Effective Jaynes-Cummings model from on-chip filter in a dc SQUID phase qubit

We examine a dc SQUID phase qubit with an on-chip low-pass resonant LC filter that transforms the line impedance, improving the qubit lifetime. Unusual features in the spectroscopy suggest dynamics more complicated than a simple two-level system. To model this behavior, we consider a lumped-element circuit model of the SQUID that includes the filter as part of the quantum system to be modeled. We show this model reduces to an effective Jaynes-Cummings Hamiltonian, in analogy with circuit QED.