Fig. 2: Superfluid phase and critical current in a JJN.

a Phase gain δϕ across each junction as a function of n, Eq. (4), where \(f(\tilde{w},n)\) and ρ_bulk are obtained from GPE calculations in a 1D JJN. Symbols refer to \(\tilde{w}=1.44\) (downward triangles), \(\tilde{w}=2.15\) (squares) and \(\tilde{w}=3\) (upward triangles). These correspond to the maximum values of \(\tilde{w}\), for n = 1, 3 and 5, respectively, for which a stable solution of the GPE can be found. For larger values of \(\tilde{w}\), the system is unstable, with the nucleation of solitons being observed in dynamical GPE simulation. Lines are guides to the eye. In particular, the solid black line connects maxima of δϕ obtained for different w, separating the stable (blue) from the unstable (orange) region. The inset shows the superfluid phase ϕ as a function of the angle θ along the ring, for n = 1 (dotted green line) and n = 6 (solid blue line). b Critical current as a function of the number n of junctions. The analytic formula, Eq. (5) (large black dots), reproduces the numerical calculation of the maximum current \({\tilde{J}}_{c}\). Small white dots show the current \(\tilde{J}\) calculated for Ω = 0 and different values of w, ranging from w = 1 (bottom) to w = 8 (top). Solid and dotted lines are guides to the eye. The orange region corresponds to values of the current above \({\tilde{J}}_{c}\) and are thus inaccessible in the system. Inset: δϕ as a function of \(\tilde{J}\) calculated for the stationary states of the 1D GPE, for n = 1 (green squares) and n = 6 (blue circles). The dotted lines are the current-phase relations \(\delta \phi={\sin }^{-1}(\tilde{J}/{\tilde{J}}_{c})+2\pi L \tilde{J}\)⁷⁶ without free fitting parameters: the kinetic inductance L is calculated from the relation \(L=(\delta {\phi }_{c}-\pi /2)/(2\pi {\tilde{J}}_{c})\), \({\tilde{J}}_{c}\) is the numerical maximum current and δϕ_c is the corresponding value of the phase gain.