PhD thesis in English

4. Mean-field description of an interacting BECthe limit T → 0, however it is very often used for all temperatures. The topic isexplored in detail in Ref. [84] where the consequences of the approximation are thoroughlydiscussed. Additionally, we neglect the possible depletion of the condensateat the zero temperature, i.e. the depletion that arises due to interactions, which isa reasonable approximation in the case of a weakly interacting gas.Finally, after implementing all the described steps, we arrive at the finite-temperatureHF description of a bosonic gas, which is given by the following system ofequations:[ ∂]∂τ − 22M △ + V (⃗r) + g|ψ(⃗r, τ)|2 + 2 gh(⃗r, τ;⃗r, τ) ψ(⃗r, τ) = µψ(⃗r, τ) , (4.9)h(⃗r, τ;⃗r, τ) = ∑ ψ ⃗k (⃗r)ψ ∗ 1⃗ k(⃗r)e β(E ⃗ −µ) k − 1 , (4.10)⃗ k][− 22M △ + V (⃗r) + 2 g|ψ(⃗r, τ)|2 + 2 gh(⃗r, τ;⃗r, τ) ψ ⃗k (⃗r) = E ⃗k ψ ⃗k (⃗r) , (4.11)where ψ ⃗k (⃗r) are effective single-particle wave-functions, and E ⃗k are the correspondingeigenvalues. More details on the derivation can be found in Ref. [78]. Althoughformally the HF equations depend on the imaginary time τ, physically is only relevantthe equilibrium case, when the macroscopic wave-function of the condensateψ(⃗r, τ) does not depend on τ anymore (∂ψ(⃗r, τ)/∂τ = 0), but only on the position⃗r. The above set of equations has to be solved self-consistently, taking into accountthat the total number of particles is fixed to N, and leads to the solution that consistsof the effective single-particle eigenfunctions ψ ⃗k and eigenvalues E ⃗k , the Hartreefunction h, and the condensate wave-function ψ. From Eq. (4.10) we immediatelysee the physical interpretation of the Hartree function h, which represents the densityof the thermal cloud, n th (⃗r). Effectively, within the mean-field description, thegas of bosons is split into the condensate and thermal component. We note the closeanalogy with the noninteracting gas description presented in Chapter 1, with theimportant exception that the two components now mutually interact. By varyingthe total number of particles N and the temperature T, a complete N −T phase diagramcan be explored. Before considering a BEC phase transition in the mean-fieldapproximation, we first present the zero-temperature limit of the HF approximation.87