Figure 1

(a) Predicted $I-V$ characteristics of an unshunted Josephson junction. In that simple system, evaluating the dc mobility $\mu ={\lim }_{V\to 0^{+}}(I/V)=0$ erroneously suggests the junction is insulating, because it completely misses the ac Josephson supercurrent and the dc supercurrent branch, the most remarkable features of that system. (b) Sketch of the top right quadrant of the $I-V$ characteristic of a RSJJ, close to the origin (for $R<R_Q$, see Ref. [47] in our article and, for $R>R_Q$, Fig. S15 in Supplemental Material in Ref. [29] of our article). In contrast to the dissipationless unshunted case (a), one can now have a finite dc current at finite voltage, because the resistor can absorb the corresponding power. The total dc current has a contribution from the time evolution of the supercurrent not averaging to zero in the steady state, leading to a ground state inductive contribution $\propto E_J⟨\cos \phi ⟩$ at zero bias (green area) and a contribution arising from the dissipative Kubo term proportional to $E_J^2$ (orange area). Most theory papers on the SB transition evaluate the dc mobility which captures the orange contribution only, overlooking [very much as discussed in (a)] the junction’s dynamical inductive (superconducting) response which dominates transport properties at equilibrium and close to it. Such dc mobility calculations indeed predict a transition, with a low-voltage suppression of the orange contribution at $R>R_Q$, as shown in the inset [10]. However, the green contribution has no dissipative transition, because the equilibrium inductive response $\propto E_J⟨\cos \phi ⟩$ remains finite for all values of the resistance (although it is reduced at large $R$), so that the RSJJ is always superconducting. Equilibrium transport properties are accurately characterized by dc mobility calculations only in open normal-state systems where one is certain to never have any supercurrent.