Ultra-small-capacitance Josephson junction: Inductive coupling to a voltage source

We consider a voltage-biased ultra-small-capacitance Josephson junction, with the coupling to the external source containing both resistive and inductive elements. In addition we include a phenomenological coupling to an external heat bath. Our goal is to extend and generalize previous studies of current-biased ultra-small junctions. Charging effects, due to the presence of discrete charge carriers in the junction, play a crucial role. In particular we find an ‘‘infinite-resistance'' branch in the I-V characteristic for a dc bias, and resistive steps in the I-V curve when the external bias contains an additional ac component. These effects are reminiscent of the ‘‘Coulomb blockade'' and the inverse Shapiro steps, respectively, predicted earlier in the context of current-biased circuits. As a response to an ac voltage bias, we also predict spikes of the voltage across the junction and a noisy background when this voltage is plotted as a function of either the external dc biasing voltage or the external frequency. Our analysis shows that various circuitry components may qualitatively affect the response of the junction to an external bias.