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

TL;DR

This paper theoretically investigates the quantum phases of frustrated Josephson junction arrays embedded in a transmission line, mapping the system to an effective long-range XX spin chain and analyzing collective quantum states.

Contribution

It introduces a variational approach to map the superconducting circuit to an effective XX spin model with long-range interactions, revealing new collective quantum phases.

Findings

Quantum beats are suppressed in long arrays.

Various collective quantum phases such as paramagnetic and superfluid states are identified.

Long-range interactions influence the emergence of different quantum states.

Abstract

We study theoretically a variety of collective quantum phases occurring in frustrated saw-tooth 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 $\pi$- Josephson junctions characterized by $E_J$ and $\alpha E_J$ 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/antivortex). The direct embedding of $\pi$-Josephson junctions in a transmission line allows to establish a short/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 $x^{-\beta}$, and the local $\hat \sigma_{x,n}$-terms corresponding to the coherent quantum beats between vortex and antivortex in a single cell. We obtain that in long arrays as $N \gg \ell_0 \simeq \sqrt{C/C_0}$, where $C$ and $C_0$ are capacitances of $0$-Josephson junction and transmission line, accordingly, 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.