Floquet theory of multi-level Landau-Zener transitions in topologically trivial Josephson junctions

We study multi-level Landau-Zener (LZ) transitions in periodically driven quantum systems, both numerically and analytically using Floquet theory. In the high frequency regime, we perform a nondegenerate Floquet perturbation theory and calculate transition probabilities between the different adiabatic states. In the low frequency regime, we employ a degenerate Floquet perturbation theory that treats Floquet resonances systematically. We apply our results to investigate multi-modal scattering in ac Josephson junctions hosting topologically trivial high-transparency modes. A single pair of high-transparency modes has been shown to mimic the 4π - periodicity in the current-phase relation and missing Shapiro steps expected of topological Majorana modes. We find that transitions between multiple pairs of high-transparency modes can result in additional structure in the current-phase relation with various periodic patterns.