Exact and approximate fluxonium array modes

TL;DR

This paper provides an exact and an approximate analytical solution for the modes of fluxonium superconducting qubit arrays, applicable to various configurations and array sizes, facilitating better understanding and design of these quantum circuits.

Contribution

It introduces a novel exact solution for fluxonium array modes and a simple approximate method, applicable to arrays of any length and configuration, enhancing analytical tools in quantum circuit design.

Findings

Exact mode energies are roots of convex combinations of Chebyshev polynomials.

Array mode profiles are plane waves.

The approximate solution effectively estimates mode properties across diverse parameters.

Abstract

We present an exact solution for the linearized junction array modes of the superconducting qubit fluxonium in the absence of array disorder. This solution holds for arrays of any length and ground capacitance, and for both differential and grounded devices. Array mode energies are determined by roots of convex combinations of Chebyshev polynomials, and their spatial profiles are plane waves. We also provide a simple, approximate solution, which estimates array mode properties over a wide range of circuit parameters, and an accompanying Mathematica file that implements both the exact and approximate solutions.