Effect of a tunnel barrier on the scattering from a Majorana bound state in an Andreev billiard

We calculate the joint distribution P(S,Q) of the scattering matrix S and time-delay matrix Q=−iℏS†dS/dE of a chaotic quantum dot coupled by point contacts to metal electrodes. While S and Q are statistically independent for ballistic coupling, they become correlated for tunnel coupling. We relate the ensemble averages of Q and S and thereby obtain the average density of states at the Fermi level. We apply this to a calculation of the effect of a tunnel barrier on the Majorana resonance in a topological superconductor. We find that the presence of a Majorana bound state is hidden in the density of states and in the thermal conductance if even a single scattering channel has unit tunnel probability. The electrical conductance remains sensitive to the appearance of a Majorana bound state, and we calculate the variation of the average conductance through a topological phase transition