Tunable fractional Josephson effect in the topological superconducting junction with embedded quantum dots

We investigate the fractional Josephson effect in the junction of two Majorana zero modes contributed by the topological superconducting wires. It shows that when the Majorana zero modes couple indirectly via one quantum dot, shifting the dot level does not change the phase of the Josephson current. However, if the junction is a double-channel geometry, abundant Josephson phase transitions can be observed. Such a result arises from the fermion parity conservation which is related to the quantum interference. Moreover, in the presence of appropriate structural parameters, destructive interference will occur between the two channels, which induces the appearance of the normal Josephson effect. These results will be helpful for describing the fractional Josephson effect modulated by quantum dots