Negative critical current in superconductor-1D metal-superconductor Josephson Junctions

We study the Joshepson effect in superconductor-1D metal-superconductor junctions. In the presence of Coulomb interaction, the number of electrons in the normal metal part of the SNS junction is controlled by the voltage on the gate. We show that the sign of the critical current alternates as a function of the number of electrons in the 1D normal region. For an odd number of electrons the π-contact is realized. The alternation of the sign of the critical current can be traced to the node theorem for one-dimensional wave functions.