Chladni figures in Andreev billiards

We study wave functions and their nodal patterns in Andreev billiards consisting of a normal-conducting (N) ballistic quantum dot in contact with a superconductor (S). The bound states in such systems feature an electron and a hole component which are coherently coupled by the scattering of electrons into holes at the S-N interface. The wave function "lives" therefore on two sheets of configuration space, each of which features, in general, distinct nodal patterns. By comparing the wave functions and their nodal patterns for holes and electrons detailed tests of semiclassical predictions become possible. One semiclassical theory based on ideal Andreev retroreflection predicts the electron- and hole eigenstates to perfectly mirror each other. We probe the limitations of validity of this model both in terms of the spectral density of the eigenstates and the shape of the wavefunctions in the electron and hole sheet. We identify cases where the Chladni figures for the electrons and holes drastically differ from each other and explain these discrepancies by limitations of the retroreflection picture.