Resonance Hamiltonian Research Articles

Dynamical Structures Associated with High-Order and Secondary Resonances in the Spin–Orbit Problem

In our solar system, spin–orbit resonances are common under Sun–planet, planet–satellite, and binary asteroid configurations. In this work, high-order and secondary spin–orbit resonances are investigated by taking numerical and analytical approaches. Poincaré sections as well as two types of dynamical maps are produced, showing that there are complicated structures in the phase space. To understand numerical structures, we adopt the theory of perturbative treatments to formulate resonant Hamiltonian for describing spin–orbit resonances. The results show that there is an excellent agreement between analytical and numerical structures. It is concluded that the main V-shape structure arising in the parameter space (θ̇,α) is sculpted by the synchronous primary resonance, those minute structures inside the V-shape region are dominated by secondary resonances and those structures outside the V-shape region are governed by high-order resonances. Finally, the analytical approach is applied to binary asteroid systems (65803) Didymos and (4383) Suruga to reveal their phase-space structures.

Compatibility relationships in van der Waals quasicrystals

The incommensurate twist structure and the interlayer coupling induce van der Waals quasicrystals (vdW-QCs). Replacing conventional band theory requiring translational symmetry, the resonant coupling Hamiltonian endowing the quasiband structure in $\mathbit{k}$ space is adopted to describe electronic properties of vdW-QCs [Moon et al., Phys. Rev. B 99, 165430 (2019)]. Here we investigate the symmetries of the resonant coupling Hamiltonians in dodecagonal and octagonal vdW-QCs. Through symmetry analyses we derive compatibility relationships (CRs) between $\mathrm{\ensuremath{\Gamma}}$ point and other irreducible pathways and predict the symmetry changes and band splits. Especially, we find that from $\mathrm{\ensuremath{\Gamma}}$ point to Brillouin zone corner points of monolayers, arbitrary twofold degenerate states are split into one ${A}^{\ensuremath{'}}$ and one ${A}^{\ensuremath{''}}$ state, and from $\mathrm{\ensuremath{\Gamma}}$ point to the intersection points of two Brillouin zones of monolayers, arbitrary twofold degenerate states are split into one $A$ and one $B$ state. Instead of projection operation analyses, we discuss the CRs of different point groups between the coupled bilayers and uncoupled monolayers to construct the interlayer hybridization selection rules (IHSRs) [Yu et al., Phys. Rev. B 105, 125403 (2022)], which govern how the interlayer states interact with each other in the resonant coupling systems of dodecagonal and octagonal vdW-QCs. These derived IHSRs indicate that the first two main resonant couplings allow the nonequivalent hybridizations only between ${B}_{1}$ and ${B}_{2}$ states and the equivalent hybridizations for $A$, ${A}_{i}$, ${A}^{\ensuremath{'}}$, ${A}^{\ensuremath{''}}$, $E$, or ${E}_{i}$ states.

Quadrupole and octupole order resonances in non-restricted hierarchical planetary systems

ABSTRACT Non-restricted hierarchical three-body configurations are common in various scales of astrophysical systems. Dynamical structures of the quadrupole-order resonance (the von Zeipel–Lidov–Kozai resonance) and the octupole-order resonance (the apsidal resonance) under the non-restricted hierarchical planetary systems are investigated in this work by taking advantage of perturbative treatments. Under the quadrupole-order Hamiltonian model, the distribution of libration and circulation regions as well as the distribution of flipping region are analytically explored in the parameter space spanned by the conserved quantities. The fundamental frequencies of system are produced and then the nominal location of octupole-order resonance is identified. From the viewpoint of perturbative theory, the quadrupole-order Hamiltonian determines the unperturbed dynamical model and the octupole-order Hamiltonian plays a role of perturbation to the quadrupole-order dynamics. The resonant Hamiltonian for octupole-order resonances is formulated by means of averaging theory, giving rise to a new constant of motion. Phase portraits are produced to analyse dynamical structures of octupole-order resonance, including resonant centres, saddle points, dynamical separatrices and islands of libration. By analysing phase portraits, it is found that there are four branches of libration centre and eight libration zones in the considered space. Applications to orbit flips show that there are five flipping regions.

The Clebsch–Gordan Coefficients and Their Application to Magnetic Resonance

The Clebsch–Gordan coefficients are extremely useful in magnetic resonance theory, yet have an infamous perceived level of complexity by many students. The Clebsch–Gordan coefficients are used to determine both the matrix elements of the spherical tensor operators and the total angular momentum states of a system of component angular momenta. Full derivations of these coefficients are rarely worked through step by step. Instead, students are provided with tables accompanied by little or no explanation of where the values in it originated from. This lack of direction is often a source of confusion for students. For this reason, we work through two common examples of the application of the Clebsch–Gordan coefficients to magnetic resonance experiments. In the first, we determine the components of the magnetic resonance Hamiltonian of ranks 0, 1, and 2 and use these to identify the secular portion of the static, heteronuclear dipolar Hamiltonian. In the second, we derive the singlet and triplet states that arise from the interaction of two identical spin- 1 / 2 particles.

Response of Quasi-Integrable and Resonant Hamiltonian Systems to Fractional Gaussian Noise

Abstract The major difficulty in studying the response of multi-degrees-of-freedom (MDOF) nonlinear dynamical systems driven by fractional Gaussian noise (fGn) is that the system response is not Markov process diffusion and thus the diffusion process theory cannot be applied. Although the stochastic averaging method (SAM) for quasi Hamiltonian systems driven by fGn has been developed, the response of the averaged systems still needs to be predicted by using Monte Carlo simulation. Later, noticing that fGn has rather flat power spectral density (PSD) in certain frequency band, the SAM for MDOF quasi-integrable and nonresonant Hamiltonian system driven by wideband random process has been applied to investigate the response of quasi-integrable and nonresonant Hamiltonian systems driven by fGn. The analytical solution for the response of an example was obtained and well agrees with Monte Carlo simulation. In the present paper, the SAM for quasi-integrable and resonant Hamiltonian systems is applied to investigate the response of quasi-integrable and resonant Hamiltonian system driven by fGn. Due to the resonance, the theoretical procedure and calculation will be more complicated than the nonresonant case. For an example, some analytical solutions for stationary probability density function (PDF) and stationary statistics are obtained. The Monte Carlo simulation results of original system confirmed the effectiveness of the analytical solutions under certain condition.

Multiharmonic Hamiltonian models with applications to first-order resonances

ABSTRACT In this work, two multiharmonic Hamiltonian models for mean motion resonances are formulated and their applications to first-order resonances are discussed. For the kp:k resonance, the usual critical argument φ = kλ − kpλp + (kp − k)ϖ is taken as the resonant angle in the first model, while the second model is characterized by a new critical argument σ = φ/kp. Based on canonical transformations, the resonant Hamiltonians associated with these two models are formulated. It is found that the second Hamiltonian model holds two advantages in comparison with the first model: (i) providing a direct correspondence between phase portraits and Poincaré sections, and (ii) presenting new phase-space structures where the zero-eccentricity point is a visible saddle point. Then, the second Hamiltonian model is applied to the first-order inner and outer resonances, including the 2:1, 3:2, 4:3, 2:3, and 3:4 resonances. The phase-space structures of these first-order resonances are discussed in detail and then the libration centres and associated resonant widths are identified analytically. Simulation results show that there are pericentric and apocentric libration zones where the libration centres diverge away from the nominal resonance location as the eccentricity approaches zero and, in particular, the resonance separatrices do not vanish at arbitrary eccentricities for both the inner and outer (first-order) resonances.

Transfers of energy through fast diffusion channels in some resonant PDEs on the circle

<p style='text-indent:20px;'>In this paper we consider two classes of resonant Hamiltonian PDEs on the circle with non-convex (respect to actions) first order resonant Hamiltonian. We show that, for appropriate choices of the nonlinearities we can find time-independent linear potentials that enable the construction of solutions that undergo a prescribed growth in the Sobolev norms. The solutions that we provide follow closely the orbits of a nonlinear resonant model, which is a good approximation of the full equation. The non-convexity of the resonant Hamiltonian allows the existence of <i>fast diffusion channels</i> along which the orbits of the resonant model experience a large drift in the actions in the optimal time. This phenomenon induces a transfer of energy among the Fourier modes of the solutions, which in turn is responsible for the growth of higher order Sobolev norms.</p>

KAM Tori for Completely Resonant Hamiltonian Derivative Beam Equations on $$\mathbb {T}^2$$

In this paper, we study a class of completely resonant beam equations with derivative nonlinearities on $$\mathbb {T}^2$$ $$\begin{aligned} u_{tt}+\Delta ^2 u+u|\nabla u|^2+u^2\Delta u=0. \end{aligned}$$ We will prove the existence of small-amplitude, quasi-periodic solutions for the above equations via KAM method.

Resonant Hamiltonian systems associated to the one-dimensional nonlinear Schrödinger equation with harmonic trapping

We study two resonant Hamiltonian systems on the phase space L 2 ( R → C ) : the quintic one-dimensional continuous resonant equation, and a cubic resonant system that has appeared in the literature as a modified scattering limit for an NLS equation with cigar shaped trap. We prove that these systems approximate the dynamics of the quintic and cubic one-dimensional NLS with harmonic trapping in the small data regime on long times scales. We then pursue a thorough study of the dynamics of the resonant systems themselves. Our central finding is that these resonant equations fit into a larger class of Hamiltonian systems that have many striking dynamical features: non-trivial symmetries such as invariance under the Fourier transform and the flow of the linear Schrödinger equation with harmonic trapping, a robust well-posedness theory, including global well-posedness in L 2 and all higher L 2 Sobolev spaces, and an infinite family of orthogonal, explicit stationary wave solutions in the form of the Hermite functions.

Long time stability of small finite gap solutions of the cubic nonlinear Schrödinger equation on [formula omitted

In this paper we study long time stability of a class of nontrivial, quasi-periodic solutions depending on one spacial variable of the cubic defocusing non-linear Schrödinger equation on the two dimensional torus. We prove that these quasi-periodic solutions are orbitally stable for finite but long times, provided that their Fourier support and their frequency vector satisfy some complicated but explicit condition, which we show holds true for most solutions.The proof is based on a normal form result. More precisely we expand the Hamiltonian in a neighborhood of a quasi-periodic solution, we reduce its quadratic part to diagonal constant coefficients through a KAM scheme, and finally we remove its cubic terms with a step of nonlinear Birkhoff normal form. The main difficulty is to impose second and third order Melnikov conditions; this is done by combining the techniques of reduction in order of pseudo-differential operators with the algebraic analysis of resonant quadratic Hamiltonians.

Singular reduction of resonant Hamiltonians

We investigate the dynamics of resonant Hamiltonians with n degrees of freedom to which we attach a small perturbation. Our study is based on the geometric interpretation of singular reduction theory. The flow of the Hamiltonian vector field is reconstructed from the cross sections corresponding to an approximation of this vector field in an energy surface. This approximate system is also built using normal forms and applying reduction theory obtaining the reduced Hamiltonian that is defined on the orbit space. Generically, the reduction is of singular character and we classify the singularities in the orbit space, getting three different types of singular points. A critical point of the reduced Hamiltonian corresponds to a family of periodic solutions in the full system whose characteristic multipliers are approximated accordingly to the nature of the critical point.

Multi-pulse Orbits and Homoclinic Trees in a Non-autonomous Resonant Hamiltonian System

In this study, we develop the energy-phase method to deal with the high-dimensional non-autonomous nonlinear dynamical systems. Our generalized energy-phase method applies to integrable, two-degree-of freedom non-autonomous resonant Hamiltonian systems. As an example, we investigate the multi-pulse orbits and homoclinic trees for a parametrically excited, simply supported rectangular thin plate of two-mode approximation. In both the Hamiltonian and dissipative case we find homoclinic trees, which describe the repeated bifurcations of multi-pulse solutions, and we present visualizations of these complicated structures.

Fourth body gravitation effect on the resonance orbit characteristics of the restricted three-body problem

In this paper, the gravitational effect of a fourth body on the resonance orbit defined in the restricted three-body problem (RTBP) is considered. In this regard, Resonance Hamiltonian of the RTBP and the Hamiltonian associated with the fourth gravitational body that perturbs the resonance orbit are computed. The Melnikov approach is utilized as a mean for the detection of chaos in resonance orbit under the influence of the fourth gravitation body. In addition, the numerical simulation of RTBP and bicircular four-body model, time–frequency analysis (TFA), and fast Lyapunov indicator (FLI) are performed to verify the results of the Melnikov approach. The results indicate that for the (2:1) resonance orbit, the Melnikov integral computed over outer loop of separatrix does not cross the zero line, and consequently chaos is unexpected. On the other hand, the Melnikov integral computed over the inner sepratrix loop crosses the zero line indicating a potential for chaos. Similarly, it is shown that inclusion of the fourth body gravitation leads the (3:1) as well as the (4:1) resonance orbits to chaos. Additionally, simulation results indicate that for some initial conditions on the separatrix, the fourth body effect bounds the amplitude of the resonance orbits while diffusing its corresponding trajectory in the bounded phase space. TFA and the FLI verify similar results.

Stochastic Averaging of Quasi-Integrable and Resonant Hamiltonian Systems Under Combined Gaussian and Poisson White Noise Excitations

A stochastic averaging method for quasi-integrable and resonant Hamiltonian systems subject to combined Gaussian and Poisson white noise excitations is proposed. The case of resonance with α resonant relations is considered. An (n + α)-dimensional averaged Generalized Fokker–Plank–Kolmogorov (GFPK) equation for the transition probability density of n action variables and α combinations of phase angles is derived from the stochastic integrodifferential equations (SIDEs) of original quasi-integrable and resonant Hamiltonian systems by using the jump-diffusion chain rule. The reduced GFPK equation is solved by using finite difference method and the successive over relaxation method to obtain the stationary probability density of the system. An example of two nonlinearly damped oscillators under combined Gaussian and Poisson white noise excitations is given to illustrate the proposed method. The good agreement between the analytical results and those from digital simulation shows the validity of the proposed method.

Geometric phase accompanying SU(2) coherent states for quantum polarized light

The geometric phase is studied for quantum polarized light described by an SU(2) coherent state. This is revealed by using quantum interference. The SU(2) coherent state is constructed from the quantized Stokes parameters. The quantum interferometry is arranged by the evolution operator implemented by the Hamiltonian inspired from the birefringence of the Faraday and Kerr effect. The geometric phase is given by the coherent state path integral. In particular, by considering a resonant Hamiltonian, the geometric phase is extracted from the total factor that consists of both geometric and dynamical phases. This extraction forms the basis of an experimental way to detect the geometric phase.

Strongly interacting fermions in an optical lattice

We analyze a system of two-component fermions, which interact via a Feshbach resonance in the presence of a three-dimensional lattice potential. By expressing a two-channel model of the resonance in the basis of Bloch states appropriate for the lattice, we derive an eigenvalue equation for the two-particle bound states, which is nonlinear in the energy eigenvalue. Compact expressions for the interchannel matrix elements, numerical methods for the solution of the nonlinear eigenvalue problem, and a renormalization procedure to remove ultraviolet divergences are presented. From the structure of the two-body solutions we identify the relevant degrees of freedom, which describe the resonance behavior in the lowest Bloch band. These degrees of freedom, which we call dressed molecules, form an effective closed channel in a many-body model of the resonance, the Fermi resonance Hamiltonian (FRH). It is shown how the properties of the FRH can be determined numerically by solving a projected lattice two-channel model at the two-particle level. As opposed to single-channel lattice models such as the Hubbard model, the FRH is valid for general $s$-wave scattering length and resonance width. Hence, the FRH provides an accurate description of the BEC-BCS crossover for ultracold fermions on an optical lattice.

Spin manipulation and relaxation in spin-orbit qubits

We derive a generalized form of the Electric Dipole Spin Resonance (EDSR) Hamiltonian in the presence of the spin-orbit interaction for single spins in an elliptic quantum dot (QD) subject to an arbitrary (in both direction and magnitude) applied magnetic field. We predict a nonlinear behavior of the Rabi frequency as a function of the magnetic field for sufficiently large Zeeman energies, and present a microscopic expression for the anisotropic electron g-tensor. Similarly, an EDSR Hamiltonian is devised for two spins confined in a double quantum dot (DQD), where coherent Rabi oscillations between the singlet and triplet states are induced by jittering the inter-dot distance at the resonance frequency. Finally, we calculate two-electron-spin relaxation rates due to phonon emission, for both in-plane and perpendicular magnetic fields. Our results have immediate applications to current EDSR experiments on nanowire QDs, g-factor optimization of confined carriers, and spin decay measurements in DQD spin-orbit qubits.

Generating functions for effective hamiltonians. Symmetry, topology, combinatorics

Recent developments associated with old technique of generating functions and invariant theory which I have started to apply to molecular problems due to my collaboration with Yu.F. Smirnov about 25 years ago are discussed. Three aspects are presented: the construction of diagonal in polyad quantum number effective resonant vibrational Hamiltonians using the symmetrized Hadamard product; the decomposition of the number of state generating function into regular and oscillatory contributions and its relation with Todd polynomials and topological characterization of energy bands; qualitative aspects of resonant oscillators and fractional monodromy as one of new generalizations of Hamiltonian monodromy.

Resonance and Landesman-Lazer conditions for first order systems in R^2

The first part of the paper surveys the concept of resonance for T-periodic nonlinear problems. In the second part, some new results about existence conditions for nonlinear planar systems are presented. In particular, the Landesman-Lazer conditions are generalized to systems in R ^2 where the nonlinearity interacts with two resonant Hamiltonians. Such results apply to second order equations, generalizing previous theorems by Fabry [4] (for the undamped case), and Frederickson-Lazer [9] (for the case with friction). The results have been obtained with A. Fonda, and have been published in [8].

Electronic spectroscopy of jet-cooled HCP+: Molecular structure, phosphorus hyperfine structure, and Renner-Teller analysis

Laser-induced fluorescence spectra of jet-cooled HCP(+) and DCP(+) have been obtained with the pulsed discharge technique using HCPDCP and argon precursor mixtures. Transitions involving all of the excited state vibrations have been observed and a set of vibrational constants has been obtained. High-resolution spectra of the (2)Pi(32) components of the 0(0) (0) bands of both isotopomers have been recorded, and these spectra show resolved phosphorus hyperfine structure which allowed the determination of the excited state Fermi contact parameter. The B values were used to obtain the ground and excited state effective geometric parameters as r(0) (")(CH)=1.077(2) A, r(0) (")(CP)=1.6013(3) A, r(0) (')(CH)=1.082(2) A, and r(0) (')(CP)=1.5331(3) A. A Renner-Teller analysis of the ground state vibrational energy levels obtained from the literature was attempted. All of the observed levels of DCP(+) and the majority of those of HCP(+) were satisfactorily fitted with a standard Renner-Teller model, but three HCP(+) levels showed large systematic deviations which could not be accommodated by reassignments or improvements in the Fermi resonance Hamiltonian. Further improvements in the theory or in the experimental data will be needed to resolve this discrepancy.