Resonant frequencies and spatial correlations in frustrated arrays of Josephson type nonlinear oscillators

Abstract

We present a theoretical study of resonant frequencies and spatial correlations of Josephson phases in frustrated arrays of Josephson junctions. Two types of one-dimensional arrays, namely, the diamond and sawtooth chains, are discussed. For these arrays in the linear regime the Josephson phase dynamics is characterized by multiband dispersion relation $\omega (k)$, and the lowest band becomes completely $flat$ at a critical value of frustration, $f=f_c$ . In a strongly nonlinear regime such critical value of frustration determines the crossover from non-frustrated ($0<f<f_c$) to frustrated ($f_c<f<1$) regimes. The crossover is characterized by the thermodynamic spatial correlation functions of phases on vertices, ${\phi }_i$, i.e. $C_p(i-j)=⟨\cos [p({\phi }_i-{\phi }_j)]⟩$ displaying the transition from long- to short-range spatial correlations. We find that higher-order correlations functions, e.g. $p=2$ and $p=3$, restore the long-range behavior deeply in the frustrated regime, $f\simeq 1$. Monte-Carlo simulations of the thermodynamics of frustrated arrays of Josephson junctions are in good agreement with analytical results.

9 pages, 10 figures

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