Circular Billiard Research Articles

Self-similarity among energy eigenstates.

In a quantum system, different energy eigenstates have different properties or features, allowing us to define a classifier to divide them into different groups. We find that the ratio of each type of energy eigenstate in an energy shell [E_{c}-ΔE/2,E_{c}+ΔE/2] is invariant with changing width ΔE or Planck constant ℏ as long as the number of eigenstates in the shell is statistically large enough. We give an argument that such self-similarity in energy eigenstates is a general feature for all quantum systems, which is further illustrated numerically with various quantum systems, including circular billiard, double top model, kicked rotor, and Heisenberg XXZ model.

Realization of geodesic flows with a linear first integral by billiards with slipping

An arbitrary geodesic flow on the projective plane or Klein bottle with an additional, linear in the momentum, first integral is modelled using billiards with slipping on table complexes. The requisite table of a circular topological billiard with slipping is constructed algorithmically. Furthermore, linear integrals of geodesic flows can be reduced to the same canonical integral of a circular planar billiard. Bibliography: 36 titles.

Spherical billiards with almost complete escape.

A dynamical billiard consists of a point particle moving uniformly except for mirror-like collisions with the boundary. Recent work has described the escape of the particle through a hole in the boundary of a circular or spherical billiard, making connections with the Riemann Hypothesis. Unlike the circular case, the sphere with a single hole leads to a non-zero probability of never escaping. Here, we study variants in which almost all initial conditions escape, with multiple small holes or a thin strip. We show that equal spacing of holes around the equator is an efficient means of ensuring almost complete escape and study the long time survival probability for small holes analytically and numerically. We find that it approaches a universal function of a single parameter, hole area multiplied by time.

Chaotic Motion in the Breathing Circle Billiard

We consider the free motion of a point particle inside a circular billiard with periodically moving boundary, with the assumption that the collisions of the particle with the boundary are elastic so that the energy of the particle is not preserved. It is known that if the motion of the boundary is regular enough then the energy is bounded due to the existence of invariant curves in the phase space. We show that it is nevertheless possible that the motion of the particle is chaotic, also under regularity assumptions for the moving boundary. More precisely, we show that there exists a class of functions describing the motion of the boundary for which the billiard map has positive topological entropy. The proof relies on variational techniques based on the Aubry–Mather theory.

Quantizing neutrino billiards: an expanded boundary integral method

With the pioneering fabrication of graphene the field of relativistic quantum chaos emerged. We will focus on the spectral properties of massless spin-1/2 particles confined in a bounded two-dimensional region, named neutrino billiards by Berry and Mondragon in 1987. A commonly used method for the determination of the eigenvalues is based on a boundary integral equation originating from Green’s theorem. Yet, in the quantization one might face problems similar to those occurring for non-relativistic quantum billiards. Especially in cases where the eigenvalue spectrum contains near degeneracies the identification of complete sequences of eigenvalues might be extremely elaborate, if not unfeasible. We propose an expanded boundary integral method, which yields complete eigenvalue sequences with a considerably lower numerical effort than the standard one. Actually, it corresponds to an extension of the method introduced in Veble et al (2007 New J. Phys. 9 15) to relativistic quantum billiards. To demonstrate its validity and its superior efficiency compared to the standard method, we apply both methods to a circular billiard of which the eigenvalues are known analytically and exhibit near degeneracies. Finally, we employ it for the quantization of a neutrino billiard with a hole, of which the spectrum contains many close lying levels and exhibits unusual fluctuation properties.

Exact solution to Lippmann-Schwinger equation for a circular billiard

We present an exact solution to the Lippmann-Schwinger equation for a two-dimensional circular billiard. After diagonalizing an integral operator whose kernel is a zeroth order Hankel function of first kind, we use its eigenfunctions and eigenvalues to obtain in a straightforward way the exact wavefunctions of the referred Lippmann-Schwinger equation.

Two-dimensional quantum scattering by non-isotropic interactions localized on a circle, applications to open billiards

Two-dimensional quantum scattering by isotropic and non-isotropic interactions localized on a circle is considered. The expansion of the interaction on the circle in a Fourier series allows us to express basic objects of scattering theory (resolvent, T operator, differential cross length, cross length, and cross length averaged over all orientations of the incident particle), in terms of operations on matrices. For numerical applications, these matrices are truncated to a given order. If the interaction is isotropic, this general formulation reduces to the usual one, and the resonances in the isotropic cases are studied because they allow us to interpret resonances in some non-isotropic cases. Applications to open circular billiards are given. A first approach to the open quantum circular billiard is an interaction equal to zero on some parts of the circle and to a great value λ on the other parts. Then, it turns out that the limit λ → +∞ does not lead to an explicit result in this approach. A second approach to the open circular billiard is given by replacing apertures (where the interaction is zero) by pseudo-apertures (where the interaction is small but non-zero), and the limit λ → +∞ then leads to an explicit result. In the high energy regime, the two approaches give similar results and allow interpretation of numerous results, in particular, for differential cross lengths, in terms of classical mechanics.

Nekhoroshev’s approach to Hamiltonian monodromy

Using the hyperbolic circular billiard, introduced in [31] by Delos et al. as possibly the simplest system with Hamiltonian monodromy, we illustrate the method developed by N. N. Nekhoroshev and coauthors [48] to uncover this phenomenon. Nekhoroshev’s very original geometric approach reflects his profound insight into Hamiltonian monodromy as a general topological property of fibrations. We take advantage of the possibility of having closed form elementary function expressions for all quantities in our system in order to provide the most explicit and detailed explanation of Hamiltonian monodromy and its relation to similar phenomena in other domains.

Open quantum billiard in a magnetic field: A perfect spin filter

The transport properties of an open system, a circular billiard with attached channels, are studied in the presence of Rashba spin-orbit coupling. The inclusion of even a weak magnetic field in the structure plane is shown to cause the conductance to become dependent on the spin state of charge carriers. By choosing the system’s parameters, this property allows a spin filter based on it to be realized.

Time-dependent quantum circular billiard

The motion of a quantum particle in a time-dependent circular billiard is studied on the basis of the Schrodinger equation with time-dependent boundary conditions. The cases of monotonically expanding (contracting), non-harmonically, harmonically breathing circles the case when billiard wall suddenly disap- pears are explored in detail. The exact analytical solutions for monotonically expanding and contracting circles are obtained. For all cases, the time-dependence of the quantum average energy is calculated. It is found that for an harmonically breathing circle, the average energy is time-periodic in the adiabatic regime with the same period as that of the oscillation. For intermediate frequencies which are comparable with the initial frequency of the particle in unperturbed billiard, such periodicity is broken. However, for very high frequencies, the average energy once again becomes periodic. A qualitative analysis of the border between adiabatic and non-adiabatic regimes is provided.

Survival probability for open spherical billiards.

We study the survival probability for long times in an open spherical billiard, extending previous work on the circular billiard. We provide details of calculations regarding two billiard configurations, specifically a sphere with a circular hole and a sphere with a square hole. The constant terms of the long-time survival probability expansions have been derived analytically. Terms that vanish in the long time limit are investigated analytically and numerically, leading to connections with the Riemann hypothesis.

Effect of a weak magnetic field on resonant features of conductance in an open circular billiard with Dresselhaus spin-orbit interaction

The effect of a weak magnetic field on the transport properties of an open circular billiard with attached channels in the presence of Dresselhaus spin-orbit interaction are considered. It is shown that the application of a magnetic field leads to splitting of the Fano resonances, which were found earlier on the energy dependence of conductance, into pairs of resonances with half lower amplitudes. The relationship between the energy values to which these resonances collapse when the spin-orbit interaction is absent, and the levels of the energy spectrum in the corresponding closed billiard has been established. It is shown also that the applied magnetic field induces a qualitative change in the spin polarization of a wave in the output channel.

Chaos and ergodicity of two hard disks within a circular billiard

We investigate dynamical properties of the system of two interacting hard disks within a circular billiard numerically in the case of zero total angular momentum. Varying the radius of two identical disks, we examine chaotic irregularity and ergodicity of the system. Single-particle configuration and velocity distributions are obtained from dynamical trajectory calculations and compared with those in the microcanonical ensemble. We also analyze properties of trajectories by calculating the finite-time maximum Lyapunov exponent and clarify the existence of sticky motions around Kolmogorov-Arnold-Moser (KAM) tori even for small radii of disks. It is shown that the present system is almost ergodic in spite of the existence of tori for small radii of disks since the ratio of tori to the whole phase space is extremely small. On the other hand, a number of tori increase abruptly as the radius of disks increases beyond some value and tori prevent trajectories to run over the phase space uniformly, which makes the ergodicity of the system broken down.

Quantum flux and reverse engineering of quantum wave functions

An interpretation of the probability flux is given, based on a derivation of its eigenstates and relating them to coherent-state projections on a quantum wave function. An extended definition of the flux operator is obtained using coherent states. We present a “processed Husimi” representation, which makes decisions using many Husimi projections at each location. The processed Husimi representation reverse engineers or deconstructs the wave function, yielding the underlying classical ray structure. Our approach makes possible interpreting the dynamics of systems where the probability flux is uniformly zero or strongly misleading. The new technique is demonstrated by the calculation of particle flow maps of the classical dynamics underlying a quantum wave function in simple model systems such as a circular billiard with and without a magnetic field.

Two-particle circular billiards versus randomly perturbed one-particle circular billiards

We study a two-particle circular billiard containing two finite-size circular particles that collide elastically with the billiard boundary and with each other. Such a two-particle circular billiard provides a clean example of an "intermittent" system. This billiard system behaves chaotically, but the time scale on which chaos manifests can become arbitrarily long as the sizes of the confined particles become smaller. The finite-time dynamics of this system depends on the relative frequencies of (chaotic) particle-particle collisions versus (integrable) particle-boundary collisions, and investigating these dynamics is computationally intensive because of the long time scales involved. To help improve understanding of such two-particle dynamics, we compare the results of diagnostics used to measure chaotic dynamics for a two-particle circular billiard with those computed for two types of one-particle circular billiards in which a confined particle undergoes random perturbations. Importantly, such one-particle approximations are much less computationally demanding than the original two-particle system, and we expect them to yield reasonable estimates of the extent of chaotic behavior in the two-particle system when the sizes of confined particles are small. Our computations of recurrence-rate coefficients, finite-time Lyapunov exponents, and autocorrelation coefficients support this hypothesis and suggest that studying randomly perturbed one-particle billiards has the potential to yield insights into the aggregate properties of two-particle billiards, which are difficult to investigate directly without enormous computation times (especially when the sizes of the confined particles are small).

Hopf rigidity for convex billiards on the hemisphere and hyperbolic plane

This paper deals with Hopf type rigidity for convex billiards onsurfaces of constant curvature. I prove that the only convexbilliard without conjugate points on the hyperbolic plane or on thehemisphere is a circular billiard.

Resonance features of the conductance of open billiards: Spin-orbit coupling effects

Transport properties of an open circular billiard with attached channels in the presence of the spin-orbit coupling are studied. It is shown that the inclusion of the spin-orbit coupling causes the occurrence of additional narrow Fano resonances on the energy dependence of the conductance. It is established that the resonance regions are associated with the energy levels in a corresponding closed billiard. The effect of small finite temperatures on the observability of the conductance resonances is estimated.

Generation of large orbital angular momentum from superposed Bessel beams corresponding to resonant geometric modes

We theoretically verify that a coherent superposition of nearly degenerate Bessel beams characterizes the family of resonant geometric modes in circular billiards. With the ray-wave correspondence, we experimentally exploit a large-aperture cylindrical waveguide to generate the resonant geometric modes with large orbital angular momentum. Furthermore, the free-space propagation of the geometric modes emerging from the cylindrical waveguide is demonstrated to display the transient dynamics of the coherent states released from a circular quantum billiard.

Resonance features of the conductance of open billiards with the spin-orbit interaction

The transport properties of a circular billiard with attached channels, which is an open system, have been studied in the presence of the Dresselhaus and Rashba spin-orbit interactions. It has been shown that this interaction leads to the appearance of additional Fano resonances in the energy dependence of the conductance, the width of which is proportional to the fourth power of the spin-orbit coupling constant.

Chaos and stability in a two-parameter family of convex billiard tables

We study, by numerical simulations and semi-rigorous arguments, a two-parameter family of convex, two-dimensional billiard tables, generalizing the one-parameter class of oval billiards of Benettin–Strelcyn (1978 Phys. Rev. A 17 773–85). We observe interesting dynamical phenomena when the billiard tables are continuously deformed from the integrable circular billiard to different versions of completely chaotic stadia. In particular, we conjecture that a new class of ergodic billiard tables is obtained in certain regions of the two-dimensional parameter space, when the billiards are close to skewed stadia. We provide heuristic arguments supporting this conjecture, and give numerical confirmation using the powerful method of Lyapunov-weighted dynamics.