Appendix K: this appendix is an introduction to another very active and promising domain of quantum physics, named circuit quantum electrodynamics (cQED), dealing with the quantum properties of macroscopic objects consisting of superconducting electrical circuits. In an LC circuit the energy is quantized, and charge and flux are two quantum canonical conjugate quantities that do not commute. Their quantum fluctuations are bound by a Heisenberg inequality. A Josephson junction inserted in the circuit introduces strong nonlinearities in the system, which breaks the equidistance between the energy levels and makes the circuit look like a qubit, called a transmon. Cooling at mK temperatures is necessary to have quantum effects dominate over thermal effects. The circuit is embedded in a resonant cavity, and the system bears many analogies with cavity QED and Jaynes–Cummings formalism for coupled photons and atoms. One can perform nondestructive read-out and control of the transmon, as well as phase-sensitive, quantum-limited amplification, with nonlinearities that are much stronger than the ones used in quantum optics.

In this chapter, we review the basic properties of superconductors, illustrating that they go beyond describing a metal without resistance. We develop London's mesoscopic theory of superconductivity as a quantum mechanical description of the charged superfluid, which accounts for the behavior of the superconductor under electromagnetic fields. Using this theory, we explain the flux quantization for supercurrents flowing in closed loops and derive the energetics of the Josephson junction from the tunneling of Cooper pairs.