Fig. 4. Stability phase diagram of an atomtronic JJN.

a Mean circulation as a function of time, for w₀ = 2 and different number of barriers, n (symbols), with averages and error bars obtained from 15 repeated measurements for each point. The dashed lines are exponential fits, $\left.⟨\right)w\left(t\right)⟩=w_f+\mathrm{\Delta }w\exp \left(-\mathrm{\Gamma }t\right)$. b Effective decay rate $\tilde{\mathrm{\Gamma }}\propto \mathrm{\Delta }w\mathrm{\Gamma }$ (colormap), extracted from the exponential fits as in (a) as a function of w₀ and n. $\tilde{\mathrm{\Gamma }}$ quantifies the stability of an initial finite-circulation state w₀. The dashed white line is the critical circulation w_c(n) as a function of n, obtained from 3D GPE simulations. The right axis shows the rescaled current $\tilde{J}$ of w₀ states in the clean ring. c Upper (dashed red line) and lower (dashed blue line) bounds to the superfluid fraction f_s, Eq. (6), as a function of the number of junctions. Bounds are obtained from the ground state density of the numerical 3D GPE. The solid lines are the bounds evaluated by including the finite resolution of the experimental imaging system. Circles are the upper bound evaluated using experimental in-situ images and averaged over 10 realizations.