# Convergence of the multimode quantum Rabi model of circuit quantum electrodynamics

## TL;DR

This paper presents a first-principles multimode circuit QED model showing convergence without cutoff, explaining how higher modes naturally decouple and enabling analysis of ultra-strong coupling regimes.

## Contribution

It introduces a novel multimode circuit QED model that demonstrates convergence without cutoff and clarifies the physical mechanisms behind mode decoupling.

## Key findings

- Quantities like the Lamb shift do not diverge without a cutoff.
- Higher modes naturally decouple due to circuit-based rescaling.
- Framework for analyzing ultra-strong coupling in multimode circuit QED.

## Abstract

Circuit quantum electrodynamics (QED) studies the interaction of artificial atoms, open transmission lines and electromagnetic resonators fabricated from superconducting electronics. While the theory of an artificial atom coupled to one mode of a resonator is well studied, considering multiple modes leads to divergences which are not well understood. Here, we introduce a first-principles model of a multimode resonator coupled to a Josephson junction atom. Studying the model in the absence of any cutoff, in which the coupling rate to mode number $n$ scales as $\sqrt{n}$ for $n$ up to $\infty$, we find that quantities such as the Lamb shift do not diverge due to a natural rescaling of the bare atomic parameters that arises directly from the circuit analysis. Introducing a cutoff in the coupling from a non-zero capacitance of the Josephson junction, we provide a physical interpretation of the decoupling of higher modes in the context of circuit analysis. In addition to explaining the convergence of the quantum Rabi model with no cutoff, our work also provides a useful framework for analyzing the ultra-strong coupling regime of multimode circuit QED.