Induced Superconductivity Distinguishes Chaotic from Integrable Billiards

Random-matrix theory is used to show that the proximity to a superconductor opens a gap in the excitation spectrum of an electron gas confined to a billiard with a chaotic classical dynamics. In contrast, a gapless spectrum is obtained for a non-chaotic rectangular billiard, and it is argued that this is generic for integrable systems. PACS numbers: 05.45.+b, 73.23.Ps, 74.50.+r, 74.80.Fp Typeset using REVT E X The quantization of a system with a chaotic classical dynamics is the fundamental problem of the field of "quantum chaos" [1,2]. It is known that the statistics of the energy levels of a two-dimensional confined region (a "billiard") is different if the dynamics is chaotic or integrable [3--5]: A chaotic billiard has Wigner-Dyson statistics, while an integrable billiard has Poisson statistics. The two types of statistics are entirely different as far as the level correlations are concerned [6]. However, the mean level spacing is essentially the same: Particles of mass m in a bi..