Abstract
In the third lesson, we focused on flux-charge duality, contrasting the cases of two " artificial atoms " basic superconductors : the Cooper pair box and the SQUID-RF (SQUID=Superconducting QUantum Interference Device). The basic variables of these two circuits are electrically dual : in the case of the box, the charge of an isolated superconducting island, and in the case of the SQUID, the flux through a superconducting loop. In the box, the circuit's wavefunction is localized in charge, so charge fluctuations, expressed in units of the charge of a Cooper pair, are smaller than unity. In the SQUID, the wave function is localized in flux, and it is the flux fluctuations that are smaller than the flux quantum. However, there is an important difference : in the box, the charge is a discrete variable taking only values that are integer multiples of the charge 2e of a Cooper pair, whereas in the SQUID, the flux takes continuous values. We then introduced the important notion of circuit anharmonicity, which describes how far the energy levels deviate from the harmonic spectrum where they form a uniform scale. Anharmonicity is a function of the ratio between the charge energy EC, corresponding to the Coulomb energy of an electron's charge e on the capacitance associated with the tunnel junction, and the Josephson energy EJ corresponding to a superconducting phase difference of 90 degrees across the junction. We ended the lesson by outlining the basis of the semi-classical approximation for the case of a quantum system of weak anharmonicity, and demonstrated that in the " transmon " regime of the Cooper pair box corresponding to an EJ /EC ratio large in front of unity, the difference between two successive transition frequencies was given by EC /h, where h is Planck's constant.
