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We studied the energy levels of graphene based Andreev billiards consisting of a superconductor region on top of a monolayer graphene sheet. For the case of Andreev retro-reflection we show that the graphene based Andreev billiard can be mapped to the normal metal-superconducting billiards with the same geometry. We also derived a semiclassical quantization rule in graphene based Andreev billiards. The exact and the semiclassically obtained spectrum agree very well both for the case of Andreev retro-reflection and specular Andreev reflection.
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We investigate statistical aspects of the entanglement production for open chaotic mesoscopic billiards in contact with superconducting parts, known as Andreev billiards. The complete distributions of concurrence and entanglement of formation are obtained by using the Altland-Zirnbauer symmetry classes of circular ensembles of scattering matrices, which complements previous studies in chaotic universal billiards belonging to other classes of random matrix theory. Our results show a unique and very peculiar behavior: the realization of entanglement in a Andreev billiard always results in non-separable state, regardless of the time reversal symmetry. The analytical calculations are supported by a numerical Monte Carlo simulation.
We examine the density of states of an Andreev billiard and show that any billiard with a finite upper cut-off in the path length distribution $P(s)$ will possess an energy gap on the scale of the Thouless energy. An exact quantum mechanical calculation for different Andreev billiards gives good agreement with the semi-classical predictions when the energy dependent phase shift for Andreev reflections is properly taken into account. Based on this new semi-classical Bohr-Sommerfeld approximation of the density of states, we derive a simple formula for the energy gap. We show that the energy gap, in units of Thouless energy, may exceed the value predicted earlier from random matrix theory for chaotic billiards.
The energy spectrum and the eigenstates of a rectangular quantum dot containing soft potential walls in contact with a superconductor are calculated by solving the Bogoliubov-de Gennes (BdG) equation. We compare the quantum mechanical solutions with a semiclassical analysis using a Bohr--Sommerfeld (BS) quantization of periodic orbits. We propose a simple extension of the BS approximation which is well suited to describe Andreev billiards with parabolic potential walls. The underlying classical periodic electron-hole orbits are directly identified in terms of ``scar like features engraved in the quantum wavefunctions of Andreev states determined here for the first time.
An effective random matrix theory description is developed for the universal gap fluctuations and the ensemble averaged density of states of chaotic Andreev billiards for finite Ehrenfest time. It yields a very good agreement with the numerical calculation for Sinai-Andreev billiards. A systematic linear decrease of the mean field gap with increasing Ehrenfest time $tau_E$ is observed but its derivative with respect to $tau_E$ is in between two competing theoretical predictions and close to that of the recent numerical calculations for Andreev map. The exponential tail of the density of states is interpreted semi-classically.
We present a classical and quantum mechanical study of an Andreev billiard with a chaotic normal dot. We demonstrate that in general the classical dynamics of these normal-superconductor hybrid systems is mixed, thereby indicating the limitations of a widely used retracing approximation. We show that the mixed classical dynamics gives rise to a wealth of wavefunction phenomena, including periodic orbit scarring and localization of the wavefunction onto other classical phase space objects such as intermittent regions and quantized tori.
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