A method for detecting low dimensional deterministic behavior in noisy data files, based on phase plane topologies, has been developed(D. Pierson and F. Moss, Phys. Rev. Lett. 75, 2124 (1995)). The technique specifically detects the presence of unstable periodic orbits (UPOs) while discriminating against limit cycles (LCs) and, because it can operate satisfactorily on short noisy files, is suitable for searches of low dimensional behavior in biological data(X. Pei and F. Moss, Nature 379, 618 (1996)). A certain pattern - the signature of a UPO - is defined, and the probability of occurrence in the data file is determined and compared to that found in a completely randomized (surrogate) file. Here we describe a mathematical method for determining the occurrence probability in a random file.
[S17.02] Eigenmodes and Correlation functions of Microwave Quantum Chaos Experiments
Dong-Ho Wu, Ali Gokirmak, S. M. Anlage (Center for Superconductivity Research, Physics Department, University of Maryland, College Park, MD 20742)
We have mapped eigenmodes of various 2-dimensional microwave cavities to investigate the wave mechanical properties of both classically integrable and non-integrable systems in a comparative manner. At low frequencies eigenmodes of a bow-tie cavity that is a classically non-integrable system exhibit `scars', corresponding to the isolated periodic orbits of classical chaos. At high frequencies where the wavelengths are much shorter than the dimension of cavity, the eigenmode patterns are complicate and look `random'. To understand the correlations among individual eigenfunctions, we extracted the spatial density correlation <|\psi (r_1)|^2 |\psi (r_2)|^2> from the eigenmode patterns of the integrable and the nonintegrable systems. The extracted spatial density correlations are compared with theoretical models, and we find that the eigenfunctions have a long-range correlation. Discussions will include experimental procedures to measure the correct spatial density of eigenfunction, comparison of eigenmode correlations of integrable and nonintegrable systems, and eigenfunction fluctuations.
[S17.03] Classical and quantum dynamics of a free particle in a driven potential inside a rigid box
J. L. Mateos. (Instituto de Fisica, UNAM, México), Jorge .V. José (Northeastern University, Boston)
We study the classical and quantum dynamics of a free particle inside a rigid box with an internal square well potential that oscillates periodically in time. The location of the square potential is fixed but arbitrary between the two rigid walls of the box. In the classical case, as a function of the parameters of the potential, the particle can present periodic, quasiperiodic and chaotic behavior. Because of the oscillating nature of the square well potential one can define a classical ``dwell time" for the particle trapping in the oscillating potential region. A possible connection between the classical chaotic and dwell time dynamics and their quantum counterparts will be considered. Also, the properties of the classical vs quantum energy exchange between a ``large" and ``small" well, as a microscopic model for friction, will be discussed.
[S17.04] Chaos in an Acoustic Stadium
Christopher Carr, Chad Burdyshaw, Matt Forrest, John Stanfield, Roger Yu (Department of Physics, Central Washington University, Ellensburg, WA98926)
Most recently a stadium-shaped quantum dot, whose classical equivalence is chaotic, has been studied intensively1. We are reporting the chaotic wave behavior on macroscopic scale. Blume2 et al. have studied the surface wave form on water confined by a stadium-shaped container. Chineery and Humphrey3 successfully demonstrated the wavefunction of resonances within a stadium-shaped air cavity. We have studied the both eigenfrequencies and eigenfunctions of an acoustic stadium cavity using the pulse and normal mode analyzing techniques4. The acoustic cavity consists of two half circles of radius R (R=38.0 cm) and a rectangle of 2R by A (A being the separation between the half circles). The separation is made adjustable so that the transition from nonchaotic (circle) to chaotic (stadium) waves and the parametric variation of eigenvalues can be investigated by changing the ratio A/R.
1. C. Marcus, et al. Phys. Rev. Lett.69, 506(1992) 2. R. Bl=FCmel, I. Davidson, W. Reinhardt, H. Lin, and M. Sharnoff, Phys. Rev. A 45, 2641(1992) 3.P. Chinnery and V. Humphrey, Phys. Rev. E53, 272(1996) 4.Samantha Parmley, Tom Zobrist, Terry Clough, Anthony Perez-Miller,Mark Makela, and Roger Yu, Appl. Phys. Lett. 67, 777(1995)
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[S17.05] The origin of the Strong Scarring of the Wavefunctions in a Quantum Well in a Tilted Magnetic Field
A.D. Stone, E. Narimanov (Applied Physics, Yale University, P.O.Box 208284, New Haven CT 06520-8284)
Recently, an unusually strong ``scarring'' of the quantum wavefunctions due to a subset of (unstable) periodic orbits in a quantum well in a tilted magnetic field was observed both numerically (T.M.Fromhold el al \rm, Phys.Rev.Lett. 75 \rm, 1142 (1995) ) and experimentally (P.B.Wilkinson el al , Nature 380\rm, 608 (1996)). To uncover the origin of this effect, we perform a detailed analysis of the classical periodic orbits in this system. We show that the the relevant periodic orbits appear in organized sequences of bifurcations, spawned by the bifurcations of the period-one traversing orbits, for whose periods and stabilities we obtain analytical expressions. We find, that some of the most important periodic orbits exist only in finite intervals of the system parameters and disappear in tangent bifurcations with the other orbits of the organized sequence on both sides of the interval of their existence. This property does not allow a development of a strong unstability for these orbits and therefore explains the unusually strong scarring, observed recently in the quantum well in tilted magnetic field.
[S17.06] Thermodynamic Chaos and the Structure of Spin Glasses
D.L. Stein (University of Arizona)
In realistic disordered systems, such as the Edwards-Anderson (EA) spin glass, no order parameter, such as the Parisi overlap distribution, can be both translation invariant and non-self-averaging. The standard mean-field picture of the EA spin glass phase can therefore not be valid in any dimension and at any temperature(C.M.~Newman and D.L.~Stein, Phys.~Rev.~Lett.\/) 76, 515 (1996). Further analysis shows that, in general, when systems have many competing (pure) thermodynamic states, a single state which is a mixture of many of them (as in the standard mean-field picture) contains insufficient information to reveal the full thermodynamic structure. We propose a different approach(C.M.~Newman and D.L.~Stein, Phys.~Rev.~Lett.\/) 76, 4821 (1996), in which an appropriate thermodynamic description of such a system is instead based on a metastate\/, which is an ensemble\/ of (possibly mixed) thermodynamic states. This approach, modelled on chaotic dynamical systems, is needed when chaotic size dependence (of finite volume correlations) is present. Here replicas arise in a natural way, and such concepts as replica symmetry breaking, chaotic size dependence and replica non-independence are explained, connected, and unified. As an application, we classify allowed thermodynamic scenarios for the EA model, including two new pictures uncovered using this approach.
[S17.07] New type of cantorus phonons in the Frenkel-Kontorova model
Jukka Ketoja (University of Helsinki), Indubala Satija (George Mason Univ)
We study the incommensurate Frenkel-Kontorova model in the pinned (cantorus) phase where the associated area-preserving standard map has a positive Lyapunov exponent. Using an exact decimation method we show that the eigenfunctions corresponding to the minimal phonon frequency remain critical throughout the cantorus phase and their scaling properties are described by a line of renormalization limit cycles. A novel aspect of the phonons is the existence of an infinite set of parameter values where the eigenfunctions are represented by series of step functions associated with a trivial attractor of the renormalization flow. These parameter values converge geometrically to the onset of the pinning transition.
This work of IIS is supported by a grant from National Science Foundation DMR~093296.
[S17.08] Hierarchical level-clustering in two-dimensional harmonic oscillators
C. B. Whan (MIT)
We present numerical results on the statistical distribution of energy level spacings in two-dimensional (2D) harmonic oscillators with irrational frequency ratio, R \equiv ømega_1/ømega_2. Unlike the scaled level spacings, the distribution of the true energy level spacings is well behaved and directly reflects the corresponding classical quasiperiodic motion. The histogram of the energy level spacings shows sharp peaks at discontinuous values which form hierarchical rational approximations to R. The peak heights follow a characteristic inverse-square-law increase as the level spacing \Delta\cal E decreases, indicating a form of level clustering rather than level repulsion as previously believed. We believe the failure of convergence in the scaled level spacing distribution is due to the lack of proper energy scales in the system, since the average (true) level spacing vanishes in the semiclassical limit.
[S17.09] Parametric Variations and Phase Space Localization
Nicholas Cerruti, Steve Tomsovic, Julie Lefebvre (Washington State University)
We investigate the phase space localization of the eigenstates of a quantum system possessing a chaotic classical limit. The localization properties are conspicuously revealed in a system's response to parametric variation. This allows us to introduce a new, sensitive correlation measure between level velocities and certain overlap intensities in which the full phase space can be explored for the extent of localization. We apply this to the stadium and the return dynamics of the system are shown to predict the eigenstate properties. Applications of this analysis include data from
microwave cavities and quantum dots in the Coulomb-blockade regime.
[S17.10] Nonlinear Wave Packet Recurrences in Chaotic Systems
Julie H. Lefebvre, Steven Tomsovic (Washington State University)
In a highly, unstable chaotic system, it is not generally possible to construct a minimum uncertainty wave packet that produces large early recurrences. This is true even if one places the wave packet on the shortest periodic orbit of the system. By extending the wave packet to cover the full length of the orbit, early recurrences are naturally larger from a reduced energy uncertainty argument. Even taking this into account, there can be surprisingly large recurrences for some mean momentum values of the wave packet when applied to the stadium billiard. We have adapted the semiclassical theory based on homoclinic orbits (S. Tomsovic and E. J. Heller, Phys Rev E 47, 282 (1993).) for these new states to quantitatively predict their observed behavior. We investigate the role of phase space structures, such as broken separatrices and flux-crossing turnstiles, on the interference patterns generated by the homoclinic orbits.
[S17.11] Multidimensional Parameter Space in the Chaotic Behavior of a PN junction Diode
Gustavo Gutiérrez, Carlos Fehr, Duilio Valdivia (Departamento de F\'\isica, Universidad Simón Bol\'\ivar), Arnaldo Donoso (Chemistry Department, University of California at Irvine.)
Rollins and Hunt (R.W. Rollins and
E.R. Hunt, Phys. Rev. Lett., 49(18), 1295 (1982).) proposed a model to
describe the chaotic behavior of a diode in series with an inductance
and a resistor. In this model the forward bias voltage and the reverse
recovery time are required to explain the chaotic behavior of the
diode resonator. They used the amplitude of the driving voltage as a
control parameter.We have generalized this model to include the
temperature as an independent control parameter. We have found an
empirical equation for the reverse recovery time that includes the
temperature of the diode.
The geometrical and topological characteristics of the parameter space
was analyzed by constructing isoperiodic maps for the amplitude of the
driving voltage, the frequency and the temperature. Evidence of
universal features were observed in the above isoperiodic diagrams.
[S17.12] Spectral Properties of Mixed Phase Space Quantum Billiards
Donald M Pianto, David K Campbell (University of Illinois at Urbana-Champaign)
We study the quantum spectral properties of two-parameter family of classical billiards introduced by Dullin, Richter, and Wittek (H. R. Dullin, P. H. Richter, A. Wittek, Chaos 6), 43 (1996). The parameters can be varied such that paths in the shape space take the billiard smoothly from an integrable system through a mixed phase space system to a completely ergodic system. We explore the behavior of the energy difference spectra in several different regions of energy and show how the semiclassical results, expected to hold at sufficiently high energy, are altered as one moves down to the lower-lying energy levels.
[S17.13] Wavepackets, periodic orbits and semiclassical quantization.
Paul A. Houle, Chris L. Henley (LASSP, Cornell University)
By expanding classical trajectories around a periodic orbit to linear order and computing matrix elements between gaussian wavepackets, we are developing a technique of semiclassical quantization that can be generalized to systems such as spin that lack a continuous position basis. Although a set of gaussian wavepackets of fixed width (the coherent states) form a complete basis, propagating a coherent state along a classical path without changes in width(J. R. Klauder Phys.\ Rev.\ D.) 19 2349 (1979) incorrectly computes the zero-point energy of the harmonic oscillator. The width of the wavepacket must change over time to compute accurate matrix elements.(R. G. Littlejohn Physics Reports) 138 193 (1986) Inspired by the numerous numerical successes of this method in the time domain(S. Tomsovic, E. J. Heller Phys.\ Rev.\ E.) 47 282 (1993), we analytically compute the energy-dependent Green's function in the vicinity of a periodic orbit using the stationary phase-approximation. We discuss the derivation of trace formulas in the coherent state basis for integrable and non-integrable systems and the advantages of a wavepacket formalism which is manifestly covariant with respect to canonical transformations in phase space.