Quantal Andreev Maps: A New Paradigm of Quantum Chaos in Superconductivity Philippe Jacquod Leiden University During the past two decades, one-dimensional quantum mechanical maps have developed into one of two paradigms of quantum chaos, the other paradigm being the two-dimensional billiard. These two paradigms share a common set of phenomena in the fields of quantum chaos and localization. The combination of chaos and superconductivity produces an entirely new phenomenology, notably the appearance of an excitation gap as a signature of quantum chaos. The paradigm common to most of the literature is the two-dimensional billiard connected to a superconductor, a constriction usually referred to as an "Andreev billiard". My talk will be devoted to the construction of a second paradigm of quantum chaos in presence of superconductivity: quantum maps with particle-hole conversion (Andreev reflection) and particle-hole symmetry. I will show that the resulting "Andreev maps" exhibit the same excitation gap as quantum dots in the proximity to a superconductor, and check competing analytical predictions of random-matrix theory and semiclassical chaos - in particular, the logarithmic reduction of the quasiparticle gap when the Lyapunov exponent and/or the effective Planck constant are small enough, that the quantum ergodicity time - the "Ehrenfest" time - becomes comparable to the quasiparticle dwell time.