Cand . Scient . project

The spectacular demonstration of Bose-Einstein condensation (BEC) in gases of alkali atoms 87Rb, 23Na, 7Li confined in magnetic traps[1, 2, 3] has led to an explosion of interest in confined Bose systems. Of interest is the fraction of condensed atoms, the nature of the condensate, the excitations above the condensate, the atomic density in the trap as a function of Temperature and the critical temperature of BEC, Tc. The extensive progress made up to early 1999 is reviewed by Dalfovo et al.[4]. A key feature of the trapped alkali and atomic hydrogen systems is that they are dilute. The characteristic dimensions of a typical trap for 87Rb is ah0 = (h̄/mω⊥) 1 2 = 1 − 2 × 104 Å (Ref. 1). The interaction between 87Rb atoms can be well represented by its s-wave scattering length, aRb. This scattering length lies in the range 85 < aRb < 140a0 where a0 = 0.5292 Å is the Bohr radius. The definite value aRb = 100a0 is usually selected and for calculations the definite ratio of atom size to trap size aRb/ah0 = 4.33 × 10−3 is usually chosen [4]. A typical 87Rb atom density in the trap is n ' 1012 − 1014 atoms/cm3 giving an inter-atom spacing ` ' 104 Å. Thus the effective atom size is small compared to both the trap size and the inter-atom spacing, the condition for diluteness (i.e., naRb ' 10−6 where n = N/V is the number density). In this limit, although the interaction is important, dilute gas approximations such as the Bogoliubov theory[5], valid for small na3 and large condensate fraction n0 = N0/N , describe the system well. Also, since most of the atoms are in the condensate (except near Tc), the Gross-Pitaevskii equation[6, 7] for the condensate describes the whole gas well. Effects of atoms excited above the condensate have been incorporated within the Popov approximation[8]. The purpose of this Cand. Scient thesis is to go beyond the dilute limit, to test the limits of the above approximations and to explore the properties of the trapped Bose gas as na3 increases between the dilute limit and the dense limit. The tools to be used are from Variational Monte Carlo (VMC) and Diffusion Monte Carlo methods (DMC) methods. The aim is to use these methods and evaluate the ground state properties of a trapped, hard sphere Bose gas over a wide range of densities using VMC and DMCmethods with several trial wave functions. These wave functions are used to study the sensitivity of condensate and non-condensate properties to the hard sphere radius and the number of particles. The traps we will use are both a spherical symmetric (S) harmonic and an elliptical (E) harmonic trap in three dimensions given by