Ehrenfest-time-dependent excitation gap in a chaotic Andreev billiard

A semiclassical theory is developed for the appearance of an excitation gap in a ballistic chaotic cavity connected by a point contact to a superconductor. Diffraction at the point contact is a singular perturbation in the limit $\mathrm{\hslash }\to 0$, which opens up a gap $E_{\mathrm{g}\mathrm{a}\mathrm{p}}$ in the excitation spectrum. The time scale $\mathrm{\hslash }\mathrm{/}{E}_{\mathrm{g}\mathrm{a}\mathrm{p}}\propto {\alpha }^{-1}\ln \mathrm{\hslash }$ (with $\alpha$ the Lyapunov exponent) is the Ehrenfest time, the characteristic time scale of quantum chaos.