Abstract
Having described the principle of the superfluid-insulator phase transition in an optical lattice, we now turn to the experimental study of this phenomenon. We have analyzed a time-of-flight experiment, in which the coherence associated with the superfluid state manifests itself as "Bragg peaks" corresponding to an accumulation of atoms around specific pulse classes.
As most of these experiments are carried out with a harmonic trap superimposed on the optical lattice, this has led us to refine our theoretical description. After introducing the notion of incompressibility of the insulating state, we adopted the variable associated with the grand-canonical point of view, namely the chemical potential, and introduced the local density approximation to describe the transition. We then showed that the transition to the insulating state manifests itself as constant-density plateaus, which have indeed been observed in recent "atomic microscope" experiments.
In the final part of this lecture, we returned to the nature of the superfluid-insulator phase transition, starting from the well-known Landau-Ginsburg model for second-order transitions. We discussed a specific feature of this transition, namely the symmetrical role played by particles and holes. We explained why this makes it possible to observe a collective mode in which the amplitude of the order parameter oscillates. This mode, which is formally very close to the Higgs mode of particle physics, was absent from the dynamics based on the Gross-Pitaevskii equation that we had previously studied for a uniform gas.
