Phases of Josephson junction ladders.

We study a Josephson junction ladder in a magnetic field in the absence of charging effects via a transfer matrix formalism. The eigenvalues of the transfer matrix are found numerically, giving a determination of the different phases of the ladder. The spatial periodicity of the ground state exhibits a devil’s staircase as a function of the magnetic flux filling factor f . If the transverse Josephson coupling is varied a continuous superconducting-normal transition in the transverse direction is observed, analogous to the breakdown of the KAM trajectories in dynamical systems. 64.70.Rh, 05.20.-y, 74.50.+r Typeset using REVTEX 1 Two-dimensional arrays of Josephson junctions have attracted much recent theoretical and experimental attention [1]. Interesting physics arises as a result of competing vortexvortex and vortex-lattice interactions. It is also considered to be a convenient experimental realization of the frustrated XY models. In this paper, we discuss the simplest such system, namely the Josephson junction ladder (JJL) [2–4] shown in the inset of Fig. 1. To construct the system, superconducting elements are placed at the ladder sites. Below the bulk superconducting-normal transition temperature, the state of each element is described by its charge and the phase of the superconducting wave function [5]. In this paper we neglect charging effects, which corresponds to the condition that 4e/C ≪ J , with C being the capacitance of the element and J the Josephson coupling. Let θj (θ ′ j) denote the phase on the upper (lower) branch of the ladder at the j’th rung. The Hamiltonian for the array [6] can be written in terms the gauge invariant phase differences, γj = θj−θj−1−(2π/φ0) ∫ j j−1Axdx, γ j = θ ′ j − θ ′ j−1 − (2π/φ0) ∫ j j′−1Axdx, and αj = θ ′ j − θj − (2π/φ0) ∫ j j Aydx: