Quantum dynamics of frustrated Josephson junction arrays embedded in a transmission line: An effective XX spin chain with long-range interaction

Abstract

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 αEJ 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 XX spin model with an exchange spin-spin interaction decaying with the distance x as x−β, and the local σ̂x,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 (CS), and weakly compressible superfluid (w−CS) states.

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