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 $j(\delta \varphi )$, which tends to $j~\cos (\delta \varphi /2)$ for weak barriers and to the Josephson sinusoidal relation $j~\sin (\delta \varphi )$ 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.

Figure 1

Typical solutions for a barrier with width $2d=4{\xi }_{\infty }$, $V_0=0.4{\mathit{gn}}_{\infty }$, and $j=0.3c_{\infty }{n}_{\infty }$. Density (upper panel) and phase (lower panel) as a function of position are shown for both the upper and the lower solution, corresponding to black and green (light gray) solid lines, respectively. Dashed lines in the lower panel correspond to the phase accumulated by a plane wave in absence of barrier.

Figure 2

Current-phase relation for a barrier with half width $d=0.5{\xi }_{\infty }$ (left panel) and $d=3{\xi }_{\infty }$ (right panel) for different barrier heights $V_0$ (see insets). The arrows sketch the behavior of the maximum of the curves on increasing $V_0$.

Figure 3

Current-phase relation in the reentrant regime $d=20{\xi }_{\infty }$ for different barrier heights $V_0$ (see uppermost inset). Middle and lowermost insets show in more detail the shape of the diagram close to the critical point. The arrows sketch the behavior of the maximum of the curves on increasing $V_0$.

Figure 4

Typical density profiles in the reentrant regime $d=20{\xi }_{\infty }$ for the upper (upper panel) and lower (lower panel) solutions. Here $V_0=0.1{\mathit{gn}}_{\infty }$, and $j=0.5c_{\infty }{n}_{\infty }$.

Figure 5

Comparison between the LDA + soliton model (black solid lines) and the exact results (see inset).

Figure 6

Comparison between the Deaver-Pierce model (black solid lines) and the exact results (see inset).