Bose—Einstein condensation in magnetic traps. Introduction to the theory

The recent realization of Bose—Einstein condensation in atomic gases opens new possibilities for the observation of macroscopic quantum phenomena. There are two important features of these systems — weak interaction and significant spatial inhomogeneity. Because of this a non-trivial 'zeroth-order' theory exists, compared to the 'first-order' Bogolubov theory. The zeroth-order theory is based on the mean-field Gross—Pitaevskii equation for the condensate ψ-function. The equation is classical in its essence but contains the constant [IMG align=ABSMIDDLE alt=$\hbar$]hplank[/IMG] explicitly. Phenomena such as collective modes, interference, tunneling, Josephson-like current and quantized vortex lines can be described using this equation. Elementary excitations define the thermodynamic behavior of the system and result in a Landau-type damping of collective modes. Fluctuations of the phase of the condensate wave function restrict the monochromaticity of the Josephson current. Fluctuations of the numbers of quanta result in quantum collapse-revival of the collective oscillations.