Impact of evanescent scattering modes and finite dispersion in superconducting junctions

Superconducting junctions are essential building blocks for quantum hardware, and their fundamental behavior remains a highly active research field. The behaviour of generic junctions is conveniently described by Beenakker's determinant formula, linking the subgap energy spectrum to the scattering matrix characterizing the junction. In particular, the gap closing between bound and continuum states in short junctions follows from unitarity of the scattering matrix, and thus, from probability conservation. In this work, we critically reassess two assumptions: that scattering in short junctions is approximately energy-independent and dominated by planar channels. We argue that strongly energy-dependent scattering follows from finite dispersion of the conductor electrons even when they spend little time within the scattering region, and show that evanescent modes play a central role when cross-channel scattering is important. By generalizing Beenakker's equation and performing a mapping to an effective Hamiltonian, we show that the gap closing is linked to a chiral symmetry. While finite energy-dependence in the scattering breaks the chiral symmetry, we show two distinct mechanisms preserving the gap closing, each connected to new types of constraints on energy-dependent scattering matrices beyond unitarity. If the dispersive mode is planar, the gap closing is still preserved through a time-dependent probability conservation analysis of the scattering process. If the dispersive channel is evanescent, we derive a constraint which, notably, cannot follow from probability conservation. We thus demonstrate that Andreev physics reveal a much wider variety of properties of normal metal scattering than commonly expected. We expect that our findings will have an impact on the dissipative behavior of driven junctions, and offer a new perspective on fundamental properties of scattering matrices.