Quantum degeneracy and Bose-Einstein condensation in low-dimensional trapped gases = Dégénérescence quantique et condensation de Bose-Einstein dans des gaz piégés de basse dimensionalité

At present, there are significant efforts to create 2D or 1D trapped gases by (tightly) confining the particle motion to zero point oscillations in one or two directions. The goal of this article is to show that the reduction of the spatial dimensionality in trapped Bose gases drastically changes the nature of the Bose-condensed state. In 2D and 1D gases, one can get a peculiar Bose-condensed state (quasicondensate) where the density fluctuations are suppressed, but the phase still fluctuates. The quasicondensate has the same density profile and local correlation properties as true condensates. However, it is very different with regard to phase coherence properties. We discuss how the phase coherence can be studied in experiments with 2D and 1D trapped gases.