Superconductor/normal-metal/superconductor junction of topological superconductors revisited: Fractional Josephson current, fermion parity, and oscillating wavefunctions

Abstract: The fractional Josephson effect is known to be a characteristic phenomenon of topological Josephson junctions hosting Majorana zero modes (MZMs), where the Josephson current has a $4\pi$ (rather than a $2\pi$) periodicity in the phase difference between the two topological superconductors. We introduce a one-dimensional model of a topological superconductor/normal-metal/superconductor (SNS) junction with the normal-metal (N) region of finite length, which is intermediate regime between the short- and long-junction limits. Assuming weak tunneling at the SN interfaces, we investigate resonance and finite-size effects on the fractional Josephson effect due to the existence of several discrete energy levels in the N region in which wavefunctions have oscillating nodal structure. Through careful analysis of the sign change in the transmission amplitudes through the junction and the fermion parity of the two MZMs, we find that the fractional Josephson current is proportional to the parity of total fermion numbers including both filled normal levels and two MZMs. Furthermore, we elucidate drastic enhancement of the Josephson current due to the resonance between a discrete level in the N region and MZMs.