Quasiclassical fluctuations of the superconductor proximity gap in a chaotic system

We calculate the sample-to-sample fluctuations in the excitation gap of a chaotic dynamical system coupled by a narrow lead to a superconductor. Quantum fluctuations on the order of magnitude of the level spacing, predicted by random-matrix theory, apply if $\tau_E\ll\hbar/E_T$ (with $\tau_E$ the Ehrenfest time and $E_T$ the Thouless energy). For $\tau_E\agt\hbar/ E_T$ the fluctuations are much greater than the level spacing. We demonstrate the quasiclassical nature of the gap fluctuations in the large-$\tau_E$ regime by correlating them to an integral over the classical dwell-time distribution.