Bose-Einstein condensates in dilute trapped atomic gases

The static and dynamic behavior of dilute trapped Bose-Einstein condensates at low temperature follows from the Gross-Pitaevskii equation for the condensate and the Bogoliubov equations for the linearized small-amplitude normal modes. The uniform system serves to illustrate the theoretical methods and much of the basic physics. The principal new effect of the confining trap is to introduce an additional length scale (the size of the single-particle ground state) and energy scale (the single-particle ground-state energy). Most recent experiments use large condensates, when the repulsive interactions expand the condensate considerably and thus reduce the kinetic energy associated with the nonuniform density. In this regime (known as the Thomas-Fermi limit), the system can be treated as locally uniform, which greatly simplifies the analysis. When the condensate contains one or more vortex lines, the nonuniform trap potential and local line curvature drive the resulting vortex motion. Experiments have confirmed various predicted precessional motions in considerable detail. Mixtures of two distinct bosonic species allow for new coupled dynamical motions that alter the topology of the original single complex order parameter. In particular, application of near-resonant electromagnetic fields yields a coupled system that no longer has quantized circulation. Such experimental techniques created the first vortex line by spinning up one of the components. The introduction of optical traps has allowed the study of what are called spinor condensates. In this case, all hyperfine states are trapped, in contrast to the more common magnetic traps that confine only a subset of the various hyperfine states. The rotational invariance of the interparticle interactions significantly restricts the allowed states of these spinor condensates.