Topological transconductance quantization in a four-terminal Josephson junction

Recently we predicted that the Andreev bound state spectrum of 4-terminal Josephson junctions may possess topologically protected zero-energy Weyl singularities, which manifest themselves in a quantized transconductance in units of $4e^2/h$ when two of the terminals are voltage biased [R.-P. Riwar et al., Nature Commun. 7, 11167 (2016)]. Here, using the Landauer-B\"uttiker scattering theory, we compute numerically the currents flowing through such a structure in order to assess the conditions for observing this effect. We show that the voltage below which the transconductance becomes quantized is determined by the interplay of non-adiabatic transitions between Andreev bound states and inelastic relaxation processes. We demonstrate that the topological quantization of the transconductance can be observed at voltages of the order of $10^{-2} \Delta/e$, $\Delta$ being the superconducting gap in the leads.