Nearly closed ballistic billiard with random boundary transmission

A variety of mesoscopic systems can be represented as billiard with a random coupling to the exterior at the boundary. Examples include quantum dots with multiple leads, quantum corrals with different kinds of atoms forming the boundary, and optical cavities with random surface refractive index. We study an electronic billiard with no internal impurities weakly coupled to the exterior by a large number of leads. We construct a supersymmetric nonlinear $\sigma$ model by averaging over the random coupling strengths between bound states and channels. The resulting theory can be used to evaluate the statistical properties of any physically measurable quantity. As an illustration, we present results for the local density of states.

Figure 1

LDOS plotted as a function of $\kappa$ in units of $m∕\pi$ for two sets of parameters: $g=50$, $\mathcal{M}=1∕12$ (dashed line) and $g=200$, $\mathcal{M}=1∕3$ (solid line). In both cases the rest of the parameters were fixed $x=0.1$, $\overline{\gamma }=0.2$, $v=100$.