Josephson junction dynamics in the presence of 2π - And 4π -periodic supercurrents

We investigate theoretically the dynamics of a Josephson junction in the framework of the resistively shunted junction model. We consider a junction that hosts two supercurrent contributions: a 2π and a 4π periodic in phase, with intensities I2π and I4π, respectively. We study the size of the Shapiro steps as a function of the ratio of the intensity of the mentioned contributions, i.e., I4π/I2π. We provide detailed explanations where to expect clear signatures of the presence of the 4π-periodic contribution as a function of the external parameters: the intensity ac bias Iac and frequency ωac. On the one hand, in the low ac-intensity regime (where Iac is much smaller than the critical current Ic), we find that the nonlinear dynamics of the junction allows the observation of only even Shapiro steps even in the unfavorable situation where I4π/I2π1. On the other hand, in the opposite limit (IacIc), even and odd Shapiro steps are present. Nevertheless, even in this regime, we find signatures of the 4π supercurrent in the beating pattern of the even step sizes as a function of Iac.