We study a system of
A identical interacting bosons trapped by an external field by solving ab initio the many-body Schr\"odinger 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 $3(A\ensuremath{-}1)$-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 $(3A\ensuremath{-}4)$ 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.
A two-body correlated basis set is used to develop a many-body theory which is valid for any number of bosons in the trap. The formalism incorporates the van der Waals interaction and two-body correlations in an exact way. The theory has successfully been applied to Bose-Einstein condensates---dilute weakly interacting and also dilute but having a large scattering length. Even in the extreme dilute condition, we observe the breakdown of the shape-independent approximation and the interatomic correlation plays an important role in the large particle-number limit. This correlated many-body calculation can handle, within the two-body correlation approximation, the entire range of atom number of experimentally achieved condensates. Next we successfully push the basis function for large scattering lengths where the mean-field results are manifestly bad. The sharp increase in correlation energy clearly shows the beyond-mean-field effect. We also calculate one-particle densities for various scattering lengths and particle numbers. Our many-body calculation exhibits the finite-size effect in the one-body density.
Shorttitle: The first experimental achievement of Bose Einstein condensation (BEC) in dilute alkali gases in 1995 has developed a renewed interest both theoretical and experimental. Lot of work has already been done over the last decade to study the different aspects of this condensation, mainly based on mean-field theories, hyperspherical approaches and quantum Monte Carlo methods. This thesis work presents an approximate many-body approach to investigate mainly the static (ground state) properties of the condensate, incorporating realistic inter-atomic interactions. Potential harmonic expansion method (PHEM), which includes two-body correlations only and disregards three or higher body correlations, has been adopted to analyze this condensate. This is ideally suited for BEC, which is required to be extremely dilute for its experimental realization. This treatment and analysis should be useful for the physics community especially working with weakly interacting ultra cold atoms, bosons and fermions as well.
We investigate systems of identical bosons with the focus on two-body correlations and attractive finite-range potentials. We use a hyperspherical adiabatic method and apply a Faddeev type of decomposition of the wave function. We discuss the structure of a condensate as a function of particle number and scattering length. We establish universal scaling relations for the critical effective radial potentials for distances where the average distance between particle pairs is larger than the interaction range. The correlations in the wave function restore the large-distance mean-field behavior with the correct two-body interaction. We discuss various processes limiting the stability of condensates. With correlations we confirm that macroscopic tunneling dominates when the trap length is about half of the particle number times the scattering length.
We study the properties of Bose-Einstein condensates with the realistic van der Waals two-body interaction for large numbers of trapped atoms, solving the many-body Schr\"odinger equation by the potential harmonic expansion method. The effect of different
C6 parameters has been critically examined, starting from very few to 14 000 atoms to analyze and justify the idea of a shape-independent approximation. It is found that the condensate properties almost remain unchanged when the number of atoms are quite small $(\ensuremath{\sim}100)$, in good agreement with earlier results of Blume and Greene [Phys. Rev. A 63, 063601 (2001)], but we observe the appreciable effect of a long attractive tail when
N is large, even in the low-density limit. The above reference considered only 20 atoms which is far from a real experimental situation. Our calculation gives a realistic scenario which justifies the use of a shape-dependent two-body interaction in many-body theories.
We investigate the structure and stability of Bose-Einstein condensates of
7Li atoms with realistic van der Waals interactions by using the potential harmonic expansion method. Besides the known low-density metastable solution with a contact $\ensuremath{\delta}$-function interaction, we find a stable branch at a higher density which corresponds to the formation of an atomic cluster. Comparison with the results of a nonlocal effective interaction is also presented. We analyze the effect of trap size on the transition between the two branches of solutions. We also compute the loss rate of a Bose condensate due to two- and three-body collisions.