Higher-order Exceptional Points Research Articles

Characterizing and tuning exceptional points using Newton polygons

The study of non-Hermitian degeneracies—called exceptional points (EPs)—has become an exciting frontier at the crossroads of optics, photonics, acoustics, and quantum physics. Here, we introduce the Newton polygon method as a general algebraic framework for characterizing and tuning EPs. Newton polygons, first described by Isaac Newton, are conventionally used in algebraic geometry, with deep roots in various topics in modern mathematics. We propose and illustrate how the Newton polygon method can enable the prediction of higher-order EPs, using a recently experimentally realized optical system. Using the paradigmatic Hatano-Nelson model, we demonstrate how our method can predict the presence of the non-Hermitian skin effect. As further application of our framework, we show the presence of tunable EPs of various orders in PT-symmetric one-dimensional models. We further extend our method to study EPs in higher number of variables and demonstrate that it can reveal rich anisotropic behaviour around such degeneracies. Our work provides an analytic recipe to understand exceptional physics.

Symmetry-protected third-order exceptional points in staggered flatband rhombic lattices

Higher-order exceptional points (EPs), which appear as multifold degeneracies in the spectra of non-Hermitian systems, are garnering extensive attention in various multidisciplinary fields. However, constructing higher-order EPs still remains a challenge due to the strict requirement of the system symmetries. Here we demonstrate that higher-order EPs can be judiciously fabricated in parity–time ( PT )-symmetric staggered rhombic lattices by introducing not only on-site gain/loss but also non-Hermitian couplings. Zero-energy flatbands persist and symmetry-protected third-order EPs (EP3s) arise in these systems owing to the non-Hermitian chiral/sublattice symmetry, but distinct phase transitions and propagation dynamics occur. Specifically, the EP3 arises at the Brillouin zone (BZ) boundary in the presence of on-site gain/loss. The single-site excitations display an exponential power increase in the PT -broken phase. Meanwhile, a nearly flatband sustains when a small lattice perturbation is applied. For the lattices with non-Hermitian couplings, however, the EP3 appears at the BZ center. Quite remarkably, our analysis unveils a dynamical delocalization-localization transition for the excitation of the dispersive bands and a quartic power increase beyond the EP3. Our scheme provides a new platform toward the investigation of the higher-order EPs and can be further extended to the study of topological phase transitions or nonlinear processes associated with higher-order EPs.

Localization of higher order exceptional points from finite element model and their applications to duct acoustics.

This work reviews the state of the art for high order perturbation method for parametric eigenvalue problems and propose some extensions for the multiparameters case. This approach allows to locate high order exceptional points (EP) arising in eigenvalue problems. EP correspond to a particular tuning of some complex-valued parameters which render the problem degenerate. These non-Hermitian degeneracies have raised considerable attention in the scientific community as these can have a great impact in a variety of physical problems (PT-symmetry, thermo-acoustic or fluid-structure instability, etc.) and their numerical solution. For applications dealing with dissipative acoustic waveguides, strong modal attenuation can be achieved close to EP and a maximum of attenuation occurs at EP of high order corresponding to the coalescence of more than two modes. The method is based on the automatic computation of the successive derivatives of some selected eigenpairs with respect to the parameters so that, after recombination, regular functions can be constructed. This algebraic manipulations permit to build a reduced order model allowing i) to quickly solve the eigenvalue problem for other parameters values, ii) to follow modal branches, iii) to locate higher order EPs. The method is applied to the particular case of a circular duct with a locally reacting liner at its wall which admittance varies with azimuthal position.

Non-Hermiticity induced exceptional points and skin effect in the Haldane model on a dice lattice

The interplay of topology and non-Hermiticity has led to diverse, exciting manifestations in a plethora of systems. In this work, we systematically investigate the role of non-Hermiticity in the Chern insulating Haldane model on a dice lattice. Due to the presence of a non-dispersive flat band, the dice-Haldane model hosts a topologically rich phase diagram with the non-trivial phases accommodating Chern numbers $\pm 2$. We introduce non-Hermiticity into this model in two ways -- through balanced non-Hermitian gain and loss, and by non-reciprocal hopping in one direction. Both these types of non-Hermiticity induce higher-order exceptional points of order three. We substantiate the presence and the order of these higher-order exceptional points using the phase rigidity and its scaling. Further, we construct a phase diagram to identify and locate the occurrence of these exceptional points in the parameter space. Non-Hermiticity has yet more interesting consequences on a finite-sized lattice. Unlike for balanced gain and loss, in the case of non-reciprocal hopping, the nearest-neighbour dice lattice system under periodic boundary conditions accommodates a finite, non-zero spectral area in the complex plane. This manifests as the non-Hermitian skin effect when open boundary conditions are invoked. In the more general case of the dice-Haldane lattice model, the non-Hermitian skin effect can be caused by both gain and loss or non-reciprocity. Fascinatingly, the direction of localization of the eigenstates depends on the nature and strength of the non-Hermiticity. We establish the occurrence of the skin effect using the local density of states, inverse participation ratio and the edge probability, and demonstrate its robustness to disorder. Our results place the dice-Haldane model as an exciting platform to explore non-Hermitian physics.

Revisiting the hierarchical construction of higher-order exceptional points

Higher-order exceptional points in the spectrum of non-Hermitian Hamiltonians describing open quantum or wave systems have a variety of potential applications, in particular in optics and photonics. However, the experimental realization is notoriously difficult. Recently, Zhong et al. [Q. Zhong, J. Kou, \ifmmode \mbox{\c{S}}\else \c{S}\fi{}. K. \"Ozdemir, and R. El-Ganainy, Phys. Rev. Lett. 125, 203602 (2020)] have introduced a robust construction where a unidirectional coupling of two subsystems having exceptional points of the same order leads generically to a single exceptional point of twice the order. Here, we investigate this scheme in a different manner by exploiting the nilpotency of the traceless part of the involved Hamiltonians. We generalize the scheme and derive a simple formula for the spectral response strength of the composite system hosting a higher-order exceptional point. Its relation to the spectral response strengths of the subsystems is discussed. Moreover, we investigate nongeneric perturbations. The results are illustrated with an example.

Harnessing Dynamical Encircling of an Exceptional Point in Anti-PT-Symmetric Integrated Photonic Systems.

Dynamically encircling an exceptional point in a non-Hermitian system can lead to chiral behaviors, but this process is difficult for on-chip PT-symmetric devices which require accurate control of gain and loss rates. Here, we experimentally demonstrated encircling an exceptional point with a fixed loss rate in a compact anti-PT-symmetric integrated photonic system, where chiral mode switching was achieved within a length that is an order of magnitude shorter than that of a PT-symmetric system. Based on the experimental demonstration, we proposed a topologically protected mode (de)multiplexer that is robust against fabrication errors with a wide operating wavelength range. With the advantages of simplified fabrication and characterization processes, the demonstrated system can be used for studying higher-order exceptional points and for exotic light manipulation.

Nonlinearity-enabled higher-order exceptional singularities with ultra-enhanced signal-to-noise ratio.

Higher-order exceptional points (HOEPs) with extraordinary responsivity are expected to exhibit a vastly improved performance in detection-related applications. However, over the past few years, such an approach has been questioned due to several potential drawbacks, including the stringent parameter requirements, fundamental resolution limits and noise. Here, exploring the consequence of nonlinear gain saturation in exceptional singularities of non-Hermitian systems, we offer a feasible scheme to overcome all the above difficulties. We provide a simple and intuitive example by demonstrating with both theory and circuit experiments an 'exceptional nexus' ('EX'), a HOEP with an ultra-enhanced signal-to-noise ratio (SNR), in only two coupled resonators with the aid of nonlinear gain. The tedious parameter tuning in a six-dimensional hyper-dimensional space is reduced to two dimensions. The feedback mechanism of nonlinear saturable gain can give a solution to the ongoing debate on the SNR of EPs in other linear systems. Our findings advance the fundamental understanding of the peculiar topology of nonlinear non-Hermitian systems, significantly reduce the practical difficulty in EP sensing and possibly open new avenues for applications.

Resonance energy transfer near higher-order exceptional points of non-Hermitian Hamiltonians

Exceptional points (EPs) of both eigenvalue and eigenvector degeneracy offer remarkable properties of the non-Hermitian systems based on the Jordanian form of Hamiltonians at EPs. Here we propose the perturbation theory able to underpin the physics in the vicinity of the higher-order EPs. The perturbation theory unveils lifting of degeneracy and origin of the different phases merging at the EP. It allows us to analyze the photonic local density of states and resonance energy transfer, determining their spectral behaviors in a general form. Resonant energy transfer is investigated in analytical and numerical examples. We analytically find the resonance energy transfer rate near the third-order EP occurring in the system of three coupled cavities and reveal singularities caused by the interplay of the perturbation and frequency detuning from degenerate eigenfrequency. Numerical simulation of the coupled-resonator system reveals the vital role of a mirror for switching to the EP of the doubled order and corresponding enhancement of the resonance energy transfer rate. Our investigation sheds light on the behavior of nanophotonic systems in non-Hermitian environments.

Higher-order exceptional point in a blue-detuned non-Hermitian cavity optomechanical system

Higher-order exceptional points (EPs) in non-Hermitian systems have attracted great interest due to their advantages in sensitive enhancement and distinct topological features. However, realization of such EPs is still challenged because more fine-tuning parameters is generically required in quantum systems, compared to the second-order EP (EP2). Here, we propose a non-Hermitian three-mode optomechanical system in the blue-sideband regime for predicting the third-order EP (EP3). By deriving the pseudo-Hermitian condition for the proposed system, one cavity with loss and the other one with gain must be required. Then we show EP3 or EP2 can be observed when the mechanical resonator (MR) is neutral, loss or gain. For the neutral MR, we find both two degenerate or two non-degenerate EP3s can be predicted by tuning system parameters in the parameter space, while four non-degenerate EP2s can be observed when the system parameters derivate from EP3s, which is distinguished from the previous study in the red-detuned optomechanical system. For the gain (loss) MR, we find only two degenerate EP3s or EP2s can be predicted by tuning enhanced coupling strength. Our proposal provides a potential way to predict higher-order EPs or multiple EP2s and study multimode quantum squeezing around EPs using the blue-detuned non-Hermitian optomechanical systems.

Higher order exceptional points in infinite lattices

One of the hallmarks of non-Hermitian photonics is the existence of unique degeneracies, the so-called higher order exceptional points (HEPs). So far, HEPs have been examined mostly in finite coupled systems. In this paper, we present a systematic way to construct infinite optical waveguide lattices that exhibit exceptional points of higher order. The spectral properties and the sensitivity of these lattices around such points are investigated by employing the method of pseudospectra.

Two-way enhancement of sensitivity by tailoring higher-order exceptional points

Higher-order exceptional points in non-Hermitian systems have recently been used as a tool to engineer high-sensitivity devices, attracting tremendous attention from multidisciplinary fields. Here, we present a simple yet effective scheme to enhance the device sensitivity by slightly deviating the gain-neutral-loss linear configuration to a triangular one, resulting in an abrupt phase transition from third-order to second-order exceptional points. Our analysis demonstrates that the exceptional points can be tailored by a judicious tuning of the coupling parameters of the system, resulting in enhanced sensitivity to a small perturbation. The tunable coupling also leads to a sharp change in the sensitivity slope, enabling the perturbation to be measured precisely as a function of coupling. This two-way detection of the perturbation opens up a rich landscape toward ultrasensitive measurements, which could be applicable to a wide range of non-Hermitian ternary platforms.

Landau-Zener transitions through a pair of higher-order exceptional points

Non-Hermitian quantum systems with explicit time dependence are of ever increasing importance. There are only a handful of models that have been analytically studied in this context. Here, a $\mathcal{PT}$-symmetric non-Hermitian $N$-level Landau-Zener type problem with two exceptional points of $N\mathrm{th}$ order is introduced. The system is Hermitian for asymptotically large times, far away from the exceptional points, and has purely imaginary eigenvalues between the exceptional points. The full Landau-Zener transition probabilities are derived, and found to show a characteristic binomial behavior. In the adiabatic limit the final populations are given by the ratios of binomial coefficients. It is demonstrated how this behavior can be understood on the basis of adiabatic analysis, despite the breakdown of adiabaticity that is often associated with non-Hermitian systems.

Non-Hermitian Sensing in Photonics and Electronics: A Review.

Recently, non-Hermitian Hamiltonians have gained a lot of interest, especially in optics and electronics. In particular, the existence of real eigenvalues of non-Hermitian systems has opened a wide set of possibilities, especially, but not only, for sensing applications, exploiting the physics of exceptional points. In particular, the square root dependence of the eigenvalue splitting on different design parameters, exhibited by 2 × 2 non-Hermitian Hamiltonian matrices at the exceptional point, paved the way to the integration of high-performance sensors. The square root dependence of the eigenfrequencies on the design parameters is the reason for a theoretically infinite sensitivity in the proximity of the exceptional point. Recently, higher-order exceptional points have demonstrated the possibility of achieving the nth root dependence of the eigenfrequency splitting on perturbations. However, the exceptional sensitivity to external parameters is, at the same time, the major drawback of non-Hermitian configurations, leading to the high influence of noise. In this review, the basic principles of PT-symmetric and anti-PT-symmetric Hamiltonians will be shown, both in photonics and in electronics. The influence of noise on non-Hermitian configurations will be investigated and the newest solutions to overcome these problems will be illustrated. Finally, an overview of the newest outstanding results in sensing applications of non-Hermitian photonics and electronics will be provided.

Light transfer transitions beyond higher-order exceptional points in parity-time and anti-parity-time symmetric waveguide arrays.

We propose two non-Hermitian arrays consisting of N = 2l + 1 waveguides and exhibiting parity-time (P T) or anti-P T symmetry for investigating light transfer dynamics based on Nth-order exceptional points (EPs). The P T-symmetric array supports two Nth-order EPs separating an unbroken and a broken phase with real and imaginary eignvalues, respectively. Light transfer dynamics in this array exhibits radically different behaviors, i.e. a unidirectional oscillation behavior in the unbroken phase, an edge-towards localization behavior in the broken phase, and a center-towards localization behavior just at Nth-order EPs. The anti-P T-symmetric array supports also two Nth-order EPs separating an unbroken and a broken phase, which refer however to imaginary and real eigenvalues, respectively. Accordingly, light transfer dynamics in this array exhibits a center-towards localization behavior in the unbroken phase and an origin-centered oscillation behavior in the broken phase. These nontrivial light transfer behaviors and their controlled transitions are not viable for otherwise split lower-order EPs and depend on the underlying SU(2) symmetry of spin-l matrices.

Higher-order exceptional point in a pseudo-Hermitian cavity optomechanical system

Higher-order exceptional points (EPs), resulting from non-Hermitian degeneracies, have shown greater advantages in sensitive enhancement than second-order EPs (EP2s). Therefore, seeking higher-order EPs in various quantum systems is important for quantum information science. Here we propose a benchmark cavity optomechanical (COM) system consisting of a mechanical resonator coupled to two cavities via radiation pressure for predicting the third-order exceptional point (EP3). We first give the pseudo-Hermitian condition for the non-Hermitian COM system by taking the bath effects into account. Then we consider the mechanical gain effect, and we find that the pseudo-Hermitian COM system without $\mathcal{PT}$ symmetry can host both EP3 and EP2 for symmetric and asymmetric cavities. In the symmetric case, only EP3 or EP2 can be predicted in the parameter space, but EP3 and EP2 can be transformed into each other by tuning the optomechanical coupling strength in the asymmetric case. We further consider the case of one cavity with gain. For this case, the pseudo-Hermitian COM system is $\mathcal{PT}$ symmetric and can also host EP3 or EP2. The influence of system parameters on them is discussed. Our proposal provides a potential way to realize sensitive detection and study other physical phenomena around higher-order EPs in non-Hermitian COM systems.

Asymmetric guidance of multiple hybrid modes through a gain-loss-assisted planar coupled-waveguide system hosting higher-order exceptional points

The gain-loss-assisted optical systems have revealed an unconventional light manipulation tool by hosting exceptional points (EPs), which has lately a great deal of interest in the photonics community. Here, the emergence of higher-order EPs is investigated in a planar coupled-waveguide geometry, which has the potential to assemble additional degrees of freedom to explore EP-induced light dynamics. A planar form of waveguide array consisting of four high-indexed guides is designed to support multiple quasiguided hybrid modes. An inhomogeneous customized gain-loss profile to couple at least three chosen modes is introduced in such a way that the waveguide hosts an EP of order three with the simultaneous presence of two second-order EPs in the gain-loss parameter plane. We investigate the dynamical parametric encirclement schemes of embedded EPs of different orders and exclusively reveal both the chiral and nonchiral dynamics of light assisted by asymmetric conversions of hybrid modes. The implementation of our findings in discrete photonic systems hosting higher-order EPs indeed opens up a fertile platform to meet a wide range of enticing applications in the context of manipulation of modes including conversion and switching.

Observation of higher-order exceptional points in a non-local acoustic metagrating

Higher-order exceptional points have attracted increased attention in recent years due to their enhanced sensitivity and distinct topological features. Here, we show that non-local acoustic metagratings enabling precise and simultaneous control over their multiple orders of diffraction can serve as a robust platform for investigating higher-order exceptional points in free space. The proposed metagratings, not only could advance the fundamental research of arbitrary order exceptional points, but could also empower unconventional free-space wave manipulation for applications related to sensing and extremely asymmetrical wave control.

Transient growth and dissipative exceptional points.

In the context of non-Hermitian photonics, we study the physics of transient growth in coupled waveguide systems that exhibit higher-order exceptional points. We demonstrate the counterintuitive effect of transient growth despite the decaying spectrum, which is a direct consequence of the underlying modal nonorthogonality. Eigenvalue analysis fails to capture the power dynamics and thus we have to rely on methods of nonmodal stability theory, namely singular value decomposition and pseudospectra. The relation between the order of the exceptional point and transient growth is also examined. Our work provides a general methodology that can be applied to any non-Hermitian system that contains complex elements with more loss than gain, and exploits the boundaries of transient amplification in dissipative environments.

Symmetry and Higher-Order Exceptional Points.

Exceptional points (EPs), at which both eigenvalues and eigenvectors coalesce, are ubiquitous and unique features of non-Hermitian systems. Second-order EPs are by far the most studied due to their abundance, requiring only the tuning of two real parameters, which is less than the three parameters needed to generically find ordinary Hermitian eigenvalue degeneracies. Higher-order EPs generically require more fine-tuning, and are thus assumed to play a much less prominent role. Here, however, we illuminate how physically relevant symmetries make higher-order EPs dramatically more abundant and conceptually richer. More saliently, third-order EPs generically require only two real tuning parameters in the presence of either a parity-time (PT) symmetry or a generalized chiral symmetry. Remarkably, we find that these different symmetries yield topologically distinct types of EPs. We illustrate our findings in simple models, and show how third-order EPs with a generic ∼k^{1/3} dispersion are protected by PT symmetry, while third-order EPs with a ∼k^{1/2} dispersion are protected by the chiral symmetry emerging in non-Hermitian Lieb lattice models. More generally, we identify stable, weak, and fragile aspects of symmetry-protected higher-order EPs, and tease out their concomitant phenomenology.

Robust exceptional point of arbitrary order in coupled spinning cylinders.

Exceptional points (EPs), i.e., non-Hermitian degeneracies at which eigenvalues and eigenvectors coalesce, can be realized by tuning the gain/loss contrast of different modes in non-Hermitian systems or by engineering the asymmetric coupling of modes. Here we demonstrate a mechanism that can achieve EPs of arbitrary order by employing the non-reciprocal coupling of spinning cylinders sitting on a dielectric waveguide. The spinning motion breaks the time-reversal symmetry and removes the degeneracy of opposite chiral modes of the cylinders. Under the excitation of a linearly polarized plane wave, the chiral mode of one cylinder can unidirectionally couple to the same mode of the other cylinder via the spin-orbit interaction associated with the evanescent wave of the waveguide. The structure can give rise to arbitrary-order EPs that are robust against spin-flipping perturbations, in contrast to conventional systems relying on spin-selective excitations. In addition, we show that higher-order EPs in the proposed system are accompanied by enhanced optical isolation, which may find applications in designing novel optical isolators, nonreciprocal optical devices, and topological photonics.