Signatures of Bloch oscillations in a transmon shunted by a high-impedance transmission line

A transmon qubit shunted by a high-impedance transmission line acts in a way dual to a conventional Josephson junction. In analogy to the AC Josephson effect, biasing of the transmon by a direct current (DC) leads to the oscillations of voltage across the transmon. These oscillations (also known as the Bloch oscillations) emit photons into the transmission line. We find the radiation spectrum, and show that the zero-point fluctuations of charge in the line make the spectrum broad-band. Despite being a source of a broad-band radiation, Bloch oscillations can be brought in resonance with an alternating current superimposed on the DC component. The resonances lead to steps in the voltage-current relation, which are dual to the conventional Shapiro steps. We find how the shape of the steps depends on the impedance of the transmission line, parameters of the transmon, and the power of the AC signal.