Waves and rays in open billiards

Abstract: Semiconductor systems where the deBroglie wavelength of the electron is comparable to the size of the system are called mesoscopic. Such systems are readily produced in laboratories, and have been studied extensively. These systems offer possibilities to study the borderline between quantum and classical physics, chaos and more. The continuous shrink in size of semiconductor devices will make their operation quantum mechanical by the year 2010, this offers both problems and possibilities; so called quantum computing is a very hot research subject.A class of systems called billiards are studied in this thesis. A billiard is a two-dimensional system with a constant inner potential and hard wall confinement, just like the table in a game of billiards. The billiards can be mesoscopic (then often called quantum dots) or macroscopic. The introduction of this thesis shortly describes experimental two-dimensional semiconductor and microwave billiards. Theory is described and methods for computing conductance through billiards classically, semi-classically and quantum mechanically are then developed in detail. The main emphasis in this thesis has been on development of methods for semi-classical calculations.The following papers are included:In paper I a semi-classical interpretation is provided for an experimental billiard, explaining the characteristic frequencies in its measured conductance in terms of pairs of classical trajectories. Further, the phase coherence length in the billiard is extracted from these frequencies and the relation between conductance oscillations and the common picture of periodic orbits is outlined.In paper II the conductance and weak localisation corrections in a triangularbilliard in magnetic field are calculated using both the semi-classical method and quantum mechanics. The semi-classical method is however shown to not reproduce the quantum mechanical results, break current conservation and symmetry of conductance with respect to direction of current and magnetic fields. The reason for the discrepancies is traced to the topology of the classical trajectories. The findings pinpoint some limitations of the semi-classical theory and rise question to what extent one can rely on some of the statistical predictions of the semi-classical theory, e.g. weak localisation line shapes and fractal conductance.Paper III presents experimental studies as well as theoretical quantum mechanicaand semi-classical studies, of a square microwave billiard. The quantum mechanical calculations show excellent agreement with experimental results. By means of semi-classical simulations, the characteristic oscillations in transmission and reflection amplitudes are shown to be related to the length of trajectories in the billiard. Oscillations in the transmission and reflection probabilities arc also shown to be related to length differences in pairs of trajectories.In paper IV the effects of the choice of boundary conditions at the lead mouths, are studied. A new formula for the S-matrix elements is derived, and shown to yield superior results when comparing semi-classical and quantum mechanical simulations. Paper V studies the time-resolved dynamics in classically regular and chaotic billiards, comparing classical trajectories and a quantum mechanical wave packets. For short times (up to approx. the Heisenberg time), the features in the quantum mechanical current is directly related to classical trajectories. In contrast, the longtime asymptotics of the quantum mechanical currents decay exponentially, and is not sensitive to the nature of the classical asymptotics, i.e. exponential for chaotic systems and power law for regular.