Circular Billiard Research Articles

Periodic orbit theory analysis of a continuous family of quasi-circular billiards

We compute the Fourier transform (ρ(L)) of the quantum mechanical energy level density for the problem of a particle in a two-dimensional circular infinite well (or circular billiard) as well as for several special generalizations of that geometry, namely the half-well, quarter-well, and the circular well with a thin, infinite wall along the positive x-axis (hereafter called a circular well plus baffle). The resulting peaks in plots of |ρ(L)|2 versus L are compared to the lengths of the classical closed trajectories in these geometries as a simple example of the application of periodic orbit (PO) theory to a billiard or infinite well system. We then solve the Schrödinger equation for the general case of a circular well with infinite walls both along the positive x-axis and at an arbitrary angle Θ (a circular “slice”) for which the half-well (Θ=π), quarter-well (Θ=π/2), and circular well plus baffle (Θ=2π) are then all special cases. We perform a PO theory analysis of this general system and calculate |ρ(L)|2 for many intermediate values of Θ to examine how the peaks in ρ(L) attributed to periodic orbits change as the quasi-circular wells are continuously transformed into each other. We explicitly examine the transitions from the half-circular well to the circle plus baffle case (half-well to quarter-circle case) as Θ changes continuously from π to 2π (from π to π/2) in detail. We then discuss the general Θ→0 limit, paying special attention to the cases where Θ=π/2n, as well as deriving the formulae for the lengths of closed orbits for the general case. We find that such a periodic orbit theory analysis is of great benefit in understanding and visualizing the increasingly complex pattern of closed orbits as Θ→0.

Shell structure of a circular quantum dot in weak magnetic fields

The conductance of a circular quantum dot in a two-dimensional electron gas of a GaAlAs/GaAs heterostructure has been measured. Conductance oscillations as functions both of the magnetic field B and of the size of a dot confining about 1000 electrons are related to the formation of electronic shell structure. Modeling the dot by a circular billiard, we interpret the results semiclassically in terms of periodic orbit theory, providing a simple explanation of the B-periodic oscillations. A comparison to a harmonic confinement suitable for smaller quantum dots is given.

Periodic orbit theory of a circular billiard in homogeneous magnetic fields

We present a semiclassical description of the level density of a two-dimensional circular quantum dot in a homogeneous magnetic field. We model the total potential (including electron-electron interaction) of the dot containing many electrons by a circular billiard, i.e., a hard-wall potential. Using the extended approach of the Gutzwiller theory developed by Creagh and Littlejohn, we derive an analytic semiclassical trace formula. For its numerical evaluation we use a generalization of the common Gaussian smoothing technique. In strong fields orbit bifurcations, boundary effects (grazing orbits) and diffractive effects (creeping orbits) come into play, and the comparison with the exact quantum mechanical result shows major deviations. We show that the dominant corrections stem from grazing orbits, the other effects being much less important. We implement the boundary effects, replacing the Maslov index by a quantum-mechanical reflection phase, and obtain a good agreement between the semiclassical and the quantum result for all field strengths. With this description, we are able to explain the main features of the gross-shell structure in terms of just one or two classical periodic orbits.

Semiclassical calculations of conductance in ballistic open quantum dots

Conductance of a ballistic quantum dot with convex boundary and two leads in arbitrary orientations is considered. The semiclassical conductance formula of Jalabert, Baranger, and Stone ( Phys. Rev. Lett., 1990, 65, 2442) [1] for collinear-lead dots is generalized to arbitrary lead orientation. The monodromy matrix for a convex billiard is derived, based on which a computation procedure is developed that allows for explicit numerical calculations of various quantities needed in the formula. This computation technique is applied to a circular billiard with three different lead orientations. Coherent backscattering as well as weak localization effects are obtained. Fluctuations in the reflection coefficients as a function of Fermi energy show high degree of regularities and strong autocorrelations reflecting the underlying regular classical dynamics.

Semiclassical analysis of response functions for integrable systems

We derive a semiclassical expansion of quantum spectral functions for integrable systems. This complements a previous analysis of chaotic billiards [B. Mehlig et al., Phys. Rev. Lett. 75, 57 (1995)]. We discuss similarities and differences between the integrable and the chaotic case. As a numerical example, we consider a two-dimensional circular billiard.

Superconductor-proximity effect in chaotic and integrable billiards

We explore the effects of the proximity to a superconductor on the level density of a billiard for the two extreme cases that the classical motion in the billiard is chaotic or integrable. In zero magnetic field and for a uniform phase in the superconductor, a chaotic billiard has an excitation gap equal to the Thouless energy. In contrast, an integrable (rectangular or circular) billiard has a reduced density of states near the Fermi level, but no gap. We present numerical calculations for both cases in support of our analytical results. For the chaotic case, we calculate how the gap closes as a function of magnetic field or phase difference.

The pendulum as a dynamical billiard

We consider a simple example of a dynamical billiard consisting of a mass point moving in a circle under the influence of a homogeneous gravitational field. The point reflects by the mirror elastic law when it encounters the circular boundary. The problem is integrable between one collision and another, and also when the particle moves on the bounding circle. This makes it possible to build the conditions of existence and stability (in a linear and, at times, in a nonlinear sense, too) of the families of basic periodic trajectories determining the phase space topology for a fixed energy level. The numerical implementation of the Poincaré mapping offers a means of describing the phase pictures with regular and chaotic regions in more detail as well as their evolution as the energy changes. In a weak gravitational field, numerical experiments reveal only periodic trajectories that are symmetric about the vertical diameter of the circle. An analytic proof is given that the imposition of a weak gravitational field causes the disappearance of nonsymmetric two-, three-, four-, and six-link trajectories. The phenomenon arises from the superposition of two factors: the gravitation and the perfect symmetry of the circular billiard. We also consider motion evolution in the special case of the perfectly inelastic reflection law.

Trace Formula for Broken Symmetry

We derive a trace formula for systems that exhibit an approximate continuous symmetry. It interpolates between the sum over continuous families of periodic orbits that holds in the case of exact continuous symmetry, and the discrete sum over isolated orbits that holds when the symmetry is completely broken. It is based on a simple perturbation expansion of the classical dynamics, centered around the case of exact symmetry, and gives an approximation to the usual Gutzwiller formula when the perturbation is large. We illustrate the computation with some 2-dimensional examples: the deformation of the circular billiard into an ellipse, and anisotropic and anharmonic perturbations of a harmonic oscillator.

Circular quantum billiard with a singular magnetic flux line.

We discuss the application of Gutzwiller's semiclassical theory to a circular billiard with a singular magnetic flux line added at its center. The Aharonov-Bohm effect manifests itself through the cancellation of periodic orbits for particular flux strengths. Diffraction phenomena affect the gross-shell structure of the level density and require corrections of higher order in \ensuremath{\Elzxh}. The full quantization of the level spectrum, however, is much less affected. \textcopyright{} 1996 The American Physical Society.

Failure of semiclassical methods to predict individual energy levels

The authors argue that semiclassical methods quite generally cannot predict the individual energy levels not even in the semiclassical limit of small but finite and when the number of energy levels goes to infinity. By this they mean that the average relative error of the semiclassical eigenvalues in units of the mean level spacing typically increases indefinitely as the energy goes to infinity or is at least bounded from below. This they show for the case of the integrable circular billiard and the one-dimensional potential Uo/cos2( alpha x) by comparing the torus quantized semiclassical eigenenergies with the exact results. Since all the various semiclassical methods such as Gutzwiller's and Bogomolny's are reduced to the torus quantization in integrable cases they believe that their conclusion is generally valid. They have theoretical arguments and strong numerical evidence (for the case of the circular billiard) that nevertheless the statistical properties of the exact energy spectra are correctly reproduced by the semiclassical approximations. It is numerically found that the energy level spacing distribution and the spectral rigidity for the exact spectrum and for the semiclassical spectrum are in excellent agreement even for finite spectra where they both deviate from the limiting Poissonian behaviour, so they suggest that the nonuniversal approach to the limiting energy level statistics is also correctly described by the semiclassical theory. They discuss the validity of semiclassical methods in the light of their negative and positive findings. In addition they find the surprising result for the previously mentioned special cases that the error distribution of the semiclassical approximation is stationary, i.e. it is independent of the energy.

Heat kernel expansion for fermionic billiards in an external magnetic field

Using Seeley's heat kernel expansion, the authors compute the asymptotic density of states of the Dirac operator coupled to a magnetic field on a two-dimensional manifold with boundary ('fermionic billiard'). Local boundary conditions compatible with vector current conservation depend on a free parameter alpha . It is shown that the perimeter correction identically vanishes for alpha =0. In that case, the next-order constant term is found to be proportional to the Euler characteristic of the manifold. These results are independent of the external magnetic field and of the shape of the billiard, provided the boundary is sufficiently smooth. For the flat circular billiard, the constant term is found to be -1/2, in agreement with a numerical result of Berry and Mondragon.

Spectral Sum Rules for the Circular Aharonov-Bohm Quantum Billiard

This work presents the results of a quantitative investigation of the “sum rule method” recently proposed by the author for calculating low-lying energy levels. The system considered in detail is the circular Aharonov-Bohm quantum billiard recently introduced by Berry and Robnik. Exact, expressions are derived for the spectral zeta function at positiveinteger values as a function of the magnetic flux. Using the zeta function for fixed angular momentum, we observe a very fast convergence to the exact ground state energy (“precocious convergence”).