Stability analysis of Shapiro steps in Josephson-junction arrays

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Abstract
The coupled set of evolution equations of the individual resistively shunted junctions in an array is written in a form that directly reveals the existence of a single-junction solution (SJS) for specific values of the external magnetic field. We formulate a condition that determines these field strengths in terms of the corresponding ground-state configurations. By studying the linearized evolution equations, around this coherent-phase solution, we show that the array is phase locked to the external source when the SJS is stable. Only in this case does it have relevance in practice. Our analysis clearly indicates that there are only regular Shapiro steps in the regions of stability, and in between these regions our simulations confirm the absence of fractional steps, reported earlier. Furthermore, simulations, also in fields with no SJS, demonstrate a degree of insensitivity to the field strength, which is only partly understood.