Spectral Singularities Research Articles

Direct observation of exceptional points in photonic crystal by cross-polarization imaging in momentum space

This study explores exceptional points (EPs) in photonic crystals (PhCs) and introduces an experimental technique for their single-shot observation. Exceptional points are spectral singularities found in non-Hermitian systems, such as leaky PhC slabs. However, directly observing EPs in PhC systems using regular reflectivity spectroscopy is a considerable challenge due to interference between guided resonances and background signals. In this work, we present a simple, nondestructive technique that employs crossed polarizations to directly observe EPs in momentum-resolved resonant scattering. This approach effectively suppresses the background signal, enabling exclusive probing of the guided resonances where EPs manifest. Our results demonstrate the formation of EPs in both energy-momentum mapping and isofrequency imaging. All experimental findings align seamlessly with numerical simulations and analytical models. Our approach holds great potential as a robust tool for studying non-Hermitian physics in the PhC platform.

Spectrum and Spectral Singularities of a Quadratic Pencil of a Schrödinger Operator with Boundary Conditions Dependent on the Eigenparameter

Spectrum and Spectral Singularities of a Quadratic Pencil of a Schrödinger Operator with Boundary Conditions Dependent on the Eigenparameter

Engineering low-temperature proximity effect in clean metals by spectral singularities

The present study investigates the behavior of the Cooper pair wave function in a normal metal (NM) near superconductor-NM-junctions, specifically focusing on the ballistic regime at zero temperature. It is widely assumed that the wave function follows a power-law decay, with the decay exponents dependent on the system’s dimensionality. Our work reveals that the multiband nature of a compound significantly influences the damping degree of pair amplitudes in an NM, rendering it sensitive to the position of the Fermi level. To explore this phenomenon, we employ the numerical method of self-consistent Bogoliubov–de Gennes equations, utilizing a nanowire as a model for an electronic multiband system. By analyzing the obtained pair amplitudes, we extract relevant lengths and exponents that characterize the leakage of superconducting correlations. We further examine this phenomenon by varying the sample’s cross-sectional size and the superconducting coupling constant. Consequently, our findings demonstrate that the properties of a superconducting/NM junction’s proximity effect can be manipulated not only through temperature, total impurity and defect density, but also by controlling the position of the Fermi level. This tunability enables the transition from a long-range regime to a short-range one, providing valuable insights for designing and understanding such junctions in practical applications.

Emergence of chimeras: An impetus by exceptional points.

A nonconservative system with nonreciprocal interaction has been found to reveal exotic features where sudden phase transitions can occur. In this paper, the emanation of a chimera in a network of Stuart-Landau oscillators with nonreciprocal interaction is reported. Note that the spins follow the random discrete distribution. In other words, we pick a random oscillator to rotate clockwise or anticlockwise. At the transition points, we find the spectral singularities in the eigenplane, where eigenvalues coalesce, commonly known as exceptional points. We find that the counterrotational symmetry breaking induced by exceptional points antecedents the occurrence of the chimera. We numerically attest to the findings for two cases of initial conditions, namely, bipartite and random. We also extend our study to a two-dimensional array of nonreciprocally interacting, distributed spins. The findings could have pragmatic implications in the areas of active matter, networks, and photonics.

Nonlinear Exceptional Points with a Complete Basis in Dynamics.

Exceptional points (EPs) are special spectral singularities at which two or more eigenvalues, and their corresponding eigenvectors, coalesce and become identical. In conventional wisdom, the coalescence of eigenvectors inevitably leads to the loss of completeness of the eigenbasis. Here, we show that this scenario breaks down in general at nonlinear EPs (NEPs). As an example, we realize a fifth-order NEP (NEP_{5}) within only three coupled resonators with both a theoretical model and simulations in circuits. One stable and another four auxiliary steady eigenstates of the nonlinear Hamiltonian coalesce at the NEP_{5}, and the response of eigenfrequency to perturbations demonstrates a fifth-order root law. Intriguingly, the biorthogonal eigenbasis of the Hamiltonian governing the system dynamics is still complete, and this fact is corroborated by a finite Petermann factor instead of a divergent one at conventional EPs. Consequently, the amplification of noise, which diverges at other EPs, converges at our NEP_{5}. Our finding transforms the understanding of EPs and shows potential for miniaturizing various key applications operating near EPs.

High-order spectral singularity

Exceptional point and spectral singularity are two types of singularity that are unique to non-Hermitian systems. Here, we report the high-order spectral singularity as a high-order pole of the scattering matrix for a non-Hermitian scattering system, and the high-order spectral singularity is a unification of the exceptional point and spectral singularity. At the high-order spectral singularity, the scattering coefficients have high-order divergence and the scattering system stimulates high-order lasing. The wave emission intensity is polynomially enhanced, and the order of the growth in the polynomial intensity linearly scales with the order of the spectral singularity. Furthermore, the coherent input controls and alters the order of the spectral singularity. Our findings provide profound insights into the fundamentals and applications of high-order spectral singularities.

Spectral phase singularity and topological behavior in perfect absorption

Spectral phase singularity and topological behavior in perfect absorption

Mechanism and tuning sensitivity of symmetry-protected resonances in high-contrast gratings.

We develop a theory of refractive index tuning for symmetry-protected optical bound states (SP-BICs) in high-contrast gratings (HCGs). A compact analytical formula for tuning sensitivity is derived and verified numerically. We also discover a new type of SP-BIC in HCGs that has an accidental nature with a spectral singularity, which is explained in terms of hybridization and strong coupling among the odd- and even-symmetric waveguide-array modes. Our work elucidates the physics of tuning SP-BICs in HCGs and significantly simplifies their design and optimization for dynamic applications in light modulation, tunable filtering, and sensing.

Spherical Induced Ensembles with Symplectic Symmetry

We consider the complex eigenvalues of the induced spherical Ginibre ensemble with symplectic symmetry and establish the local universality of these point processes along the real axis. We derive scaling limits of all correlation functions at regular points both in the strong and weak non-unitary regimes as well as at the origin having spectral singularity. A key ingredient of our proof is a derivation of a differential equation satisfied by the correlation kernels of the associated Pfaffian point processes, thereby allowing us to perform asymptotic analysis.

Polar magneto-optic Kerr and Faraday effects in finite periodic PT -symmetric systems

We discuss the anomalous behavior of the Faraday (transmission) and polar Kerr (reflection) rotation angles of the propagating light, in finite periodic parity-time ($\mathcal{P}\mathcal{T}$) symmetric structures, consisting of $N$ cells. The unit cell potential is two complex $\delta$-potentials placed on both boundaries of the ordinary dielectric slab. It is shown that, for a given set of parameters describing the system, a phase transition-like anomalous behavior of Faraday and Kerr rotation angles in a parity-time symmetric systems can take place. In the anomalous phase the value of one of the Faraday and Kerr rotation angles can become negative, and both angles suffer from spectral singularities and give a strong enhancement near the singularities. We also shown that the real part of the complex angle of KR, $\theta^{R}_1$, is always equal to the $\theta^{T}_1$ of FR, no matter what phase the system is in due to the symmetry constraints. The imaginary part of KR angles $\theta^{R^{r/l}}_2$ are related to the $\theta^{T}_2$ of FR by parity-time symmetry. Calculations based on the approach of the generalized nonperturbative characteristic determinant, which is valid for a layered system with randomly distributed delta potentials, show that the Faraday and Kerr rotation spectrum in such structures has several resonant peaks. Some of them coincide with transmission peaks, providing simultaneous large Faraday and Kerr rotations enhanced by an order one or two of magnitude. We provide a recipe for funding a one-to-one relation in between KR and FR.

Negatif Yoğunluk Fonksiyonuna Sahip Kendine Eşlenik Olmayan Schrödinger Operatörü Üzerine Bir Çalışma

This study focuses on the spectral features of the non-selfadjoint singular operator with an out-of-the-ordinary type weight function. Take into consideration the one-dimensional time-dependent Schrödinger type differential equation
 -y^''+q(x)y=μ^2 ρ(x)y,x∈[0,∞),
 holding the initial condition
 y(0)=0,
 and the density function defined with a completely negative value as
 ρ(x)=-1.
 There is an enormous number of the papers considering the positive values of ρ(x) for both continuous and discontinuous cases. The structure of the density function affects the analytical properties and representations of the solutions of the equation. Unlike the classical literature, we use the hyperbolic type representations of the equation’s fundamental solutions to obtain the operator’s spectrum. Additionally, the requirements for finiteness of eigenvalues and spectral singularities are addressed. Hence, Naimark’s and Pavlov’s conditions are adopted for the negative density function case.

Chiral spectral singularities spawning from quasi-bound states in the continuum in PT-symmetric dielectric metasurfaces

We investigate the chiral spectral singularities, i.e., laser threshold modes, in PT-symmetric dielectric metasurfaces originating from quasi-bound states in the continuum. The poles, referred to as the quasi-bound states in the continuum, of scattering matrix can move to the upper complex frequency plane from the lower half-plane by increasing the balanced gain and loss. The maximal intrinsic optical chirality of laser threshold mode has been theoretically demonstrated via the large transmission circular dichroism spectra. Our work paves the way for studying the enhanced optical chirality in non-Hermitian nanophotonics empowered by quasi-bound states in the continuum, together with various intriguing applications, such as chiral coherent perfect absorber and laser.

Signature of exceptional point phase transition in Hermitian systems

Exceptional point (EP) is a spectral singularity in non-Hermitian systems. The passing over the EP leads to a phase transition, which endows the system with unconventional features that find a wide range of applications. However, the need of using the dissipation and amplification limits the possible applications of systems with the EP. In this work, we demonstrate an existence of signature of exceptional point phase transition in Hermitian systems that are free from dissipation and amplification. We consider a composite Hermitian system including both two coupled oscillators and their environment consisting only of several tens of degrees of freedom. We show that the dynamics of such a Hermitian system demonstrate a transition, which occurs at the coupling strength between oscillators corresponding to the EP in the non-Hermitian system. This transition manifests itself even in the non-Markovian regime of the system dynamics in which collapses and revivals of the energy occur. Thus, we demonstrate that the phase transition occurring at the passing over the EP in the non-Hermitian system manifests itself in the Hermitian system at all time. We discuss the experimental scheme to observe the signature of EP phase transition in the non-Markovian regime.

Engineering Higgs dynamics by spectral singularities

We generalize the dynamical phase diagram of a Bardeen-Cooper-Schrieffer condensate, considering attractive to repulsive, i.e., critical quenches (CQ) and a non-constant density of states (DOS). We show that different synchronized Higgs dynamical phases can be stabilized, associated with singularities in the density of states (DOS) and different quench protocols. In particular, the CQ can stabilize an overlooked high-frequency Higgs dynamical phase related to the upper edge of the fermionic band. For a compensated Dirac system we find a Dirac-Higgs mode associated with the cusp singularity at the Fermi level, and we show that synchronized phases become more pervasive across the phase diagram. The relevance of these remarkable phenomena and their realization in ensembles of fermionic cold atoms confined in optical lattices is also discussed.

On the Spectrum of the Non-Selfadjoint Differential Operator with an Integral Boundary Condition and Negative Weight Function

In this paper, we shall study the spectral properties of the non-selfadjoint operator in the space $L_{\varrho }^{2}\left(\mathbb{R}_{+}\right) $ generated by the Sturm-Liouville differential equation \begin{equation*} -y^{^{\prime \prime }}+q\left( x\right) y=\omega ^{2}\varrho \left( x\right) y, \quad x \in \mathbb{R}_{+} \end{equation*} with the integral type boundary condition \begin{equation*} \int \limits_{0}^{\infty }G\left( x \right) y\left( x\right) dx+ \gamma y^{\prime }\left( 0\right) -\theta y\left( 0\right) =0 \end{equation*} and the non-standard weight function \begin{equation*} \varrho \left( x\right) =-1 \end{equation*} where $\left \vert \gamma \right \vert +\left \vert \theta \right \vert \neq 0$. There are an enormous number of papers considering the positive values of $ \varrho \left( x\right) $ for both continuous and discontinuous cases. The structure of the weight function affects the analytical properties and representations of the solutions of the equation. Differently from the classical literature, we used the hyperbolic type representations of the fundamental solutions of the equation to obtain the spectrum of the operator. Moreover, the conditions for the finiteness of the eigenvalues and spectral singularities were presented. Hence, besides generalizing the recent results, Naimark's and Pavlov's conditions were adopted for the negative weight function case.

Tunneling time in PT -symmetric systems

In the present work, we propose a generalization of tunneling time in parity and time $(\mathcal{P}\mathcal{T})$-symmetric systems. The properties of tunneling time in $\mathcal{P}\mathcal{T}$-symmetric systems are studied with a simple contact interaction periodic finite-size diatomic $\mathcal{P}\mathcal{T}$-symmetric model. The physical meaning of negative tunneling time in $\mathcal{P}\mathcal{T}$-symmetric systems and its relation to spectral singularities is discussed.

Some Spectral Properties of Schrödinger Operators on Semi Axis

The main aim of this work is to investigate some spectral properties of Schrödinger operators on semi axis. We first present the Schrödinger equation with a piecewise continuous potential function q so that the problem differs from the classical Schrödinger problems. Then, we get the Wronskian of two specific solution of the given equation which helps us to create the sets of eigenvalues and spectral singularities. The rest of the paper deals with eigenvalues and spectral singularities. By the help of the analytical properties of Jost solutions and resolvent operator of the Schrödinger operators, we provide sufficient conditions guarenteeing finiteness of eigenvalues and spectral singularities with finite multiplicities.

On the Bifurcation of Thresholds of the Essential Spectrum with a Spectral Singularity

We consider the Schrödinger operator on the plane with bounded potential, where is a real potential, and are compactly supported complex potentials, andis a small parameter, assuming that the lower part of the spectrum of the one-dimensional Schrödinger operator consists of a pair of isolated eigenvalues and the essential spectrum of the operator has a virtual level at its lower edge and a spectral singularity inside. Additionally, we assume that there is a certain superposition of eigenvalues of the operator with the virtual level and spectral singularity of the operator; this leads to the emergence of a special threshold in the essential spectrum of the perturbed operator, with the perturbation leading to a bifurcation of this threshold into eigenvalues and resonances with multiplicity doubling. The bifurcation scenario described in this paper is qualitatively different from the previously known ones.

Spectral Singularities of the Photoelectric Effect in a Zinc Phthalocyanine–Fullerene (ZnPc:C70) Donor–Acceptor Blend

Spectral singularities of the ampere–watt sensitivity of photoelectric structures consisting of a transparent indium–tin oxide electrode, a photosensitive organic layer, and an aluminum electrode have been studied. The structures have been formed on a quartz glass substrate. The photosensitive layer has been vacuum-evaporated either from zinc phthalocyanine ZnPc (exhibiting donor properties) and C70 fullerene (acceptor) organic precursors or from a ZnPc:Cr70 donor–acceptor blend. Using computer simulation, the structure of absorption bands has been determined in a wide spectral range for all three above systems. This has made it possible to calculate the absorbed and reflected fractions of radiation incident on the sample and explain the singular spectral behavior of the ampere–watt sensitivity of the ZnPc:C70 blend. It has been shown that the photosensitivity of the blend reaches a maximum near the overlap of the absorption bands of donor and acceptor molecules

On the Bifurcation of Thresholds of the Essential Spectrum with a Spectral Singularity

On the Bifurcation of Thresholds of the Essential Spectrum with a Spectral Singularity