Hysteresis from Nonlinear Dynamics of Majorana Zero Modes in Topological Josephson Junctions

We reveal that topological Josephson junctions provide a natural platform for the interplay between the Josephson effect and the Landau-Zener effect through a two-level system formed by coupled Majorana modes. We build a quantum resistively shunted (RSJ) junction model by modifying the standard textbook RSJ model to take account of the two-level system from the Majorana modes at the junction. We show that the dynamics of the two-level system is governed by a nonlinear Schroedinger equation and solve the equations analytically via a mapping to a classical dynamical problem. This nonlinear dynamics leads to hysteresis in the I-V characteristics, which can give a quantitative explanation to recent experiments. We also predict coexistence of two interference patterns with periods h/e and h/2e in topological superconducting quantum interference devices.