Anomalous periodicity of supercurrent in long frustrated Josephson-junction rhombi chains

We consider a chain of Josephson-junction rhombi (proposed originally in B. Dou {{c}} {c}{}ot and J. Vidal [Phys. Rev. Lett. 88, 227005 (2002)] in quantum regime, and in the realistic case when charging effects are determined by junction capacitances. In the maximally frustrated case when magnetic flux through each rhombi Φᵣ is equal to one half of superconductive flux quantum Φ₀, Josephson current is due to correlated transport of pairs of Cooper pairs, i.e., charge is quantized in units of $4e$. Sufficiently strong deviation δΦ≡Φᵣ-Φ₀∕2>δΦᶜ from the maximally frustrated point brings the system back to usual $2e$-quantized supercurrent. We present detailed analysis of Josephson current in the fluctuation-dominated regime (sufficiently long chains) as function of the chain length, EJ∕EC ratio and flux deviation δΦ. We provide estimates for the set of parameters optimized for the observation of $4e$-supercurrent.