Josephson current in finite-lenght nanowire SNS junctions with Majorana fermions

The dc Josephson effect (JE) through infinite-lenght junctions of one-dimensional topological superconductors exhibits an anomalous $4\pi$ periodic phase ($\phi$) dependence which originates from a parity-protected level crossing of zero-energy Majorana bound states (MBS) at $\phi=\pi$. This ``fractional'' JE provides an important experimental detection tool for MBS. In this talk, I will discuss the JE in more realistic SNS junctions of arbitrary transparency and when both the normal and the nanowire regions are of finite length, namely beyond the short-junction and infinite topological superconductor limits. In general, the spectrum of Andreev bound states can become rather intricate and dense as opposed to the infinite-lenght case. Moreover, the low-energy spectrum around $\phi=\pi$ shows always anticrossings, originated from hybridization of four MBS, which may preclude the experimental observation of the fractional JE. At finite bias voltages, Landau-Zener dynamics involving the MBS and quasi-continuum Andreev levels gives rise to a nontrivial ac Josephson current. Interestingly, the ac current phase diagram as a function of the Josephson frequency/normal transmission shows fractional JE regions which are tunable through bias/gate voltages.