Normal modes of a superconducting transmission-line resonator with embedded lumped element circuit components

TL;DR

This paper introduces a method to analyze the normal modes of superconducting transmission lines with embedded circuits, accounting for nonlinearities and qubit interactions, supported by experimental comparisons.

Contribution

It provides a quantitative framework for identifying coupled modes and evaluating nonlinearities in superconducting circuits with embedded elements.

Findings

Accurately models high Kerr nonlinearities observed experimentally.

Quantifies effects of embedded circuit modifications on mode properties.

Supports analysis of qubit-field Rabi coupling in transmission lines.

Abstract

We present a method to identify the coupled, normal modes of a superconducting transmission-line with an embedded lumped element circuit. We evaluate the effective transmission-line non-linearities in the case of Kerr-like Josephson interactions in the circuit and in the case where the embedded circuit constitutes a qubit degree of freedom, which is Rabi coupled to the field in the transmission-line. Our theory quantitatively accounts for the very high and positive Kerr non-linearities observed in a recent experiment [M. Reh\'ak et.al., Appl. Phys. Lett. 104, 162604], and we can evaluate the accomplishments of modified versions of the experimental circuit.