Martin M A Sieber

Semiclassical theory of chaotic quantum transport

We present a refined semiclassical approach to the Landauer conductance and Kubo conductivity of clean chaotic mesoscopic systems. We demonstrate for systems with uniformly hyperbolic dynamics that including off-diagonal contributions to double sums over classical paths gives a weak-localization correction in quantitative agreement with results from random matrix theory. We further discuss the magnetic-field dependence. This semiclassical treatment accounts for current conservation.

Classical and quantum mechanics of a strongly chaotic billiard

Abstract We study the hyperbola billiard, a strongly chaotic system whose classical dynamics is the free motion of a particle within the region D = l {( x , y )| x ≥0∧ y ≥0∧ y ≤1/ x } r with elastic reflections on the boundary ∂D. The corresponding quantum mechanical problem is to determine the bound state energies as eigenvalues of the Dirichlet Laplacian on D. It is shown that the classical periodic orbits of the hyperbola billiard can be effectively enumerated by a ternary code. Combining this code with an extremum principle, we are able to determine with high precision more than 500 000 primitive periodic orbits together with their lengths, multiplicities and Lyapunov exponents. The statistical properties of the length spectrum of the periodic orbits are found to be consistent with a random walk model, which in turn predicts asymptotically an exponential proliferation of long periodic orbits and leads to a novel formula for the topological entropy τ, whose value turns out to be approximately 0.6. The periodic orbits are used for a quantitative test of Gutzwillers periodic-orbit theory, which plays the role of a semiclassical quantization rule. We find that the predictions of the periodic-orbit theory for the Gaussian level density agree at low energies surprisingly well with the “true” results obtained from a numerical solution of the Schrodinger equation.

SEMICLASSICAL QUANTIZATION OF BILLIARDS WITH MIXED BOUNDARY CONDITIONS

The semiclassical theory for billiards with mixed boundary conditions is developed and explicit expressions for the smooth and the oscillatory parts of the spectral density are derived. The parametric dependence of the spectrum on the boundary condition is shown to be a very useful diagnostic tool in the semiclassical analysis of the spectrum of billiards. It is also used to check in detail some recently proposed parametric spectral statistics. The methods are illustrated in the analysis of the spectrum of the Sinai billiard and its parametric dependence on the boundary condition on the dispersing arc.

Leading off-diagonal approximation for the spectral form factor for uniformly hyperbolic systems

We consider the semiclassical approximation to the spectral form factor K(τ) for two-dimensional uniformly hyperbolic systems with time-reversal symmetry, and derive the first off-diagonal correction for small τ. The result agrees with the τ2-term of the form factor for the GOE random matrix ensemble.

Generalized periodic-orbit sum rules for strongly chaotic systems

Abstract The periodic-orbit theory of Gutzwiller is the only known semiclassical quantization scheme that can be applied to non-integrable systems. We present a generalized version of this theory that leads to absolutely convergent periodic-orbit sum rules, using as an example strongly chaotic billiard systems.

Unidirectional emission from circular dielectric microresonators with a point scatterer

Circular microresonators are micron-sized dielectric disks embedded in material of lower refractive index. They possess modes of extremely high Q-factors (low-lasing thresholds), which makes them ideal candidates for the realization of miniature laser sources. They have, however, the disadvantage of isotropic light emission caused by the rotational symmetry of the system. In order to obtain high directivity of the emission while retaining high Q-factors, we consider a microdisk with a pointlike scatterer placed off-center inside of the disk. We calculate the resulting resonant modes and show that some of them possess both of the desired characteristics. The emission is predominantly in the direction opposite to the scatterer. We show that classical ray optics is a useful guide to optimizing the design parameters of this system. We further find that exceptional points in the resonance spectrum influence how complex resonance wave numbers change if system parameters are varied.

Semiclassical structure of chaotic resonance eigenfunctions

We study the resonance (or Gamow) eigenstates of open chaotic systems in the semiclassical limit, distinguishing between left and right eigenstates of the nonunitary quantum propagator and also between short-lived and long-lived states. The long-lived left (right) eigenstates are shown to concentrate as variant Plancks over 2pi-->0 on the forward (backward) trapped set of the classical dynamics. The limit of a sequence of eigenstates [psi(variant Plancks over)] 2pi-->0 is found to exhibit a remarkably rich structure in phase space that depends on the corresponding limiting decay rate. These results are illustrated for the open bakers map, for which the probability density in position space is observed to have self-similarity properties.

Quantum chaos in the hyperbola billiard

Abstract A few hundred energy levels of the hyperbola billiard, a strongly chaotic system, are computed using a boundary-element method. The level statistics is investigated and found to be consistent with the predictions of random-matrix theory for the Gaussian orthogonal ensemble. The energy spectrum is used for an application of Gutzwillers periodic orbit theory, which nicely demonstrates that the quantal energies “know” the classical periodic orbits.

Bifurcations of periodic orbits and uniform approximations

We derive uniform approximations for contributions to Gutzwillers periodic-orbit sum for the spectral density which are valid close to bifurcations of periodic orbits in systems with mixed phase space. There, orbits lie close together and give collective contributions, while the individual contributions of Gutzwillers type would diverge at the bifurcation. New results for the tangent, the period-doubling and the period-tripling bifurcation are given. They are obtained by going beyond the local approximation and including higher-order terms in the normal form of the action. The uniform approximations obtained are tested on the kicked top and are found to be in excellent agreement with exact quantum results.

Semiclassical transition from an elliptical to an oval billiard

Semiclassical approximations often involve the use of stationary phase approximations. This method can be applied when is small in comparison with relevant actions or action differences in the corresponding classical system. In many situations, however, action differences can be arbitrarily small and then uniform approximations are more appropriate. In this paper we examine different uniform approximations for describing the spectra of integrable systems and systems with mixed phase space. This is done on the example of two billiard systems, an elliptical billiard and a deformation of it, an oval billiard. We derive a trace formula for the ellipse which involves a uniform approximation for the Maslov phases near the separatrix, and a uniform approximation for tori of periodic orbits close to a bifurcation. We then examine how the trace formula is modified when the ellipse is deformed into an oval. This involves uniform approximations for the break-up of tori and uniform approximations for bifurcations of periodic orbits. Relations between different uniform approximations are discussed.