Devil's Staircases and Continued Fractions in the Josephson Junctions

TL;DR

This paper investigates the appearance of devil's staircases and continued fractions in the IV-characteristics of Josephson junctions under electromagnetic radiation, revealing precise mathematical structures and their relation to experimental observations.

Contribution

It introduces a numerical simulation approach showing that Shapiro steps form continued fractions and proposes an algorithm for detecting subharmonics as radiation amplitude increases.

Findings

Devil's staircases correspond to continued fractions.

Radiation amplitude shifts staircases to higher steps.

Experimental subharmonics also form continued fractions.

Abstract

The detailed numerical simulations of the IV-characteristics of Josephson junction under external electromagnetic radiation show devil's staircases within different bias current intervals. We have found that the observed steps form very precisely continued fractions. Increasing of the amplitude of radiation shifts the devil's staircases to higher Shapiro steps. The algorithm of appearing and detection of the subharmonics with increasing radiation amplitude is proposed. We demonstrate that subharmonic steps registered in the famous experiments by A. H. Dayem and J. J. Wiegand [Phys. Rev 155, 419 (1967)] and J. Clarke [Phys. Rev. B 4, 2963 (1971)] also form continued fractions.