Quantum theory of Bloch oscillations in a resistively shunted transmon

TL;DR

This paper develops a quantum theoretical framework for Bloch oscillations in a resistively shunted transmon, revealing how charge fluctuations broaden the spectrum and how these oscillations can resonate with microwaves, producing voltage steps analogous to Shapiro steps.

Contribution

It introduces a quantum model for Bloch oscillations in a resistively shunted transmon, detailing the spectral properties and resonance conditions, which were not previously understood.

Findings

Bloch oscillations spectrum is broad-band due to zero-point charge fluctuations.

Resonance with microwave radiation produces voltage steps dual to Shapiro steps.

The shape of the steps depends on environment impedance, transmon parameters, and microwave amplitude.

Abstract

A transmon qubit embedded in a high-impedance environment 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 leads to the oscillations of voltage across it. These oscillations are known as the Bloch oscillations. We find the Bloch oscillations spectrum, and show that the zero-point fluctuations of charge make it broad-band. Despite having a broad-band spectrum, Bloch oscillations can be brought in resonance with an external microwave radiation. 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 environment impedance $R$, parameters of the transmon, and the microwave amplitude. The Bloch oscillations rely on the insulating state of the transmon which is realized at impedances exceeding the Schmid transition point, $R > R_Q = h / (2e)^2$.