# Quantum theory of Bloch oscillations in a resistively shunted transmon

**Authors:** Vladislav D. Kurilovich, Benjamin Remez, Leonid I. Glazman  
**arXiv:** [2403.04624](https://arxiv.org/abs/2403.04624)  
**Date:** 2024-03-07  
**Summary page:** https://www.summarizepaper.com/en/paper/2403.04624v1/

## Key points

- Authors: Vladislav D. Kurilovich, Benjamin Remez, and Leonid I. Glazman
- Study focus: Behavior of a transmon qubit in a high-impedance environment
- Bloch oscillations: Voltage oscillations resulting from biasing the transmon with direct current
- Spectrum of Bloch oscillations: Broad-band nature influenced by zero-point fluctuations of charge
- Resonance with external microwave radiation: Leads to distinct steps in voltage-current relation (Shapiro steps)
- Factors influencing steps' shape: Impedance of the environment ($R$), transmon system parameters, amplitude of microwave radiation
- Insulating state requirement for Bloch oscillations: Achieved at impedances surpassing Schmid transition point ($R > R_Q = h / (2e)^2$)
- Insights provided by the study: Quantum behavior and potential applications of Bloch oscillations in resistively shunted transmons

## Summary

In their paper titled "Quantum Theory of Bloch Oscillations in a Resistively Shunted Transmon," authors Vladislav D. Kurilovich, Benjamin Remez, and Leonid I. Glazman explore the behavior of a transmon qubit embedded in a high-impedance environment. They draw parallels between the transmon qubit and a conventional Josephson junction, highlighting how biasing the transmon with direct current results in voltage oscillations known as Bloch oscillations. The researchers delve into the spectrum of Bloch oscillations and demonstrate how zero-point fluctuations of charge contribute to its broad-band nature. Despite this broad spectrum, they show that Bloch oscillations can be brought into resonance with external microwave radiation. This resonance leads to distinct steps in the voltage-current relation, akin to conventional Shapiro steps. Furthermore, the authors investigate how the shape of these steps is influenced by various factors such as the impedance of the environment ($R$), parameters specific to the transmon system, and the amplitude of the microwave radiation. They emphasize that Bloch oscillations rely on the insulating state of the transmon, which is achieved at impedances surpassing the Schmid transition point ($R > R_Q = h / (2e)^2$). Overall, this study provides valuable insights into quantum behavior and potential applications of Bloch oscillations in resistively shunted transmons.