PhD thesis in English

system given by( ) ǫ Bog ⃗k = √ 2⃗ k 22M( 2⃗ k 22M + 2gn 0). (1.28)In the limit of large | ⃗ k|, the spectrum yields excitations of a noninteracting gas. Collectivephenomena appear in the long wavelength limit where the collective phononmode is found [30]. The microscopic derivation of the excitation spectrum (1.28)fits well with the Landau’s phenomenological description of the superfluid. Anotherrelevant quantity that can be derived within the Bogoliubov framework is thecondensate depletion due to interactions at T = 0. The number of non-condensedparticles is proportional to √ na 3 , which for the case of strongly interacting systemsuch as liquid 4 He yields the depletion as high as 90%. For this reason, the superfluidityof 4 He is considered only as an indirect manifestation of the Bose-Einsteincondensation.In general, the relation between the condensation and superfluidity is a subtleone. It can be shown that the macroscopic occupation of the ground state (1.27)introduces the off-diagonal long range order (ODLRO) into the system [31]:lim 〈 ˆψ † (⃗r) ˆψ(⃗r ′ )〉 = n 0 . (1.29)|⃗r−⃗r ′ |→∞However, long range correlations that are the manifestation of a superfluidity may bepresent even without the condensation, as for instance in two-dimensional systems[7].In order to put the phenomenon of Bose-Einstein condensation into a broadercontext, we emphasize that the decomposition (1.27) represents the spontaneousbreaking of the U(1) symmetry related to the conservation of the number of particlesin the system described by the Hamiltonian (1.24). Hence, ψ is the order parameterthat acquires nonzero value in the condensed phase [31, 32].Quantitative tests of the theoretical concepts introduced in this subsection becamepossible only with the experimental realization of Bose-Einstein condensationin dilute vapors of alkali atoms. However, the experiments introduced an importantadditional feature - the inhomogeneity of the system due to the trapping potential.The first theoretical study of a interacting bosonic system in a harmonic trap waspresented in Ref. [33] in relation to the early experiments that were trying to producehydrogen BEC. Since then, different approaches have been used to describe the15