Andreev crystals in hybrid Josephson junction arrays

Andreev bound states are superpositions of electrons and holes in a metal that form by coherent reflection from a superconductor. When the length of the superconductor is comparable to the superconducting coherence length, Andreev bound states at opposite edges hybridize by quasiparticle tunneling. In a periodic array, these hybridized bound states form a band at energies below the superconducting gap. In this paper, we derive a theoretical framework for the transport properties of these Andreev crystals. We demonstrate that at high interface transparency and constant phase bias between neighboring superconductors, these bands are {\it directional}: one band consists only of right -- while the other only of left-moving electronic states. This property enables the application of this device as a flux- and bias-voltage tunable filter that permits signal transmission in only one direction.