Abstract

Chaotic billiards (hard-walled cavities) in two or more dimensions are paradigm systems in the fields of classical and quantum chaos. We study the dissipation (irreversible heating) rate in such billiard systems due to general shape deformations which are periodic in time. We are motivated by older studies of one-body nuclear dissipation and by anticipated mesoscopic applications. We review the classical and quantum linear response theories of dissipation rate and demonstrate their correspondence in the semiclassical limit. In both pictures, heating is a result of stochastic energy spreading. The heating rate can be expressed as a frequency-dependent friction coefficient $\mu(\omega)$, which depends on billiard shape and deformation choice. We show that there is a special class of deformations for which $\mu$ vanishes as like a power law in the small-$\omega$ limit. Namely, for deformations which cause translations and dilations $\mu \sim \omega^4$ whereas for those which cause rotations $\mu \sim \omega^2$. This contrasts the generic case for which $\mu \sim \omega^0$. We show how a systematic treatment of this special class leads to an improved version of the `wall formula' estimate for $\mu(0)$.

We show that the special nature of dilation (a new result) is semiclassically equivalent to a quasi-orthogonality relation between the (undeformed) billiard quantum eigenstates on the boundary. This quasi-orthogonality forms the heart of a `scaling method' for the numerical calculation of quantum eigenstates, invented recently by Vergini and Saraceno. The scaling method is orders of magnitude more efficient than any other known billiard quantization method, however an adequate explanation for its success has been lacking until now. We explain the scaling method, its errors, and applications. We also present improvements to Heller's plane wave method.

Two smaller projects conclude the thesis. Firstly, we give a new formalism for quantum point contact (QPC) conductance in terms of scattering cross-section in the half-plane, of use in open mesoscopic and atomic systems. We derive the maximum conductance through a tunneling QPC coupled to a resonator. Secondly, we numerically model a novel design of coherent atom waveguide which uses the dipole force due to evanescent light fields leaking from an optical waveguide mounted on a substrate.