Resonance Hamiltonian Research Articles

Holonomic phases accompanying resonant Hamiltonians

We study a specific aspect of the holonomic phase accompanying the SU(1, 1) coherent states. The holonomic phase is defined as a special object of the geometric phases in a more general sense than the definition as a finite connection of the infinitesimal overlap between two nearby coherent states. We consider the Hamiltonians that have characteristic “resonance solutions”, which serve as the specific orbits that are required to determine the phases in an explicit and concise way. The possibility of experimental detection of the holonomic phase is also pointed out.

Semiclassical quantization of KAM resonances in time-periodic systems

A semiclassical theory for the quasi-energy spectrum of time-periodic systems with accidental classical resonances is presented. The primitive EBK quantum conditions for integrable systems are extended to multiply periodic flux tubes occurring in resonant systems. Replacing classical actions by appropriate differential operators in a classical resonance Hamiltonian yields a uniform quantization of states related to a classical resonance region. The derivation being general for time-periodic systems unfolds the organization of the quasi-energy spectrum, reducing it to the spectrum of a single time-independent Hamiltonian of one degree of freedom with additional rational shifts of h(cross) omega . In a first-order approximation the resonance Hamiltonian is reduced to a pendulum leading to a differential equation of the Mathieu type for the quasi-energies. It is rigorously shown how parameters of the differential equation can be drawn from classical dynamics, using the data of the 'essential' orbits in the resonance zone, i.e. stability coefficients and actions of hyperbolic and elliptic orbits as well as actions of homoclinic orbits. The quasi-energy spectrum of a forced quartic oscillator is studied numerically and evaluated. Semiclassical quasi-energies related to a resonance of period three are computed and compared with exact quantum mechanical eigenvalues.

Forced secondary resonances at the 3 : 1 commensurability

For the 3 : 1 Jovian resonance problem, the time scales of the two degrees of freedom of the resonant Hamiltonian are well-separated [5]. With the adiabatic approximation, the solution for the fast oscillations can be found in terms of the slowly varying variables. Thus the rapidly oscillating terms in the slow oscillation equations can be treated as forced terms. We refer to the resonance between the forcing and intrinsic frequencies as a forced secondary one in this paper. We discuss the forced secondary resonances in asteroidal motion at the 3 : 1 commensurability by using Wisdom's method. The results show that the orbits situated originally near the resonance will leave the neighbourhood of resonance and tend to the separatrices and critical points for different energies, respectively. We have not found any stochastic web as expected in this case. Moreover, we study the problem of validity on the approximation of a system.

Experimental determination of the Hamiltonian for synchrotron motion with rf phase modulation.

Synchrotron motion with rf phase modulation was studied experimentally. Poincaré maps in the resonant precessing frame were obtained from the experimental data and compared with the tori of the resonant Hamiltonian. Our experimental data revealed island structure in longitudinal phase space. Experimental results for synchrotron motion excited by phase modulation at the third harmonic of the synchrotron frequency are also reported.Received 28 June 1993DOI:https://doi.org/10.1103/PhysRevE.48.4678©1993 American Physical Society

The role of potential anisotropy in the dynamics of the CH chromophore in CHX 3 (C 3v) symmetric tops

The influence of the anisotropy of the potential of the CH chromophore in CHX 3 molecules is systematically investigated on the basis of ab initio results and model Hamiltonians for CD 3H and CHF 3. The coupling between levels with different vibrational angular momentum (quantum number l) introduced by typical potential anisotropies in a normal coordinate subspace for the CH chromophore is found to produce weak but possibly measurable changes in the overall structure of the overtone spectrum. The local C ∞v symmetry represented by the spectroscopic Fermi resonance Hamiltonian is shown to be retained as an approximate dynamical symmetry on the characteristic subpicosecond time scale of the chromophore dynamics. Predictions of the forbidden weak bands are presented and the time-dependent l non-conserving transitions are calculated explicitly.

Quantum and semiclassical Husimi distributions for a one-dimensional resonant system

We compare exact and semiclassical Husimi distributions for the single eigenstates of a one-dimensional resonant Hamiltonian. We find that both distributions concentrate near the unstable fixed points even when these points are made complex by suitably varying a parameter.

Numerical studies on the interactions between Fermi polyads: quantum and semiclassical chaos

Numerical studies were performed for a system of two vibrational modes coupled by two Fermi resonances in order to investigate its classical and quantum chaotic behavior. Thirteen different expressions were used for the second Fermi resonance Hamiltonian. Examination of the statistical properties of the spectra showed that, when treated quantum mechanically, this model differs substantially from random matrices because (i) the first Fermi resonance is integrable and its nondiagonal terms cannot create level repulsion, spacing correlations, etc. and (ii) for the same value of the coupling strength, each coupling Hamiltonian has a different ability to induce quantum chaos, depending on its expression and on the levels it connects. Computation of the largest Lyapunov exponent for trajectories in the classical phase space showed that the quantum and the classical systems behave similarly: the quicker a coupling Hamiltonian induces quantum chaos, the quicker the trajectories become unstable in the classical phase space.

Collapse and revival of the state vector in the Jaynes-Cummings model: An example of state preparation by a quantum apparatus.

The evolution of the atomic state in the resonant Jaynes-Cummings model (a two-level atom interacting with a single mode of the quantized radiation field) with the field initially in a coherent state is considered. It is shown that the atom is to a good approximation in a pure state in the middle of what has been traditionally called the region. This pure state exhibits no Rabi oscillations and is reached independently of the initial state of the atom. For most initial states a total or partial of the wave function takes place early during the interaction, at the conventional collapse time, following which the state vector is recreated, over a longer time scale. PACS numbers: 42.50.— p, 03.65.— w, 42.52.+x The Jaynes-Cummings model' (JCM) is perhaps the simplest nontrivial example of two interacting quantum systems: a two-level atom and a single mode of the radiation field. In addition to its being exactly solvable, the physical system that it represents has recently become experimentally realizable with Rydberg atoms in high-Q microwave cavities. Comparison of the predictions of the model with those of its semiclassical version have served to identify a number of uniquely quantum properties of the electromagnetic field; indeed, the model displays some very interesting dynamics, and the differences with the semiclassical theory are both profound and unexpected. The JCM would also appear to be an excellent model with which to explore some of the more puzzling aspects of quantum mechanics, such as the possibility (or impossibility) to describe an interacting quantum system by a state vector undergoing unitary evolution; i.e., the socalled of the wave function. In the semiclassical version, the atom interacting with the classical electromagnetic field may at all times be described by a state vector evolving unitarily. What happens, however, when it is recognized that the field is itself a quantum system (which leads inevitably to )? This is the question addressed in this Letter. It does not seem to have been addressed before in full generality, although entanglement in the JCM dynamics plays an essential role in a recent measurement-theory-related proposal of Scully and Walther, and preparation of a pure state of the field in the JCM has been the subject of several theoretical investigations and may be close to being achieved experimentally. The resonant JCM interaction Hamiltonian may be written as Ht = hg(~a&(b (a+ a'(b)(a ~ ), ' is a coupling constant (d is the atomic dipole matrix element for the transition, m is the transition frequency, and Vis the mode volume), ~a) and ~b) are the upper and lower atomic levels, respectively, and a and a are the annihilation and creation operators of the field mode, which in the semiclassical theory are simply replaced by c numbers. The solution to the Schrodinger equation for the atom initially in state y(0)).,&,~ =a~a)+ p b) and field initially in state ttt(0))fi iu g„-OC„n) is ~y(t)) = g [[aC„cos(gOn+1 t) — ipC„s+i l(gnawn +1 t)]~ a& +[ — iaC„~sin(gran t)+pC„cos(gran

On resonant classical Hamiltonians with n frequencies

On resonant classical Hamiltonians with n frequencies

Fermi resonance in the overtone spectra of the CH chromophore in bromoform

We present the results of an investigation of the infrared spectrum of CHBr 3, recorded in the range 400–12000 cm −1 on an interferometric Fourier transform spectrometer. The observed bands are interpreted in terms of a tridiagonal Fermi resonance Hamiltonian for the coupled CH stretching and bending vibrations, and in terms of variational calculations using internal and normal coordinates. The effective parameters obtained from these analyses are used to predict band frequencies and relative intensities up to the N = 7 polyad. At high levels of excitation there is sub-picosecond redistribution of energy between the CH stretching and bending modes.

Vibrational spectrum and potential energy surface of the CH chromophore in CHD3

The rovibrational spectrum of trideutero-methane has been measured at resolutions mostly close to the Doppler limit on an interferometric Fourier transform spectrometer from the lowest fundamental vibration to high overtones of the CH stretching vibration (wave numbers from 900 to 12 000 cm−1). The CH chromophore spectrum is fully assigned and interpreted by means of the tridiagonal Fermi resonance Hamiltonian for the coupled CH stretching and bending vibrations. The Hamiltonian predicts and also fits the visible spectrum up to 19 000 cm−1 measured by Scherer et al., Perry et al., and Campargue et al. The effective tridiagonal Hamiltonian is derived ab initio by means of MRD-CI and full CI calculations of the potential surface of methane, a variational vibrational calculation in a normal coordinate subspace of the coupled CH stretching and bending motions and an approximate similarity transformation to tridiagonal form. Fits of the experimental results by the tridiagonal and the variational Hamiltonian lead to equivalent spectroscopic constants. A careful experimental estimate of the main Fermi resonance coupling constant gives k′sbb ≂(30±15) cm−1 in agreement with the best current ab initio result (31 cm−1). The ab initio potential in polar normal coordinates agrees with the potential derived from spectroscopic data covering an energy range of about 220 kJ mol−1 (more than half the dissociation energy). Good predictions are obtained for the parameters of the effective Hamiltonian, the spectral patterns, intensity distributions, and rotational constants in the Fermi resonance polyads. Three alternative interpretations of the parameters of the effective Hamiltonian are investigated and rejected on the basis of the available experimental and ab initio data. The results and conclusions are discussed in relation to intramolecular vibrational redistribution on the subpicosecond time scale and the recombination–dissociation kinetics of methane.

Vibrational angular momentum as an approximate constant of motion.

The time dependence of the vibrational angular momentum in the resonant Henon-Heiles Hamiltonian is approximated in closed form using Birkhoff-Gustavson perturbation theory. An energy threshold is found to separate two drastically differing regimes.

Tridiagonal Fermi resonance structure in the vibrational spectrum of the CH chromophore in CHF3. II. Visible spectra

The near IR and visible vibrational absorption spectra of CHF3 were recorded up to wave numbers of 17 500 cm−1 providing complete frequency coverage, together with paper I, from the low frequency fundamentals to the N=6 CH stretching–bending overtone multiplet. All strong bands in the high overtone spectra could be predicted and assigned by means of the tridiagonal Fermi resonance Hamiltonian, including a few combinations with intense CF3 stretching vibrations already observed for the low overtones. Improved vibrational Fermi resonance constants are presented on the basis of a fit to 35 assigned bands. An analysis of the rotational fine structure of the 2ν4 (E) overtone component and several Fermi resonance component bands result in values for αb and αs, which allow us to determine Be. In the high overtone bands no rotational fine structure is observed. The bands can be understood by introducing additional homogeneous rovibrational structures of phenomenological widths Γ≊1 to 10 cm−1. The results are discussed in relation to the separation of time scales for mode selective vibrational redistribution and further evolution. The overtone band strengths are reported and analyzed approximately with the empirical local Mecke dipole function.

Decoupling of the local mode stretching vibrations of water through rotational excitation. I. Quantum mechanics

We demonstrate that in a previously studied model of the stretching modes of the water molecule rotational motion in the plane of the molecule tends to decouple the stretches. For rotational angular momentum near J=18 ℏ, the two local mode stretches are almost entirely decoupled. The source of this decoupling is the centrifugal distortion which stabilizes the asymmetric stretch and effectively cancels the G-matrix coupling. This cancellation is clarified using three different methods: Direct examination of the numerically computed matrix elements, exact analytic matrix elements of an approximate Hamiltonian, and solutions of a Mathieu equation formulation of a classical resonance Hamiltonian. The importance of this result is discussed in light of the fact that strong rotational excitation can occur in infrared multiple photon excitation. If such rotational decoupling occurs in real systems, then intramolecular energy transfer would be diminished thus holding open the possibility of mode specific infrared excitation. The calculations were carried out by numerically evaluating matrix elements between a basis of Morse oscillator eigenstates using an efficient Gaussian quadrature scheme based on associated Laguerre polynomials.

Structure and dynamics of the excited CH–chromophore in (CF3)3CH

The absorption spectra of gaseous (CF3) 3CH (1,1,1,3,3,3-hexafluoro, 2-trifluoromethyl propane) were recorded in the IR between 800 and 12 000 cm−1 by high resolution interferometric Fourier transform techniques and in the visible from 12 000 to 17 000 cm−1 by laser photoacoustic spectroscopy. Instead of single bands in the CH overtone region, complex multiplet structures were observed. Thirty-nine bands were assigned as arising from the interacting CH-stretching and CH-bending manifolds, which account for most of the absorption in the overtone region. The results can be understood quantitatively with an effective, tridiagonal many-level Fermi resonance Hamiltonian. Close agreement is obtained for the positions and intensities of the observed spectral features using only seven spectroscopic parameters. The experimental and theoretical results are summarized in Tables II, III, and IV. The Hamiltonian can be used to calculate and understand the time-dependent redistribution of vibrational energy between the coupled CH-stretching and CH-bending vibrations. The role of broad vibrational band shapes and the possible exponential decay of CH excitation into a background of states from low-frequency vibrations is discussed.

Tridiagonal Fermi resonance structure in the IR spectrum of the excited CH chromophore in CF3H

The absorption spectrum of trifluoromethane has been recorded between 900 and 14 000 cm−1 with resolutions between 0.004 and 0.5 cm−1 (pressure broadened). 22 bands were assigned as arising from the interacting CH stretching and bending manifolds, which account for most of the absorption in the overtone region. The results can be understood quantitatively with an effective, tridiagonal many-level Fermi resonance Hamiltonian. The experimental and theoretical results are summarized in Table II. The Hamiltonian is given in Table III and shows a very large stretching–bending interaction constant ‖ksbb‖=106 cm−1, which is even larger than the diagonal anharmonic constant for the stretching vibration ‖x′ss‖=62 cm−1. This leads to extensive vibrational redistribution between stretching and bending motions at high levels of excitation. The time dependent redistribution is calculated with the spectroscopic Hamiltonian. A rotational analysis is presented for some of the bands involved in the Fermi resonance. The effect of the Fermi resonance on hot bands is investigated using the same Hamiltonian in comparison with experiment. The results are discussed in relation to the universal local dynamics of the isolated alkyl CH-stretching chromophore and in relation to the vibrational dynamics of highly excited polyatomic molecules as a function of certain elements of molecular structure.

On the relation of Child and Lawton’s harmonically coupled anharmonic–oscillator model and Darling–Dennison couplinga)

Child and Lawton’s1 model for two identifical coupled anharmonic oscillators is further discussed. It is shown that the effective Hamiltonian of that model can be transformed into a special case of the familiar Darling-Dennison resonance Hamiltonian. (AIP)

On resonant classical Hamiltonians with two equal frequencies

This paper contains a detailed study of the flow that the classical Hamiltonian $$H = \tfrac{1}{2}(x_1^2 + y_1^2 ) - \tfrac{1}{2}(x_2^2 + y_2^2 ) + \mathcal{O}_3 $$ induces near the origin of its phase spaceR 4. Here the perturbation term\(\mathcal{O}_3 \) represents a convergent power series. In particular, criteria for the existence and stability of periodic orbits are developed and expressed in terms of canonical invariants that are extracted from the perturbation term.

Off-resonance decoupling in abx spin systems

The effects of a perturbing field in the frequency range of the A and B nuclei on the transition frequencies of the X nucleus in an ABX spin system are studied. A simple model using the effective field for nuclei A and B is employed to obtain the basic features. The quantitative aspects are studied by diagonalization of the complete double resonance Hamiltonian. The relative signs of all three coupling constants of the spin system may be obtained easily, without the strict conditions typical for other methods. It is also shown, that off resonance decoupling of protons may be used to assign carbons which bear tightly coupled protons.

Lattice vibration motional averaging in nuclear quadrupole resonance of molecular crystals. II. Consequences of translationlibration coupling

The contribution of the acoustic branch frequencies to the librational motional averaging of the nuclear quadrupole resonance Hamiltonian in molecular crystals is examined in two simple linear chain models. Coupling between the translational and librational motions of the molecules in forming the normal coordinates causes the mean square librational amplitudes 〈 θ 2〉 to depend on the frequencies in both the optical and acoustical branches of the phonon spectrum. In the models studied it is found that the acoustic modes make significant contributions to 〈 θ 2〉 and that the functional dependence of 〈 θ 2〉 on wavevector does not necessarily reflect the dispersion in the optical branch as has often been assumed in the literature. It is therefore concluded that NQR cannot unambiguously determine average optical branch frequencies in molecular crystals.