Superconductivity favors the pairing of electrons with opposite spin, but spin-orbit coupling can induce a transition to a ground state with one unpaired electron or hole. This socalled 'fermionparity switch' is the prototypical example of a topological phase transition, it is a signature of Majorana fermions, and it provides a way to store and manipulate quantum information.

We have recently discovered a correlation of subsequent switches in the phase-dependent fermion parity of a Josephson junction, analogous to the level repulsion of random matrices. The aim of the project is to develop this unexpected analogy into a theory of fermion parity switches, to identify methods to observe these experimentally, and to explore the implications for quantum information processing.

We will exploit recent advances in the statistical theory of large non-Hermitian random matrices, in particular concerning the appearance of a level spacing statistics that is a hybrid of the Wigner and Poisson statistics of Hermitian matrices. This socalled 'mermaid' statistics* is associated with a fractal structure of wave functions, and we will investigate whether this applies to the Majorana fermions in the superconducting analogue.

• majorana
• quantum physics
• superconductivity

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