Bandwidth-Limited Nonreciprocal Transport and Directional Gain from Synthetic Flux in Josephson Microwave Networks

We formulate a quantitative and falsifiable framework for nonreciprocal microwave transportin Josephson/parametric networks engineered by synthetic gauge flux. We consider a minimalthree-mode loop (two primary port-coupled modes and one auxiliary mediator) with pump-inducedfrequency-conversion couplings carrying programmable phases. In the passive conversion regime, wederive closed-form expressions for the forward and reverse scattering amplitudes S21(ω) and S12(ω)via a Schur-complement reduction, yielding an exact directionality condition S12(0) = 0 obtained bydestructive interference between direct and indirect coupling paths. We then derive a bandwidth-limited asymmetry estimate controlled by the auxiliary susceptibility scale Ωc = p(κc/2)2 + ∆2c ,leading to ABW ≈ 20 log10(Ωc/ωmax) for half-bandwidth ωmax. For a concrete design target—asymmetry X = 6 dB across 33 MHz with linewidths κ/2π ≃ 1 GHz and internal loss fractionκi/κ ≃ 1.5%—the theory predicts that dispersion alone permits ∼ 30 dB asymmetry across theband, implying that any observed ceiling near 6 dB is diagnostic of a large reciprocal leakage channelor parasitic pump-induced mixing. Finally, we extend the loop architecture to a directional amplifierby replacing one conversion edge with a two-mode squeezing (gain) interaction and presenting theBogoliubov (Nambu) input–output formalism, stability threshold, and a phase-tuned condition forcanceling reverse gain. The resulting paper is an audit-grade map from pump phases and linewidthsto measurable nonreciprocity and a clear falsification protocol.