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 $\ensuremath{\pi}$ Josephson junctions characterized by
EJ and $\ensuremath{\alpha}{E}_{J}$ 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 or antivortex). The direct embedding of $\ensuremath{\pi}$ 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 ${x}^{\ensuremath{-}\ensuremath{\beta}}$, and the local ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\sigma}}}_{x,n}$ terms corresponding to the coherent quantum beats between vortex and antivortex in a single cell. We obtain that, in long arrays such as $N\ensuremath{\gg}{\ensuremath{\ell}}_{0}\ensuremath{\simeq}\sqrt{C/{C}_{0}}$, 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 ($w\text{\ensuremath{-}}CS$) states.
Abstract We designed, fabricated and measured short one-dimensional arrays of masked ion-irradiated YBa 2 Cu 3 O 7 Josephson junctions embedded into log-periodic spiral antennas. They consist of four or eight junctions separated either by 960 nm or 80 nm long areas of pristine material. Large spacing arrays show ‘giant’ Shapiro steps in the hundreds-GHz band at 66 K and are tested as Josephson mixers with improved impedance matching. On the contrary, small spacing arrays behave as one junction with a lower superconducting transition temperature, hence forming a single weak link on distances up to 880 nm. Such design opens a new way to increase the I c R n product of the devices, and therefore the efficiency of Josephson mixers. Hints on the origin of the observed long range proximity effect are proposed.