PhD thesis in English

Chapter 4Mean-field description of an interacting BECIn the previous Chapter we have considered BEC of an ideal Bose gas, and,despite neglecting the interactions, we had to apply sophisticated numerical methodsfor characterization of the BEC phase transition and properties of the condensate.Now we extend the description of a Bose gas to include interactions. For an exactmicroscopic description of an interacting Bose gas, we need to consider a full manybodyHamiltonian. For systems with a dominant two-body contact interaction, theHamiltonian has the form (1.24):∫Ĥ =d⃗r(− ˆψ † (⃗r) 22M ∇2 ˆψ(⃗r) + V (⃗r) ˆψ† (⃗r) ˆψ(⃗r) + g )2 ˆψ † (⃗r) ˆψ † (⃗r) ˆψ(⃗r) ˆψ(⃗r) . (4.1)In the case of ultra-cold atomic gases, the bosons are alkali atoms, which are notelementary particles. However, due to very low temperatures, the atoms can beconsidered to be always in their ground states and their internal structure can beneglected. The exact treatment of the system described by the Hamiltonian (4.1)is possible only by numerical simulations based on Monte Carlo approach [76, 77].Such simulations are ubiquitously time-consuming and significantly limit the scopeof problems that can be addressed. For this reason, a number of approximativemethods have been developed. Simplified approximative approaches are much easierfor the numerical implementation and sometimes even provide analytical insightsinto the studied problem. Furthermore, very often, information on the parametersof the system, such as the temperature or condensate fraction, are extracted fromexperimental data indirectly, by fitting to expressions derived within one of simplifiedschemes.In this Chapter we review the mean-field Hartree-Fock (HF) approximation fora many-body Hamiltonian (4.1). The HF approximation is the most-widely usedapproach to study finite-temperature properties of weakly interacting BECs [78, 5].We will show within this mean-field scheme how the presence of the interactioninfluences properties of the condensate, both in the zero-temperature limit, and in84