Almost all superconducting quantum technologies are built using a combination of qubits and microwave resonators. In this chapter, we develop the theory to study coherent qubit–photon interaction in such devices. We start with the equivalent of an atom in free space, studying a qubit in an open waveguide. We develop the spin-boson Hamiltonian, with specific methods to solve its dynamics in the limits of few excitations. Using these tools, we can study how an excited qubit can relax to the ground state, producing a photon, and how a propagating photon can interact with a qubit. We then move to closed environments where the photons are confined in cavities or resonators, developing the theory of cavity-QED. Using this theory, we study the Purcell enhancement of interactions, the Jaynes–Cummings model, Rabi oscillations, and vacuum Rabi splitting. We close the chapter illustrating some limits in which cavities can be used to control and measure qubits.

This appendix provide a self-contained presentation of the open systems and quantum optics methods used in other parts of the book (e.g., studying the relaxation of a microwave cavity or a qubit, the driving of a quantum amplifier, etc.). Half of the appendix is devoted to the derivation of master equations for small systems that are in contact with Markovian environments. The other half of the appendix is devoted to the development of an alternative input–output description of how those systems absorb information from the environment and reflect it back.