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.

Frequency analysis of the rf emission of oscillating Josephson supercurrent is a powerful passive way of probing properties of topological Josephson junctions. In particular, measurements of the Josephson emission enable the detection of topological gapless Andreev bound states that give rise to emission at half the Josephson frequency f_j rather than conventional emission at f_j. Here, we report direct measurement of rf emission spectra on Josephson junctions made of HgTe-based gate-tunable topological weak links. The emission spectra exhibit a clear signal at half the Josephson frequency f_j=2. The linewidths of emission lines indicate a coherence time of 0.3-4 ns for the f_j=2 line, much shorter than for the f_j line (3-4 ns). These observations strongly point towards the presence of topological gapless Andreev bound states and pave the way for a future HgTe-based platform for topological quantum computation.