Beyond the Gross-Pitaevskii Equation: A Perturbative Approach

Abstract

We propose and implement a beyond Gross-Pitaevskii approach based on Many-Body Perturbation theory (MBPT) for the study of Bose-Einstein Condensate (BECs) ground states. Two partitions of the system Hamiltonian are considered. Firstly, a bosonic analogue to Møller-Plesset simply dubbed RSPT, and secondly an approach based on the Epstein-Nesbet partitioning scheme labeled ENPT. We consider one-dimensional BECs and work in the Harmonic Oscillator (HO) basis. Both third order RSPT and ENPT show overall good agreement with Full Configuration Interaction (FCI) in the low-particle number regime for a harmonically trapped BEC. For the same system, fast convergence is also seen towards the GP energy in the mean-field limit, as expected. Moreover, third order ENPT is seen to consistently produce lower ground state energies with better accuracy, compared to RSPT. Finally, we explore more complicated systems. Firstly, a BEC trapped in a double-well potential, where the mean-field ground state exhibits symmetry breaking. For low enough particle counts, when far away from the mean-field limit, we found that third order ENPT applied to a symmetric mean-field state results in lower energies compared to starting from the asymmetric mean-field ground state. Further studies examining the perturbed wave functions are necessary to determine whether or not this ground state is symmetric. Lastly, we study self-bound BEC droplet states, and although our methods are not able to correctly reproduce the characteristic energy minima they do hold promise in the study of self-bound states. Future studies exploring other starting points such as the extended GP equation are proposed.

Popular Abstract

Bose-Einstein Condensates, an intriguing type of ultra-cold gas where all par- ticles behave as one, and microscopic quantum effects are brought to light. They are notoriously difficult to describe and previous approach are either too simplified or too time consuming. Is there a better way?
All particles in nature appear in one of two flavors, either they are called bosons or fermions. These two flavors differ in one major way, bosons, in contrast to fermions, are allowed by the laws of nature to exist in the same state. This property of bosons is very peculiar, imagine having 10 special tennis balls that can all be placed in exactly the same spot, only taking up the space of a single tennis ball. This is essentially what happens when a gas of bosons is cooled down far enough, the bosons occupy the same state and behave as one, resulting in the formation of a Bose-Einstein Condensate (BEC).
Unfortunately, describing BECs theoretically in an attempt to better understand them, is no easy feat. Simplifications are necessary and typically one assumes that a large amount of bosons occupy exactly the same state, and only collide very weakly as to not bump bosons to other states. This approach results in the Gross-Pitaevskii (GP) equation which has proved instrumental over the years, accounting for tons of properties of BECs. Sadly, the GP equation fails when considering stronger collisions or BECs with smaller amounts of bosons, where impact of each boson plays a larger role. The GP equation has also failed to account for more delicate states such as the formation of water like BEC droplets, which were recently discovered.
Therefore, a lot of effort has been put towards building upon the GP equation to study more complicated BECs. This is usually done by allowing for bosons to occupy other states, which allows for stronger collisions and fewer bosons in the BEC. One such ap- proach, is known as Full Configuration Interaction (FCI), which considers a large amount of particles and states. FCI is known to be accurate but is sadly very time consuming and is therefore only useful for a tiny amount of bosons.
This thesis explores one particular approach to build upon the GP equation by allowing for one or two bosons to occupy different states. We found great agreement with both FCI and the GP equation, verifying that our method works! Our method was also applied to more complicated systems such as the droplet BECs mentioned earlier, and although we were not able to correctly describe these systems, it did however show great promise for future research.

• author: Johansson, Anton
• supervisor: Jakob Bengtsson; Stephanie Reimann
• organization: Department of Physics; Engineering Physics (M.Sc.Eng.)
• course: PHYM01 20202
• year: 2023
• type: H2 - Master's Degree (Two Years)
• subject: Physics and Astronomy
• keywords: Gross-Pitaevskii, Beyond mean-field, Many-body Perturbation Theory, Bose-Einstein Condensate, BEC
• publication/series: Lund Reports on Atomic Physics
• report number: 591
• language: English
• id: 9056137
• date added to LUP: 2023-08-18 16:47:18
• date last changed: 2023-08-18 16:47:18