Tunneling Hamiltonian analysis of DC Josephson currents in a weakly-interacting Bose-Einstein condensate

Atomtronics experiments with ultracold atomic gases allow us to explore quantum transport phenomena of a weakly-interacting Bose-Einstein condensate (BEC). Here, we focus on two-terminal transport of such a BEC in the vicinity of zero temperature. By using the tunnel Hamiltonian and Bogoliubov theory, we obtain a DC Josephson current expression in the BEC and apply it to experimentally relevant situations such as quantum point contact and planar junction. Due to the absence of Andreev bound states but the presence of couplings associated with condensation elements, a current-phase relation in the BEC is found to be different from one in an s-wave superconductor. In addition, it turns out that the DC Josephson current in the BEC depends on the sign of tunneling elements, which allows to realize the so-called $\pi$ junction by using techniques of artificial gauge fields.