Exceptional Points Of Order Research Articles

General Conditions to Realize Exceptional Points of Degeneracy in Two Uniform Coupled Transmission Lines

We present the general conditions to realize a fourth order exceptional point of degeneracy (EPD) in two uniform (i.e., invariant along z) lossless and gainless coupled transmission lines (CTLs), namely, a degenerate band edge (DBE). Until now the DBE has been shown only in periodic structures. In contrast, the CTLs considered here are uniform and subdivided into four cases where the two TLs support combinations of forward propagation, backward propagation and evanescent modes (when neglecting the mutual coupling). We demonstrate for the first time that a DBE is supported in uniform CTLs when there is proper coupling between: (i) propagating modes and evanescent modes, (ii) forward and backward propagating modes, or (iii) four evanescent modes (two in each direction). We also show that the loaded quality factor of uniform CTLs exhibiting a fourth order EPD at k=0 is robust to series losses due to the fact that the degenerate modes do not advance in phase. We also provide a microstrip possible implementation of a uniform CTL exhibiting a DBE using periodic series capacitors with very sub-wavelength unit-cell length. Finally, we show an experimental verification of the existence DBE for a microstrip implementation of a CTL supporting coupled propagating and evanescent modes.

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.

Topological dynamics of an adiabatically varying Hamiltonian around third order exceptional points

We report an open three-state perturbed system that depends on the topological parameters, where the underlying Hamiltonian is varying quasi-statically. The effective system hosts two second order exceptional points (EP2s). Here a third order exceptional point (EP3) is encountered with simultaneous encirclement of two EP2s by adiabatic variation of topological parameters. We study the robust successive state-exchange around the EP3. Maintaining adiabaticity, we estimate the evolution of total phase accumulated by each of the interacting states during encirclement; where interestingly, the state common to the pairs of coupled state picks up three times phase shift of 2π as a signature of EP3. Such an exclusively reported scheme once implemented in an anisotropic optical waveguide can be exploited in potential applications of mode conversion, switching and lasing by manipulating topological parameters.

Topological states at exceptional points

We consider an N-level non-Hermitian Hamiltonian with an exceptional point of order N. We define adiabatic equivalence in such systems and explore topological phase. We show that the topological exceptional states appear at the interface of topologically distinct systems. We discuss that topological states appear even in closed systems. We explore dynamical robustness of exceptional edge states.

Encounter of higher order exceptional singularities and towards cascaded state conversion

The appearance of topological singularities, namely exceptional points (EPs) is an intriguing feature of parameter-dependent open quantum or wave systems. EPs are the special type of non-Hermitian degeneracies where two (or more) eigenstates of the underlying system coalesce. In this paper, we present a three level non-Hermitian Hamiltonian which hosts three interacting eigenstates. The matrix elements are optimized in such a way that the intermediate eigenstate interacts with both the other states and the underlying system hosts at least two different second order EPs. This coupling scheme has proven to be highly effective to encounter higher order EPs. The impact of quasi-static parameter variation along a cyclic contour around the embedded EPs on the dynamics of interacting eigenvalues has been investigated comprehensively in the context of cascaded state conversion. Such dynamics of the eigenvalues show a clear signature of the third order EP. Moreover, we examine the accumulation of phases around the identified EPs and study the hallmark of phase exchange during cascaded state conversions accompanied by the parametric encirclement of the third order EP. A device level implementation of the scheme is possible exploiting multimode anisotropic waveguides or coupled waveguide systems.

Robust exceptional points in disordered systems

We construct a theory to introduce the concept of topologically robust exceptional points (EPs). Starting from an ordered system with N elements, we find the necessary condition to have the highest-order exceptional point, namely the N-th–order EP. Using symmetry considerations, we show that an EP associated with an ordered system is very sensitive to the disorder. Specifically, if the EP associated with the ordered system occurs at the fixed degree of non-Hermiticity , the disordered system will not have an EP at the same which puts an obstacle in front of the observation and applications of EPs. To overcome this challenge, by incorporating an asymmetric coupling we propose a disordered system that has a robust EP. The mode associated with the robust EP is extended all over the space. While our approach can be easily realized in electronic circuits and acoustics, we propose a simple experimentally feasible photonic system to realize our robust EP. Our results will open a new direction to search for topologically robust extended states (as opposed to topological localized states) and find considerable applications in direct observation of EPs, realizing topological sensors and designing robust devices for metrology.

Arbitrary order exceptional point induced by photonic spin\u2013orbit interaction in coupled resonators

Many novel properties of non-Hermitian systems are found at or near the exceptional points—branch points of complex energy surfaces at which eigenvalues and eigenvectors coalesce. In particular, higher-order exceptional points can result in optical structures that are ultrasensitive to external perturbations. Here we show that an arbitrary order exceptional point can be achieved in a simple system consisting of identical resonators placed near a waveguide. Unidirectional coupling between any two chiral dipolar states of the resonators mediated by the waveguide mode leads to the exceptional point, which is protected by the transverse spin–momentum locking of the guided wave and is independent of the positions of the resonators. Various analytic response functions of the resonators at the exceptional points are experimentally manifested in the microwave regime. The enhancement of sensitivity to external perturbations near the exceptional point is also numerically and analytically demonstrated.

Complex symmetric Hamiltonians and exceptional points of order four and five

In the broad context of physics ranging from classical experimental optics to quantum mechanics of unitary as well as non-unitary systems there emerge interesting phenomena related to the presence of the so called Kato's exceptional points in the space of parameters. An elementary linear-algebraic method of their localization is proposed and applied to the class of tridiagonal $N$ by $N$ complex symmetric toy-model generators of evolution $H=H(\gamma)$. The implementation of the method is shown to provide new models with the exceptional points $\gamma=\gamma^{(EP)}$ of higher orders. Two distinct areas of applicability are expected to lie (1) in quantum mechanics of non-Hermitian (open as well as closed) systems, and (2) in the experiments using the coupled classical optical waveguides simulating the EP-related effects in the laboratory.

Non-Hermitian lattices with a flat band and polynomial power increase [Invited

In this work we first discuss systematically three general approaches to construct a non-Hermitian flat band, defined by its dispersionless real part. They resort to, respectively, spontaneous restoration of non-Hermitian particle-hole symmetry, a persisting flat band from the underlying Hermitian system, and a compact Wannier function that is an eigenstate of the entire system. For the last approach in particular, we show the simplest lattice structure where it can be applied, and we further identify a special case of such a flat band where every point in the Brillouin zone is an exceptional point of order 3. A localized excitation in this "EP3 flat band" can display either a conserved power, quadratic power increase, or even quartic power increase, depending on whether the localized eigenstate or one of the two generalized eigenvectors is initially excited. Nevertheless, the asymptotic wave function in the long time limit is always given by the eigenstate, and in this case, the compact Wannier function or its superposition in two or more unit cells.

Theory of coupled resonator optical waveguides exhibiting high-order exceptional points of degeneracy

We present a novel approach and a theoretical framework for generating high order exceptional points of degeneracy (EPD) in photonic structures based on periodic coupled resonators optical waveguides (CROWs). Such EPDs involve the coalescence of Floquet-Bloch eigenwaves in CROWs, without the presence of gain and loss, which is in contrast to the requirement of Parity-Time (PT) symmetry to develop exceptional points based on gain and loss balance. The EPDs arise here by introducing symmetry breaking in a conventional chain of coupled resonators through coupling the chain of resonators to an adjacent uniform optical waveguide, which leads to unique modal characteristics that cannot be realized in conventional CROWs. Such remarkable characteristics include high quality factors (Q-factor) and strong field enhancement, even without any mirrors at the two ends of a cavity. We show for the first time the capability of CROWs to exhibit EPDs of various order; including the degenerate band edge (DBE) and the stationary inflection point (SIP). The proposed CROW of finite length shows enhanced quality factor when operating near the DBE, and the Q-factor exhibits an anomalous scaling with the CROW's length. We develop the theory of EPDs in such unconventional CROW using coupled-wave equations, and we derive an analytical expression for the dispersion relation. The proposed unconventional CROW concepts have various potential applications including Q-switching, nonlinear devices, lasers, and extremely sensitive sensors.

A model of three coupled wave guides and third order exceptional points

A -symmetric model for three interacting wave guides is investigated. Each wave guide is represented by an attractive δ-function potential being in equidistant positions. The two outer potentials are complex describing loss and gain, respectively. The real parts of the outer potentials are assumed to be equal. The major focus of the study lies on the occurrence of an exceptional point of third order and the physical effects of such singularity. While some results resemble those from similar studies with two wave guides, the three wave guides appear to have a richer structure. Emphasis is placed on the fine tuning in the approach of the EP3 as this appears to be a particular challenge for an experimental realization.

Light transport inPT-invariant photonic structures with hidden symmetries

We introduce a recursive bosonic quantization technique for generating classical PT photonic structures that possess hidden symmetries and higher order exceptional points. We study light transport in these geometries and we demonstrate that perfect state transfer is possible only for certain initial conditions. Moreover, we show that for the same propagation direction, left and right coherent transports are not symmetric with field amplitudes following two different trajectories. A general scheme for identifying the conservation laws in such PT-symmetric photonic networks is also presented.

Harmonic inversion analysis of exceptional points in resonance spectra

The spectra of, e.g. open quantum systems are typically given as the superposition of resonances with a Lorentzian line shape, where each resonance is related to a simple pole in the complex energy domain. However, at exceptional points two or more resonances are degenerate and the resulting non-Lorentzian line shapes are related to higher order poles in the complex energy domain. In the Fourier-transform time domain an nth order exceptional point is characterized by a non-exponentially decaying time signal given as the product of an exponential function and a polynomial of degree n − 1. The complex positions and amplitudes of the non-degenerate resonances can be determined with high accuracy by application of the nonlinear harmonic inversion method to the real-valued resonance spectra. We extend the harmonic inversion method to include the analysis of exceptional points. The technique yields, in the energy domain, the amplitudes of the higher order poles contributing to the spectra, and, in the time domain, the coefficients of the polynomial characterizing the non-exponential decay of the time signal. The extended harmonic inversion method is demonstrated on two examples, viz. the analysis of exceptional points in resonance spectra of the hydrogen atom in crossed magnetic and electric fields, and an exceptional point occurring in the dynamics of a single particle in a time-dependent harmonic trap.

Exceptional points for cocompact Fuchsian groups

Let G be a cocompact Fuchsian group acting on the hyperbolic plane H. If G covers a compact hyperbolic surface of genus g ≥ 2, then almost every Dirichlet region for G has 12g −6 sides. In this article, we study the exceptional points for G, i.e., the points in H associated to Dirichlet regions for G with strictly less than 12g − 6 sides. More specifically, we show that uncountably many exceptional points exist for any cocompact group. We also define and prove the existence of higher order exceptional points for any such group.

A non-Hermitian symmetric Bose–Hubbard model: eigenvalue rings from unfolding higher-order exceptional points

We study a non-Hermitian symmetric generalization of an N-particle, two-mode Bose–Hubbard system, modeling for example a Bose–Einstein condensate in a double well potential coupled to a continuum via a sink in one of the wells and a source in the other. The effect of the interplay between the particle interaction and the non-Hermiticity on characteristic features of the spectrum is analyzed drawing special attention to the occurrence and unfolding of exceptional points (EPs). We find that for vanishing particle interaction there are only two EPs of order N + 1 which under perturbation unfold either into [(N + 1)/2] eigenvalue pairs (and in the case of N + 1 odd, into an additional zero-eigenvalue) or into eigenvalue triplets (third-order eigenvalue rings) and (N + 1) mod 3 single eigenvalues, depending on the direction of the perturbation in parameter space. This behavior is described analytically using perturbational techniques. More general EP unfoldings into eigenvalue rings up to (N + 1)th order are indicated.