Signatures of Majorana assisted tunneling in 1D Josephson junction rings.

Majorana zero modes in condensed matter settings have drawn immense attention recently and the search for unequivocal signatures of Majorana modes is underway. In this respect the presence of 4$\pi$ periodic tunneling between topological superconductors, which is assisted by the Majorana modes is a promising candidate. In this work, we study how the inclusion of Majorana assisted tunneling modifies physics in a ring of Josephson junction coupled islands. In addition, we study parity breaking processes in a controlled manner by coupling the ring to a two-level electronic system, such as found in quantum dots. For the ring alone, we find that the effects of the 2$\pi$ periodic terms are suppressed by increasing the number of islands in the ring. This allows a clearer distinction of the 4$\pi$ periodicity. In the combined system, we find that the periodicity can be tuned by controlling the occupation energy of the two level system. We also discuss the signatures of Majoranas that remain after parity breaking effects restore the $2\pi$ periodicity.