the Dirac points in reciprocal space are extended into dispersionless local flat bands, with two shared eigenstates: low-loss (high-loss) one with fields confined at sublattice B (A). When these local flat bands are broad enough due to larger loss difference, the incident beam with its tangential wave vector being at the K point in reciprocal space is effectively localized at sublattice B with weaker absorption, namely, the commonly seen power exchange between adjacent channels in photonic lattices is effectively prohibited. The current work unlocks a new capability from non-Hermitian two-dimensional photonic lattices and provides an alternative route for engineering tunable local flat bands in photonic structures.

Condensation phases in a non-Hermitian optical lattices system

We explore the condensate phase of single-species bosons in an optical lattice through the use of dissipative Bose–Hubbard models. The ground state of system possesses a real-energy, which corresponds to non-Hermitian Hamiltonians with non-reciprocal hopping phase factors. Through the application of inhomogeneous dynamical Gutzwiller mean-field theory (DGMFT), we predict a fascinating phenomenon: the ground state displays a periodic arrangement of superfluid islands amidst a Mott insulator sea. Additionally, the number of superfluid islands is proportional to the spatial period squared.

Experimental characterization of three-band braid relations in non-Hermitian acoustic lattices

Experimental characterization of three-band braid relations in non-Hermitian acoustic lattices

Non-Hermitian photonic lattices: tutorial

Non-Hermitian photonic lattices combine the peculiar consequences of energy non-conservation with the physics of bandstructures, giving rise to a variety of exotic properties not found in conventional materials or photonic metamaterials. In this tutorial, we introduce the key concepts in the design and implementation of non-Hermitian photonic lattices, including the general features of non-Hermitian lattice Hamiltonians and their bandstructures, the role of non-Hermitian lattice symmetries, and the topological chracterization of non-Hermitian bandstructures. We survey several important non-Hermitian lattice designs, as well as the photonics platforms on which they can be realized. Finally, we discuss the possibilities for future developments in the field.

Non-Hermitian synthetic lattices with light-matter coupling

We propose that light-matter coupling can be used to realize synthetic lattices. In particular, we consider a one-dimensional chain of exciton-photon sites to create a comb lattice that exhibits a transition from a flat band to a finite mass dispersion by tuning site-dependent light-matter coupling. Moreover, in a non-Hermitian system with gain and loss, the flat-band phase is much more robust, and the transition is accompanied by the appearance of exceptional points in the complex energy spectrum. We demonstrate that by engineering the light-matter coupling in the synthetic lattice, one can explore various phases in the lasing regime. Our proposal paves the way for studying non-Hermitian systems in higher dimensions.

Complex Berry phase and imperfect non-Hermitian phase transitions

In many classical and quantum systems described by an effective non-Hermitian Hamiltonian, spectral phase transitions, from an entirely real-energy spectrum to a complex spectrum, can be observed as a non-Hermitian parameter in the system is increased above a critical value. A paradigmatic example is provided by systems possessing parity-time ($\mathcal{PT}$) symmetry, where the energy spectrum remains entirely real in the unbroken $\mathcal{PT}$ phase while a transition to complex energies is observed in the broken $\mathcal{PT}$ phase. Such spectral phase transitions are universally sharp. However, when the system is slowly and periodically cycled, the phase transition can become smooth, i.e., imperfect, owing to the complex Berry phase associated to the cyclic adiabatic evolution of the system. This remarkable phenomenon is illustrated by considering the spectral phase transition of the Wannier-Stark ladders in a $\mathcal{PT}$-symmetric class of two-band non-Hermitian lattices subjected to an external dc field, revealing that a nonvanishing imaginary part of the Zak phase---the Berry phase picked up by a Bloch eigenstate evolving across the entire Brillouin zone---is responsible for imperfect spectral phase transitions.

Non-Hermitian squeezed polarons

Recent experimental breakthroughs in non-Hermitian ultracold atomic lattices have dangled tantalizing prospects in realizing exotic, hitherto unreported, many-body non-Hermitian quantum phenomena. In this work, we discover and propose an experimental platform for a radically different non-Hermitian phenomenon dubbed polaron squeezing. It is marked by a dipole-like accumulation of fermions arising from an interacting impurity in a background of non-Hermitian reciprocity-breaking hoppings. We computed their spatial density and found that, unlike Hermitian polarons which are symmetrically localized around impurities, non-Hermitian squeezed polarons localize asymmetrically in the direction opposite to conventional non-Hermitian pumping and non-perturbatively modify the entire spectrum, despite having a manifestly local profile. We investigated their time evolution and found that, saliently, they appear almost universally in the long-time steady state, unlike Hermitian polarons which only exist in the ground state. In our numerics, we also found that, unlike well-known topological or skin localized states, squeezed polarons exist in the bulk, independently of boundary conditions. Our findings could inspire the realization of many-body states in ultracold atomic setups, where a squeezed polaron can be readily detected and characterized by imaging the spatial fermionic density.

Robustness versus sensitivity in non-Hermitian topological lattices probed by pseudospectra

Non-Hermitian topological systems simultaneously possess two antagonistic features: ultrasensitivity due to exceptional points and robustness of topological zero-energy modes, and it is unclear which one prevails under different perturbations. We study that question by applying the pseudospectrum theory on the prototypical non-Hermitian Su-Schrieffer-Heeger lattice. Topological modes around the underlying third-order exceptional point (EP3) are robust with respect to chiral perturbations but sensitive to diagonal perturbations. In fact, exactly at the EP3 the chiral symmetry leads to a suppressed sensitivity, that corresponds to an EP2. Finally, for nonlinearly induced perturbations we provide a connection between the pseudospectrum approach and a nonlinear phase shift, which is relevant for experiments.

Propagation of partially coherent light in non-Hermitian lattices

Band theory for partially coherent light is introduced by using the formalism of second-order classical coherence theory. It is demonstrated that the cross-spectral density function can have bands and gaps and form a correlation band structure. The propagation of a partially coherent beam in non-Hermitian periodic structures is considered to elucidate the interplay between the degree of coherence and the gain or loss present in the lattice. We apply the formalism to study partially coherent Bloch oscillations in lattices having parity-time symmetry.

Non-Hermitian Bloch-Zener phase transition.

Bloch-Zener oscillations (BZO), i.e., the interplay between Bloch oscillations and Zener tunneling in two-band lattices under an external direct current (DC) force, are ubiquitous in different areas of wave physics, including photonics. While in Hermitian systems such oscillations are rather generally aperiodic and only accidentally periodic, in non-Hermitian (NH) lattices BZO can show a transition from aperiodic to periodic as a NH parameter in the system is varied. Remarkably, the phase transition can be either smooth or sharp, contrary to other types of NH phase transitions which are universally sharp. A discrete-time photonic quantum walk on a synthetic lattice is suggested for an experimental observation of smooth BZO phase transitions.