On-chip ac driving for dual Shapiro steps

A single Josephson junction in the phase-slip regime exhibits Bloch oscillations in the voltage when biased with a dc current ${I}_{\text{dc}}$. The frequency of the oscillation is given by $\ensuremath{\pi}{I}_{\text{dc}}/e$, with $e$ the elementary charge, linking the current to the frequency via fundamental constants of nature. If an additional ac drive is applied, the Bloch oscillations may synchronize with the external drive. This leads to the emergence of dual Shapiro steps at fixed current in the $IV$ characteristics of the device. For applications as a current standard, frequencies of the order of 10 GHz are required. These are challenging to implement experimentally without detrimental effects due to stray capacitances. Here, we propose to employ an additional Josephson junction with a dc voltage bias as an on-chip ac source due to the ac Josephson effect. We study the back action of the Bloch oscillations on the Josephson oscillations and identify a parameter regime in which it is minimized. Furthermore, we find that the back action can even be utilized to further enhance the driving signal, which can lead to increased widths of the resulting dual Shapiro steps. Finally, we show dual Shapiro steps for a set of realistic experimental parameters at finite temperatures.