Discrete Nonlinear Dynamics of Bose-Einstein Condensates

Since the first experimental observation of the Bose-Einstein condensation of a gas of alkali atoms, six years ago, there has been an increasing interest on this new, highly interdisciplinary, branch of physics. It was realized, from the very beginning that, despite of the diluteness of such systems, the nonlinearity was crucial on the understanding of both the equilibrium and dynamical properties. In this Thesis we point out that there is a further key aspect that must be taken into account when the condensates are trapped in deep optical lattices: discreteness. The interplay between discreteness and nonlinearity raises a new class of deeply non-trivial phenomena which disappear in the continuum limit of the Gross-Pitaevskii equation (GPE). Recognizing and exploiting such interplay is the main topic of this Thesis. We think that the dynamics of a Bose-Einstein condensate (BEC) in optical lattices is going to become of central interest in the next future. Optical potentials provide BEC's waveguides, which prevent the spatial spreading of the BEC wave packets and preserve their coherence over long distances. This has allowed to build high sensitive interferometers, including gyroscopes and gravitometers. From the theoretical point of view, the big challenge is to reconsider the meaning of phenomena which have been well understood in homogeneous systems: superfluidity, for instance, is in the top list. A further example is given by quantum phase transitions. The connection and the interplay with the language and insights of the nonlinear physics community is a second challenge. We believe that the results obtained in the BEC context can provide a new prospective view into old problems. The understanding of the modulational instability in terms of phase randomization and a (classical) dynamical phase transition are among this line, as well as the possibility to generalize the Landau criteria for superfluidity to discrete systems. On the other hand, it is clear that nonlinear physicists can have much to say about BEC. With this Thesis we hope we have provided, together with original results, some links between these different languages and views.