Calculation of Bose-Einstein condensations and characteristic features of fluctuations for systems with and without a vortex in two-component alkali atom gases

Characteristic features in phases of Bose-Einstein condensations (BEC) for systems of two-components in gas phases of Alkali-metal atoms trapped by harmonic potentials are discussed on the basis of numerical calculations by the use of Gross-Pitaevskii equation. Condensations with and without vortices are considered in order to make clear spatial distributions. The shape of spatial distribution is essentially dependent on the ratio of interaction strengths between intra-components and inter-components, We emphasize the possibility of coexistence of vortex-free state (where the angular momentum l is 0.) for the one component in BEC and vortex state (l(z) = 1) for another component whose shape is the donut-like spatial distribution. Furthermore; fluctuations of these states are discussed. In partiucular, we concentrate our attention to the appearance of the core modes with l(z) = 0 symmetry for the component with a vortex. In connection with the behavior of the core modes, the contribution of the component without a vortex to the stability of the above vortex is discussed on the basis of numerical calculation.