Origin and implications of an A2 -like contribution in the quantization of circuit-QED systems

By placing an atom into a cavity, the electromagnetic mode structure of the cavity is modified. In cavity QED, one manifestation of this phenomenon is the appearance of a gauge-dependent diamagnetic term, known as the A2 contribution. Although in atomic cavity QED, the resulting modification in the eigenmodes is negligible, in recent superconducting circuit realizations, such corrections can be observable and may have qualitative implications. We revisit the canonical quantization procedure of a circuit-QED system consisting of a single superconducting transmon qubit coupled to a multimode superconducting microwave resonator. A complete derivation of the quantum Hamiltonian of an open circuit-QED system consisting of a transmon qubit coupled to a leaky transmission line cavity is presented. We introduce a complete set of modes that properly conserves the current in the entire structure and present a sum rule for the dipole transition matrix elements of a multilevel transmon qubit coupled to a multimode cavity. Finally, an effective multimode Rabi model is derived with coefficients that are given in terms of circuit parameters.