Beyond Mean-Field Description of Bose-Einstein Condensate

Sammanfattning: This thesis work is on the theoretical description of dilute Bose-Einstein condensates (BEC) that are non-uniform, meaning that more than one single-particle state is occupied. This phenomenon is usually called quantum depletion. A common model for the ground state of a dilute and weakly interacting BEC is the mean-field (MF) approximation leading to the Gross-Pitaevskii (GP) equation. However, for more strongly correlated systems, a beyond MF method is required. We propose a method for finding the energies of BECs through deriving the expressions up to fourth order in many-body perturbation theory using the GP equation in both Rayleigh-Schrödinger perturbation theory (RSPT) and Epstein-Nesbet perturbation theory (ENPT). The two perturbation theories are similar in terms of computational cost and a comparison of the accuracy of the two is thus of interest. We implement RSPT and ENPT computationally and apply these methods in the analysis of a one-dimensional quantum ring system using contact interaction. The outputs of the two methods are benchmarked against Configuration Interaction in the low particle-number regime. Excellent agreement was found for both RSPT and ENPT for weak interaction strengths. For very high repulsive interaction strengths, RSPT starts to deviate from the correct solution. ENPT, however, continued to show good agreement in this regime. ENPT is of particular interest for further research since it provides a better description of systems with higher interaction strengths. This method could be analysed using other kinds of systems, for example systems in a harmonic confinement, to see if it continues to provide accurate results.