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We investigate the Josephson effects in the junction formed by the indirect coupling between  DIII -class topological and  s -wave superconductors via an embedded quantum dot. Due to the presence of two kinds of superconductors, three dot-superconductor coupling manners are considered, respectively. As a result, the Josephson current is found to oscillate in period 2 π . More importantly, the presence of Majorana doublet in the  DIII -class superconductor renders the current finite at the case of zero phase difference, with its sign determined by the fermion parity of such a junction. In addition, the dot-superconductor coupling plays a nontrivial role in adjusting the Josephson current. When the  s -wave superconductor couples to the dot in the weak limit, the current direction will have an opportunity to reverse. It is believed that these results will be helpful for understanding the transport properties of the  DIII -class superconductor.

Josephson effects in the junction formed by DIII-class topological and s-wave superconductors with an embedded quantum dot