Asymptotic Expressions for Charge Matrix Elements of the Fluxonium Circuit

TL;DR

This paper derives analytical formulas for charge matrix elements in fluxonium circuits, revealing new selection rules in certain parameter regimes, aiding understanding of experimental results.

Contribution

It provides the first analytical expressions for fluxonium charge matrix elements, highlighting new selection rules in the asymptotic limit, improving upon prior numerical methods.

Findings

Analytical expressions for charge matrix elements derived.

New selection rules identified in large Josephson energy limit.

Results facilitate qualitative understanding of experimental data.

Abstract

In charge-coupled circuit QED systems, transition amplitudes and dispersive shifts are governed by the matrix elements of the charge operator. For the fluxonium circuit, these matrix elements are not limited to nearest-neighbor energy levels and are conveniently tunable by magnetic flux. Previously, their values were largely obtained numerically. Here, we present analytical expressions for the fluxonium charge matrix elements. We show that new selection rules emerge in the asymptotic limit of large Josephson energy and small inductive energy. We illustrate the usefulness of our expressions for the qualitative understanding of charge matrix elements in the parameter regime probed by previous experiments.