Higher-order Exceptional Points Research Articles

Symmetry-induced higher-order exceptional points in two dimensions

Exceptional points of order n (EPns) appear in non-Hermitian systems as points where the eigenvalues and eigenvectors coalesce. They emerge if 2(n−1) real constraints are imposed, such that EP2s generically appear in two dimensions (2D). Local symmetries have been shown to reduce this number of constraints. In this work, we provide a complete characterization of the appearance of symmetry-induced higher-order EPs in 2D parameter space. We find that besides EP2s only EP3s, EP4s, and EP5s can be stabilized in 2D. Moreover, these higher-order EPs must always appear in pairs with their dispersion determined by the symmetries. Upon studying the complex spectral structure around these EPs, we find that depending on the symmetry, EP3s are accompanied by EP2 arcs, and two- and three-level open Fermi structures. Similarly, EP4s and closely related EP5s, which arise due to multiple symmetries, are accompanied by exotic EP arcs and open Fermi structures. For each case, we provide an explicit example. We also comment on the topological charge of these EPs, and discuss similarities and differences between symmetry-protected higher-order EPs and EP2s. Published by the American Physical Society 2024

Higher-order exceptional points in parity-time symmetry and the optical gyroscope

The practical application of integrated gyroscopes in engineering has not yet been fully realized due to the linear correlation between the Sagnac effect and dimensions. In recent demonstrations, gyroscopes operating near exceptional points (EPs) under parity-time (PT) symmetry have shown significant potential in enhancing their response to rotational rates. However, constructing higher-order EPs with refined physical properties poses a considerable challenge. Additionally, current methods for constructing higher-order EPs with robustness primarily rely on passive cavities, with almost no reports on constructing robust EPs using PT-symmetric systems that encompass both gain and loss. Here, we propose a robust design for a scalable fabrication of higher-order EP gyroscopes with PT-symmetric structure. We investigate the influence of perturbations on the frequency splitting of the higher-order EP gyroscope and demonstrate that it is possible to achieve a resonance splitting eight orders of magnitude higher than that obtained through the classical Sagnac effect. In comparison to the previously proposed PT-symmetric gyroscope, our solution allows a tunable frequency splitting by adjusting the phase shift, making it more measurable at the output power spectrum.

Enhanced sensitivity with nonlinearity-induced exceptional points degeneracy lifting

Bifurcation of exceptional points (EPs), particularly higher-order EPs, can offer applications in metrology by amplifying sensitivity, but this method suffers from a tradeoff between sensitivity and robustness. To break this constraint, we experimentally introduce nonlinearity into the EP degeneracy lifting at the coupled electric resonators and observe a sixth-order nonlinear bifurcation which amplifies the sensitivity elevenfold compared to the conventional EP-based approach operating in the linear regime, while maintaining the degrees of freedom, thereby without cost in robustness. Moreover, we discover a chaotic dynamics near the EP due to the nonlinear contribution, which constitutes a distinct difference from the EP degeneracy lifting in the linear regime with random noise. Our study expands the scope of EP degeneracy lifting into nonlinearity, providing a paradigm to exploit the benefit of EPs.

Topological phases and non-Hermitian topology in tunable nonreciprocal cyclic three-mode optical systems

We propose a method for simulating a 1D non-Hermitian Su-Schrieffer-Heeger model with modulated nonreciprocal hopping using a cyclic three-mode optical system. The current system exhibits different localization of topologically nontrivial phases, which can be characterized by the winding number. We find that the eigenenergies of such a system undergo a real-complex transition as the nonreciprocal hopping changes, accompanied by a non-Bloch parity-time symmetry breaking. We explain this phase transition by considering the evolution of saddle points on the complex energy plan and the ratio of complex eigenenergies. Additionally, we demonstrate that the skin states resulting from the non-Hermitian skin effect possess higher-order exceptional points under the critical point of the non-Bloch parity-time phase transition. Furthermore, we investigate the non-Hermitian skin phase transition by the directional mean inverse participation ratio and the generalized Brillouin zone. This work provides an alternative way to investigate the novel topological and non-Hermitian effects in nonreciprocal optical systems.

Uncovering exceptional contours in non-Hermitian hyperbolic lattices

Hyperbolic lattices are starting to be explored in search of novel phases of matter. At the same time, non-Hermitian physics has come to the forefront in photonic, optical, phononic, and condensed matter systems. In this work, we introduce non-Hermitian hyperbolic lattices and elucidate its exceptional properties in depth. We use hyperbolic Bloch theory to investigate band structures of hyperbolic lattices in the presence of non-Hermitian on-site gain and loss as well as non-reciprocal hopping. Using various analytical and numerical approaches we demonstrate widely accessible and tunable exceptional points (EPs) and contours in {10,5} tessellations, which we characterize using phase rigidity, energy scaling, and vorticity. We further demonstrate the occurrence of higher-order EPs and contours in the {8,4} tessellations using the method of Newton polygons, supported by vorticity and phase rigidity computations. Finally, we investigate the open boundary spectra and densities of states to compare with results from band theory, along with a demonstration of boundary localisation. Our results unveil an abundance of exceptional degeneracies in hyperbolic non-Hermitian matter.

Gain and loss induced higher-order exceptional points in a non-Hermitian electrical circuit

Abstract Non-Hermitian Hamiltonians effectively describing the physics of open systems have become an important tool in various physical platforms, such as photonics, mechanical systems, and electric circuits. The study of non-Hermitian states in electric circuits is developing rapidly recently and forming the field of topoelectrical circuits. Here, we report on the theoretical realization of a higher-order exceptional points (EPs) topological circuit induced by gain and loss. It is shown that, by tuning the value of the positive and negative resistors in the circuit, EPs can collide and merge leading to higher-order singularities. We observe the different energy-difference conserving dynamics near the EPs and the enhanced sensitivity at higher-order EPs. Our results show a way to get higher sensitivity in a non-Hermitian electrical circuit.

Refractive index sensing near exceptional point of a system of triple microcavity

Degeneracies of non-Hermitian Hamiltonian i.e., exceptional points (EPs) of parity-time (PT)-symmetric systems have received considerable research attention due to their various possible applications in optical devices. At EPs, at least two eigenvalues as well as their eigenvector coalesce. Recently, the effect of the eigenfrequency splitting on transfer function near EP was studied for an optical system consisting of two micro ring resonators, which led to complex splitting in PT‐symmetric and real splitting in anti‐PT‐symmetric sensors. In present work, we propose a simple system of three coupled ring resonators to show real splitting in both PT-symmetric and anti-PT-symmetric parameter domains by exploiting higher-order EPs. We indirectly couple two rings with equal amount of gain and loss via an intermediate neutral ring. This system is then tested for refractive index (RI) sensing by modulating the cladding index and we numerically show a huge enhancement in sensitivity as compared to those reported in previous studies of micro ring resonators. Importantly, the enhancement is found to be of the order of 108. Further, we have found that the order of EP can be tuned by perturbating the cladding. The outcomes of this study may set up a wide range of applications in non-Hermitian triplet cavity systems.

Realization and topological properties of third-order exceptional lines embedded in exceptional surfaces

As the counterpart of Hermitian nodal structures, the geometry formed by exceptional points (EPs), such as exceptional lines (ELs), entails intriguing spectral topology. We report the experimental realization of order-3 exceptional lines (EL3) that are entirely embedded in order-2 exceptional surfaces (ES2) in a three-dimensional periodic synthetic momentum space. The EL3 and the concomitant ES2, together with the topology of the underlying space, prohibit the evaluation of their topology in the eigenvalue manifold by prevailing topological characterization methods. We use a winding number associated with the resultants of the Hamiltonian. This resultant winding number can be chosen to detect only the EL3 but ignores the ES2, allowing the diagnosis of the topological currents carried by the EL3, which enables the prediction of their evolution under perturbations. We further reveal the connection between the intersection multiplicity of the resultants and the winding of the resultant field around the EPs and generalize the approach for detecting and topologically characterizing higher-order EPs. Our work exemplifies the unprecedented topology of higher-order exceptional geometries and may inspire new non-Hermitian topological applications.

Multiple exceptional points and phase transitions of a one-dimensional PT-symmetric Lieb photonic lattice

Exceptional points (EPs) in non-Hermitian systems have attracted enormous attention and spawned intriguing prospects for the manipulation of waves. Despite many efforts focusing on the exotic behaviors about EPs, there are only a few studies of phase transitions involving multiple EPs. Here, by employing staggered couplings as well as two pairs of on-site gain/loss, we propose a one-dimensional parity-time (PT)-symmetric Lieb photonic lattice and demonstrate diverse phase transitions of such a multiband structure. Owing to the non-Hermitian chiral symmetry, symmetry-protected higher-order EPs are constructed, and the system exhibits PT symmetry breaking beyond a certain threshold. More importantly, both the relative couplings and the on-site gain/loss can be flexibly reconfigured on demand, which yields the degeneracy of different bands, i.e., the emergence of multiple EPs. We also unveil that the EPs will no longer exist in the presence of a non-Hermitian diagonal disorder. In contrast, the spectrum remains symmetric and the EPs, along with the flatband, are robust against the off diagonal disorder due to the preserved non-Hermitian particle-hole symmetry. Our work not only provides a controllable platform for studying EPs but also sheds light on the exciting non-Hermitian physics based on exceptional degeneracies.

Higher-order exceptional points in waveguide-coupled microcavities: perturbation induced frequency splitting and mode patterns

Exceptional points are degeneracies in the spectrum of non-Hermitian open systems where at least two eigenfrequencies and simultaneously the corresponding eigenstates of the Hamiltonian coalesce. Especially, the robust construction of higher-order exceptional points with more than two degenerate eigenfrequencies and eigenstates is challenging but yet worthwhile for applications. In this paper, we reconsider the formation of higher-order exceptional points through waveguide-coupled microring cavities and asymmetric backscattering. In this context, we demonstrate the influence of perturbations on the frequency splitting of the system. To generate higher-order exceptional points in a simple and robust way, a mirror-induced asymmetric backscattering approach is used. In addition to the exceptional-point enhanced sensing capabilities of such systems, also a cavity-selective sensitivity is achieved for particle sensing. The results are motivated by an effective Hamiltonian description and verified by full numerical simulations of the dielectric structure.

Experimental simulation of symmetry-protected higher-order exceptional points with single photons.

Exceptional points (EPs) of non-Hermitian (NH) systems have recently attracted increasing attention due to their rich phenomenology and intriguing applications. Compared to the predominantly studied second-order EPs, higher-order EPs have been assumed to play a much less prominent role because they generically require the tuning of more parameters. Here, we experimentally simulate two-dimensional topological NH band structures using single-photon interferometry, and observe topologically stable third-order EPs obtained by tuning only two real parameters in the presence of symmetry. In particular, we explore how different symmetries stabilize qualitatively different third-order EPs: the parity-time symmetry leads to a generic cube-root dispersion, while a generalized chiral symmetry implies a square-root dispersion coexisting with a flat band. Additionally, we simulate fourfold degeneracies, composed of the non-defective twofold degeneracies and second-order EPs. Our work reveals the abundant and conceptually richer higher-order EPs protected by symmetries and offers a versatile platform for further research on topological NH systems.

Correlations at higher-order exceptional points in non-Hermitian models

Correlations at higher-order exceptional points in non-Hermitian models

Enhanced optical forces on coupled chiral particles at arbitrary order exceptional points.

Exceptional points (EPs)-non-Hermitian degeneracies at which eigenvalues and eigenvectors coalesce-can give rise to many intriguing phenomena in optical systems. Here, we report a study of the optical forces on chiral particles in a non-Hermitian system at EPs. The EPs are achieved by employing the unidirectional coupling of the chiral particles sitting on a dielectric waveguide under the excitation of a linearly polarized plane wave. Using full-wave numerical simulations, we demonstrate that the structure can give rise to enhanced optical forces at the EPs. Higher order EPs in general can induce stronger optical forces. In addition, the optical forces exhibit an intriguing "skin effect": the force approaches the maximum for the chiral particle at one end of the lattice. The results contribute to the understanding of optical forces in non-Hermitian systems and can find applications in designing novel optical tweezers for on-chip manipulations of chiral particles.

Higher-order exceptional points using lossfree negative-index materials

Negative-index materials (NIMs) support optical anti-parity-time (anti-) symmetry even when they are lossless. Here we prove the feasibility in achieving higher-order exceptional points (EPs) in lossfree waveguide arrays by utilizing the anti- symmetry induced by NIM. Numerical simulation about a third-order EP fits well with the coupled-mode theory. A scheme of achieving fourth-order EPs is also discussed. This work highlights the great potential of NIM in overcoming the obstacles of ordinary non-Hermitian optics, and the possibilities of combining anti-, , and Hermitian couplings for various purposes.

Exceptional points and non-Hermitian photonics at the nanoscale.

Exceptional points (EPs) arising in non-Hermitian systems have led to a variety of intriguing wave phenomena, and have been attracting increased interest in various physical platforms. In this Review, we highlight the latest fundamental advances in the context of EPs in various nanoscale systems, and overview the theoretical progress related to EPs, including higher-order EPs, bulk Fermi arcs and Weyl exceptional rings. We peek into EP-associated emerging technologies, in particular focusing on the influence of noise for sensing near EPs, improving the efficiency in asymmetric transmission based on EPs, optical isolators in nonlinear EP systems and novel concepts to implement EPs in topological photonics. We also discuss the constraints and limitations of the applications relying on EPs, and offer parting thoughts about promising ways to tackle them for advanced nanophotonic applications.

A tropical geometric approach to exceptional points

Non-Hermitian systems have been widely explored in platforms ranging from photonics to electric circuits. A defining feature of non-Hermitian systems is exceptional points (EPs), where both eigenvalues and eigenvectors coalesce. Tropical geometry is an emerging field of mathematics at the interface between algebraic geometry and polyhedral geometry, with diverse applications to science. Here, we introduce and develop a unified tropical geometric framework to characterize different facets of non-Hermitian systems. We illustrate the versatility of our approach using several examples and demonstrate that it can be used to select from a spectrum of higher-order EPs in gain and loss models, predict the skin effect in the non-Hermitian Su-Schrieffer-Heeger model, and extract universal properties in the presence of disorder in the Hatano-Nelson model. Our work puts forth a framework for studying non-Hermitian physics and unveils a connection of tropical geometry to this field.

Theoretical investigation of dynamics and concurrence of entangled {{mathcal {P}}}{{mathcal {T}}} and anti-{{mathcal {P}}}{{mathcal {T}}} symmetric polarized photons

Non-Hermitian systems with parity-time mathcal {(PT)} symmetry and anti-parity-time mathcal {(APT)} symmetry have exceptional points (EPs) resulting from eigenvector co-coalescence with exceptional properties. In the quantum and classical domains, higher-order EPs for {{mathcal {P}}}{{mathcal {T}}} symmetry and mathcal {APT}-symmetry systems have been proposed and realized. Both two-qubits mathcal {APT}-mathcal {APT} and {{mathcal {P}}}{{mathcal {T}}}-{{mathcal {P}}}{{mathcal {T}}} symmetric systems have seen an increase in recent years, especially in the dynamics of quantum entanglement. However, to our knowledge, neither theoretical nor experimental investigations have been conducted for the dynamics of two-qubits entanglement in the {{mathcal {P}}}{{mathcal {T}}}-mathcal {APT} symmetric system. We investigate the {{mathcal {P}}}{{mathcal {T}}}-mathcal {APT} dynamics for the first time. Moreover, we examine the impact of different initial Bell-state conditions on entanglement dynamics in {{mathcal {P}}}{{mathcal {T}}}-{{mathcal {P}}}{{mathcal {T}}}, mathcal {APT}-mathcal {APT} and {{mathcal {P}}}{{mathcal {T}}}-mathcal {APT} symmetric systems. Additionally, we conduct a comparative study of entanglement dynamics in the {{mathcal {P}}}{{mathcal {T}}}-{{mathcal {P}}}{{mathcal {T}}} symmetrical system, mathcal {APT}-mathcal {APT} symmetrical system, and {{mathcal {P}}}{{mathcal {T}}}-mathcal {APT} symmetrical systems in order to learn more about non-Hermitian quantum systems and their environments. Entangled qubits evolve in a {{mathcal {P}}}{{mathcal {T}}}-mathcal {APT} symmetric unbroken regime, the entanglement oscillates with two different oscillation frequencies, and the entanglement is well preserved for a long period of time for the case when non-Hermitian parts of both qubits are taken quite away from the exceptional points.

Enhanced eigenvector sensitivity and algebraic classification of sublattice-symmetric exceptional points

Exceptional points (EPs) are degeneracy of non-Hermitian Hamiltonians, at which the eigenvalues, along with their eigenvectors, coalesce. Their orders are given by the Jordan decomposition. Here, we focus on higher-order EPs arising in fermionic systems with a sublattice symmetry, which restricts the eigenvalues of the Hamitlonian to appear in pairs of $\lbrace E, -E\rbrace $. Thus, a naive prediction might lead to only even-order EPs at zero energy. However, we show that odd-order EPs can exist and exhibit enhanced sensitivity in the behaviour of eigenvector-coalescence in their neighbourhood, depending on how we approach the degenerate point. The odd-order EPs can be understood as a mixture of higher- and lower-valued even-order EPs. Such an anomalous behaviour is related to the irregular topology of the EPs as the subspace of the Hamiltonians in question, which is a unique feature of the Jordan blocks. The enhanced eigenvector sensitivity can be described by observing how the quantum distance to the target eigenvector converges to zero. In order to capture the eigenvector-coalescence, we provide an algebraic method to describe the conditions for the existence of these EPs. This complements previous studies based on resultants and discriminants, and unveils heretofore unexplored structures of higher-order exceptional degeneracy.

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.