Channel-Resolved Perturbative Analysis of Parameter-Spread Effects in rf-SQUID Traveling-Wave Parametric Amplifiers

Abstract Josephson traveling-wave parametric amplifiers (JTWPAs) are well suited to broadband amplification in superconducting quantum readout, but their long nonlinear transmission lines are sensitive to fabrication-induced parameter spread. We develop a channel-resolved perturbative analysis for three-wave-mixing JTWPAs based on radio-frequency superconducting quantum interference devices (rf-SQUIDs), separating the effects of parameter variations into linear scattering, phase mismatch, and direct change in the quadratic coupling. A physical-cell Bloch projection resolves the local phase and scattering terms, while the rf-SQUID expansion gives the direct coupling variation. Their effects are quantified through average gain, gain ripple, and passive reflection. For the periodically loaded rf-SQUID JTWPA studied here, critical-current spread mainly affects the average gain through the nonlinear coupling, whereas loop-inductance and shunt-capacitance spreads contribute more strongly to ripple and reflection through linear scattering. Their relative importance also changes with the spatial correlation of the parameter variations. These results show that fabrication tolerances should be considered together with the relevant amplifier metric, clarifying how parameter variations influence the broadband gain response and reflection through distinct physical channels.