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 RSJ model. We consider a junction that hosts two supercurrrent contributions: a $2\pi$- and a $4\pi$-periodic in phase, with intensities $I_{2\pi}$ and $I_{4\pi}$ respectively. We study the size of the Shapiro steps as a function of the ratio of the intensity of the mentioned contributions, i.e. $I_{4\pi}/I_{2\pi}$. We provide detailed explanations where to expect clear signatures of the presence of the $4\pi$-periodic contribution as a function of the external parameters: the intensity AC-bias $I_\text{ac}$ and frequency $\omega_\text{ac}$. On the one hand, in the low AC-intensity regime (where $I_\text{ac}$ is much smaller than the critical current, $I_\text{c}$), we find that the non-linear dynamics of the junction allows the observation of only even Shapiro steps even in the unfavorable situation where $I_{4\pi}/I_{2\pi}\ll 1$. On the other hand, in the opposite limit ($I_\text{ac}\gg I_\text{c}$), even and odd Shapiro steps are present. Nevertheless, even in this regime, we find signatures of the $4\pi$-supercurrent in the beating pattern of the even step sizes as a function of $I_\text{ac}$.