Beyond MeanField Description of BoseEinstein Condensate
(2021) FYSK02 20211Mathematical Physics
Department of Physics
 Abstract
 This thesis work is on the theoretical description of dilute BoseEinstein condensates (BEC) that are nonuniform, meaning that more than one singleparticle 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 meanfield (MF) approximation leading to the GrossPitaevskii (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 manybody perturbation theory using the GP equation in both RayleighSchrödinger perturbation theory (RSPT) and EpsteinNesbet perturbation theory (ENPT). The two... (More)  This thesis work is on the theoretical description of dilute BoseEinstein condensates (BEC) that are nonuniform, meaning that more than one singleparticle 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 meanfield (MF) approximation leading to the GrossPitaevskii (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 manybody perturbation theory using the GP equation in both RayleighSchrödinger perturbation theory (RSPT) and EpsteinNesbet 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 onedimensional quantum ring system using contact interaction. The outputs of the two methods are benchmarked against Configuration Interaction in the low particlenumber 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. (Less)  Popular Abstract
 The meanfield 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 meanfield approximation often gives reasonable results if... (More)  The meanfield 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 meanfield 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 meanfield solution. The meanfield approach with the inclusion of an estimation of this difference is part of what is called perturbation theory. When the meanfield approach with the inclusion of a perturbation is applied to many particles it is called manybody perturbation theory or MBPT for short.
A BoseEinstein 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 BoseEinstein 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 BoseEinstein condensates. (Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/studentpapers/record/9060450
 author
 Larsson Persson, Daniel ^{LU}
 supervisor

 Jakob Bengtsson ^{LU}
 Stephanie Reimann ^{LU}
 organization
 course
 FYSK02 20211
 year
 2021
 type
 M2  Bachelor Degree
 subject
 keywords
 BoseEinstein condensate, BEC, dilute, nonuniform, quantum depletion, second quantisation, mean field, meanfield approximation, the GrossPitaevskii equation, GrossPitaevskii, perturbation theory, manybody perturbation theory, RayleighSchrödinger perturbation theory, RayleighSchrödinger, EpsteinNesbet perturbation theory, EpsteinNesbet, quantum ring
 language
 English
 id
 9060450
 date added to LUP
 20210707 21:02:26
 date last changed
 20210707 21:02:26
@misc{9060450, abstract = {{This thesis work is on the theoretical description of dilute BoseEinstein condensates (BEC) that are nonuniform, meaning that more than one singleparticle 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 meanfield (MF) approximation leading to the GrossPitaevskii (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 manybody perturbation theory using the GP equation in both RayleighSchrödinger perturbation theory (RSPT) and EpsteinNesbet 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 onedimensional quantum ring system using contact interaction. The outputs of the two methods are benchmarked against Configuration Interaction in the low particlenumber 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.}}, author = {{Larsson Persson, Daniel}}, language = {{eng}}, note = {{Student Paper}}, title = {{Beyond MeanField Description of BoseEinstein Condensate}}, year = {{2021}}, }