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One Majorana doublet can be realized at each end of the time-reversal-invariant Majorana nanowires. We investigate the Josephson effect in the Majorana-doublet-presented junction modified by different inter-doublet coupling manners. It is found that when the Majorana doublets couple indirectly via a non-magnetic quantum dot, only the normal Josephson effect occurs, and the fermion parity in the system just affects the current direction and amplitude. However, one magnetic field applied on the dot can induce the fractional Josephson effect in the odd-parity case. Next if the direct and indirect couplings between the Majorana doublets coexist, no fractional Josephson effect takes place, regardless of the presence of magnetic field. Instead, there almost appears the  π -period-like current in some special cases. All the results are clarified by analyzing the influence of the fermion occupation in the quantum dot on the parity conservation in the whole system. We ascertain that this work will be helpful for describing the dot-assisted Josephson effect between the Majorana doublets.

Influence of an embedded quantum dot on the Josephson effect in the topological superconducting junction with Majorana doublets