Multiple Exceptional Points Research Articles

Proximity-encirclement of exceptional points in a multimode optomechanical system

Dynamic encirclement of second-order exceptional points (EPs) exhibits chiral state transfer; however, investigations into the dynamics involving multiple and higher-order EPs remain sparse. Here, we study the proximity-encirclement of EPs within a multimode optomechanical system to elucidate the closed-path evolution in high-order non-Hermitian systems. Our optomechanical framework presents three distinct EP scenarios: absence of EPs, presence of a pair of second-order EPs, and the existence of a third-order EP. We meticulously analyze the system’s dynamic behavior, considering variables such as initial state, loop orientation and velocity, loop starting point variance, and the number and order of encircled EPs during state transfer processes. The findings reveal that chiral or non-reciprocal state transfer can be achieved when a loop encircles a second-order EP with varying radii. Encircling two second-order EPs results exclusively in chiral state transfer. Furthermore, both chiral and non-reciprocal state transfers are observed within a single loop encircling a third-order EP. These phenomena in the context of multimode optomechanical systems provide a new approach for manipulating state transfer in higher-order non-Hermitian systems.

Dynamical encircling of multiple exceptional points in anti-PT symmetry system

Exceptional points (EPs) in non-Hermitian systems have turned out to be at the origin of many intriguing effects with no counterparts in Hermitian cases. A typically interesting behavior is the chiral mode switching by dynamically winding the EP. Most encircling protocols focus on the two-state or parity-time (PT) symmetry systems. Here, we propose and investigate the dynamical encircling of multiple EPs in an anti-PT-symmetric system, which is constructed based on a one-dimensional lattice with staggered lossy modulation. We reveal that dynamically encircling the multiple EPs results in the chiral dynamics via multiple non-Hermiticity-induced nonadiabatic transitions, where the output state is always on the lowest-loss energy sheet. Compared with the PT-symmetric systems that require complicated variation of the gain/loss rate or on-site potentials, our system only requires modulations of the couplings which can be readily realized in various experimental platforms. Our scheme provides a route to study non-Hermitian physics by engineering the EPs and implement novel photonic devices with unconventional functions.

Regulation of multiple exceptional points in a plasmonic quadrumer

Exceptional points (EPs), which signify the singularity of eigenvalues and eigenstates in non-Hermitian systems, have garnered considerable attention in two-state systems, revealing a wealth of intriguing phenomena. However, the potential of EPs in multi-state systems, particularly their interaction and coalescence, has been underexplored, especially in the context of electromagnetic fields where far-field coupling can revolutionize spatial wave control. Here, we theoretically and computationally explore the coalescence of multiple EPs within a designer surface plasmonic quadrumer system. The coupled mode model shows that the multiple EPs can emerge and collide as the system parameters vary, leading to higher-order singularities. Numerically calculated results showcase that multiple EPs with different orders have special far-field responses. This pioneering strategy heralds a new era of wavefront engineering in non-Hermitian photonic structures, presenting a transformative class of radiative systems that transcend the conventional frequency spectrum from microwave to optical realms.

Resolving the topology of encircling multiple exceptional points

Non-Hermiticity has emerged as a new paradigm for controlling coupled-mode systems in ways that cannot be achieved with conventional techniques. One aspect of this control that has received considerable attention recently is the encircling of exceptional points (EPs). To date, most work has focused on systems consisting of two modes that are tuned by two control parameters and have isolated EPs. While these systems exhibit exotic features related to EP encircling, it has been shown that richer behavior occurs in systems with more than two modes. Such systems can be tuned by more than two control parameters, and contain EPs that form a knot-like structure. Control loops that encircle this structure cause the system’s eigenvalues to trace out non-commutative braids. Here we consider a hybrid scenario: a three-mode system with just two control parameters. We describe the relationship between control loops and their topology in the full and two-dimensional parameter space. We demonstrate this relationship experimentally using a three-mode mechanical system in which the control parameters are provided by optomechanical interaction with a high-finesse optical cavity.

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.

Gain-loss-induced non-Abelian Bloch braids

Onsite gain-loss-induced topological braiding principle of non-Hermitian energy bands is theoretically formulated in multiband lattice models with Hermitian hopping amplitudes. Braid phase transition occurs when the gain-loss parameter is tuned across exceptional point degeneracy. Laboratory realizable effective-Hamiltonians are proposed to realize braid groups B2 and B3 of two and three bands, respectively. While B2 is trivially Abelian, the group B3 features non-Abelian braiding and energy permutation originating from the collective behavior of multiple exceptional points. Phase diagrams with respect to lattice parameters to realize braid group generators and their non-commutativity are shown. The proposed theory is conducive to synthesizing exceptional materials for applications in topological computation and information processing.

Asymmetric Switching of Edge Modes by Dynamically Encircling Multiple Exceptional Points

Dynamically encircling exceptional points (EPs) in the parameter space of a non-Hermitian system has drawn widespread attention for realizing asymmetric mode switching and related applications. While previous works have been restricted to two-level systems, the switching between edge states in a multistate system that contains more complex dynamics remains to be explored. Herein, asymmetric switching of edge modes in multistate systems is demonstrated by encircling multiple EPs. A multistate non-Hermitian system containing multiple EPs is constructed based on a one-dimensional lattice, which can support four edge-localized modes. Dynamically encircling the EPs in parameter space leads to a specific edge mode surviving to the end, determined by the encircling direction. Correspondingly, optical waveguide arrays are designed to investigate mode evolution through refractive index modulation. Simulated results show that clockwise (counterclockwise) encirclement leads to an output mode localized at the left (right) side of the waveguide array with a phase difference of 0 (\ensuremath{\pi}), regardless of the input modes; this excellently illustrates the asymmetric switching of edge modes in a multistate system. Moreover, such an effect can be extended to the multistate system with an arbitrary number of EPs. This work enables investigation of interesting effects in non-Hermitian physics by engineering the EPs and topological properties, especially for complex multistate systems, which will be useful for developing functional nanophotonic devices.

All-dielectric apodized photonic crystals: A nondissipative pseudo-Hermitian system hosting multiple exceptional points

Optical systems obeying non-Hermitian dynamics have been the subject of intense and concerted investigation over the last two decades owing to their broad implications in photonics, acoustics, electronics as well as atomic physics. A vast majority of such investigations rely on a dissipative, balanced loss-gain system which introduces unavoidable noise and consequently, this limits the coherent control of propagation dynamics. Here, we show that an all-dielectric, non-dissipative photonic crystal (PC) could host, at least two exceptional points in its eigenvalue spectrum. By introducing optimum apodization in the PC architecture, namely 1D-APC, we show that such a configuration supports a spectrum of exceptional points which distinctly demarcates the PT-symmetric region from the region where PT -symmetry is broken in the parameter space. The analytical framework allows us to estimate the geometric phase of the reflected beam and derive the constraint that governs the excitation of topologically-protected optical Tamm-plasmon modes in 1D-APCs.

Exceptional Non-Abelian Topology in Multiband Non-Hermitian Systems.

Defective spectral degeneracy, known as exceptional point (EP), lies at the heart of various intriguing phenomena in optics, acoustics, and other nonconservative systems. Despite extensive studies in the past two decades, the collective behaviors (e.g., annihilation, coalescence, braiding, etc.) involving multiple exceptional points or lines and their interplay have been rarely understood. Here we put forward a universal non-Abelian conservation rule governing these collective behaviors in generic multiband non-Hermitian systems and uncover several counterintuitive phenomena. We demonstrate that two EPs with opposite charges (even the pairwise created) do not necessarily annihilate, depending on how they approach each other. Furthermore, we unveil that the conservation rule imposes strict constraints on the permissible exceptional-line configurations. It excludes structures like Hopf link yet permits novel staggered rings composed of noncommutative exceptional lines. These intriguing phenomena are illustrated by concrete models which could be readily implemented in platforms like coupled acoustic cavities, optical waveguides, and ring resonators. Our findings lay the cornerstone for a comprehensive understanding of the exceptional non-Abelian topology and shed light on the versatile manipulations and applications based on exceptional degeneracies in nonconservative systems.

Proximity‐Encirclement of Multiple Exceptional Points in Non‐Hermitian Photonic Waveguides

AbstractConventionally, dynamical encirclement of exceptional points in non‐Hermitian systems is known to manifest a counterintuitive chiral state conversion. However, the prerequisite of such traits enclosing an exceptional point is broken when only encircling its proximity, preserving a still chiral switching. Research on the proximity‐encirclement in multistate systems is lacking. In this paper, a photonic‐waveguide‐array non‐Hermitian system is proposed to investigate the dynamics by encircling two exceptional points or their proximity. A series of encircling trajectories defined by the parametric equations are designed to steer the evolution of photonic modes in waveguides. The wave propagating along the waveguides is also simulated to capture this non‐Hermitian physics. The chiral behavior in proximity‐encirclement contrasts with the familiar encirclement of one exceptional point and exhibits the unexpected occurrence of nonadiabatic transitions. Furthermore, if two exceptional points are sufficiently encircled, the system will evolve to a stable final state earlier, as a symbol of the occurrence of the nonadiabatic transition. Such novel chiral conversion is maintained only if the encircling trajectories are located at adequate proximity.

Multiple Exceptional Points in APT–Symmetric Cantor Multilayers

In this study, we explore the anisotropic reflection of light waves around the exceptional points (EPs) in anti-parity-time−symmetric (APT−symmetric) Cantor dielectric multilayers. This one-dimensional fractal structure governed by the Cantor substitution law is modulated to satisfy APT symmetry. The Cantor multilayers are aperiodic and support optical fractal resonances. The optical fractal effect combined with APT symmetry can induce multiple exceptional points (EPs) in the parameter space by modulating the loss coefficient of materials and optical frequency. Reflection anisotropy for light waves incident from two opposite directions presents unidirectional suppression and enhancement around EPs. This study can be utilized for multiple wavelengths of photonic suppressors and reflectors.

Topology of multipartite non-Hermitian one-dimensional systems

The multipartite non-Hermitian Su-Schrieffer-Heeger model is explored as a prototypical example of one-dimensional systems with several sublattice sites for unveiling intriguing insulating and metallic phases with no Hermitian counterparts. These phases are characterized by composite cyclic loops of multiple complex-energy bands encircling single or multiple exceptional points (EPs) on the parametric space of real and imaginary energy. We show the topology of these composite loops is similar to well-known topological objects like M\"obius strips and Penrose triangles, and can be quantified by a nonadiabatic cyclic geometric phase which includes contributions only from the participating bands. We analytically derive a complete phase diagram with the phase boundaries of the model. We further examine the connection between encircling of multiple EPs by complex-energy bands on parametric space and associated topology.

General Rules Governing the Dynamical Encircling of an Arbitrary Number of Exceptional Points.

Dynamically encircling an exceptional point in non-Hermitian systems has drawn great attention recently, since a nonadiabatic transition process can occur and lead to intriguing phenomena and applications such as the asymmetric switching of modes. While all previous experiments have been restricted to two-state systems, the dynamics in multistate systems where more complex topology can be formed by exceptional points, is still unknown and associated experiments remain elusive. Here, we propose an on-chip photonic system in which an arbitrary number of exceptional points can be encircled dynamically. We reveal in experiment a robust state-switching rule for multistate systems, and extend it to an infinite-period system in which an exceptional line is encircled with outcomes being located at the Brillouin-zone boundary. The proposed versatile platform is expected to reveal more physics related to multiple exceptional points and exceptional lines, and give rise to applications in multistate non-Hermitian systems.

Exceptional Points through Variation of Distances between Four Coaxial Dielectric Disks

By variation of a refractive index and aspect ratio of the isolated disk, we achieved exceptional points (EPs) at which the resonant frequencies and resonant modes coalesce. However, in practice, that kind of variation presents a technological problem. We considered the method to avoid the problem by substitution of two disk’s dimers. In each dimer, variation of the distance between disks was equivalent to a variation of the aspect ratio of the dimer. Moreover, the variation of the distance between dimers provides the second parameter that gives rise to a vast number of EPs. We recovered the initial resonant eigenmode by encircling multiple EPs two, three, and four times in the two-dimensional parametric space of distances.

Floquet higher-order exceptional points and dynamics in PT-symmetric quadrimer waveguides

We investigate the effect of periodic modulation on PT-symmetry recovery, higher-order exceptional points (EPs) and dynamics for a periodically modulated PT-symmetric quadrimer waveguide system. It is analytically and numerically revealed that the periodic modulation not only can restore the broken PT symmetry, but also can manipulate the PT symmetry by tuning modulation amplitude. Furthermore, we found that multiple EPs can emerge, and as the modulation amplitude vary, these EPs can collide and merge, leading to higher-order Floquet singularities. By defining average relative light intensity distribution, we explore the dynamical characteristics nearby these EPs. It is found that we can control the light tunneling direction of optical quadrimer waveguide via periodic modulation and proper choice of gain/loss parameters.

Broadband unidirectional near-zero reflection induced by logical combination of parity-time symmetric photonic crystal

The logical combined aperiodic photonic crystal is derived by the “OR” Boolean algorithm of two periodic photonic crystals with slightly different periods, which shows broadband reflection features. Inspired by the this, here we propose an aperiodic photonic crystal with PT-symmetric configuration to investigate the broadband unidirectional reflection behavior. By analyzing the scattering matrix of the proposed structure, we reveal that the broadband near-zero reflection phenomenon is induced by the interaction of multiple exceptional points. The underlying physics of this phenomenon is exploited by investigating the electric field distribution at exceptional points. Furthermore, an interface state is also found locating at the loss/gain boundary of PT-symmetric materials. The broadband (∼1200 nm) near-zero reflection phenomenon with unidirectional property is first discussed in the PT-symmetric structure, which gives us hints for further researches of unidirectional reflection in non-Hermitian systems.

Waveguide-induced coalescence of exceptional points

In non-Hermitian systems, multiple eigenstates can merge into one state at so-called exceptional points (EPs). This paper presents the coalescence of the multiple EPs based on a waveguide. For the ring cavities located at the second-order EPs with desired chiral modes, the waveguide provides unidirectional couplings among the modes of the second-order EPs, which merges the second-order EPs into one high-order EP. The principle of this framework and properties of this system are presented by analyzing the effective Hamiltonian of the system, a mode-coupling diagram, and numerical simulations.

Nonlinearity-induced anomalous mode collapse and nonchiral asymmetric mode switching around multiple exceptional points

The dynamical encirclement around a second order exceptional point (EP) and corresponding chirality driven nonadiabatic modal dynamics have attracted enormous attention in the topological study of various non-Hermitian systems. However, dynamical encirclement around multiple second-order EPs in a multi-state system is yet to be explored. Here, exploiting an exclusive design of a planar gain-loss assisted three-mode supported optical waveguide with local Kerr-nonlinearity, we encounter multiple second-order EPs. Judiciously, choosing a specific parameter space by varying the unbalanced gain-loss profile, we encircle multiple EPs simultaneously, and explore the beam-dynamics toward corresponding chiral or non-chiral aspects of the device. While propagating through the designed waveguide, three coupled modes are collapsed into a specific dominating mode, owing to corresponding nonadiabatic corrections around multiple EPs. Even in the absence of chirality, here, the same amount of focusing and de-focusing type nonlinearity gives different dominating output, irrespective of the choice of inputs, for the same topological structure of the waveguide. This exclusive topologically robust compact scheme of nonlinearity induced asymmetric and non-chiral light dynamics should provide a promising opportunity to switch or retrieve a selective mode from a multi-mode signal in integrated devices.

Chirality breakdown in the presence of multiple exceptional points and specific mode excitation.

The dynamical parametric encirclement around a second-order exceptional point (EP) enables the time-asymmetric nonadiabatic evolution of light, which follows the chirality of the underlying system. Such light dynamics in the presence of multiple EPs and the corresponding chiral aspect is yet to be explored. In this Letter, we report a gain-loss assisted four-mode-supported optical waveguide that hosts a parameter space to dynamically encircle multiple EPs. In the presence of multiple EPs, we establish a unique nonadiabatic behavior of light, where beyond the chiral aspect of the system, light is switched to a particular mode, irrespective of the choice of the input mode. Proposed scheme certainly opens a step-forward approach in light manipulation to facilitate next-generation integrated photonic systems.

Multiple exceptional points in bulk parity-time symmetric optical media

We study bulk optical modes in media with a parity-time symmetric permittivity tensor via the k-surface approach. We show that k-surfaces support multiple exceptional points (EPs), and the off-axial EPs can be manipulated by tuning the tensor elements. They merge into diabolic points if they have opposite handednesses, and annihilate each other otherwise. The underlying physical mechanisms and the potential applications are discussed.