Summary
In the second lesson, we introduced the general notions of Quantum Condensed Matter Physics essential to understanding the Josephson effect, which plays an essential role in the operation of superconducting qubits. In its simplest form, this qubit consists of two superconducting islands separated by an insulating barrier. A qualitative analysis of electron pairing in Cooper pairs in superconducting metal has been developed. These pairs, considered as composite bosons, tunnel through the barrier constituting the junction. The Josephson effect thus appears as a property of a double condensate whose two parts are coupled by a pair-exchange term. We have established the Cooper pair Hamiltonian in this system, which is very similar to the one describing the dynamics of an atomic Bose-Einstein double condensate, the subject of previous lectures in 2005-2006. The Hamiltonian equations of this system allow us to recover the well-known continuous and alternating Josephson effects. The canonically conjugate variables of this Hamiltonian are the junction charge (or the difference in the number of pairs between the two islands) and the difference in the macroscopic phase between the two parts. Considering these variables as quantum operators, we obtain a non-linear oscillator Hamiltonian whose two deepest levels are those of a qubit resonating at a few gigahertz. This Hamiltonian can be interpreted as that of a fictitious particle in a potential well, the kinetic and potential energies of the particle being associated respectively with the capacitive and inductive energy of the junction. We then briefly describe the magnetic effects associated with the dynamics of a superconducting junction by analyzing the operation of a SQUID, an interferential system including additional Josephson junctions enabling the qubit state to be read out.
