Billiard Geometries Research Articles

The geometry of billiards in ellipses and their poncelet grids

The goal of this paper is an analysis of the geometry of billiards in ellipses, based on properties of confocal central conics. The extended sides of the billiards meet at points which are located on confocal ellipses and hyperbolas. They define the associated Poncelet grid. If a billiard is periodic then it closes for any choice of the initial vertex on the ellipse. This gives rise to a continuous variation of billiards which is called billiard motion though it is neither a Euclidean nor a projective motion. The extension of this motion to the associated Poncelet grid leads to new insights and invariants.

Differential geometry of rigid bodies collisions and non-standard billiards

The configuration manifold $M$ of a mechanical system consisting of two unconstrained rigid bodies in $\mathbb{R}^n$is a manifold with boundary (typically with singularities.) A full descriptionof the system requires boundary conditions specifying how orbits should be continued after collisions, that is,the assignment of a collision map at each tangent spaceon the boundary of $M$ giving the post-collision state of the system for each pre-collision state.We give a complete description of the space of linear collision maps satisfying energy and (linear and angular) momentum conservation, time reversibility,and the natural requirement that impulse forces only act at the point of contact of the colliding bodies.These assumptions are stated geometrically in terms of a family of vector subbundlesof the tangent bundle to $\partial M$: the diagonal, non-slipping, and impulse subbundles.Collision maps are shown to be the isometric involutions that restrict to the identity on the non-slipping subspace.We then make a few observations of a dynamical nature about non-standard billiard systems, among which is a sufficient condition for the billiard map on the space of boundary states to preserve the canonical measure onconstant energyhypersurfaces.

Effect of geometry on the classical entanglement in a chaotic optical fiber.

The effect of boundary deformation on the classical entanglement which appears in the classical electromagnetic field is considered. A chaotic billiard geometry is used to explore the influence of the mechanical modification of the optical fiber cross-sectional geometry on the production of classical entanglement within the electromagnetic fields. For the experimental realization of our idea, we propose an optical fiber with a cross section that belongs to the family of Robnik chaotic billiards. Our results show that a modification of the fiber geometry from a regular to a chaotic regime can enhance the transverse mode classical entanglement.

Weak localization with nonlinear bosonic matter waves

We investigate the coherent propagation of dilute atomic Bose-Einstein condensates through irregularly shaped billiard geometries that are attached to uniform incoming and outgoing waveguides. Using the mean-field description based on the nonlinear Gross–Pitaevskii equation, we develop a diagrammatic theory for the self-consistent stationary scattering state of the interacting condensate, which is combined with the semiclassical representation of the single-particle Green function in terms of chaotic classical trajectories within the billiard. This analytical approach predicts a universal dephasing of weak localization in the presence of a small interaction strength between the atoms, which is found to be in good agreement with the numerically computed reflection and transmission probabilities of the propagating condensate. The numerical simulation of this quasi-stationary scattering process indicates that this interaction-induced dephasing mechanism may give rise to a signature of weak antilocalization, which we attribute to the influence of non-universal short-path contributions.

The conformal geometry of billiards

In the absence of friction and other external forces, a billiard ball on a rectangular billiard table follows a predictable path. As it reflects off the sides of the table, the ball will either (1) return to its original position and direction, or (2) travel all over the table, spending equal time in regions with equal area. Because of this dichotomy, we say the rectangular table is optimal: billiard trajectories define an easily understood dynamical system on the phase space of positions and directions. In this article, we ask:

Dynamical chaos: systems of classical mechanics

This article is a methodological manual for those who are interested in chaotic dynamics. An exposition is given on the foundations of the theory of deterministic chaos that originates in classical mechanics systems. Fundamental results obtained in this area are presented, such as elements of the theory of nonlinear resonance and the Kolmogorov–Arnol'd–Moser theory, the Poincaré–Birkhoff fixed-point theorem, and the Mel'nikov method. Particular attention is given to the analysis of the phenomena underlying the self-similarity and nature of chaos: splitting of separatrices and homoclinic and heteroclinic tangles. Important properties of chaotic systems — unpredictability, irreversibility, and decay of temporal correlations — are described. Models of classical statistical mechanics with chaotic properties, which have become popular in recent years — billiards with oscillating boundaries — are considered. It is shown that if a billiard has the property of well-developed chaos, then perturbations of its boundaries result in Fermi acceleration. But in nearly-integrable billiard systems, excitations of the boundaries lead to a new phenomenon in the ensemble of particles, separation of particles in accordance their velocities. If the initial velocity of the particles exceeds a certain critical value characteristic of the given billiard geometry, the particles accelerate; otherwise, they decelerate.

On configuration spaces of plane polygons, sub-Riemannian geometry and periodic orbits of outer billiards

Following a recent paper by Baryshnikov and Zharnitsky, we consider outer billiards in the plane possessing invariant curves consisting of periodic orbits. We prove the existence and abundance of such tables using tools from sub-Riemannian geometry. We also prove that the set of 3-periodic outer billiard orbits has empty interior.

Experimental Investigation of the Breakdown of the Onsager-Casimir Relations

We use magnetoconductance fluctuation measurements of phase-coherent semiconductor billiards to quantify the contributions to the nonlinear electric conductance that are asymmetric under reversal of magnetic field. We find that the average asymmetric contribution is linear in magnetic field (for magnetic flux much larger than 1 flux quantum) and that its magnitude depends on billiard geometry. In addition, we find an unexpected asymmetry in the power spectrum of the magnetoconductance with respect to reversal of magnetic field and bias voltage.

Geometry of integrable billiards and pencils of quadrics

We study the deep interplay between geometry of quadrics in d-dimensional space and the dynamics of related integrable billiard systems. Various generalizations of Poncelet theorem are reviewed. The corresponding analytic conditions of Cayley's type are derived giving the full description of periodical billiard trajectories; among other cases, we consider billiards in arbitrary dimension d with the boundary consisting of arbitrary number k of confocal quadrics. Several important examples are presented in full details proving the effectiveness of the obtained results. We give a thorough analysis of classical ideas and results of Darboux and methodology of Lebesgue; we prove their natural generalizations, obtaining new interesting properties of pencils of quadrics. At the same time, we show essential connections between these classical ideas and the modern algebro-geometric approach in the integrable systems theory.

Three key questions on fractal conductance fluctuations: Dynamics, quantization, and coherence

Recent investigations of fractal conductance fluctuations (FCF) in electron billiards reveal crucial discrepancies between experimental behavior and the semiclassical Landauer-Buttiker (SLB) theory that predicted their existence. In particular, the roles played by the billiard's geometry, potential profile and the resulting electron trajectory distribution are not well understood. We present measurements on two custom-made devices - a 'disrupted' billiard device and a 'bilayer' billiard device - designed to probe directly these three characteristics. Our results demonstrate that intricate processes beyond those proposed in the SLB theory are required to explain FCF.

Quantum mechanics of the two-dimensional circular billiard plus baffle systemand half-integral angular momentum

We examine the quantum mechanical eigensolutions of the two-dimensionalinfinite-well or quantum billiard system consisting of a circular boundarywith an infinite barrier or baffle along a radius. Because of the changein boundary conditions, this system includes quantized angularmomentum values corresponding to half-integral multiples of ℏ/2. Wediscuss the resulting energy eigenvalue spectrum and visualize some of the novel energyeigenstates found in this system. We also discuss the density of energy eigenvalues,N(E), comparing thissystem to the standard circular well. These two billiard geometries have the same area (A = πR2), but different perimeters(P = 2πR versus (2π + 2)R), and we compareboth cases to fits of N(E)which make use of purely geometric arguments involving onlyA andP. Wealso point out connections between the angular solutions of this system and thefamiliar pedagogical example of the one-dimensional infinite-well plusδ-functionpotential.

Measuring billiard eigenfunctions with arbitrary trajectories.

We propose a method of measuring approximate quantum eigenfunctions in polygonalized billiard geometries, based on a quasiclassical evolution operator having a (smoothened) Perron-Frobenius kernel modulated by a phase arising from quantum considerations. Using a plane wave ansatz, we show that the condition under which this is an eigenfunction of the quasiclassical operator is identical to the condition for it to be an eigenfunction of the Schrödinger equation for polygonalized billiards. Finally, we demonstrate this technique by determining the quasiclassical eigenfunctions of the polygonalized stadium billiard using arbitrary trajectories and comparing this with the exact quantum stadium eigenfunctions.

Effective Hamiltonian for a microwave billiard with attached waveguide.

In a recent work the resonance widths in a microwave billiard with attached waveguide were studied in dependence on the coupling strength [E. Persson et al., Phys. Rev. Lett. 85, 2478 (2000)], and resonance trapping was experimentally found. In the present paper an effective Hamiltonian is derived that depends exclusively on billiard and waveguide geometry. Its eigenvalues give the poles of the scattering matrix provided that the system and environment are defined adequately. Further, we present the results of resonance trapping measurements where, in addition to our previous work, the position of the slit aperture within the waveguide was varied. Numerical simulations with the derived Hamiltonian qualitatively reproduce the experimental data.

Electronic states in a cylindrical quantum lens: quantum chaos for decreasing system symmetry.

The eigenvalue problem in a cylindrical lens geometry is studied. Using a conformal mapping method, the shape of the boundary and the Hamiltonian for a free particle are reduced to those of a two-dimensional problem with circular symmetry. The wave functions are separated into two independent Hilbert subspaces due to the inherent symmetry of the problem. For small geometry deformations, the solutions are found by a specially designed perturbation approach. Comparisons between exact and perturbative solutions are made for different lens parameters. As the symmetry of the lens is reduced, the characteristics of the spectrum and the corresponding spatial properties of the wave functions are studied. Our results provide a family of billiard geometries in which the electronic level spectrum is well characterized. In analyzing the level spacing distribution of the spectrum, a strong deviation from the Poisson and Wigner limiting distributions is found as the boundary geometry changes. This intermediate distribution is indicative of a mixed phase space, also revealed explicitly in the classical Poincaré maps we present.

Semiclassical theory of integrable and rough Andreev billiards

We study the effect on the density of states in mesoscopic ballistic billiards to which a superconducting lead is attached. The expression for the density of states is derived in the semiclassical S-matrix formalism shedding light onto the origin of the differences between the semiclassical theory and the corresponding result derived from random matrix models. Applications to a square billiard geometry and billiards with boundary roughness are discussed. The saturation of the quasiparticle excitation spectrum is related to the classical dynamics of the billiard. The influence of weak magnetic fields on the proximity effect in rough Andreev billiards is discussed and an analytical formula is derived. The semiclassical theory provides an interpretation for the suppression of the proximity effect in the presence of magnetic fields as a coherence effect of time reversed trajectories. It is shown to be in good agreement with quantum mechanical calculations.

Quantum correlations and classical resonances in an open chaotic system

We show that the autocorrelation of quantum spectra of an open chaotic system is well described by the classical Ruelle-Pollicott resonances of the associated chaotic strange repeller. This correspondence is demonstrated utilizing microwave experiments on 2-D n-disk billiard geometries, by determination of the wave-vector autocorrelation C(κ) from the experimental quantum spectra S21(k). The correspondence is also established via "numerical experiments" that simulate S21(k) and C(κ) using periodic orbit calculations of the quantum and classical resonances. Semiclassical arguments that relate quantum and classical correlation functions in terms of fluctuations of the density of states and correlations of particle density are also examined and support the experimental results. The results establish a correspondence between quantum spectral correlations and classical decay modes in an open system.

Quantum chaos, localization and tunnelling: Microwave experiments on model geometries

Microwave experiments using two-dimensional billiard geometries are a precise test of basic issues in quantum chaos, localization and tunnelling. In closed chaotic geometries, analysis of eigenvalue statistics yields good agreement with random-matrix theory. A unique aspect of the experiments is the ability to measure eigenfunctions directly. The influence of periodic orbit scarring in chaotic eigenfunctions is directly demonstrated. Disordered microwave billiards are a textbook model system for studying the quantum properties of a single particle in a disordered potential. Localization is directly observed in eigenfunctions of the disordered billiards. Statistical properties of disordered eigenfunctions deviate from universal behaviour due to localization. These statistical properties are in good agreement with predictions from nonlinear-sigma models, although many challenges for further theoretical understanding remain. The experiments can also probe open systems, in terms of the quantum resonances and escape rate of a fractal repeller.

Exact and statistical self-similarity in magnetoconductance fluctuations: A unified picture

Self-similarity (the invariance of an intrinsic pattern under scaling) in the fractal magnetotransport properties of mesoscopic billiards is highly topical. Employing three billiard geometries, we investigate the relationship between the two classes of observed scaling behavior---exact and statistical self-similarity. We employ a correlation analysis to determine the conditions required for their observation and to show that the two forms of self-similarity can be understood within a common framework.

Experimental and Theoretical Investigations of Electron Dynamics in a Semiconductor Sinai Billiard

We describe a surface gate-defined mesoscopic semiconductor billiard with a square geometry which we can evolve continuously to a Sinai geometry by adjusting the bias on a circular gate located at the centre of the square. We concentrate on clusters of magneto-conductance structure formed on two magnetic field scales, which emerge during the transition from the square to the Sinai geometry. This change in shape is accompanied by a transition from non-chaotic to chaotic electron dynamics, which has been a topic of considerable interest for many years. The observed structure is due to quantum chaotic processes induced by the presence of the circle. Our experimental results agree with classical dynamics simulations which suggest the two field scales are determined by two families of electron trajectories which sample the full geometry and corner sub-geometry. Finally, we report the observation of striking similarities in the structures observed on the two different field scales. This indicates the emergence of self-similarity with the introduction of the circle to the Sinai billiard geometry.

Visualizing classical periodic orbits from the quantum energy spectrum via the Fourier transform: Simple infinite well examples

The Fourier transform of the density of quantized energy levels for a quantum mechanical particle in a two-dimensional (2-D) infinite well (or billiard geometry) is known to exhibit δ-function-like spikes at distance values (L) corresponding to the lengths of periodic orbits or closed trajectories. We show how these Fourier transforms can be rather easily calculated numerically for simple infinite well geometries including the square and rectangular well in 2 D, the cubical well in three dimensions, as well as the circular infinite well (and variations) in two dimensions. Such calculations provide a novel, well-motivated, and relatively straightforward example of numerical Fourier transform techniques and make interesting connections between quantum energy levels and classical trajectories in a way which is seldom stressed in the undergraduate curriculum.