Degenerate Oscillator Research Articles

Response Solutions for Degenerate Reversible Harmonic Oscillators with Zero-average Perturbation

Response Solutions for Degenerate Reversible Harmonic Oscillators with Zero-average Perturbation

Photorefractive four-wave mixing cannot be treated perturbatively. Revisiting the undepleted-pump approximation

We theoretically consider photorefractive degenerate four-wave mixing oscillators and show that, for this particular system, the linear stability analysis technique leads to incorrect results. The reason for this astonishing failure lies in the unphysical predictions of the undepleted-pump approximation, which appear naturally during the linearization process of the full model. As a consequence, photorefractive four-wave mixing does not seem to permit a perturbative treatment that allows the derivation of mean-field models, and one is forced to use the full model in order to make sensible predictions.

Response Solutions for Completely Degenerate Oscillators Under Arbitrary Quasi-Periodic Perturbations

Response Solutions for Completely Degenerate Oscillators Under Arbitrary Quasi-Periodic Perturbations

Response Solutions of Quasi-periodically Forced Degenerate Oscillator Equations with Small Parameters

Response Solutions of Quasi-periodically Forced Degenerate Oscillator Equations with Small Parameters

Etalon-Assisted Determination of the Complex Index of Refraction of a Solution for the Study of Strong Cavity-Vibrational Coupling of PF6- in Acetonitrile.

A new method is established using an etalon cavity to assist in the determination of the wavelength-dependent complex index of refraction of a solution throughout the mid-infrared range. The results are used to study the cavity-vibration polaritons of PF6- in acetonitrile. Mixed states are formed by placing solution inside a pair of parallel plate mirrors with a wavelength-scale spacing, i.e., within an etalon, such that there are cavity states that are angle-tuned into resonance with the strong P-F vibrations. The dominant ν3 vibrations of PF6- consist of nearly triply degenerate oscillations of the partial-positively charged phosphorous against antisymmetric concerted motions of different sets of fluorine atoms with partial negative charges. These vibrations are dominant even though the solute is 29 times less concentrated than the solvent on a molar basis. The first part of the paper describes the method of determining the complex index of refraction of the solution from a combination of etalon transmission maxima and the attenuated total reflection (ATR) absorption spectrum of the solution. The results are presented as an analytical function including a sum of 37 vibrational contributions. Absolute integrated isolated band intensities were determined to be 463 ± 4, 462 ± 7, and 266 ± 4 km/mol for the three ν3 PF6- vibrations at 841.4, 847.4, and 854.0 cm-1, respectively, which sum to 1191 ± 9 km/mol for the ν3 band. Then, the results are used to simulate the measured etalon transmission using the transfer matrix (TM) method with and without the ν3 target vibrations. The etalon transmission simulations reconstruct the position of cavity modes in the absence of target vibrations. They provide input data for the testing of simple quantum mechanical models for the interaction of vibrations with cavity modes and the interactions of vibrations with other vibrations within the molecule and between solute and solvent. The model shows that the nearly degenerate ν3 vibrations interact with each other with a vibration-vibration coupling of 33 ± 5 cm-1. This is comparable to the cavity-vibration coupling of 30.4 ± 2.9 cm-1 of the two strongest vibrations of PF6-.

ЗБУДЖЕННЯ ХВИЛІ TЕ01 У КРУГЛОМУ ХВИЛЕВОДІ ЗА ДОПОМОГОЮ ВИЩИХ ТИПІВ КОЛИВАНЬ ВІДКРИТОГО РЕЗОНАТОРА

Subject and Purpose. Peculiarities of the TE01 wave excitation in a circular waveguide located in the center of the flat mirror of a hemispherical open resonator (OR) are studied using the OR oscillation type TEM30q and the OR degenerate oscillation type TEM in the extremely high frequency (EHF) range. Methods and Methodology. The efficiency of the circular-waveguide TE01 wave excitation using the TEM30q and TEM11q types of OR oscillations is evaluated through the factor of antenna surface utilization. Amplitude distributions of the OR oscillation fields were measured by the trial body method. Results. It has been established that the efficiency, h, of the TE01 wave excitation in the circular waveguide is not high when it is by use of the OR oscillation ТЕМ30q. Things are different when the TE01 wave is excited with the inner ring of the TEM11q degenerate oscillation field whereby the h value shoots up to 95.5%, the circular waveguide section radius being a = 0.993w0, where w0 is the radius of the field spot of the main oscillation type TEM00q on the OR flat mirror. The experimental studies have confirmed that the TE01 wave excitation in the waveguide has high efficiency. The attachment of the circular oversized waveguide section worsens the OR loss by no more than 0.9 dB. The presence of the circular waveguide makes the TEM3012 oscillation type transform into the TEM1112 degenerate oscillation. Conclusions. The considered OR can be used as a storage resonator in the construction of electromagnetic pulse compressors in the EHF range, since the resonant system loss increases insignificantly, and the circular waveguide itself is oversized. It is easy to implement an interference switch in such a waveguide.

Highly Sensitive Coupled Oscillator Based on an Exceptional Point of Degeneracy and Nonlinearity

We propose a scheme for obtaining highly-sensitive oscillators in a coupled-resonator system with an exceptional point of degeneracy (EPD) and a small instability. The oscillator with the exceptional degeneracy is realized by using two coupled resonators with an almost balanced small-signal gain and loss, that saturates due to nonlinear effects of the active component, resulting in an oscillation frequency that is very sensitive to a perturbation of the circuit. Two cases are investigated, with two parallel LC resonators with balanced small-signal gain and loss that are either coupled wirelessly by mutual inductance or coupled-wired by a capacitor. This paper demonstrates theoretically and experimentally the conditions to obtain a second-order EPD oscillator and analyzes the ultrasensitivity of the oscillation frequency to components' perturbation, including the case of asymmetric perturbation that breaks PT-symmetry. We discuss the effects of nonlinearity on the performance of the oscillator and how the proposed scheme improves the sensing's sensitivity of perturbations. In contrast to previous methods, our proposed degenerate oscillator can sense positive or negative changes of a circuit component. The degenerate oscillator circuit may find applications in various areas such as ultrasensitive sensors, tunable oscillators and modulators.

Cascade of isospin phase transitions in Bernal-stacked bilayer graphene at zero magnetic field

Emergent phenomena arising from the collective behaviour of electrons is expected when Coulomb interactions dominate over the kinetic energy, and one way to create this situation is to reduce the electronic bandwidth. Bernal-stacked bilayer graphene intrinsically supports saddle points in the band structure that are predicted to host a variety of spontaneous symmetry-broken states1,2,3,4,5,6,7. Here we show that bilayer graphene displays a cascade of symmetry-broken states with spontaneous spin and valley isospin ordering at zero magnetic field. We independently tune the carrier density and electric displacement field to explore the phase space of isospin order. Itinerant ferromagnetic states emerge near the conduction and valence band edges with complete spin and valley polarization. At larger hole densities, twofold degenerate quantum oscillations manifest in an additional symmetry-broken state that is enhanced by the application of an in-plane magnetic field. Both symmetry-broken states display enhanced layer polarization, suggesting a coupling to the layer degree of freedom1,7. These states occur in the absence of a moiré superlattice and are intrinsic to natural graphene bilayers. Therefore, we demonstrate that bilayer graphene represents a related but distinct approach to produce collective behaviour from flat dispersion, complementary to engineered moiré structures.

Response Solutions in Degenerate Oscillators Under Degenerate Perturbations

Response Solutions in Degenerate Oscillators Under Degenerate Perturbations

Parabolic invariant tori in quasi-periodically forced skew-product maps

We consider the existence of parabolic invariant tori for a class of quasi-periodically forced analytic skew-product maps φ:Rn×Td→Rn×Td:φ(zθ)=(z+ϕ(z)+h(z,θ)+ϵf(z,θ)θ+ω), where ϕ:Rn→Rn is a homogeneous function of degree l with l≥2 and h=O(|z|l+1). We obtain the following results: (a) For n=1, l being odd and ϵ sufficiently small, parabolic invariant tori exist if ω satisfies the Brjuno-Rüssmann's non-resonant condition. (b) For n=1, and ϵ sufficiently small, parabolic invariant tori also exist if one of the following conditions holds: (i) First order average is non-zero, first order non-average part is small enough and the forcing frequency ω does not need any arithmetic condition. (ii) First order average is non-zero and ω satisfies the Brjuno-type weak non-resonant condition; (iii) l=2, first order average is zero, both first and second order non-average parts are small enough and ω satisfying Brjuno-type weak non-resonant condition; (iv) l>2, first order average is zero, the second order average is non-zero, both first and second order non-average parts are small enough and ω satisfies the Brjuno-type weak non-resonant condition. (c) In the case n>1, if first order average belongs to the interior of the range of ϕ, Spec(Dϕ)∩iR=∅, first order non-average part is small enough and the forcing frequency ω does not need any arithmetic condition, then the quasi-periodically forced skew-product maps above admit parabolic invariant tori for ϵ sufficiently small. The main methods of this paper are KAM theory and fixed point theorem, which are finally shown that it can be directly applied to the existence problem of quasi-periodic response solutions of degenerate harmonic oscillators.

Completely degenerate responsive tori in Hamiltonian systems**The first author was partially supported by a CSC scholarship and Fundamental Research Funds of Shandong University. The second author was partially supported by NSERC discovery Grant 1257749, PIMS CRG Grant, a faculty development grant from the University of Alberta, and a Scholarship from Jilin University.

We consider the existence of responsive tori for the completely degenerate Hamiltonian system with the following Hamiltonian which is associated with the standard symplectic structure, where λ = ±1 and n > 2, n ⩾ m ⩾ 2 are integers. With P satisfying certain non-degenerate conditions, we obtain the following results: (1) For λ = −1 and ϵ sufficiently small, responsive tori exist for each ω satisfying a weak non-resonant condition; (2) For λ = 1 and ϵ* sufficiently small, there exists a Cantor set with almost full Lebesgue measure such that responsive tori exist for each if ω satisfies a Diophantine condition. Non-existence of responsive tori are also discussed when P fails to satisfy the non-degenerate condition. Our results are directly applicable to the existence problem of quasi-periodic responsive solutions of degenerate harmonic oscillators.

Oscillator realisations associated to the D-type Yangian: Towards the operatorial Q-system of orthogonal spin chains

We present a family of novel Lax operators corresponding to representations of the RTT-realisation of the Yangian associated with D-type Lie algebras. These Lax operators are of oscillator type, i.e. one space of the operators is infinite-dimensional while the other is in the first fundamental representation of so(2r). We use the isomorphism between the first fundamental representation of D3 and the 6 of A3, for which the degenerate oscillator type Lax matrices are known, to derive the Lax operators for r=3. The results are used to generalise the Lax matrices to arbitrary rank for representations corresponding to the extremal nodes of the simply laced Dynkin diagram of Dr. The multiplicity of independent solutions at each extremal node is given by the dimension of the fundamental representation. We further derive certain factorisation formulas among these solutions and build transfer matrices with oscillators in the auxiliary space from the introduced degenerate Lax matrices. Finally, we provide some evidence that the constructed transfer matrices are Baxter Q-operators for so(2r) spin chains by verifying certain QQ-relations for D4 at low lengths.

Ultrafast Quantum Interference in the Charge Migration of Tryptophan.

Extreme-ultraviolet-induced charge migration in biorelevant molecules is a fundamental step in the complex path leading to photodamage. In this work we propose a simple interpretation of the charge migration recently observed in an attosecond pump-probe experiment on the amino acid tryptophan. We find that the decay of the prominent low-frequency spectral structure with increasing pump-probe delay is due to a quantum beating between two geometrically distinct, almost degenerate charge oscillations. Quantum beating is ubiquitous in these systems, and at least on the few-to-tens of femtosecond time scales, it may dominate over decoherence the line intensities of time-resolved spectra. We also address the experimentally observed phase shift in the charge oscillations of two different amino acids, tryptophan and phenylalanine. Our results indicate that a beyond mean-field treatment of the electron dynamics is necessary to reproduce the correct behavior.

Sideband cooling of nearly degenerate micromechanical oscillators in a multimode optomechanical system

Multimode optomechanical systems are an emerging platform for studying fundamental aspects of matter near the quantum ground state and are useful in sensitive sensing and measurement applications. We study optomechanical cooling in a system where two nearly degenerate mechanical oscillators are coupled to a single microwave cavity. Due to an optically mediated coupling the two oscillators hybridize into a bright mode with strong optomechanical cooling rate and a dark mode nearly decoupled from the system. We find that at high coupling, sideband cooling of the dark mode is strongly suppressed. Our results are relevant to novel optomechanical systems where multiple closely-spaced modes are intrinsically present.

Electron-Beam-Driven Devices With Synchronous Multiple Degenerate Eigenmodes

We provide here a comprehensive exposition of a novel dispersion engineering framework for high-power electron-beam-driven devices based on multiple degenerate eigenwave synchronization in slow-wave structures (SWSs). These degenerate eigenwaves, related to an exceptional point of degeneracy (EPD) in the parameter space of the SWSs, lead to an enhanced interaction with the electron beam associated with a vanishing group velocity, which depends on the EPD order. Therefore, a “supersynchronous” mechanism based on multiple degenerate eigenwaves, associated with any order of EPD, provides advantages in terms of power generation efficiency at microwave frequencies, millimeter-wave frequencies, and terahertz frequencies. We present a physical description of EPDs in SWSs using a generalized Pierce theory developed for multimodal interaction. We show unique characteristics related to EPDs in “hot” (coupled with the electron beam) SWSs with third- and fourth-order EPDs for the sake of demonstration, but this can be extended to any order of degeneracy. We demonstrate advantages of the proposed regimes compared to conventional single-mode high-power devices such as backward-wave oscillators and traveling-wave tube amplifiers. In addition to that we show a possibility of high-power oscillators at millimeter-wave and terahertz frequencies based on a fourth-order EPD. In particular, we demonstrate a degenerate band-edge oscillator at 638 GHz based on particle-in-cell simulations. Apart from the examples provided herein, the concept of EPD can be used to design a wide variety of devices including gyrotrons, free electron lasers, and high-power isolators.

Exact solution of degenerate and nondegenerate optical parametric oscillator coupled with a squeezed thermal bath

We solve exactly the master equation of the degenerate and nondegenerate parametric oscillator in the presence of a squeezed reservoir. We show that the formal solutions of both systems can be obtained in terms of the density operator by applying squeeze transformations and using the superoperators formalism.

Connection between the su(3) algebraic and configuration spaces: bending modes of linear molecules

ABSTRACTAn approach to connect the dynamical group – used to describe the bending modes of linear molecules – with configuration space is discussed. The group may be seen as a consequence of adding a scalar boson to the space of two degenerate harmonic oscillators. The resulting group becomes the dynamical group for 2D systems The connection of the model with configuration space however is not obvious. Our approach is based on establishing a mapping between the algebraic and configuration states. The identification in the algebraic space of coordinates and momenta leads to a new identification of the group chains: they are associated with energy, coordinate and momentum representations. In addition an analysis of local-to-normal mode transition is presented. This provides a criterion to decide between a local or normal mode descriptions in an spectroscopic description. As an example we consider the situation of the bending modes of acetylene taking a Hamiltonian up to quartic order. The different results for the force constants obtained from the local and normal mode treatments leads to the conclusions that previous applications of the bending modes of acetylene dealing with a local mode treatment is on the verge of applicability.

Raman Scattering by Crystals of Rare-Earth Hexaborides with Different Isotopes of Boron

Monocrystals of lanthanum hexaboride LaB6 containing both natural boron and its isotopes 10B and 11B have been produced using the solution-melt method. Polyelement hexaboride rare-earths have been grown and the corresponding ceramics have been synthesized for the first time. All these crystals have been studied by means of various techniques, generally using Raman scattering. The Raman spectra attributed to various spectral lines corresponding to nonanalyzable representations have been obtained and interpreted. Frequencies and half-widths of spectral lines have been obtained, the removal of degeneracy and the development of respective splitting of degenerate oscillations induced by defects, mainly by boron isotope inclusions, have been identified. The influence of defects on the Raman spectra has been determined.

Рамановское рассеяние кристаллами гексаборидов редких земель с различными изотопами бора

AbstractMonocrystals of lanthanum hexaboride LaB_6 containing both natural boron and its isotopes ^10B and ^11B have been produced using the solution-melt method. Polyelement hexaboride rare-earths have been grown and the corresponding ceramics have been synthesized for the first time. All these crystals have been studied by means of various techniques, generally using Raman scattering. The Raman spectra attributed to various spectral lines corresponding to nonanalyzable representations have been obtained and interpreted. Frequencies and half-widths of spectral lines have been obtained, the removal of degeneracy and the development of respective splitting of degenerate oscillations induced by defects, mainly by boron isotope inclusions, have been identified. The influence of defects on the Raman spectra has been determined.

Stepwise relaxation and stochastic precession in degenerate oscillators dispersively coupled to particles

By numerical integration, we study the relaxation dynamics of degenerate harmonic oscillator modes dispersively coupled to particle positions. Depending on whether the effective inertial potential induced by the oscillators keep the particles confined, or if the particle trajectories traverse the system, the local oscillator energy dissipation rate changes drastically. The inertial trapping, release and retrapping of particles results in a characteristic step-wise relaxation process, with alternating regions of fast and slow dissipation. To demonstrate this phenomenon we consider first a one-dimensional minimal prototype model which displays these characteristics. We then treat the effect of dispersive interaction in a model corresponding to an adsorbate diffusing on a circular membrane interacting with its three lowest vibrational modes. In the latter model, stepwise relaxation appears only in the presence of thermal noise, which also causes a slow-in-time stochastic precession of the mixing angle between the degenerate eigenmodes.