Abstract
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.
Highlights
For describing the Josephson effect governed by Majorana doublets, the parameter order should be much smaller than the superconducting gap ΔTS, we assume the parameter unit to be 0.1ΔTS without loss of generality
It has been found that an embedded quantum dot (QD) in this junction plays a nontrivial role in modifying the Josephson currents, since the tunable fermion occupation in the QD re-regulates the fermion parity (FP) of the Majorana doublets for conserving the FP in whole system
To be concrete, when the Majorana doublets couple indirectly via a non-magnetic QD, the normal Josephson effects occur, and the FP change just leads to the reversal of current direction and the variation of current amplitude
Summary
The that in the system matrix form of T can be deduced on the basis of |nsnsn f n f 〉 with Majorana bound states, only FP is the good quantum number, so we should build the Fock state according to FP. Fock + b7 state 1101 can + b8 be writt 1110 and en the as Ψo matrix of= bT(1o)0t0ak01es the form as. For the extreme case of strong magnetic-field limit, if εsis in the finite-energy region, εs will be empty, and only one level contributes to the Josephson effects, respectively. In such a case, the matrixes of T(e) and T(o) will be halved, i.e., HT(e). In which EG(eS/o) are the ground-state (GS) energies in the even- and odd-FP cases, respectively
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