Workshop on

Quantum Nonintegrability: Molecular Systems and General Theory

April 30 - May 2, 1998

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Abstracts

Participants


Eugene Bogomolny

Division de Physique Theorique
Institut de Physique Nucleaire
91406 Orsay, Cedex, France
bogomolny@ipncls.in2p3.fr
 


Prof. Paul Brumer

Chemical Physics Theory Group
University of Toronto
Toronto, Ontario M5S 3H6, Canada
pbrumer@tikva.chem.utoronto.ca
 

Stephen C. Creagh

Division de Physique Theorique
Orme des Merisiers
CEA/Saclay
F91191 Gif-sur-Yvette CEDEX, France
creagh@spht.saclay.cea.fr


Dr. Dominique Delande

Laboratoire Kastler-Brossel
Tour 12 - Etage 1
Universite Pierre et Marie Curie
4 Place Jussieu
75252 Paris Cedex 05, France
delande@spectro.jussieu.fr


Prof. Gregory S. Ezra

Baker Laboratory of Chemistry
Cornell University
Ithaca, NY 14853-1301
gse1@cornell.edu

Prof. Robert W. Field

Department of Chemistry
Massachusetts Institute of Technology
Cambridge, MA 02139
rwfield@mit.edu
 

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Prof. Martin C. Gutzwiller

370 Riverside Drive, Apt. 14B
New York, NY 10025-2179
MoonGutz@aol.com


Prof. Eric J. Heller

Department of Physics
Harvard University
Cambridge, MA 02138
heller@physics.harvard.edu


Mr. Matthew P. Jacobson

Department of Chemistry
Massachusetts Institute of Technology
Cambridge, MA 02139
jacobson@mit.edu


Charlie Jaffe

Department of Chemistry
PO Box 6045
West Virginia University
Morgantown, WV 26506-6045
u0d96@wvnvm.wvnet.edu


Jon Keating

Hewlett Packard Laboratories
Filton Road
Bristol, BS126QZ, United Kingdom
J.P.Keating@bristol.ac.uk
 
 

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Michael E. Kellman

Department of Chemistry
University of Oregon
Eugene, OR 97403
kellman@oregon.voregon.edu


Prof. Robert G. Littlejohn

Department of Physics
University of California
Berkeley, CA 94720
robert@wigner.lbl.gov


Prof. Wing-Ki Liu

PhysicsDepartment
University of Waterloo
200 University Ave. W
Waterloo, ON, N2L 3G1, Canada
wkliu@uwaterloo.ca


Dr. Jörg Main

Theoretische Physik I
Ruhr-Universität Bochum
D-44780 Bochum,Germany
main@tp1.ruhr_uni_bochum.de


Dr. Romanas Narevich

Department of Physics
University of Maryland
College Park, MD 20742
romas@katherine.physics.umd.edu
 
 

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Prof. Edward Ott

Institute for Plasma Research
University of Maryland
College Park, MD 20742
kynam@Glue.umd.edu


Arjendu Pattanayak

Chemical Physics Theory Group
University of Toronto
Toronto, Ontario M5S 3H6, Canada


Prof. Richard E. Prange

Physics Department
University of Maryland
College Park, MD 20742
prange@quantum.umd.edu


Prof. Herschel Rabitz

Department of Chemsitry
Princeton University
Princeton, NJ 08544
hrabitz@chemvax.princeton.edu
 

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Prof. Mark G. Raizen

Department of Physics
University of Texas at Austin
Austin, TX 78712-1081
raizen@physics.utexas.edu


Prof. Linda E. Reichl

Physics Department
University of Texas at Austin
Austin, TX 78712-1081
reichl@mail.utexas.edu


Prof. Rex Skodje

Department of Chemistry and Biochemistry
University of Colorado
Boulder, CO 80309-0215
skodje@spot.colorado.edu


Prof. Srinivas Sridhar

Physics Department
Notheastern University
360 Huntington Ave
Boston, MA 02115
srinivas@neu.edu
 

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Dr. Allan Tameshtit

Department of Physics
Bryn Mawr College
Bryn Mawr, PA 19010
atamesht@brynmawr.edu


Prof. Howard S. Taylor

Department of Chemistry, SSC 704
University of South California
920 W. 37th St.
Los Angeles, CA 90089.
taylor@chem1.usc.edu


Prof. Steven Tomsovic

Department of Physics
Washington State University
Pullman, WA 99164-2814
tomsovic@wsu.edu


Masa Tsuchiya

Baker Laboratory of Science
Cornell University
Ithaca, NY 14853-1301

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Denis Ullmo

Division de Physique Theorique
Institut de Physique Nucleaire
91406 Orsay, Cedex, France
ullmo@ipno.in2p3.fr

Michel Vallieres

Department of Physics
Drexel University
Philadelphia, PA 19104-2875
vallieres@einstein.drexel.edu

Prof. Jian-Min Yuan

Department of Physics
Drexel University
Philadelphia, PA 19104-2875
Fax: (215) 895-5934
yuan@molecule.physics.drexel.edu

 

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Abstracts

Bogomolny

Brumer

Creagh

Delande

Ezra

Field*

Jaffé

Keating

Littlejohn

Liu

Main

Ott

Rabitz

Raizen

Taylor

Tomsovic

Ullmo

*Not available

 Intermediate Spectral Statistics

Eugene Bogomolny

Division de Physique Theorique

Institut de Physique Nucleaire

91406 Orsay, Cedex, France

The spectral statistics of pseudo-integrable billiards in the form of right triangle with one angle pi/n is investigated. It is conjectured that it has a special form (depending on n) which is different from standard random matrix ensembles but in many respects is similar to the statistics of the Anderson model at the transition point. A few other models with analogous behaviour are considered. In particular, the eigenvalue statistics of a Poisson-distributed matrix perturbed by a rank one matrix, which is a good model for spectral statistics of a singular billiard, is found analytically.

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Quantum Chaos: Exponential Divergence,

Correspondence And Decoherence

Arjendu Pattanayak and Paul Brumer

Chemical Physics Theory Group

University of Toronto

Three recent developments in Quantum Chaos are described. Two of these advances resolve major longstanding problems in Quantum Chaos, i.e. we demonstrate exponential divergence in quantum mechanics, and provide a semi-formal derivation of classical chaos from quantum quasiperiodic dynamics. The third result links a new diagnostic for quantum and classical chaos in phase space to the rate of decoherence of systems which are chaotic in the classical limit.

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Tunnelling and Chaos

S. C. Creagh* and N. D. Whelan

*Division de Physique theorique

Orme des Merisiers

CEA/Saclay

Gif-sur-Yvette CEDEX, France

Tunnelling from a quantum well is intimately connected with complexified dynamics in the classical limit. This has long been understood quantitatively in one-dimensional and other simple systems. When the classical limit is nonintegrable, the mechanisms of complex dynamics which underly tunnelling become qualitatively richer and remain unexplored to a surprising degree. We offer solutions to this problem in the case where the classical limit is dominated by chaos, adapting methods from the field of quantum chaos. At the core is a calculation using complex periodic orbits to explain quantitatively the average tunnelling rate and dominant fluctuations therein. Effects very similar to those predicted have been seen in recent experiments with resonant tunnelling diodes. At shorter energy scales, explicit formulas are given for an individual tunnelling rate a simple matrix element measuring the weight of the wavefunction around a specific real classical trajectory. This classical trajectory is often periodic, in which case tunnelling is quantitatively related to "scarring".

 

Spectral Properties of Non-Hydrogenic Atoms in Weak External Fields

Dominique Delande

Laboratoire Kastler-Brossel

Universite Pierre et Marie Curie

We study how the ionic core in a non-hydrogenic atom modifies the dynamics of a Rydberg electron in the presence of a weak static external field. We show that such a system is neither regular nor chaotic: its energy levels display unusual statistical properties, intermediate between the standard Poisson and Random Matrix ones, which are well described by a simple model. The reason for these intermediate statistics is the breakdown of the semiclassical approximation when the Rydberg electron approaches the nucleus. The ionic core acts as a scatterer whose size is comparable to the de Broglie wavelength of the electron, inducing specific quantum effects.

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Periodic Orbit Analysis of Molecular Vibrational Spectra

Masateru Tsuchiya and Gregory S. Ezra

Department of Chemistry

Baker Laboratory

Cornell University

Ithaca, NY 14853

We present some recent results on the periodic orbit analysis of molecular vibrational spectra.

Semiclassical periodic orbit theory is used to analyze the quantum spectrum of a model vibrational Hamiltonian consisting of a pair of Morse oscillators coupled by two resonant terms. High-resolution energy-action plots are obtained using the z-scaling approach of Main and Taylor (Fourier transform with respect to 1/h[bar] at constant E), and the passage from the integrable (single-resonance) limit to nonintegrability studied. Nonintegrability manifests itself in the presence of ubiquitous tangent bifurcations. The bifurcation route from the integrable limit is elucidated.

Reaction Paths and Transition States

Charles Jaffé

Department of Chemistry

West Virginia University

Morgantown, WV 26506-6045

I will discuss some of the problems inherent in the traditional definitions of the reaction path and the transition state. These difficulties center on the fact that the effects of dynamical barriers are not included in the traditional approach and on the fact that the definitions require the existence of time-reversal symmetry. I will present two examples: the collinear H + H2 reaction and the ionization of hydrogen in crossed electric and magnetic fields. I will conclude with some thoughts on the extension of these concepts to systems having more than two degrees of freedom.

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 Orbit Bifurcations and Spectral Statistics

Jon Keating

Hewlett Packard Laboratories

Bristol, United Kingdom

Systems whose phase space is mixed have been conjectured to exhibit quantum spectral correlations that are, in the semiclassical limit, a combination of Poisson and random-matrix, with relative weightings determined by the corresponding measures of regular and chaotic orbits. In this talk I will discuss an additional component in long-range spectral statistics, associated with periodic orbit bifurcations, which can be semiclassically large. This is illustrated for a family of pertubed cat maps.

The results I will present have been published in:

MV Berry, JP Keating & SD Prado, J. Phys. A 31, L245-L254 (1998).

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Gauge Theories, Fiber Bundles, and Molecular Dynamics

Robert Littlejohn

 

University of California, Berkeley

 

A basic problem in the dynamics of polyatomic molecules is to eliminate analytically the five degrees of freedom which are ignorable due to translational and rotational invariance. The molecular literature on this subject for the last 70 years is exclusively coordinate based, but in fact the problem is ideally suited to modern geometrical methods. These are the coordinate-free methods of differential geometry and topology which concentrate on invariance, covariance, and geometrical modes of thinking about multidimensional spaces. These methods are well known in particle physics and relativity theory, but have not been used much in molecular physics. These geometrical methods reveal new insights into the traditional formulation of molecular dynamics, and also yield new applications.

Examples of new insights: Coriolis forces in a polyatomic molecule are described by an SO(3)-type Yang-Mills field, in which configuration space is the fiber bundle. The Coriolis field itself is source-free, except for singularities of the monopole type. Both Dirac and t'Hooft-Polyakov monopoles occur in different applications. The geometrical structure of configuration space is the same as in Kaluza-Klein theories. The Eckart frame, well known in traditional molecular dynamics, is constructed from radial geodesics of zero angular momentum, and has much in common with Riemann normal coordinates in Riemannian geometry.

New applications are concentrated in areas where coordinate-based methods are inadequate, and where deep insight into multidimensional spaces is required. These include Hamiltonians and problems of ro-vibrational coupling in molecules and molecular clusters; understanding internal spaces in four- or more-body problems; problems of frames and frame singularities in four-body scattering calculations; construction of hyperspherical harmonics; basis sets, basis set contractions, and boundary conditions in polyatomic internal spaces; and others.

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Infrared Chirped Pulse Excitation and Dissociation of Molecules

Wing-Ki Liu

Department of Physics

Guelph-Waterloo Program for Graduate Work in Physics (GWP)(2)

University of Waterloo

Waterloo, Ontario, Canada N2L 3G1

 

Jian-Min Yuan

Department of Physics and Atmospheric Science

Drexel University

Philadelphia, Pennsylvania 19104

 

The possibility of controlling the vibrational excitation of molecular bonds by lasers has been a subject of immense interest for many years. A prototype of such studies is the multiphoton excitation and dissociation of a diatomic molecule, in which the threshold laser intensity for dissociation has been found theoretically to be lowered by two order of magnitude when a chirped pulse is used(1). Classical nonlinear dynamics analysis of both the fixed-frequency excitation and the chirped pulse excitation show that nonlinear resonances play an important role. In the chirped case, the route to dissociation can be described in terms of "bucket dynamics" in which trajectories can be trapped in the resonance zone of a field-dressed potential well moving up in phase space. Such excitation is thus due to convection in phase space, which is very different from the fixed-frequency case where chaotic diffusion is responsible for dissociation(2). The creation of 'buckets' in the former case requires a much lower laser intensity than driving the system into chaos in the latter case, and this explains the efficiency of the chirped process. The classical results have been found to compare well with quantum mechanical calculations based on the split-operator method(3). We have further extended our classical studies to the case of a linear triatomic molecule, and the possibility of selective bond excitation and dissociation by chirped laser pulses will be discussed.

1. Chelkowski, A.D. Bandrauk, and P.B. Corkum, Phys. Rev. Lett. 65, 2355 (1990).

2. W.-K. Liu, B. Wu, and J.-M. Yuan, Phys. Rev. Lett. 75, 1292 (1995).

3. J.-M. Yuan and W.-K. Liu, Phys. Rev. A (1998) in press.

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Large Semiclassical Quantization by Harmonic Inversion of

Cross-Correlated Periodic Orbit Sums

Jörg Main

Institut für Theoretische Physik I

Ruhr-Universität Bochum,

D-44780 Bochum, Germany

Recently, harmonic inversion has been introduced as a general method for periodic orbit quantization of bound and open systems with an underlying regular or chaotic classical dynamics [1,2]. The method requires as input the orbits up to a certain maximum period, which depends on the density of states. Here we report an extension of the method, which allows to reduce the maximum period to shorter than the Heisenberg time and thus to significantly reduce the number of orbits required for semiclassical quantization. The extension is based on the harmonic inversion of cross-correlated periodic orbit sums, which are defined as the semiclassical analogue of the quantum mechanical density of states weighted with products of diagonal matrix elements of various smooth operators.

[1] J. Main, V. A. Mandelshtam, and H. S. Taylor, Phys. Rev. Lett. 79, 825 (1997).

[2] J. Main, V. A. Mandelshtam, G. Wunner, and H. S. Taylor, Nonlinearity, to be published       (Preprint: chao-dyn/9709009).

Chaotic Scattering in Higher Dimensions

Edward Ott

University of Maryland

Chaotic scattering implies infinitely sensitive dependence of an output variable (e.g., scattering angle) on an input variable (e.g., impact parameter) for a fractal set of input variables. By now this phenomenon is well understood in the lowest dimensional systems [1] (i.e., two degree of freedom time independent systems). The situation for chaotic scattering in higher dimensional phase spaces (e.g., three degrees of freedom) is much less clear. In this talk we present results on the structure and fractal dimension of chaotic scattering problems describable via four dimensional symplectic maps. In particular, we discuss the relationship of Lyapunov exponents and decay times to dimension [2], the fundamentally different topological structures that can result depending upon whether or not the scattering problem possesses distinct escape modes [3], and methods for numerical investigation of invariant chaotic scattering sets [3]. As specific examples [3], we consider scattering from a system of four specularly reflecting hard spheres located at the corners of a tetrahedron, the same system when placed inside a square cross-section pipe, and a model for chemical reaction among three atoms.

1. E. Ott, Chaos in Dynamical Systems Chapt. 5.

2. B. R. Hunt, E. Ott and J. A. Yorke, Phys. Rev. E 54, 4819 (1996). M. Ding and E. Ott, N. Y. Acad. Sci 751, 182 (1995).

3. D. Sweet, E. Ott and J. A. Yorke, in preparation.

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Altering the Court of Quantum Dynamics

Herschel Rabitz

Department of Chemistry

Princeton University

Efforts at achieving the active manipulation of quantum dynamics are receiving increasing attention. By the introduction of coherent optical fields, it is possible to alter the pathway of molecular (quantum) dynamics in the most intimate fashion. Establishing this capability could have a number of applications, including the creation of usual dynamical states. Recent interest in the physics community towards creating molecular-scale quantum computers and atom lasers would also fall into the same domain of active manipulation of quantum dynamics. The underlying physical principles for quantum dynamical control will be discussed, along with some recent illustrations.

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 Experimental Study of Decoherence in Quantum Chaos

M. G. Raizen

The University of Texas at Austin

Department of Physics

Austin, Texas 78712-1081

We report an experimental study of decoherence in quantum chaos. Our system consists of ultra-cold cesium atoms in a pulsed standing wave of a far-detuned laser, and is an experimental realization of the quantum kicked rotor. In earlier experiments with sodium atoms we observed dynamical localization, a quantum suppression of classicalchaos. The larger mass of cesium relative to sodium greatly reduces the effect of finite pulse duration on the momentum boundary in phase space. We study the effects of external noise and dissipation on dynamical localization of cesium atoms. We observe delocalization as the noise level is increased, approaching the classical limit. The sensitivity to different types of noise and dissipation will be discussed.

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The Bending Dynamics of Acetylene

C. Jung* and H.S. Taylor**

*UNAM Instituto de Matematicas

Unidad Cuernavaca

Av. Universidad s/n,

62191 Cuernavaca, Mexico

 

**Department of Chemistry

University of Southern California,

Los Angeles, CA 90089-0482, USA

The effective four degree freedom Hamilitonian fitted to 15000 cm-1 by successive diagonalization and parameter adjustment to absorption and dispersed fluorescence spectra,(1) is converted to a four degree of freedom classical Hamiltonian in action angle variables. A specific canonical transform is then used to convert to an effective Hamiltonian in which for the first time the two effective constants of the motion are seen explicitly and which now is a function of two abstract action variables and two abstract angle variables. The constants of the motion are the polyad quantum number and the total bending angular momentum (here eventually set to zero as it is in the experiment); the former measures the total occupation of the cis and trans bend modes here considered as basis states. The 2D abstract phase space Hamiltonian has symmetries which suggest the existence of two key periodic orbits that will be shown to organize the phase space dynamics of the Hamiltonian over its fitted energy range. These periodic orbits are numerically followed in energy. Additionally, surface of sections are generated to observe the surrounding tori or chaos as the case may be. The generic motions of these periodic orbits which for a given P and changing energy, are librations and rotations in abstract angle configuration space, show how the organized abstract motions change with P and within a polyad.

To obtain chemically recognizable motions, we start with the general solution of a four degree of freedom adopted model that envisions the "bending" acetylene as two opposite outward facing, bottom to bottom attached cups. Each cup contains a ball representing the Hydrogen atom. This is a two 2D spherical harmonic oscillator model. The constants of the general solution, whose values specify the motions of the H atoms in parallel face to face planes on each end of the acetene, are shown to be determined and correlated by choosing a trajectories in the abstract space. When the organizing periodic orbits or their nearby tori based 2D trajectories are each separately plugged into the constants in the general solution, 4D trajectories are obtained that specify the various ways the hydrogens move in their respective planes.

Below P=9, the motion is regular with cis and trans motions dominating the top and bottom of the polyads, respectively. In the middle a beat motion (linear combinations of cis and trans) appear. Above P=9, cis and trans no longer exist and now the bottom and top of the polyads are regular with local and counter rotating motions, respectively. The middle of the polyads have a mixed phase space. The regular regions embedded in the chaos, show beautiful correlated motions whose pictures will be shown. The fraction of phase space with chaotic motion increases up to about P=16 and then decreases, in agreement with the regularity of the energy level patterns seen when comparing level diagrams of polyads. In general in regular regions the levels are separated by the frequency of the organizing periodic orbit taken halfway between the levels. This alone shows a quantum connection. We have quantized (by diagonalization) the 2D effective semi-classical actions-angle Hamiltonian. We get the same eigenvalues as when the spectra was fitted using a diagonalization in 4D space. We see that the 2D wavefunctions in the two angle variables lie on top of the abstract stable periodic orbits, allowing us to count nodes and complete the assignment in the regular region. "Chaotic" states, scarred by one or more unstable periodic orbits, also appear and explain the mixings that occur at the given energy in the polyad.

(1). M.P. Jacobson, J.P. O'Brien, R.J. Silby and R. Field, Preprint. 1998

Can Wave Packet Revivals Occur in Chaotic Quantum Systems?

Steven Tomsovic

Department of Physics

Washington State University,

Pullman, WA99164-2814, USA

 

The short time revivals of initially localized wave packets are well known in simple, closed, one-degree-of-freedom (1d) systems. In 2d or higher, if the system is integrable or has exclusively periodic dynamics, a generalization is possible. If the dynamics are chaotic, revivals have not been previously seen and are, a priori, not expected. Nevertheless, we have found that some stretched wave packets in a chaotic system experience very early, surprisingly large recurrences. We extend a semiclassical theory founded on summing over heteroclinic orbits to determine a set of necessary conditions. The most important one is an EBK-like quantization of classical flux crossing the turnstile formed by the stable and unstable manifolds of the initial wave packet's underlying central orbit.

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 Trace Formula for Nearly Integrable Systems : Resonances.

Denis Ullmo

Division de Physique Theorique

Institute de Physique Nucleaire

91406 Orsay, Cedex, France

Trace formulas relate the quantum density of states to the properties of the underlying classical system. The resulting expression depend critically on the nature of the dynamics and on whether the orbits are stable or unstable. Several open questions exist for the class of system that are near integrability. I will derive uniform expressions appropriate for resonances and and apply them on a system that can be taken as a paradigm for the transition from regular to chaotic dynamics.

 

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