Dual Shapiro steps and fundamental transconductance in dc driven Bloch transistor

We propose a superconducting circuit based on the Bloch transistor, a quantum device consisting of two small-capacitance Josephson junctions connected in series and having a small island in between. This device is driven by two dc electrical sources controlling Josephson oscillations of frequency $f_J=2e\overline{V_J}\mathrm{/}h$, related to the average voltage $\overline{V_J}$ on the transistor, and Bloch oscillations of frequency $f_B=\overline{I_B}\mathrm{/}2e$, related to the average current $\overline{I_B}$ injected into the transistor island. Due to the Bloch transistor properties, these two types of oscillations can mutually phase lock, i.e., $f_J=f_B$. This leads to formation of current steps on the current-voltage curve at ${\bar{I}}_B=2e{f}_J$, which are similar to the dual Shapiro steps appearing at current $\bar{I}=2ef$ under microwave irradiation of frequency $f$. Moreover, transconductance $\overline{I_B}\mathrm{/}\overline{V_J}$ takes the fundamental value of $1\mathrm{/}{R}_Q$, where $R_Q=h\mathrm{/}4e^2$ is the resistance quantum. The obtained results pave the way to the alternative quantum standard of resistance, based on the superconducting circuit and operating without applying strong magnetic field.