Josephson Junction Dynamics In The Presence Of 2 Pi- And 4 Pi-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 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-ac and frequency omega(ac). On the one hand, in the low ac-intensity regime (where I-ac is much smaller than the critical current I-c), we find that the nonlinear dynamics of the junction allows the observation of only even Shapiro steps even in the unfavorable situation where I-4 pi/I-2 pi << 1. On the other hand, in the opposite limit (I-ac >> I-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-ac.