Potential harmonics expansion method for trapped interacting bosons: Inclusion of two-body correlation

We study a system of A identical interacting bosons trapped by an external field by solving ab initio the many-body Schrodinger equation. A complete solution by using, for example, the traditional hyperspherical harmonics (HH) basis develops serious practical problems due to the large degeneracy of HH basis. Symmetrization of the wave function, calculation of the matrix elements, etc., become an immensely formidable task as A increases. Instead of the HH basis, here we use a new basis, called "potential harmonics" (PH) basis, which is a subset of HH basis. We assume that the contribution to the orbital and grand orbital [in 3(A-1)-dimensional space of the reduced motion] quantum numbers comes only from the interacting pair. This implies inclusion of two-body correlations only and disregard of all higher-body correlations. Such an assumption is ideally suited for the Bose-Einstein condensate (BEC), which is required, for experimental realization of BEC, to be extremely dilute. Hence three and higher-body collisions are almost totally absent. Unlike the (3A-4) hyperspherical variables in HH basis, the PH basis involves only three active variables, corresponding to three quantum numbers-the orbital l, azimuthal m, and the grand orbital 2K+l quantum numbers for any arbitrary A. It drastically reduces the number of coupled equations and calculation of the potential matrix becomes tremendously simplified, as it involves integrals over only three variables for any A. One can easily incorporate realistic atom-atom interactions in a straightforward manner. We study the ground and excited state properties of the condensate for both attractive and repulsive interactions for various particle number. The ground state properties are compared with those calculated from the Gross-Pitaevskii equation. We notice that our many-body results converge towards the mean field results as the particle number increases.