  Hybrid Josephson junctions realized on a two-dimensional electron gas are
considered promising candidates for developing topological elements that are
easily controllable and scalable. Here, we theoretically study the possibility
of the detection of topological superconductivity via the non-local
spectroscopy technique. We show that the non-local conductance is related to
the system band structure, allowing probe of the gap closing and reopening
related to the topological transition. We demonstrate that the topological
transition induces a change in the sign of the non-local conductance at zero
energy due to the change in the quasiparticle character of the dispersion at
zero momentum. Importantly, we find that the tunability of the superconducting
phase difference via flux in hybrid Josephson junctions systems is strongly
influenced by the strength of the Zeeman interaction, which leads to
considerable modifications in the complete phase diagram that can be measured
under realistic experimental conditions.
