Resonant suppression of the $π$-type Josephson component in a Kitaev-ladder junction

Abstract

We study how discrete normal-segment levels control equilibrium Josephson transport between two Kitaev-ladder superconductors. Away from resonance, detuning opens a selective-clamping window: inter-leg hybridization removes the phase dependence of a $0$-favoring low-energy ground-state contribution while retaining a competing $π$-favoring contribution, allowing a smooth negative first harmonic even for bulk-topologically trivial leads. At resonance and $ϕ=π$, an exact collective-clamping identity makes both low-energy ground-state contributions phase independent above a threshold, although individual quasiparticle levels can remain dispersive. Finite-energy quasiparticles then produce a positive residual first harmonic in a controlled sweet-point regime. For fixed $0<ϕ<π$, helicity mixing instead yields a positive resonant low-energy harmonic in the weak-coupling limit. Exact identities, fourth-order sign rules, and recursive Green-function calculations establish how resonant alignment changes the spectral contributions governing Josephson phase selection.