Resonances control when multiterminal Josephson currents reduce to two-terminal couplings
Multiterminal Josephson junctions, in which one weak link couples three or more superconductors, are studied as hosts of topological Andreev bands, multi-pair supercurrents and tunable circuit elements. Their currents are modeled both as pairwise networks of two-terminal couplings and through genuinely multiterminal processes such as quartets. We ask how accurate the pairwise description is and what controls its error. In a microscopic scattering model of planar junctions, we compare exact currents with the best pairwise approximation, which allows arbitrary nonsinusoidal couplings. In ensembles of disordered three- and four-terminal junctions, the pairwise terms capture nearly all of the energy variation, yet the median current error ranges from 8.0 to 21.2 percent, about twice the energy error, because currents weigh the multiterminal harmonics more strongly. Designed devices extend the comparison to sixteen terminals. We identify a mechanism that controls this error. A normal-region mode near the Fermi level that couples to three or more terminals produces large nonpairwise currents, and detuning it with a gate suppresses them. Shifting only this mode, selected from normal-state properties, predicts the gate dependence of the error in eight three- and four-terminal devices without fitting currents. Finite-gap calculations in two clean devices confirm a drop from tens of percent near resonance to below one percent. An analytic single-level model gives a sufficient detuning for pairwise accuracy at any terminal count. Planar junctions thus inherit the resonant and cotunneling regimes known from quantum dots, and normal-state properties identify the mode that selects between them.