Probing Fractional Josephson Junction with a Quantum Dot

We consider theoretically a junction between two topological superconducting wires, mediated by a quantum dot. The wires are modelled by the Kitaev chains tuned into topological phase, which possess unpaired Majorana states at their ends. We derive the low energy Hamiltonian of the model. The Majorana states closer to the dot convert into the Dirac fermion inside the dot, forming fractional Josephson junction. The dot is additionally weakly coupled to the normal tunneling probe allowing transport measurement through the dot. When the topological wires are short, the unpaired Majorana end-states can hybridize inside the wire forming an extended Dirac fermionic state. It yields the destruction of the extended state in the dot. We discuss the dependence of the spectral density of the dot and its conductance on superconducting phase. We show that the conservation of parity of the junction, crucial for successful measurement of the fractional effect, can be assured by the gate voltage manipulation of the dot level position and that in case of an unpaired Majorana state in the junction a half conductance quantum can be observed.