Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1204.2701 · Physical Review A

Perturbative analysis of spectral singularities and their optical realizations

Ali Mostafazadeh, Saber Rostamzadeh

Published Aug 2, 2012• 25 citations
arXivHTMLPDF

Abstract

We develop a perturbative method of computing spectral singularities of a Schr\"odinger operator defined by a general complex potential that vanishes outside a closed interval. These can be realized as zero-width resonances in optical gain media and correspond to a lasing effect that occurs at the threshold gain. Their time-reversed copies yield coherent perfect absorption of light that is also known as antilasing. We use our general results to establish the exactness of the $n$th-order perturbation theory for an arbitrary complex potential consisting of $n$ delta functions, obtain an exact expression for the transfer matrix of these potentials, and examine spectral singularities of complex barrier potentials of arbitrary shape. In the context of optical spectral singularities, these correspond to inhomogeneous gain media.

DOI
10.1103/physreva.86.022103

Related papers

Nonlinear spectral singularities of a complex barrier potential and the lasing threshold condition

Ali Mostafazadeh

A spectral singularity is a mathematical notion with an intriguing physical realization in terms of certain zero-width resonances. In optics it manifests as lasing at the threshold gain. We explore the recently developed nonlinear generalization of spectral singularities for a complex barrier potential and study the associated resonance effect for an infinite planar slab of gain medium. In particular, for a Kerr nonlinearity, we show that the first-order perturbative equation that determines the nonlinear spectral singularities provides an explicit expression for the intensity of the emitted waves from the slab. This is a mathematical derivation of the known linear relationship between the output intensity of the slab and its gain coefficient. We also discuss the implications of our results for the time-reversed system which acts as a coherent perfect absorber.

arXiv:1303.4874•Physical Review A•Jun 24, 2013•38 citations

Spectral Singularities of Complex Scattering Potentials and Infinite Reflection and Transmission Coefficients at Real Energies

Alí Mostafazadeh

Spectral singularities are spectral points that spoil the completeness of the eigenfunctions of certain non-Hermitian Hamiltonian operators. We identify spectral singularities of complex scattering potentials with the real energies at which the reflection and transmission coefficients tend to infinity, i.e., they correspond to resonances having a zero width. We show that a waveguide modeled using such a potential operates like a resonator at the frequencies of spectral singularities. As a concrete example, we explore the spectral singularities of an imaginary PT-symmetric barrier potential and demonstrate the above resonance phenomenon for a certain electromagnetic waveguide.

arXiv:0901.4472•Physical Review Letters•Jun 5, 2009•493 citations

Coherent scattering from semi-infinite non-Hermitian potentials

Zafar Ahmed, D. C. Ghosh, Sachin Kumar

When two identical (coherent) beams are injected at a semi-infinite non-Hermitian medium from left and right, we show that both reflection $({r}_{L},{r}_{R})$ and transmission $({t}_{L},{t}_{R})$ amplitudes are nonreciprocal. In a parametric domain, there exists spectral singularity (SS) at a real energy $E={E}_{*}={k}_{*}^{2}$ and the determinant of the time-reversed two port scattering matrix, i.e., $|det(S(\ensuremath{-}k))|=|{t}_{L}(\ensuremath{-}k){t}_{R}(\ensuremath{-}k)\ensuremath{-}{r}_{L}(\ensuremath{-}k){r}_{R}(\ensuremath{-}k)|$, vanishes sharply at $k={k}_{*}$, displaying the phenomenon of coherent perfect absorption (CPA). In the complementary parametric domain, the potential becomes either left or right reflectionless at $E={E}_{z}$. We rule out the existence of invisibility despite ${r}_{R}({E}_{i})=0$ and ${t}_{R}({E}_{i})=1$ but $T({E}_{i})\ensuremath{\ne}1$, in this avenue. We present two simple exactly solvable models where expressions for ${E}_{*}$, ${E}_{z}$, ${E}_{i}$, and parametric conditions on the potential have been obtained in explicit and simple forms. Earlier, the phenomena of SS and CPA have been found to occur only in the scattering complex potentials which are spatially localized (vanish asymptotically) and have ${t}_{L}={t}_{R}$.

arXiv:1709.08513•Physical Review A•Feb 16, 2018•7 citations

Designing lasing and perfectly absorbing potentials

V. V. Konotop, Evgeny Lakshtanov, Boris Rufimovich Vainberg

Existence of a spectral singularity (SS) in the spectrum of a Schr\"odinger 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, $\mathcal{PT}$-symmetric potentials. We illustrate all these opportunities with examples. We also describe splitting of a second-order SS under change of the potential parameters, and discuss possibilities of experimental observation of SSs of different orders.

arXiv:1812.05499•Physical Review A•Apr 26, 2019•20 citations

Engineering coherent perfect absorption and lasing

V. V. Konotop, Evgeny Lakshtanov, Boris Rufimovich Vainberg

If spectrum of a Schr\{o}dinger oparator with a non-Hermitian potential contains a spectral singularity (SS), the latter requires exact matching of the parameters characterizing 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 SS 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 or with SSs of the second order and 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, $\mathcal{PT}-$symmetric potential. We illustrate all opportunities with examples. We also describe splitting of the SSs of the second order, under change of the potential parameters, and discuss possibilities of experimental observation of SSs of different orders.

•Dec 13, 2018•0 citations

Transfer matrix in scattering theory: A survey of basic properties and recent developments

Ali Mostafazadeh (14357856)

We give a pedagogical introduction to time-independent scattering theory in one dimension focusing on the basic properties and recent applications of transfer matrices. In particular, we begin surveying some basic notions of potential scattering such as transfer matrix and its analyticity, multi-delta-function and locally periodic potentials, Jost solutions, spectral singularities and their time-reversal, and unidirectional reflectionlessness and invisibility. We then offer a simple derivation of the Lippmann-Schwinger equation and Born series, and discuss the Born approximation. Next, we outline a recently developed dynamical formulation of time-independent scattering theory in one dimension. This formulation relates the transfer matrix and therefore the solution of the scattering problem for a given potential to the solution of the time-dependent Schr\"odinger equation for an effective non-unitary two-level quantum system. We provide a self-contained treatment of this formulation and some of its most important applications. Specifically, we use it to devise a powerful alternative to the Born series and Born approximation, derive dynamical equations for the reflection and transmission amplitudes, discuss their application in constructing exact tunable unidirectionally invisible potentials, and use them to provide an exact solution for single-mode inverse scattering problems. The latter, which has important applications in designing optical devices with a variety of functionalities, amounts to providing an explicit construction for a finite-range complex potential whose reflection and transmission amplitudes take arbitrary prescribed values at any given wavenumber.

arXiv:2009.10507•Nov 30, 2023•10 citations
We develop a perturbative method of computing spectral singularities of a Schr"odinger operator defined by a general complex potential that vanishes outside a closed interval. These can be realized as zero-width resonances in optical gain media and correspond to a lasing effect that occurs at the threshold gain. Their time-reversed copies yield coherent perfect absorption of light that is also known as antilasing. We use our general results to establish the exactness of the nnnth-order perturbation theory for an arbitrary complex potential consisting of nnn delta functions, obtain an exact expression for the transfer matrix of these potentials, and examine spectral singularities of complex barrier potentials of arbitrary shape. In the context of optical spectral singularities, these correspond to inhomogeneous gain media.
A spectral singularity is a mathematical notion with an intriguing physical realization in terms of certain zero-width resonances. In optics it manifests as lasing at the threshold gain. We explore the recently developed nonlinear generalization of spectral singularities for a complex barrier potential and study the associated resonance effect for an infinite planar slab of gain medium. In particular, for a Kerr nonlinearity, we show that the first-order perturbative equation that determines the nonlinear spectral singularities provides an explicit expression for the intensity of the emitted waves from the slab. This is a mathematical derivation of the known linear relationship between the output intensity of the slab and its gain coefficient. We also discuss the implications of our results for the time-reversed system which acts as a coherent perfect absorber.
Spectral singularities are spectral points that spoil the completeness of the eigenfunctions of certain non-Hermitian Hamiltonian operators. We identify spectral singularities of complex scattering potentials with the real energies at which the reflection and transmission coefficients tend to infinity, i.e., they correspond to resonances having a zero width. We show that a waveguide modeled using such a potential operates like a resonator at the frequencies of spectral singularities. As a concrete example, we explore the spectral singularities of an imaginary PT-symmetric barrier potential and demonstrate the above resonance phenomenon for a certain electromagnetic waveguide.
When two identical (coherent) beams are injected at a semi-infinite non-Hermitian medium from left and right, we show that both reflection (rL,rR)({r}_{L},{r}_{R})(rL​,rR​) and transmission (tL,tR)({t}_{L},{t}_{R})(tL​,tR​) amplitudes are nonreciprocal. In a parametric domain, there exists spectral singularity (SS) at a real energy E=E∗=k∗2E={E}_{*}={k}_{*}^{2}E=E∗​=k∗2​ and the determinant of the time-reversed two port scattering matrix, i.e., $|det(S(\ensuremath{-}k))|=|{t}_{L}(\ensuremath{-}k){t}_{R}(\ensuremath{-}k)\ensuremath{-}{r}_{L}(\ensuremath{-}k){r}_{R}(\ensuremath{-}k)|$, vanishes sharply at k=k∗k={k}_{*}k=k∗​, displaying the phenomenon of coherent perfect absorption (CPA). In the complementary parametric domain, the potential becomes either left or right reflectionless at E=EzE={E}_{z}E=Ez​. We rule out the existence of invisibility despite rR(Ei)=0{r}_{R}({E}_{i})=0rR​(Ei​)=0 and tR(Ei)=1{t}_{R}({E}_{i})=1tR​(Ei​)=1 but $T({E}_{i})\ensuremath{\ne}1$, in this avenue. We present two simple exactly solvable models where expressions for E∗{E}_{*}E∗​, Ez{E}_{z}Ez​, Ei{E}_{i}Ei​, and parametric conditions on the potential have been obtained in explicit and simple forms. Earlier, the phenomena of SS and CPA have been found to occur only in the scattering complex potentials which are spatially localized (vanish asymptotically) and have tL=tR{t}_{L}={t}_{R}tL​=tR​.
Existence of a spectral singularity (SS) in the spectrum of a Schr\"odinger 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\mathcal{PT}PT-symmetric potentials. We illustrate all these opportunities with examples. We also describe splitting of a second-order SS under change of the potential parameters, and discuss possibilities of experimental observation of SSs of different orders.
If spectrum of a Schr\{o}dinger oparator with a non-Hermitian potential contains a spectral singularity (SS), the latter requires exact matching of the parameters characterizing 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 SS 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 or with SSs of the second order and 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−\mathcal{PT}-PT−symmetric potential. We illustrate all opportunities with examples. We also describe splitting of the SSs of the second order, under change of the potential parameters, and discuss possibilities of experimental observation of SSs of different orders.
We give a pedagogical introduction to time-independent scattering theory in one dimension focusing on the basic properties and recent applications of transfer matrices. In particular, we begin surveying some basic notions of potential scattering such as transfer matrix and its analyticity, multi-delta-function and locally periodic potentials, Jost solutions, spectral singularities and their time-reversal, and unidirectional reflectionlessness and invisibility. We then offer a simple derivation of the Lippmann-Schwinger equation and Born series, and discuss the Born approximation. Next, we outline a recently developed dynamical formulation of time-independent scattering theory in one dimension. This formulation relates the transfer matrix and therefore the solution of the scattering problem for a given potential to the solution of the time-dependent Schr\"odinger equation for an effective non-unitary two-level quantum system. We provide a self-contained treatment of this formulation and some of its most important applications. Specifically, we use it to devise a powerful alternative to the Born series and Born approximation, derive dynamical equations for the reflection and transmission amplitudes, discuss their application in constructing exact tunable unidirectionally invisible potentials, and use them to provide an exact solution for single-mode inverse scattering problems. The latter, which has important applications in designing optical devices with a variety of functionalities, amounts to providing an explicit construction for a finite-range complex potential whose reflection and transmission amplitudes take arbitrary prescribed values at any given wavenumber.