Calculating Tunneling Matrix Elements to Majorana Zero Modes Using the Barrier-Integrated Mahaux-Weidenmüller Formula

Hybrid superconductor-semiconductor nanowires are considered a prime candidate for hosting Majorana zero modes. For their creation, readout and manipulation, one relies on tunable tunnel gates. More specifically, these gates govern the coupling between a nanowire and another system, including quantum dots, leads or another nanowire. Constructing a fault-tolerant topological quantum computer relies on having fine-tuned tunnel couplings between all the sub-systems. These couplings are probed in transport, by controlling the conductance of an island using tunnel gates. We employ the Mahaux-Weidenmüller formula to calculate the scattering matrix for an island connected to leads. Splitting off the tunnel junctions, one retrieves the tunneling matrix elements between the leads and system. Hence, we utilize this Barrier-Integrated Mahaux-Weidenmüller formula to estimate tunnel couplings to Majorana zero modes in a range of geometries, including realistic devices.