Conventional and dual Shapiro steps in small Josephson junctions

We propose a model describing the formation of both conventional and dual Shapiro steps in small Josephson junctions. According to our model, dual Shapiro steps occur at relatively low frequency of the microwave signal and low microwave power, whereas conventional steps occur in the opposite limit of high frequency and power. The crossover between these two regimes is controlled by a single parameter—the effective relaxation time of the environment. Our model takes into account the effect of a large inductor in the bias circuit, which has been used in recent experiments to protect the junction from the high-frequency noise of the environment. We predict the possibility of observing both types of steps in the same sample. Our model describes the - curves observed in the experiments with reasonable accuracy, thus opening the possibility of quantitatively fitting experimental data.