Dynamical Reduction of Two Series Josephson Junctions to a Synthetic High-Transparency Josephson Element

Abstract

Two conventional Josephson junctions connected in series can reproduce, in the static limit in which the currents through the capacitive and resistive channels are negligible, the current-phase relation of a single effective weak link with tunable transparency. Therefore, the two-junction series can be treated as a single synthetic high-transparency element. Here, we investigate to what extent this mapping remains valid under finite-frequency drive and retaining the junctions' resistive and capacitive terms. The full resistively and capacitively shunted junction equations are compared with an effective synthetic element with tunable transparency that retains the synthetic tunable-transparency current-phase relation together with effective capacitive and dissipative terms, thus reducing the two second order degree of freedom system to a single second order degree of freedom. The resulting single-element dynamics is compared with the complete two-junction system under ac excitation. The agreement is quantified through a normalized root-mean-square error between the full and effective voltage waveforms. A broad low-error region is found at low drive frequency, while pronounced deviations emerge as the drive frequency approaches the relevant plasma-frequency scale and at larger drive amplitudes. The results provide a quantitative dynamical criterion for using the reduced single-element description of a synthetic high-transparency Josephson element in superconducting circuits.