The dc Josephson current in a long multi-channel quantum wire

The dc Josephson current across a multi-channel SNS junction is computed by summing contributions from sub-gap Andreev bound states, as well as from continuum states propagating within the superconducting leads. We show that, in a long multi-channel SNS-junction, at low temperatures, all these contributions add up, so that the current can be entirely expressed in terms of single-particle normal- and Andreev reflection amplitudes at the Fermi level at both SN interfaces. Our derivation applies to a generic number of channels in the normal region and/or in the superconducting leads, without assumptions about scattering processes at the SN interfaces: if the channels within the central region have the same dispersion relation, it leads to simple analytical formulas for the current at low temperatures; if the channels within the central region have different dispersion relations, it allows for expressing the current in terms of a simple integral involving only scattering amplitudes at the Fermi level. Our result motivates using a low energy effective boundary Hamiltonian formalism for computing the current, which is crucial for treating Luttinger liquid interaction effects.