Quantum dynamics of frustrated Josephson junction arrays embedded in a transmission line: An effective𝑋⁢𝑋spin chain with long-range interaction

We study theoretically a variety of collective quantum phases occurring in frustrated sawtooth chains of Josephson junctions embedded in a dissipationless transmission line. The basic element of a system, i.e., the triangular superconducting cell, contains two 0 and one 𝜋 Josephson junctions characterized by 𝐸𝐽 and 𝛼⁢𝐸𝐽 Josephson energies, accordingly. In the frustrated regime the low-energy quantum dynamics of a single cell is determined by anticlockwise or clockwise flowing persistent currents (vortex or antivortex). The direct embedding of 𝜋 Josephson junctions in a transmission line allows us to establish a short or long-range interaction between (anti)vortices of well-separated cells. By making use of the variational approach, we map the superconducting circuit Hamiltonian to an effective 𝑋⁢𝑋 spin model with an exchange spin-spin interaction decaying with the distance 𝑥 as 𝑥−𝛽, and the local ˆ𝜎𝑥,𝑛 terms corresponding to the coherent quantum beats between vortex and antivortex in a single cell. We obtain that, in long arrays such as 𝑁≫ℓ0≃√𝐶/𝐶0, where 𝐶 and 𝐶0 are capacitances of a 0 Josephson junction and a transmission line, respectively, the amplitude of quantum beats is strongly suppressed. By means of exact numerical diagonalization, we study the interplay between the coherent quantum beats and the exchange spin-spin interaction leading to the appearance of various collective quantum phases such as the paramagnetic (𝑃), compressible superfluid (𝐶⁢𝑆), and weakly compressible superfluid (𝑤−𝐶⁢𝑆) states.