Low-energy spectrum of double-junction superconducting circuits in the Born-Oppenheimer approximation

TL;DR

This paper models the low-energy spectrum of double-junction superconducting circuits using a Born-Oppenheimer approximation, deriving an effective single-mode model that accounts for internal modes and their effects.

Contribution

It introduces a new effective single-mode model for double-junction circuits that includes corrections from internal modes, validated against experimental parameter regimes.

Findings

The model accurately describes the low-energy spectrum of the qubit.

Eliminating internal degrees of freedom affects boundary conditions and non-uniqueness.

The harmonic content of the element is sensitive to charge noise.

Abstract

The superconductor-insulator-superconductor Josephson junction is the fundamental nonlinear element of superconducting circuits. Connecting two junctions in series gives rise to higher-harmonic content in the total energy-phase relation, enabling new design opportunities in multimode circuits. However, the double-junction element hosts an internal mode whose spectrum is set by the finite capacitances of the individual junctions. Using a Born-Oppenheimer approximation that treats the additional mode as fast compared to the qubit mode, we analyze the double-junction circuit element shunted by a large capacitor. Here, we derive an effective single-mode model of the qubit containing a correction term owing to the presence of the internal mode. We explore experimentally relevant parameter regimes and find that our model accurately describes the low-energy spectrum of the qubit. We further discuss how eliminating the internal degree of freedom affects the system's periodic boundary conditions and how this leads to non-uniqueness in performing the Born-Oppenheimer approximation. Finally, we analyze the harmonic content of the double-junction element and discuss its sensitivity to charge noise.