Coalescence Of Eigenvalues Research Articles

Clarification of the Exceptional-Point Contribution to Photonic Sensing

Exceptional points, with simultaneous coalescence of eigenvalues and eigenvectors, can be realized with non-Hermitian photonic systems. With the enhanced response, exceptional points have been proposed to improve the performance of photonic sensing. Recently, there are intense debates about the actual sensing advantage of exceptional points. The major concern is that intrinsic noise is also increased at exceptional points. Here, we aim to clarify the contribution of exceptional points for photonic sensing. This is achieved by analyzing the condition to realize divergent quantum Fisher information in linear non-Hermitian photonic systems. We show that the divergence of quantum Fisher information is the result of lasing threshold, instead of exceptional points. However, exceptional points correspond to the condition that lasing threshold is simultaneously achieved across multiple photonic modes. Therefore, exceptional points can further improve the sensitivity on top of lasing threshold. On the other hand, exceptional points alone cannot provide sensing advantage.

Subharmonic eigenvalue orbits in the spectrum of pulsating Poiseuille flow

Spectral degeneracies where eigenvalues and eigenvectors simultaneously coalesce, also known as exceptional points, are a natural consequence of the strong non-normality of the Orr–Sommerfeld operator describing the evolution of infinitesimal disturbances in parallel shear flows. While the resonances associated with these points give rise to algebraic growth, the development of non-modal stability theory exploiting specific perturbation structures with much larger potential for transient energy growth has led to waning interest in spectral degeneracies. The appearance of subharmonic eigenvalue orbits, recently discovered in the periodic spectrum of pulsating Poiseuille flow, can be traced back to the coalescence of eigenvalues at exceptional points. We present a thorough analysis of the spectral properties of the linear operator to identify exceptional points and accurately map the prevalence of subharmonic eigenvalue orbits for a large range of pulsation amplitudes and frequencies. This information is then combined with solutions of the linear initial value problem to analyse the impact of the appearance of these orbits on the temporal evolution of linear disturbances in pulsating Poiseuille flow. The periodic amplification phases are shown to be heralded by repeated non-normal growth bursts that are intensified by the formation of subharmonic orbits involving the leading eigenvalues. These bursts are associated with the change of alignment of the perturbation from the decaying towards the amplified branch of the subharmonic eigenvalue orbits in a so-called branch transition process.

Exceptional degeneracy in a waveguide periodically loaded with discrete gain and radiation loss elements

We demonstrate that a periodic waveguide comprising of uniform lossless segments together with discrete gain and radiating elements supports exceptional points of degeneracy (EPDs). We provide analytical expressions for all possible conditions that guarantee the occurrence of an EPD, i.e., the coalescence of eigenvalues and eigenvectors. We show that EPDs are not only achieved using symmetric gain and radiation periodic loading, but they are also obtained using asymmetric gain and radiation loss conditions. We illustrate the characteristics of the degenerate electromagnetic modes, showing the dispersion diagram and discussing the tunability of the EPD frequency. We show a special condition, and we refer to it as a parity-time-glide symmetry, which leads to a degeneracy that is occurring at all frequencies of operation. The class of EPDs proposed in this work is very promising for many applications that incorporate discrete-distributed coherent sources and radiation loss elements; operating in the vicinity of such special degeneracy conditions leads to a potential performance enhancement in a variety of microwave and optical resonators, antennas, and devices and can be extended to a new class of active integrated antenna arrays and radiating laser arrays.

Spectra of PT-symmetric fractional Schrödinger equations with multiple quantum wells

We evaluate the eigenvalues of a type of one-dimensional PT-symmetric fractional Schrodinger equation with multiple quantum wells potential profile. By using a finite-difference scheme, we solve the fractional Schrodinger equation and present the algorithm. We study the effects of different parameters on the pairwise coalescence of eigenvalues. We show that by using the mentioned parameters, we can tune the position of the pairwise coalescence of the eigenvalues and the surface area between the two eigenvalues that intersect. An interesting phenomenon is that a small value of the fractionality as much as 0.15 can destroy the pairwise coalescence of eigenvalues and produce a single energy level. We also, consider the Hofstadter butterfly of a PT-symmetric one-dimensional system and show that by increasing the intensity of the potential imaginary part, we can kill the butterfly.

Solitonic coalescence and rogue waves for the coupled nonlinear Schrödinger system with the negative coherent coupling in a weakly birefringent fiber

Coupled nonlinear Schrödinger system with negative coherent coupling in a weakly birefringent fiber is hereby studied. For the slowly varying envelopes of two interacting optical modes, analytic soliton solutions with the coalescence of eigenvalues are derived. Based on the asymptotic analyses, we find three kinds of the counterpropagating solitons, indicating that they consist of the so-called nondegenerate or degenerate solitons. From the plane-wave seed solutions with the certain eigenvalues, single-peak rogue waves are exhibited as well.

Coalescing points for eigenvalues of banded matrices depending on parameters with application to banded random matrix functions

In this work, we develop and implement new numerical methods to locate generic degeneracies (i.e., isolated parameters’ values where the eigenvalues coalesce) of banded matrix valued functions. More precisely, our specific interest is in two classes of problems: (i) symmetric, banded, functions A(x) ∈ ℝn×n, smoothly depending on parameters x ∈ Ω ⊂ ℝ2 and (ii) Hermitian, banded, functions A(x) ∈ ℂn×n, smoothly depending on parameters x ∈Ω⊂ ℝ3. The computational task of detecting coalescing points of banded parameter-dependent matrices is very delicate and challenging and cannot be handled using existing eigenvalues’ continuation approaches. For this reason, we present and justify new techniques that will enable continuing path of eigendecompositions and reliably decide whether or not eigenvalues coalesce, well beyond our ability to numerically distinguish close eigenvalues. As important motivation, and illustration, of our methods, we perform a computational study of the density of coalescing points for random ensembles of banded matrices depending on parameters. Relatively to random matrix models from truncated GOE and GUE ensembles, we will give computational evidence in support of power laws for coalescing points, expressed in terms of the size and bandwidth of the matrices.

Exact description of coalescing eigenstates in open quantum systems in terms of microscopic Hamiltonian dynamics

At the exceptional point where two eigenstates coalesce in open quantum systems, the usual diagonalization scheme breaks down and the Hamiltonian can only be reduced to the Jordan block form. Most of the studies on the exceptional point appearing in the literature introduce a phenomenological effective Hamiltonian that essentially reduces the problem to that of a finite non-Hermitian matrix for which it is straightforward to obtain the Jordan form. In this paper, we demonstrate how the microscopic total Hamiltonian of an open quantum system reduces to the Jordan block form at an exceptional point in an exact manner that treats the continuum without any approximation by extending the problem to include eigenstates with complex eigenvalues that reside outside the Hilbert space. Our method relies on the Brillouin-Wigner-Feshbach projection method according to which we can obtain a finite-dimensional effective Hamiltonian that shares the discrete sector of the spectrum with the total Hamiltonian. Because of the eigenvalue dependence of the effective Hamiltonian due to the dynamical nature of the coupling between the discrete states via the continuum states, a coalescence of eigenvalues results in the coalescence of the corresponding eigenvectors of the total Hamiltonian, which means that the system is at an exceptional point. We also introduce an extended Jordan form basis away from the exceptional point, which provides an alternative way to obtain the Jordan block at an exceptional point. The extended Jordan block connects continuously to the Jordan block exactly at the exceptional point implying that the observable quantities are continuous at the exceptional point.

Nearby states in non-Hermitian quantum systems I: Two states

In part I, the formalism for the description of open quantum systems (that are embedded into a common well-defined environment) by means of a non-Hermitian Hamilton operator $\ch$ is sketched. Eigenvalues and eigenfunctions are parametrically controlled. Using a 2$\times$2 model, we study the eigenfunctions of $\ch$ at and near to the singular exceptional points (EPs) at which two eigenvalues coalesce and the corresponding eigenfunctions differ from one another by only a phase. In part II, we provide the results of an analytical study for the eigenvalues of three crossing states. These crossing points are of measure zero. Then we show numerical results for the influence of a nearby ("third") state onto an EP. Since the wavefunctions of the two crossing states are mixed in a finite parameter range around an EP, three states of a physical system will never cross in one point. Instead, the wavefunctions of all three states are mixed in a finite parameter range in which the ranges of the influence of different EPs overlap. We may relate these results to dynamical phase transitions observed recently in different experimental studies. The states on both sides of the phase transition are non-analytically connected.

Bound states, scattering states, and resonant states inPT-symmetric open quantum systems

We study a simple open quantum system with a PT-symmetric defect potential as a prototype to illustrate general features of PT-symmetric open quantum systems; however, the potential could be mimicked by a number of recent PT experiments. One key feature is the resonance in continuum (RIC), which appears in both the discrete spectrum and scattering spectrum. The RIC forms a standing wave extending throughout the spatial extent of the system, representing a resonance between the open environment and the central PT-symmetric potential. We illustrate that as one deforms the system parameters, the RIC may exit the continuum by splitting into a bound state and a virtual bound state at the band edge, a process that should be experimentally observable. We also study the exceptional points (EPs) at which two eigenvalues coalesce; we categorize these as either EP2As, at which two real-valued solutions coalesce before becoming complex-valued, or EP2Bs, for which the two solutions are complex on either side of the EP. The EP2As are associated with PT-symmetry breaking; we argue that these are more stable against parameter perturbation than the EP2Bs. We also study complex-valued solutions of the discrete spectrum for which the wave function is nevertheless spatially localized, something not allowed in traditional open quantum systems; we illustrate that these may form quasi-bound states in continuum (QBICs) under some circumstances. We also study the scattering properties of the system, including states that support invisible propagation and some general features of perfect transmission states. We finally construct scattering states that satisfy PT-symmetric boundary conditions; while these states do not conserve the traditional probability current, we introduce the PT-current that is preserved. The perfect transmission states appear as a special case of the PT-symmetric scattering states.

Supersonic flutter suppression of electrorheological fluid-based adaptive panels resting on elastic foundations using sliding mode control

Brief reviews on suppressing panel flutter vibrations by various active control strategies as well as utilization tunable electrorheological fluids (ERFs) for vibration control of structural systems are presented. Active suppression of the supersonic flutter motion of a simply supported sandwich panel with a tunable ERF interlayer, and coupled to an elastic foundation, is subsequently investigated. The structural formulation is based on the classical beam theory along with the Winkler–Pasternak foundation model, the ER fluid core is modeled as a first-order Kelvin–Voigt material, and the quasi-steady first-order supersonic piston theory is employed to describe the aerodynamic loading. Hamilton’s principle is used to derive a set of fully coupled dynamic equations of motion. The generalized Fourier expansions in conjunction with the Galerkin method are then employed to formulate the governing equations in the state space domain. The critical dynamic pressures at which unstable panel oscillations (coalescence of eigenvalues) occur are obtained via the p-method for selected applied electric field strengths (E = 0,2,4 kV mm−1). The classical Runge–Kutta time integration algorithm is subsequently used to calculate the open-loop aeroelastic response of the system in various basic loading configurations (i.e. uniformly distributed blast, gust, sonic boom, and step loads), with or without an interacting soft/stiff elastic foundation. Finally, a sliding mode control synthesis (SMC) involving the first six natural modes of the structural system is set up to actively suppress the closed-loop system response in supersonic flight conditions and under the imposed excitations. Simulation results demonstrate performance, effectiveness, and insensitivity with respect to the spillover of the proposed SMC-based control system. Limiting cases are considered and good agreements with the data available in the literature as well as with the computations made by using the Rayleigh–Ritz approach are obtained.

`Positon' and `Dromion' Solutions of the (2+1) Dimensional Long Wave-Short Wave Resonance Interaction Equations

`Positon' and `dromion' solutions are derived for the long wave-short wave interaction equations in a two-layer fluid. Positons are new, exact solutions of nonlinear evolution equations that exhibit algebraic decay in the far field. A positon solution can be generated by taking a special limit of multi-soliton expansion. Variation of the limiting process yields different solutions. Dromions are exact, localized solutions of (2+1) dimensional (2 spatial, 1 temporal) nonlinear evolution equations that decay exponentially in all directions. One and higher dromion solutions are investigated, and a particular case of higher dromion solutions is considered in details. By applying another limiting procedure a new solution is generated. This method of `coalescence of eigenvalues' or `wavenumbers' is thus quite universal, and can be applied to a wide variety of expansions, not just the multi-soliton type. Finally, the case of propagation solutions on a continuous wave background is also studied.

Coalescence of Wavenumbers and Exact Solutions for a System of Coupled Nonlinear Schrödinger Equations

An exact 2-soliton expression is obtained for the Manakov system, a special, coupled set of nonlinear Schrödinger equations. The solution permits different asymptotic states for the components in the far field. A `coalescence' of wavenumbers is considered from the perspective of the Hirota bilinear operator. This is roughly equivalent to a double (or in general multiple) pole solution in the language of the inverse scattering transform. Physically counterpropagating waves will occur. With the help of computer algebra software a 3-soliton solution is derived. Coalescence of eigenvalues is investigated. Temporal modulation of the amplitude is observed.

Flutter Prediction Using an Eigenvector Orientation Approach

A method for predicting the onset of coupled-mode flutter is presented. A distinguishing characteristic of this method is that it emphasizes eigenvectors rather than eigenvalues. In the popular methods based on eigenvalues, flutter is predicted as instability begins to occur, as evidenced by the movement of one or more eigenvalues from the left-hand to the right-hand side of the s-plane, or the coalescence of eigenvalues. Alternatively, the method of eigenvector orientation being presented has the potential to predict the occurence of instability. Although the eigenvectors of vibrating systems generally satisfy the orthogonality condition, there are certain cases In aeroelasticity in which they are not orthogonal. In a typical case of coupled-mode flutter, the eigenvectors Initially may be oriented orthogonally but gradually lose orthogonality as airspeed is varied. The consequence of such a progressive loss of eigenvector orthogonality on structural dynamics is a phenomenon that seems to warrant further investigation. This investigation has applications in areas such as helicopter dynamics, aeroelastic analysis of plate and shell structures, analysis of slightly mistuned bladed disk assemblies, and rotordynamics.

Branch point singularities in the energy of the delta‐function model of one‐electron diatoms

AbstractTwo different perturbation series (the polarization expansion and a united‐atom expansion) of the ground state energy of the delta‐function model for one‐electron diatoms are studied and the radii of convergence are determined. For both expansions the singularity in the energy which limits the radius of convergence is a branch point with exponent one‐half. The physical significance of the branch point is that for particular values of the perturbation parameter, two different energy eigenvalues coalesce. The positions of the branch points are computed as a function of the internuclear separation R. For all values of R, both series converge for all physical values of the perturbation parameters. A lower bound to the radius of convergence of the polarization expansion has been computed previously by Claverie. It is proved in the present paper that the lower bound calculation is in fact an exact determination of the radius of convergence. The results of the model study are applied to real one‐electron diatoms to suggest the possible location of a branch point singularity in the energy of the ground state.