Current-Phase Relation of a Bose-Einstein Condensate Flowing Through a Weak Link

We study the current-phase relation of a Bose-Einstein condensate flowing through a repulsive square barrier by solving analytically the one dimensional Gross-Pitaevskii equation. The barrier height and width fix the current-phase relation $j(\delta\phi)$, which tends to $j\sim\cos(\delta\phi/2)$ for weak barriers and to the Josephson sinusoidal relation $j\sim\sin(\delta\phi)$ for strong barriers. Between these two limits, the current-phase relation depends on the barrier width. In particular, for wide enough barriers, we observe two families of multivalued current-phase relations. Diagrams belonging to the first family, already known in the literature, can have two different positive values of the current at the same phase difference. The second family, new to our knowledge, can instead allow for three different positive currents still corresponding to the same phase difference. Finally, we show that the multivalued behavior arises from the competition between hydrodynamic and nonlinear-dispersive components of the flow, the latter due to the presence of a soliton inside the barrier region.