Spectral Singularities Research Articles

New scattering features of quaternionic point interaction in non-Hermitian quantum mechanics

Spectral singularities have been extensively studied over the last one and half decades for different non-Hermitian potentials in non-Hermitian quantum mechanics. The nature of spectral singularities has not been studied for the case of quaternionic potential. In the present work, we perform an analytical study on scattering from a quaternionic point interaction represented by a delta function. New features of spectral singularities are observed, which are different than the case of a complex (non-quaternionic) point interaction. The most notable difference is the occurrence of spectral singularity from the lossy point interaction, which is forbidden in the case of standard non-Hermitian quantum mechanics.

Attosecond spectral singularities in solid-state high-harmonic generation

Strong-field-driven electric currents in condensed-matter systems are opening new frontiers in petahertz electronics. In this regime, new challenges are arising as the roles of band structure and coherent electron–hole dynamics have yet to be resolved. Here, by using high-harmonic generation spectroscopy, we reveal the underlying attosecond dynamics that dictates the temporal evolution of carriers in multi-band solid-state systems. We demonstrate that when the electron–hole relative velocity approaches zero, enhanced constructive interference leads to the appearance of spectral caustics in the high-harmonic generation spectrum. We introduce the role of the dynamical joint density of states and identify its mapping into the spectrum, which exhibits singularities at the spectral caustics. By studying these singularities, we probe the structure of multiple unpopulated high conduction bands. High-harmonic waves are generated from a MgO crystal under experimental conditions where the simple semi-classical analysis fails. High-harmonic generation spectroscopy directly probes the strong-field attosecond dynamics over multiple bands.

Electromagnetically induced transparency at a chiral exceptional point

Electromagnetically induced transparency, as a quantum interference effect to eliminate optical absorption in an opaque medium, has found extensive applications in slow light generation, optical storage, frequency conversion, optical quantum memory as well as enhanced nonlinear interactions at the few-photon level in all kinds of systems. Recently, there have been great interests in exceptional points, a spectral singularity that could be reached by tuning various parameters in open systems, to render unusual features to the physical systems, such as optical states with chirality. Here we theoretically and experimentally study transparency and absorption modulated by chiral optical states at exceptional points in an indirectly-coupled resonator system. By tuning one resonator to an exceptional point, transparency or absorption occurs depending on the chirality of the eigenstate. Our results demonstrate a new strategy to manipulate the light flow and the spectra of a photonic resonator system by exploiting a discrete optical state associated with specific chirality at an exceptional point as a unique control bit, which opens up a new horizon of controlling slow light using optical states. Compatible with the idea of state control in quantum gate operation, this strategy hence bridges optical computing and storage.

A universal form of localized complex potentials with spectral singularities

We establish necessary and sufficient conditions for localized complex potentials in the Schrödinger equation to enable spectral singularities (SSs) and show that such potentials have the universal form , where w(x) is a differentiable function, such that , and k0 is a non-zero real. We also find that when k0 is a complex number, then the eigenvalue of the corresponding Shrödinger operator has an exact solution which, depending on k0, represents a coherent perfect absorber (CPA), laser, a localized bound state, a quasi bound state in the continuum (a quasi-BIC), or an exceptional point (the latter requiring additional conditions). Thus, k0 is a bifurcation parameter that describes transformations among all those solutions. Additionally, in a more specific case of a real-valued function w(x) the resulting potential, although not being symmetric, can feature a self-dual SS associated with the CPA-laser operation. At this moment, the complex potential has exactly two coexisting SSs at different points of the continuous spectrum. In the space of the system parameters, the transition through each self-dual SS corresponds to a bifurcation of a pair of complex-conjugate propagation constants from the continuum. The bifurcation of a first complex-conjugate pair corresponds to the phase transition from purely real to complex spectrum.

On the spectrality and spectral expansion of the non-self-adjoint mathieu-hill operator in $ L_{2}(-\infty, \infty) $

In this paper we investigate the non-self-adjoint operator \begin{document}$ H $\end{document} generated in \begin{document}$ L_{2}(-\infty, \infty) $\end{document} by the Mathieu-Hill equation with a complex-valued potential. We find a necessary and sufficient conditions on the potential for which \begin{document}$ H $\end{document} has no spectral singularity at infinity and it is an asymptotically spectral operator. Moreover, we give a detailed classification, stated in term of the potential, for the form of the spectral decomposition of the operator \begin{document}$ H $\end{document} by investigating the essential spectral singularities.

Enhanced Sensing and Nondegraded Thermal Noise Performance Based on PT-Symmetric Electronic Circuits with a Sixth-Order Exceptional Point.

An exceptional point (EP) is a non-Hermitian degeneracy where both eigenvalues and their corresponding eigenvectors coalesce. It was recently proposed and demonstrated that such spectral singularity can be utilized for enhanced sensing. Potential drawbacks of EP sensing include both fundamental resolution limit and noise effects that might mask the hypersensitive resonant splitting. Here, we address these issues by proposing a parity-time (PT)-symmetric sensing circuit bearing a sixth-order EP. By employing capacitive coupling channel as a sensing platform, we achieve an enhanced resonance shift proportional to the fourth-order root of the perturbation strength and maintain a high resolution for weak perturbation. Due to the low-pass feature of our circuit, thermal noise is mitigated down to a level comparable to its Hermitian counterpart, despite the presence of highly noisy gain and loss elements. Our EP sensing scheme offers combined enhanced sensitivity, improved resolution and nondegraded thermal noise performance, showing an exciting prospect for next-generation sensing technologies.

Spectral singularities of solutions to a boundary-value problem for the Fredholm integro-differential equation of the second order with reflection of argument

Spectral singularities of solutions to a boundary-value problem for the Fredholm integro-differential equation of the second order with reflection of argument

Solving scattering problems in the half-line using methods developed for scattering in the full line

We reduce the solution of the scattering problem defined on the half-line [0,∞) by a real or complex potential v(x) and a general homogeneous boundary condition at x=0 to that of the extension of v(x) to the full line that vanishes for x<0. We find an explicit expression for the reflection amplitude of the former problem in terms of the reflection and transmission amplitudes of the latter. We obtain a set of conditions on these amplitudes under which the potential in the half-line develops bound states, spectral singularities, and time-reversed spectral singularities where the potential acts as a perfect absorber. We examine the application of these results in the study of the scattering properties of a δ-function potential and a finite barrier potential defined in [0,∞), discuss optical systems modeled by these potentials, and explore the configurations in which these systems act as a laser or perfect absorber. In particular, we derive an explicit formula for the laser threshold condition for a slab laser with a single mirror and establish the surprising fact that a nearly perfect mirror gives rise to a lower threshold gain than a perfect mirror. We also offer a nonlinear extension of our approach which allows for utilizing a recently developed nonlinear transfer matrix method in the full line to deal with finite-range nonlinear scattering interactions defined in the half-line.

Spacing gain and absorption in a simple -symmetric model: spectral singularities and ladders of eigenvalues and resonances

We consider a parity-time () symmetric waveguide consisting of a localized gain and loss element separated by a variable distance. The situation is modeled by a Schrödinger operator with localized complex symmetric potential. Properties of the latter Hamiltonian are considered subject to the change of the gain-to-loss distance. Resonances, spectral singularities and eigenvalues are analyzed in detail and discussed in the context of the associated laser-absorber modes and symmetry breaking phase transition. Increasing gain-to-loss distance creates new resonances and spectral singularities that do not exist in the waveguide with adjacent gain and loss. In the limit of large gain-to-loss distance, the waveguide features a ladder of resonances which can be transformed to a ladder of complex eigenvalues by means of the change of the gain-and-loss amplitude.

Spectral singularity enhances transverse spin

We study the transverse spin and the Belinfante spin momentum in a gain–loss balanced waveguide, known to be one of the first and most studied examples of PT-symmetric systems. Such a guide supports the spectral singularities leading to infinite scattering amplitudes for both reflection and transmission. We show that near the spectral singularity there can be dramatic enhancement of the transverse spin and the transverse spin momentum. Note that these exotic spin and spin momentum have recently been observed experimentally despite having tiny magnitudes, making it worthy to explore ways and means to enhance these fundamentally important elusive quantities.

Surface plasmon-polaritons in graphene, embedded into medium with gain and losses

The paper deals with the theoretical consideration of surface plasmon-polaritons in the graphene monolayer, embedded into dielectric with spatially separated gain and losses. It is demonstrated, that presence of gain and losses in the system leads to the formation of additional mode of graphene surface plasmon-polaritons, which does not have its counterpart in the conservative system. When the gain and losses are mutually balanced, the position of exceptional point—transition point between unbroken and broken -symmetry—can be effectively tuned by graphene’s doping. In the case of unbalanced gain and losses the spectrum of surface plasmon-polaritons contains spectral singularity, whose frequency is also adjustable through the electrostatic gating of graphene.

Harnessing Spectral Singularities in Non- Hermitian Cylindrical Structures

Non-Hermitian systems characterized by suitable spatial distributions of gain and loss can exhibit "spectral singularities" in the form of zero-width resonances associated to real-frequency poles in the scattering operator. Here, we study this intriguing phenomenon in connection with cylindrical geometries, and explore possible applications to controlling and tailoring in unconventional ways the scattering response of sub-wavelength and wavelength-sized objects. Among the possible implications and applications, we illustrate the additional degrees of freedom available in the scattering-absorption-extinction tradeoff, and address the engineering of zero-forward-scattering, transverse scattering, and gain-controlled reconfigurability of the scattering pattern, also paying attention to stability issues. Our results may open up new vistas in active and reconfigurable nanophotonics platforms.

Principal functions of impulsive difference operators on semi axis

In this paper, we investigate the continuous spectrum and resolvent operator of a second-order difference operator with an impulsive condition. Then, under certain conditions, we prove finiteness of eigenvalues, spectral singularities. At last, we present principal functions of corresponding impulsive operator.

Optical controlling the spectral singularity in a doped slab

In this work, we have used the probe and control fields to theoretically investigate optical spectral singularity (OSS) in a doped and gained (DG) slab. This slab is composed of a gain medium modeled by two-level atoms and doped by the three-level $\Lambda$ -type atoms. This work is based on calculations and the results show that the OSS effect can be switched on and off by changing the relative phase of the fields from 0 to $\pi$ . Also, in comparison with an undoped slab, the OSS effect appears for lower values of gain. For a given value of gain coefficient at $\phi = 0$ , the DG slab produces a sharp OSS with infinite transmission but, for $\phi = \pi$ , the slab acts as an optical filter with a narrow bandwidth around the wavelength of OSS. We have also illustrated that it is possible to tune the wavelength of OSS just by tuning the relative phase.

Spectral expansion of Sturm-Liouville problems with eigenvalue-dependent boundary conditions

In this paper, we consider the operator L generated in L₂(R₊) by the differential expression l(y)=-y′′+q(x)y,x∈R₊:=[0,∞) and the boundary condition ((y′(0))/(y(0)))=α₀+α₁λ+α₂λ², where q is a complex valued function and α_{i}∈C,[mbox] \mbox{\:} i=0,1,2α₂. We have proved that spectral expansion of L in terms of the principal functions under the condition q∈AC(R₊), lim_{x→∞}q(x)=0, sup[e^{e√x}|q′(x)|] 0 taking into account the spectral singularities. We have also proved the convergence of the spectral expansion.

Scattering matrices for dissipative quantum systems

We consider a quantum system S interacting with another system S′ and susceptible of being absorbed by S′. The effective, dissipative dynamics of S is supposed to be generated by an abstract pseudo-Hamiltonian of the form H=H0+V−iC⁎C. The generator of the free dynamics, H0, is self-adjoint, V is symmetric and C is bounded. We study the scattering theory for the pair of operators (H,H0). We establish a representation formula for the scattering matrices and identify a necessary and sufficient condition to their invertibility. This condition rests on a suitable notion of spectral singularity. Our main application is the nuclear optical model, where H is a dissipative Schrödinger operator and spectral singularities correspond to real resonances.

La singularité fantôme. Hypnose, chasse aux spectres et cybernétique de la transmigration à Calcutta

Cet article est un essai d’ethnographie dickienne, en hommage au dernier roman de Philip K. Dick, La Transmigration de Timothy Archer. À Calcutta, des patients explorent leurs « vies antérieures » dans le cabinet d’une hypnothérapeute, tandis que des chasseurs de fantômes enquêtent sur des maisons délabrées et tentent d’établir une communication avec des présences via des appareils électromagnétiques. Alors que l’hypnotiseuse mène sa propre enquête sur la transmigration à travers les visions de ses patients, démêlant leurs identités troublantes et en quoi elles pourraient être liées, les chasseurs en quête de signaux font leurs propres expériences qui visent à surmonter la paréidolie pour entrer en contact avec des corps inconnus. Si les deux sont liés, ce n’est pas uniquement parce que réincarnation et hantise sont les deux composantes d’un même circuit (la cybernétique du samsâra), qui devient ici terrain d’expérience. C’est aussi parce que hypnose et chasse aux spectres visent toutes deux, bien que par des moyens expérimentaux différents, à s’approcher d’un point de catastrophe, d’un seuil critique de présence (appelé singularité fantôme), qui marque le jaillissement d’une vie spectrale foisonnante et imprévisible et d’où la matière humaine ressort à la fois entièrement questionnée et recomposée.

Designing lasing and perfectly absorbing potentials

Existence of a spectral singularity (SS) in the spectrum of {a Schr\"{o}dinger operator with} a non-Hermitian potential requires exact matching of parameters of the potential. We provide a necessary and sufficient condition for a potential to have a SS at a given wavelength. It is shown that potentials with SSs at prescribed wavelengths can be obtained by a simple and effective procedure. In particular, the developed approach allows one to obtain potentials with several SSs and with SSs of the second order, as well as potentials obeying a given symmetry, say, $\PT-$symmetric potentials. Also, the problem can be solved when it is required to obtain a potential obeying a given symmetry, say, $\PT-$symmetric potential. We illustrate all these opportunities with examples. We also describe splitting of a second-order SSs under change of the potential parameters, and discuss possibilities of experimental observation of SSs of different orders.

Exceptional-point-induced lasing dynamics in a non-Hermitian Su-Schrieffer-Heeger model

Non-Hermitian systems exhibit many peculiar dynamic behaviors that never showed up in Hermitian systems. The existence of spectral singularity for a non-Hermitian scattering center provides a ``lasing mechanism'' in the context of quantum mechanics. In this paper, we investigate the dynamics of the non-Hermitian Su-Schrieffer-Heeger model around an exceptional point, based on an analysis of parity-time and chiral-time symmetries of the Hamiltonian. We show exactly that a finite system near its exceptional point can support EP-related amplification of cyclic states. It becomes a stationary laser solution with linear-power amplification under certain conditions in an infinite limit. In contrast to the SS lasing mechanism for a scattering system, such an SSH chain acts as an active laser medium at threshold, within which a stationary particle emission can be fired anywhere, rather than a specific location at the non-Hermitian scattering center only. In addition, some relevant peculiar phenomena arising from interference between wave packets are revealed based on the analytical solutions.

Spectral singularities of a potential created by two coupled microring resonators.

Two microring resonators, one with gain and one with loss, coupled to each other and to a bus waveguide, create an effective non-Hermitian potential for light propagating in the waveguide. Due to geometry, the coupling for each microring resonator yields two counter-propagating modes with equal frequencies. We show that such a system enables implementation of many types of scattering peculiarities. The spectral singularities, which are either the second or fourth order, separate parameter regions where the spectrum is either purely real or composed of complex eigenvalues; hence, they represent the points of the phase transition. By modifying the gain-loss relation for the resonators, such an optical scatterer can act as a laser, as a coherent perfect absorber, be unidirectionally reflectionless or transparent, and support bound states either growing or decaying in time. These characteristics are observed for a discrete series of the incident-radiation wavelengths.