  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.
