Quantum information with atoms and photons in a cavity: Entanglement, complementarity and decoherence studies

We review recent experiments in which Rydberg atoms interacting, with microwave photons in a superconducting cavity are used to perform quantum information manipulations. We describe quantum phase gates using atoms and photons as "qubits", various engineered entanglement experiments involving up to three particles at a time and ideal non-destructive measurements of single photons. We also analyse an atomic interferometry experiment which illustrates the link between the notions of complementarity and entanglement. We finally recall previous experiments performed with the same set-up. in which we had generated and studied quantum superpositions of coherent fields with different phases. the so called "Schrodinger cat states" of the field. The decoherence of these states was experimentally studied. We conclude by discussing some of the perspectives opened by this line of research.