Nonabelian time evolution in a chaotic Majorana billiard

We discuss signatures of the nonabelian nature of Majorana bound states in a quantum-chaotic setting. We consider a "chaotic Majorana billiard", which consists of a chaotic cavity coupled to topological superconductors via point contacts. In the limit of vanishing transmission, each of these contacts hosts a Majorana zero mode. Close to a cavity resonance, a finite transparency of the contacts couples the Majorana modes, but a ground-state degeneracy per fermion parity subspace remains if the number of Majorana modes coupled to the cavity exceeds five. Upon varying shape-defining gate voltages while remaining close to resonance, a nontrivial evolution within the degenerate ground-state manifold can be achieved. We characterize the corresponding nonabelian Berry phase using random matrix theory and discuss a measurable signature of the nonabelian time evolution in terms of the cavity charge as well as differences between the cases of a cavity coupled to Majorana zero modes and to Andreev bound states.