Zero-width Resonances Research Articles

PT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathscr{P}\\mathscr{T}$$\\end{document}-symmetric KdV solutions and their algebraic extension with zero-width resonances

A class of complex breather and soliton solutions to both KdV and mKdV equations are identified with a Pöschl-Teller type PT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathscr{P}\\mathscr{T}$$\\end{document}-symmetric potential. However, these solutions represent only the unbroken-PT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathscr{P}\\mathscr{T}$$\\end{document} phase owing to their isospectrality to an infinite potential well in the complex plane having real spectra. To obtain the broken-PT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathscr{P}\\mathscr{T}$$\\end{document} phase, an extension of the potential satisfying the sl2,R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$sl\\left( 2,\\mathbb {R}\\right)$$\\end{document} potential algebra is mandatory that additionally supports non-trivial zero-width resonances.

Matching long and short distances in the form factors for K → πℓ+ℓ−

At order O(αGF), the amplitudes for the decays K→πℓ+ℓ− involve a form factor given by the matrix element of the time-ordered product of the electromagnetic current with the four-quark operators describing weak non-leptonic neutral-current transitions between a kaon and a pion. The short-distance behaviour of this time-ordered product, when considered at order O(αs) in the perturbative expansion of QCD, involves terms linear and quadratic in the logarithm of the Euclidean momentum transfer squared. It is shown how one can exactly match these short-distance features using a dispersive representation of the form factor, with an absorptive part given by an infinite sum of zero-width resonances following a Regge-type spectrum. Some phenomenology-related issues are briefly discussed.

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.

Zero-width resonances in the context of Fano's configuration interaction formalism

ABSTRACTWe examine how to link two approaches for resonance width calculations, in a situation of crossing of two diatomic molecular potentials. One is the semiclassical formalism of Child. The other is Fano's configuration interaction approach leading to the Fermi golden rule. We build a case where the vanishing of the width in the two formalisms can be reduced to the same conditions.

On the amplitudes for the CP-conserving K\xb1(KS) \u2192 \u03c0\xb1(\u03c00)\u2113+\u2113\u2212 rare decay modes

The amplitudes for the rare decay modes K± → π±ℓ+ℓ− and KS → π0ℓ+ℓ− are studied with the aim of obtaining predictions for them, such as to enable the possibility to search for violations of lepton-flavour universality in the kaon sector. The issue is first addressed from the perspective of the low-energy expansion, and a two-loop representation of the corresponding form factors is constructed, leaving as unknown quantities their values and slopes at vanishing momentum transfer. In a second step a phenomenological determination of the latter is proposed. It consists of the contribution of the resonant two-pion state in the P wave, and of the leading short-distance contribution determined by the operator-product expansion. The interpolation between the two energy regimes is described by an infinite tower of zero-width resonances matching the QCD short-distance behaviour. Finally, perspectives for future improvements in the theoretical understanding of these amplitudes are discussed.

Exotic states in the strong-field control of dissociation dynamics: from exceptional points to zero-width resonances

is an ideal candidate for the detailed study of strong-field coherent control strategies inspired by basic mechanisms referring to some specific photodissociation resonances. Two of them are considered in this work, namely: zero-width resonances (ZWRs) and coalescing pairs of resonances at exceptional points (EPs). An adiabatic transport theory based on the Floquet Hamiltonian formalism is developed within the challenging context of multiphoton dynamics involving nuclear continua. It is shown that a rigorous treatment is only possible for ZWRs, whereas adiabatic transport mediated by EPs is subjected to restrictions. Numerical maps of resonance widths and non-adiabatic couplings in the laser parameter plane help in optimally shaping control pulses. Full time-dependent wavepacket dynamics shows the possibility of selective, robust filtration and vibrational population transfers, within experimentally feasible criteria.

Vibrational-ground-state zero-width resonances for laser filtration: An extended semiclassical analysis

A semiclassical model supporting the destructive interference interpretation of zero-width resonances (ZWR) is extended to wavelengths inducing c_minus-type curve crossing situations in Na2 strong field dissociation. This opens the possibility to get critical couples of wavelengths lambda and field intensities I to reach ZWRs associated with the ground vibrationless level v = 0, that, contrary to other vibrational states (v > 0), is not attainable for the commonly referred c+-type crossings. The morphology of such ZWRs in the laser (I; lambda) parameter plane and their usefulness in filtration strategies aiming at molecular cooling down to the ground v = 0 state are examined within the frame of an adiabatic transport scheme.

Controlling vibrational cooling with zero-width resonances: An adiabatic Floquet approach

In molecular photodissociation, some specific combinations of laser parameters (wavelength and intensity) lead to unexpected Zero-Width Resonances (ZWR), with in principle infinite lifetimes. Their interest in inducing basic quenching mechanisms have recently been devised in the laser control of vibrational cooling through filtration strategies [O. Atabek et al., Phys. Rev. A87, 031403(R) (2013)]. A full quantum adiabatic control theory based on the adiabatic Floquet Hamiltonian is developed to show how a laser pulse could be envelop-shaped and frequency-chirped so as to protect a given initial vibrational state against dissociation, taking advantage from its continuous transport on the corresponding ZWR, all along the pulse duration. As compared with previous control scenarios actually suffering from non-adiabatic contamination, drastically different and much more efficient filtration goals are achieved. A semiclassical analysis helps in finding and interpreting a complete map of ZWRs in the laser parameter plane. In addition, the choice of a given ZWR path, among the complete series identified by the semiclassical approach, amounts to be crucial for the cooling scheme, targeting a single vibrational state population left at the end of the pulse, while all others have almost completely decayed. The illustrative example, offering the potentiality to be transposed to other diatomics, is Na2 prepared by photoassociation in vibrationally hot but translationally and rotationally cold states.

Gaussian sum rules for 0−+ glueball in the instanton vacuum model

A semi-classical expansion based on the instanton vacuum model of quantum chromodynamics is taken to obtain the correlation function of the 0−+ pseudoscalar glueball current. Working on the framework of Gaussian sum rules with a Gaussian-tail instantons size distribution, we find the contributions arising from the interaction between the classical instanton fields and quantum fluctuation ones are comparable to the pure classical and quantum ones, and thus impact the sum rule results; whereas the usual condensate contribution turned out to be negligible and may be understood as a small part of the corresponding instanton contributions. In consideration of axial U(1) anomaly, and inspired by the correlation function’s behavior, we assume that the spectral function is saturated by the three zero-width resonances (the mesons η and η′, and the pseudoscalar glueball G) plus continuum, and the corresponding Gaussian sum rules are constructed. The optimal mass and the coupling constant of the lowest pseudoscalar glueball are predicted to be consistent in a series Gaussian sum rules and almost the same as that of our previous Laplace sum rules.

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

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 application of their recently-developed nonlinear generalization in the study of the effect of a nonlinearity on the lasing threshold condition for an infinite planar slab of gain medium. In particular, for a Kerr nonlinearity, we derive an explicit expression for the intensity of the emitted waves from the slab and discuss the implications of our results for the time-reversed system that acts as a coherent perfect absorber.

Proposal for laser purification in molecular vibrational cooling using zero-width resonances

Intense-field molecular photodissociation offers the possibility to define resonances of zero width for particular intensities at given wavelengths. We show how a laser pulse can be shaped to fulfill, at all times, the conditions leading to a zero width for a particular resonance originating from a field-free vibrational state. The solution of the time-dependent Schr\odinger equation confirms that only this state survives after the pulse is off. This is the basis for a purification procedure leading to population left only in a single state chosen at will. The illustrative example is ${{\mathrm{H}}_{2}}^{+}$.

EXPLICIT MODEL REALIZING PARTON–HADRON DUALITY

We present a model that realizes both resonance-Regge (Veneziano) and parton–hadron (Bloom–Gilman) duality. We first review the features of the Veneziano model and we discuss how parton–hadron duality appears in the Bloom–Gilman model. Then we review limitations of the Veneziano model, namely that the zero-width resonances in the Veneziano model violate unitarity and Mandelstam analyticity. We discuss how such problems are alleviated in models that construct dual amplitudes with Mandelstam analyticity (so-called DAMA models). We then introduce a modified DAMA model, and we discuss its properties. We present a pedagogical model for dual amplitudes and we construct the nucleon structure function F2(x, Q2). We explicitly show that the resulting structure function realizes both Veneziano and Bloom–Gilman duality.

Spectral singularities inPT-symmetric periodic finite-gap systems

The origin of spectral singularities in finite-gap singly periodic $\mathcal{P}\mathcal{T}$-symmetric quantum systems is investigated. We show that they emerge from a limit of band-edge states in a doubly periodic finite gap system when the imaginary period tends to infinity. In this limit, the energy gaps are contracted and disappear, every pair of band states of the same periodicity at the edges of a gap coalesces and transforms into a singlet state in the continuum. As a result, these spectral singularities turn out to be analogous to those in the nonperiodic systems, where they appear as zero-width resonances. Under the change of topology from a noncompact into a compact one, spectral singularities in the class of periodic systems we study are transformed into exceptional points. The specific degeneration related to the presence of finite number of spectral singularities and exceptional points is shown to be coherently reflected by a hidden, bosonized nonlinear supersymmetry.

Perturbative analysis of spectral singularities and their optical realizations

We develop a perturbative method of computing spectral singularities of a Schreodinger 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 an antilaser. 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.

Laser cooling of the vibrational motion of Na2combining the effects of zero-width resonances and exceptional points

We propose various scenarios for molecular vibrational cooling combining the effects of two kinds of resonance states occurring during the photodissociation of Na${}_{2}$ taken as an illustrative example. Such resonances result from an appropriate sampling of laser parameters (wavelength and intensity): (a) For particular choices of intensity and wavelength, two resonance energies can be brought to complete coalescence, with their positions and widths becoming equal and leading to a so-called exceptional point (EP) in the parameter plane. Advantage can be taken from such points for very selective laser-controlled vibrational transfer strategies. (b) For specific intensities, far beyond the perturbation regime, some resonances can have a zero width (infinite lifetime). They are referred to as a zero-width resonance (ZWR) and may be used for vibrational purification purposes. We show how appropriately shaped, experimentally reachable laser pulses, encircling EPs or inducing ZWRs, may be used for a thorough and comprehensive control aiming at population transfer or purification schemes, which, starting from an initial field-free vibrational distribution, ends up in the ground vibrational level.

Optical spectral singularities as threshold resonances

Spectral singularities are among generic mathematical features of complex scattering potentials. Physically they correspond to scattering states that behave like zero-width resonances. For a simple optical system, we show that a spectral singularity appears whenever the gain coefficient coincides with its threshold value and other parameters of the system are selected properly. We explore a concrete realization of spectral singularities for a typical semiconductor gain medium and propose a method of constructing a tunable laser that operates at threshold gain.

Multiple occurrence of zero-width resonances in photodissociation: Effect of laser field intensity and frequency

Critical intensities at which the strong field photodissociation rates of a molecule are vanishing could play an important role in control scenarios to filter a given vibrational state. We have previously shown [O. Atabek, R. Lefebvre, and F. Gad\'ea, Phys. Rev. A 74, 063412 (2006)] that there is a multiplicity of such intensities when the diabatic curve crossing point is on the left of the minimum of the attractive potential in a photon dressed picture (${c}^{\ensuremath{-}}$ case). A semiclassical explanation was given. We complete this study of the multiple occurrence of zero-width resonances (ZWRs) by addressing, both numerically and semiclassically, the more common ${c}^{+}$ case of a crossing point on the right of the potential minimum. Coincidences between modified diabatic and adiabatic levels are the fingerprints of such events. They occur several times because the adiabatic energies are moving upward more rapidly than the diabatic ones. Advantage can be taken of the frequency dependence of the critical intensities to produce the phenomenon even for weak fields, this circumstance leading to more favorable control issues. This is illustrated on the example of ${\mathrm{H}}_{2}^{+}$. The wave functions associated with the ZWRs reflect the fact that the adiabatic picture is the best zeroth order description in the limit of high fields.

Intense-field zero-width resonances and control of molecular photodissociation

An efficient and selective laser-induced molecular stabilization with respect to dissociation is obtained by an appropriate shaping of an intense ultraviolet-visible laser pulse which ensures the adiabatic transport of the system onto an infinitely long-lived, zero-width Floquet resonance. Such a resonance is obtained for a frequency-dependent critical field intensity specific to each field-free initial vibrational state. This mechanism, illustrated here for the photodissociation of ${\text{H}}_{2}^{+}$ under a $\ensuremath{\lambda}=400\text{ }\text{nm}$ intense laser pulse, opens the way to a control scenario of a selective preparation of a given vibrational level that could be used in the context of molecular cooling.

Zero-width resonances in intense-field molecular photodissociation

We provide additional evidence for the existence of zero-width resonances in the intense-field photodissociation of ${{\mathrm{H}}_{2}}^{+}$. In a previous investigation [Atabek, Chrysos, and Lefebvre, Phys. Rev. A 49, R8 (1994)] the situation, in a two-channel dressed picture, corresponded to a diabatic crossing point classically reachable in both adiabatic potentials. A semiclassical explanation could be modeled after that developed for predissociation in the intermediate-coupling regime. At higher frequencies the crossing point lies between the two turning points. A numerical study shows that zero-width resonances exist also in such a case. An extension of the semiclassical approach provides an explanation for the occurrence of these resonances. It is shown that they survive even when going to a multichannel description. The associated wave functions and probability densities are studied: they are very similar to those of the upper adiabatic potential, with a minor component in the lower adiabatic channel. Some conditions for the production of such long-lived dressed molecular species are stated.

Correlators of hadron currents: The model and the ALEPH data on τ-decay

The model with the meson spectrum, consisting of zero-width equidistant resonances, is considered with connection to current correlators in coordinate space. The comparison of the explicit expressions for the correlators, obtained in this model, with the experimental data of ALEPH Collaboration on τ-decay is made and good agreement is found.