Quantum electrodynamics with Josephson junctions.

Superconducting circuits have been playing an increasing role in testing the limits of conventional quantum optics, and as a hardware platform for quantum computing, due to their malleable interaction strength and efficient monitoring schemes. When describing the interaction between these systems and microwave fields, standard theories begin with a lumped-element approach, where the geometry of the circuit and structure of the field are neglected, and can overlook important effects necessary for new circuit fabrication. Here, we present a theory of the interaction between a Josephson junction in the transmon regime and external quantum electromagnetic fields. Our framework goes beyond standard circuit models by producing an interaction Hamiltonian where the fields couple explicitly to the junction's degrees of freedom through a multipole expansion. We present examples of light-matter interactions where nontrivial effects arise due to the interplay between the fields' helicity and the junction's structure. By incorporating the vectorial nature of the fields and the geometry of the junction, our work presents a consistent and gauge-invariant extension of the theory of quantum optics with superconducting circuits, with measurable consequences using current experiments.