True Resonances Research Articles

TOI-1136 is a Young, Coplanar, Aligned Planetary System in a Pristine Resonant Chain

Convergent disk migration has long been suspected to be responsible for forming planetary systems with a chain of mean-motion resonances (MMRs). Dynamical evolution over time could disrupt the delicate resonant configuration. We present TOI-1136, a 700 ± 150 Myr old G star hosting at least six transiting planets between ∼2 and 5 R ⊕. The orbital period ratios deviate from exact commensurability by only 10−4, smaller than the ∼10−2 deviations seen in typical Kepler near-resonant systems. A transit-timing analysis measured the masses of the planets (3–8M ⊕) and demonstrated that the planets in TOI-1136 are in true resonances with librating resonant angles. Based on a Rossiter–McLaughlin measurement of planet d, the star’s rotation appears to be aligned with the planetary orbital planes. The well-aligned planetary system and the lack of a detected binary companion together suggest that TOI-1136's resonant chain formed in an isolated, quiescent disk with no stellar flyby, disk warp, or significant axial asymmetry. With period ratios near 3:2, 2:1, 3:2, 7:5, and 3:2, TOI-1136 is the first known resonant chain involving a second-order MMR (7:5) between two first-order MMRs. The formation of the delicate 7:5 resonance places strong constraints on the system’s migration history. Short-scale (starting from ∼0.1 au) Type-I migration with an inner disk edge is most consistent with the formation of TOI-1136. A low disk surface density (Σ1 au ≲ 103g cm−2; lower than the minimum-mass solar nebula) and the resultant slower migration rate likely facilitated the formation of the 7:5 second-order MMR.

Asymptotics for 2D whispering gallery modes in optical micro-disks with radially varying index

Abstract Whispering gallery modes [WGM] are resonant modes displaying special features: they concentrate along the boundary of the optical cavity at high polar frequencies and they are associated with complex scattering resonances very close to the real axis. As a classical simplification of the full Maxwell system, we consider 2D Helmholtz equations governing transverse electric or magnetic modes. Even in this 2D framework, very few results provide asymptotic expansion of WGM resonances at high polar frequency $m\to \infty $ for cavities with radially varying optical index. In this work, using a direct Schrödinger analogy, we highlight three typical behaviors in such optical micro-disks, depending on the sign of an ‘effective curvature’ that takes into account the radius of the disk and the values of the optical index and its derivative. Accordingly, this corresponds to abruptly varying effective potentials (step linear or step harmonic) or more classical harmonic potentials, leading to three distinct asymptotic expansions for ground state energies. Using multiscale expansions, we design a unified procedure to construct families of quasi-resonances and associate quasi-modes that have the WGM structure and satisfy eigenequations modulo a super-algebraically small residual ${\mathscr{O}}(m^{-\infty })$. We show using the black box scattering approach that quasi-resonances are ${\mathscr{O}}(m^{-\infty })$ close to true resonances.

Fano Lineshapes and Rabi Splittings: Can They Be Artificially Generated or Obscured by the Numerical Aperture?

Fano resonances and Rabi splittings are routinely reported in the scientific literature. Asymmetric resonance lineshapes are usually associated with Fano resonances, and two split peaks in the spectrum are often attributed to a Rabi splitting. True Fano resonances and Rabi splittings are unequivocal signatures of coherent coupling between subsystems. However, can the same spectral lineshapes characterizing Fano resonances and Rabi splittings arise from a purely incoherent sum of intensities? Here we answer this question through experiments with a tunable Fabry-Pérot cavity containing a CsPbBr3 perovskite crystal. By measuring the transmission and photoluminescence of this system using microscope objectives with different numerical aperture (NA), we find that even a modest NA = 0.4 can artificially generate Fano resonances and Rabi splittings. We furthermore show that this modest NA can obscure the anticrossing of a bona fide strongly coupled light–matter system. Through transfer matrix calculations we confirm that these spectral artifacts are due to the incoherent sum of transmitted intensities at different angles captured by the NA. Our results are relevant to the wide nanophotonics community, characterizing dispersive optical systems with high numerical aperture microscope objectives. We conclude with general guidelines to avoid pitfalls in the characterization of such optical systems.

Anharmonic Phonon Dispersion in Polyethylene.

The second-order Green's function method for anharmonic crystals has been applied to an infinite, periodic chain of polyethylene taking into account up to quartic force constants. The frequency-independent approximation to the Dyson self-energy gives rise to numerous divergent resonances, which are fortuitous. Instead, solving the Dyson equation self-consistently with a frequency-dependent self-energy resists divergences from resonances or zero-frequency acoustic vibrations. The calculated anharmonic phonon dispersion, which nonetheless displays many true resonances, and anharmonic phonon density of states furnish hitherto unknown details that explain smaller features of observed vibrational spectra.

Why static bound-state calculations of tetraquarks should be met with scepticism

Recent experimental signals have led to a revival of tetraquarks, the hypothetical q2q̄2 hadronic states proposed by Jaffe in 1976 to explain the light scalar mesons. Mesonic structures with exotic quantum numbers have indeed been observed recently, though a controversy persists as to whether these are true resonances and not merely kinematical threshold enhancements, or otherwise states not of a true q2q̄2 nature. Moreover, puzzling non-exotic mesons are also often claimed to have a tetraquark configuration. However, the corresponding model calculations are practically always carried out in pure and static bound-state approaches, ignoring completely the coupling to asymptotic two-meson states and unitarity, especially the dynamical effects thereof. In this short paper we argue that these static predictions of real tetraquark masses are highly unreliable and provide little evidence of the very existence of such states.

Towards a realistic description of hadron resonances

We report on our attempts of treating excited hadron states as true quantum resonances. Hitherto the spectroscopy of mesons, usually considered as quark-antiquark systems, and of baryons, usually considered as three-quark systems, has been treated through excitation spectra of bound states (namely, confined few-quark systems), corresponding to poles of the quantum-mechanical resolvent at real negative values in the complex energy plane. As a result the wave functions, i.e. the residua of the resolvent, have not exhibited the behaviour as required for hadron resonances with their multiple decay modes. This has led to disturbing shortcomings in the description of hadronic resonance phenomena. We have aimed at a more realistic description of hadron resonances within relativistic constituent-quark models taking into account explicitly meson-decay channels. The corresponding coupled-channels theory is based on a relativistically invariant mass operator capable of producing hadron ground states with real energies and hadron resonances with complex energies, the latter corresponding to poles in the lower half-plane of the unphysical sheet of the complex energy plane. So far we have demonstrated the feasibility of the coupled-channels approach to hadron resonances along model calculations producing indeed the desired properties. The corresponding spectral properties will be discussed in this contribution. More refined studies are under way towards constructing a coupled-channels relativistic constituent-quark model for meson and baryon resonances.

Cross section versus time delay and trapping probability

We study the behavior of the s-wave partial cross section σ(k), the Wigner–Smith time delay τ(k), and the trapping probability P(k) as function of the wave number k. The s-wave central square well is used for concreteness, simplicity, and to elucidate the controversy whether it shows true resonances. It is shown that, except for very sharp structures, the resonance part of the cross section, the trapping probability, and the time delay, reach their local maxima at different values of k. We show numerically that τ(k)>0 at its local maxima, occurring just before the resonant part of the cross section reaches its local maxima. These results are discussed in the light of the standard definition of resonance.

Guaranteed resonance enclosures and exclosures for atoms and molecules.

In this paper, we confirm, with absolute certainty, a conjecture on a certain oscillatory behaviour of higher auto-ionizing resonances of atoms and molecules beyond a threshold. These results not only definitely settle a more than 30 year old controversy in Rittby et al. (1981 Phys. Rev. A 24, 1636–1639 (doi:10.1103/PhysRevA.24.1636)) and Korsch et al. (1982 Phys. Rev. A 26, 1802–1803 (doi:10.1103/PhysRevA.26.1802)), but also provide new and reliable information on the threshold. Our interval-arithmetic-based method allows one, for the first time, to enclose and to exclude resonances with guaranteed certainty. The efficiency of our approach is demonstrated by the fact that we are able to show that the approximations in Rittby et al. (1981 Phys. Rev. A 24, 1636–1639 (doi:10.1103/PhysRevA.24.1636)) do lie near true resonances, whereas the approximations of higher resonances in Korsch et al. (1982 Phys. Rev. A 26, 1802–1803 (doi:10.1103/PhysRevA.26.1802)) do not, and further that there exist two new pairs of resonances as suggested in Abramov et al. (2001 J. Phys. A 34, 57–72 (doi:10.1088/0305-4470/34/1/304)).

Late-Time Estimation for Resonance-Based Radar Target Identification

This communication presents a novel method for determining the commencement of the late-time resonant response of a radar target, without a priori knowledge of the target geometry or orientation. Insights from Kronecker's theorem for Hankel matrices are used to determine the offset for the processing window for extraction of the true complex natural resonances by exploiting characteristic changes in the Hankel matrix, including the matrix rank and the distribution of eigenvalues. Results are presented for both simulated and measured targets.

Role of resonances in building cross sections: Comparison between the Mittag–Leffler and the T‐matrix Green function expansion approaches

AbstractPeaks in collision cross sections are often interpreted as resonances. The complex dilation method, as well as other methods relying on analytic continuation of the scattering formalism, can be used to clarify whether these structures are true resonances in the sense that they are poles of the S‐matrix and the associated Green function. The performance of the Mittag–Leffler expansion and T‐matrix Green function expansion methods are formally and computationally compared. The two methods are applied to two model potentials. Eigenenergies, s‐wave residues, and cross sections are computed with both methods. The resonance contributions to the cross sections are further analyzed by removing the residue contributions from the Mittag–Leffler and Green function expansion sums, respectively. It is suggested that the contribution of a resonance to a cross section should be defined through its S‐matrix residue. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007

Order Reduction of Parametrically Excited Nonlinear Systems: Techniques and Applications

The basic problem of order reduction of nonlinear systems with time periodic coefficients is considered in state space and in direct second order (structural) form. In state space order reduction methods, the equations of motion are expressed as a set of first order equations and transformed using the Lyapunov–Floquet (L–F) transformation such that the linear parts of new set of equations are time invariant. At this stage, four order reduction methodologies, namely linear, nonlinear projection via singular perturbation, post-processing approach and invariant manifold technique, are suggested. The invariant manifold technique yields a unique ‘reducibility condition’ that provides the conditions under which an accurate nonlinear order reduction is possible. Unlike perturbation or averaging type approaches, the parametric excitation term is not assumed to be small. An alternate approach of deriving reduced order models in direct second order form is also presented. Here the system is converted into an equivalent second order nonlinear system with time invariant linear system matrices and periodically modulated nonlinearities via the L–F and other canonical transformations. Then a master-slave separation of degrees of freedom is used and a nonlinear relation between the slave coordinates and the master coordinates is constructed. This method yields the same ‘reducibility conditions’ obtained by invariant manifold approach in state space. Some examples are given to show potential applications to real problems using above mentioned methodologies. Order reduction possibilities and results for various cases including ‘parametric’, ‘internal’, ‘true internal’ and ‘true combination resonances’ are discussed. A generalization of these ideas to periodic-quasiperiodic systems is included and demonstrated by means of an example.

Central brightening due to constructive interference with, without, and despite dielectric resonance

To aid in discussion about the mechanism for central brightening in high field magnetic resonance imaging (MRI), especially regarding the appropriateness of using the term dielectric resonance to describe the central brightening seen in images of the human head. We present both numerical calculations and experimental images at 3 T of a 35-cm-diameter spherical phantom of varying salinity both with one surface coil and with two surface coils on opposite sides, and further numerical calculations at frequencies corresponding to dielectric resonances for the sphere. With two strategically placed surface coils it is possible to create central brightening even when one coil alone excites an image intensity pattern either bright on one side only or bright on both sides with central darkening. This central brightening can be created with strategic coil placement even when the resonant pattern would favor central darkening. Results in a conductive sample show that central brightening can similarly be achieved in weakly conductive dielectric materials where any true resonances would be heavily damped, such as in human tissues. Constructive interference and wavelength effects are likely bigger contributors to central brightening in MR images of weakly conductive biological samples than is true dielectric resonance.

Three-body continuum spatial correlations in Borromean halo nuclei

Spatial correlations in the three-body continuum of Borromean (having no bound binary subsystems) three-body systems are discussed. The hyperspherical harmonics method is used to investigate low-lying resonances and the soft dipole mode in the two-neutron halo nucleus $^{6}\mathrm{He}$, which has only the $\ensuremath{\alpha}+n+n$ continuum for excitation energies below $13\phantom{\rule{0.3em}{0ex}}\text{MeV}$. The spatial correlations reveal characteristic structures for true three-body resonances, a moderate amplification in the interior region for above-barrier resonances and long-range correlations in the cases of three-body ${1}^{\ensuremath{-}}$ virtual and ${0}^{+}$ continuum states.

Infrared studies of magnetic surface modes on antiferromagnets (invited)

In contrast to ferromagnets, where low frequency surface excitations typically have frequencies in the 10 GHz range, surface excitations in antiferromagnets are often in the several hundred GHz to few THz range. Theoretical predictions for surface spin waves on antiferromagnets indicate that they should be highly nonreciprocal, i.e., the properties of a wave with wave vector +k would be very different from those with a reversed wave vector of −k. Surface spin waves on antiferromagnets have recently been measured using a high resolution Fourier transform infrared spectrometer. The results show evidence of both true surface modes and surface resonances. The nonreciprocal features of the surface modes are seen in a dramatic nonreciprocal reflection. For example, the reflectivity can be 80% for one orientation, but when the incident and reflected waves are reversed the reflectivity drops to near zero. While the initial measurements were done on a bulk antiferromagnet, we also present calculations showing the results for thin films.

Normal form singularities in area-preserving maps and analytic continuation by Padé approximants

When in the Birkhoff series the leading contributions of the main resonances (given by the continued fraction expansion of the linear frequency) are retained, the Fourier spectrum of the conjugation function is given by geometric series. Any Fourier coefficient corresponding to a main resonance is therefore approximated by a single pole in ϱ 2, where ϱ is the distance from the origin in the normal coordinates, and the residue turns out to be exponentially small in 1/|ϱ| and proportional to the resonance width. In the ϱ 2 plane the poles are on the real axis and their signs alternate since at any pole the winding number is resonant. The poles located on the real ϱ axis correspond to true resonances with island structure in the phase plane, the poles on the imaginary ϱ axis to virtual resonances with no counterpart in the phase plane. When the next to leading contribution of the main resonances are included, any Fourier coefficient is given by a series of poles with exponentially small residues. This singularity pattern is consistent with the asymptotic properties of the Birkhoff series and suggest that the Padé approximants should be used to continue them analytically. This is fully confirmed by numerical results.

INC model calculation of (π+, p), ( $$\bar p$$ , p) and (K+, p) spectra in the 1GeV/c range, p) and (K+, p) spectra in the 1GeV/c range

The proton inclusive spectra measured with 800MeV/cπ+ beam (BNL) on light nuclei, and the 600MeV/c\(\bar p\) beam (LEAR) on6Li and63Cu have been successfully interpreted in terms of the Intranuclear Cascade (INC) Model. According to the INC model predictions the\(\bar p\) induced reaction results in high pion yield for the light6Li target case. This is attributed to the nonavailability of nucleons to be knocked out by the pions produced following the\(\bar p\) annihilation. While in the case of the heavier63Cu target some of the internally produced pions are subsequently absorbed leading to thermal equilibration of the nuclear medium, resulting in a softer proton spectrum at the high-energy end. Based on the INC model predictions the K+ induced reactions could be studied to search for the «true resonances» beyond 1GeV K+ energy.

Exotic Drell-Yan processes at the tevatron and the SSC

We present a systematic analysis of various exotic sources (composite, technicolour, non-linear gauged σ and E 6 superstring-inspired models) for the Drell-Yan processes at the Tevatron and the SSC. We show that light composite bosons can be easily differentiated from others by studying the lepton pair distribution and/or the forward-backward asymmetry. For heavy bosons of mass equal to 1 TeV, E 6 bosons are the only ones which remain true resonances. Spin-zero and spin-two heavy composite bosons have characteristics distributions. The effects of contact interactions for a value of the compositeness scale of 1 TeV are important at the SSC collider.

Superresonant systems of scatterers. I

Single compact scatterers insonified at frequencies near their intrinsic radial resonance ω0 act primarily as monopole sources, amplifying the local pressure field by a factor of order (kRa)−1, a being the scatterer radius and kR the wavenumber at ω0 in the surrounding material (kRa≂1.37×10−2 for air bubbles in water). This paper deals with systems of such scatterers (bubbles, inflated balloons, thin shells in water) which, through coupling via multiple scatter, may themselves exhibit true resonant modes at frequencies close to ω0, amplifying the local pressure field by a further factor of order (kRa)−1, leading to net amplifications in excess of 103 for bubbles or balloons, i.e., of over 120 dB for the intensity. This phenomenon, referred to as superresonance, is predicted by formulas for the equivalent source strength B of each scatterer of the system, proper account being taken of multiple scatter between all elements, using a self-consistent scheme. It has long been known that for a pair of scatterers (doublet), or in periodic lattices, B exhibits maxima for certain values of x=kl (l being the distance between scatterers), which had been called resonances [Twersky, J. Opt. Soc. Am. 52, 145–171 (1962)], but are in fact only partial (quasi) resonances. It is demonstrated here that, under suitable conditions, true resonances exist; i.e., in the absence of attenuation, B exhibits real poles (the infinities are removed by introduction of acoustic damping and finite radii). The existence of such superresonances is furthered by the presence of elastic boundaries in the fluid near the system (e.g., a thin plate or a solid half-space). Multiple scatter coupling then occurs by means of guided flexural or surface waves, which offer a more efficient energy transport mechanism between scatterers than the acoustic modes (volume waves) of a homogeneous fluid fullspace. In the latter case, a doublet of resonant bubbles, or shells, exhibits no superresonances (poles), only partial ones (quasiresonances); it takes at least three scatterers to create a true superresonant system in a fullspace, and even so there is only one resonant spacing (kl=1.8955, ω=1.0022 ω0 for an equilateral triangle of bubbles in water). But near a thin plate, both doublet and triplet configurations develop spectra of resonant spacings xn and frequencies ωn. Near an elastic solid half-space, Stoneley (Scholte) wave coupling creates similar effects. The role of small interface roughness on this phenomenon is also examined.

Resonant scatterer configurations near elastic boundaries

Systems of precisely spaced bubbles, air‐filled shells or balloons in water, insonified at frequencies near their intrinsic radial resonance ω0 exhibit true resonant modes. For resonant wavenumber kR (in water) and scatterer radius a, this leads to amplification factors (kRa)−1 above and beyond the similar classical factor due to single scatter, and thus to net pressure amplification of order (kRa)−2 relative to the incident field (i.e., over 120 dB on the intensity scale, on or inside the scatterer). This effect is predicted by formulas for the equivalent source strength B of each scatterer of the system, account being taken of multiple scatter interaction. Whereas for pairs of scatterers, or periodic lattices, B is known to exhibit relatively modest maxima for selected values of x = kl (l being the spacing between scatterers), which had been called resonances [V. Twersky, J. Opt. Soc. Am. 52, 145–171 (1962)], one can show that, under certain conditions, true resonances exist— i.e., in the absence of attenuation, B exhibits real poles. This phenomenon is much enhanced for systems near elastic boundaries for which coupling between scatterers is mediated by surface waves, and doublet or triplet configurations develop spectra of resonant configurations xn at frequencies ωn (the latter being all close to ω0 for a spectrum of scatterer sizes an). [Work supported by ONR.]

Are some π−N resonances a manifestation of coupling to the π−Δ(1232) channel?

Using separable potentials for the processes πN → πN, πN → πΔ, and πΔ → πΔ, we investigate the possibility that some of the low lying resonances seen in π−N elastic scattering may be due to the opening of the threshold for π− Δ production. We find excellent fits to the P 11, D 13, D 15, and S 31 phase shifts and inelasticities without the need for introducing the P 11(1440), D 13(1520), D 15(1675), and S 31(1620) resonances explicitly. While not claiming that true particle resonances do not exist, our results indicate that coupling to the π − Δ channel should not be ignored.