2004/06/30 by C. W. J. Beenakker
Physics and Astronomy · #Cold Atom Physics and Bose-Einstein Condensates #Nonlinear Photonic Systems #Quantum chaos and dynamical systems #cond-mat.mes-hall
paper · pdf · doi:10.1007/11358817_4
published as Lect. Notes Phys. 667, 131 (2005) · 22 pages including 24 figures; [v2] new sections III, VI.C; [v3] major revision of section VIII.B, correcting the effective RMT of cond-mat/0208192
arxiv created 2004/07/09 · openalex publication_date 2005/03/04 · arxiv updated 2009/12/01 · openalex created_date 2020/07/02 · openalex updated_date 2026/07/28
This is a review of recent advances in our understanding of how Andreev reflection at a superconductor modifies the excitation spectrum of a quantum dot. The emphasis is on two-dimensional impurity-free structures in which the classical dynamics is chaotic. Such Andreev billiards differ in a fundamental way from their non-superconducting counterparts. Most notably, the difference between chaotic and integrable classical dynamics shows up already in the level density, instead of only in the level--level correlations. A chaotic billiard has a gap in the spectrum around the Fermi energy, while integrable billiards have a linearly vanishing density of states. The excitation gap Egap corresponds to a time scale h/Egap which is classical (h-independent, equal to the mean time tdwell between Andreev reflections) if tdwell is sufficiently large. There is a competing quantum time scale, the Ehrenfest time tE, which depends logarithmically on h. Two phenomenological theories provide a consistent description of the tE-dependence of the gap, given qualitatively by Egap min(h/tdwell,h/tE). The analytical predictions have been tested by computer simulations but not yet experimentally.