Floquet exponents of underdamped Josephson ladders: A comparison with predictions of the discrete sine-Gordon equation

Abstract

We calculate Floquet exponents for phase-locked solutions in ladder arrays of Josephson junctions in zero external field. We assume a resistively and capacitively shunted junction (RCSJ) model, and we allow for critical current anisotropy between the horizontal and vertical junctions. The ladders range in size from 5 to 30 plaquettes and are biased along the rungs with uniform dc bias currents. The Floquet exponents quantify the stability of the solutions and are calculated numerically for the RCSJ model as a function of junction capacitance (βc) as well as critical current anisotropy (Λ). We also model the array with the discrete sine-Gordon (DSG) equation, and we are able to calculate the exponents analytically in that case. We find the analytic results from the DSG equation agree quantitatively with the numerical results from the RCSJ model over a wide range of βc and Λ values and even agree qualitatively for βc→1 and Λ→0. Based on the analytic result we argue that perturbations in the array are damped by the small-angle phase oscillations of the underlying lattice (the “phonons” of the lattice), and like a classical harmonic oscillator with damping, each phonon mode has a crossover (as a function of decreasing βc or Λ) from underdamped to overdamped dynamics. Such crossover behavior is clearly visible in the results for the Floquet exponents and is manifested as a maximum in the Floquet exponent as a function of the junction capacitance. This intriguing result speaks to the opportunity, in principle, of tuning the capacitance such as to optimize the stability of the phase-locked solutions.

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