Beyond Mean-Field Description of Bose-Einstein Condensate

Abstract

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.

Popular Abstract

The mean-field approximation is a method of solving for properties of a system that has been around for a long time and has been useful in many areas of physics. The premise is that many effects from interacting particles is averaged to a single effect. That way the problem is often made much simpler. An example is the interaction of Pluto with the other large bodies in the solar system. To analyse the full and correct system, the gravitational pull between Pluto, every planet as well as the sun would have to be considered. In using a mean field, the gravitational pull on Pluto from the other bodies in the solar system is then replaced with an average gravitational pull.

The mean-field approximation often gives reasonable results if it’s possible to find a good mean field. For many different quantum systems, i.e. those at very small scale, perturbation theory has provided good results and was part of the foundations in describing both the electronic orbits as well as the structure of the nucleus of the atom.

Using a mean field often does not give results that are precise enough however, and thus the difference between the mean field and the true interaction needs to be estimated and added to the mean-field solution. The mean-field approach with the inclusion of an estimation of this difference is part of what is called perturbation theory. When the mean-field approach with the inclusion of a perturbation is applied to many particles it is called many-body perturbation theory or MBPT for short.

A Bose-Einstein condensate is a specific quantum system in which a system of a particular type of particle called bosons has collapsed to only occupy the lowest possible energy level. This only occurs when a low enough temperature has been reached. MBPT has given accurate results for large systems of this type when the bosons are weakly interacting. Since this success, more complicated and involved methods have been developed to try to achieve higher accuracy in modelling quantum systems. These have in large part overtaken MBPT in popularity because of the better accuracy they often provide.

Higher interaction strengths have not been analysed as much with any of these methods however, but initial tests using MBPT on smaller systems with higher interaction strengths have shown promise, even though it was expected to fail. This success calls for extending MBPT to see if it can be more precise in modelling the systems or if it fails and must be discarded for some of the more sophisticated methods. It will also be a step in implementing some of the other more advanced methods mentioned to such systems.

MBPT is thus a promising approach to Bose-Einstein condensates with higher interaction strengths that seems to be giving good results, and it can also help in implementing more sophisticated methods, all in the goal of advancing the theoretical knowledge of Bose-Einstein condensates.

• author: Larsson Persson, Daniel
• supervisor: Jakob Bengtsson; Stephanie Reimann
• organization: Mathematical Physics; Department of Physics
• course: FYSK02 20211
• year: 2021
• type: M2 - Bachelor Degree
• subject: Physics and Astronomy
• keywords: Bose-Einstein condensate, BEC, dilute, non-uniform, quantum depletion, second quantisation, mean field, mean-field approximation, the Gross-Pitaevskii equation, Gross-Pitaevskii, perturbation theory, many-body perturbation theory, Rayleigh-Schrödinger perturbation theory, Rayleigh-Schrödinger, Epstein-Nesbet perturbation theory, Epstein-Nesbet, quantum ring
• language: English
• id: 9060450
• date added to LUP: 2021-07-07 21:02:26
• date last changed: 2021-07-07 21:02:26