Dynamics of Majorana Bound States in a thin annular topological Josephson junction

The Majorana bound states formed in extended SIS Josephson junctions between p+ip superconductors have been popular both as a means of detection of Majorana zero modes as well as designing braiding operations. In this context, we study the topological Josephson junction in annular geometries. In the presence of an external magnetic field, we demonstrate the nucleation of an even number of Majorana bound states and calculate the current contribution stemming from Andreev scattering across the junction. We also discuss possible signatures of these states in critical currents across the junction, and schemes to control the dynamics of these bound states in the junctions via voltage pulses. In doing so, we compare the characteristics of the annular junction with an extended rectangular junction and establish boundary and size effects.