PhD thesis in English

4. Mean-field description of an interacting BECperature with respect to the noninteracting case [90]. To study effects of interactionwithin the TFSC model, we start from the gas phase. In that case the condensateis absent, and n 0 (r) = 0. If we insert this into Eq. (4.27), we getn th (r) = 1 (ζ ) λ 3 3/2 eβ(µ−2gn th (r)−V (r)). (4.31)TThe condensation first appears in the center of the trap, where the following relationholds:n th (0) = 1 (ζ ) λ 3 3/2 eβ(µ−2gn th (0)).TTherefore, at the onset of a BEC phase transition, we have µ c = 2gn c th (0) andn c th (0) = 1 ζλ 3 3/2 , which yields the critical value of the chemical potential is µ c =T2gζ 3/2 /λ 3 T . Now we calculate the critical number of atoms in the trap as∫ ∞N c = 4π dr r 2 1 (ζ )0 λ 3 3/2 eβ(µ c−2gn th (r)−V (r)). (4.32)TTo calculate the above integral, we use the perturbative approach and expand thezeta function to power series in the interaction constant g to linear terms. Thismeans that we consider the term µ c − 2gn th (r) as a small quantity. In this way weobtainN c ≈ 4π≈λ 3 T4πλ 3 T×= 1λ 3 T∫ ∞0∫ ∞0(2 gλ 3 Tdr r 2 [ ζ 3/2(e−βV (r) ) + βζ 1/2(e−βV (r) ) (µ c − 2gn th (r)) ]dr r 2 [ ζ 3/2(e−βV (r) ) + βζ 1/2(e−βV (r) )ζ 3/2 − 2 g (ζ ))]λ 3 3/2 e−βV (r)T[ ] 3/2 [ 2πζβMω 2 3 + 2gβ 1 ζλ 3 3/2 ζ 2 − 2gβ 1 ζTλ 3 T( 12 , 3 2 ; 3 2)], (4.33)where ζ(i, j; k) = ∑ ∞m,n=1 1/mi n j (m + n) k . From the above expression we immediatelyfind the shift of the condensation temperature [90]:δT c = T cint− T 0 cT 0 c= −1.33 a l N1/6 , (4.34)and conclude that repulsive interactions lead to the lower condensation temperaturecompared to the noninteracting case. The result (4.34) is derived by assuming that99