On the ground state energy of the inhomogeneous Bose gas

Abstract

Within the self-consistent Hartree-Fock approximation, an explicit expression for the ground state energy of inhomogeneous Bose gas is derived as a functional of the inhomogeneous density of the Bose-Einstein condensate. The results obtained are based on existence of the off-diagonal long-range order in the single-particle density matrix for systems with a Bose-Einstein condensate. This makes it possible to avoid the use of anomalous averages. The explicit form of the kinetic energy, which differs from one in the Gross-Pitaevski approach, is found. This form is valid beyond the Hartree-Fock approximation and can be applied for arbitrary strong interparticle interaction. PACS number(s): 05.30. Jp, 67.10.Fj, 67.85.Bc Experimental observation of the Bose-Einstein condensate (BEC) in ultracold gases of alkali metals [1] is a strong motivation for theoretical studies of weakly nonideal Bose systems. Due to the presence of magnetic moment the alkali metal atoms can be confined in magnetic traps. Ultralow temperatures requiring to form the BEC are achieved by use laser cooling which leads to evaporation of atoms with high energy from a magnetic trap (see [2] for more details). The ultracold gas obtained in such a way is rarefied and is characterized by strong inhomogeneity [3]. For these reasons, the Gross-Pitaevskii equation [4,5] corresponding to the "mean" field approximation and allowing the consideration of the effect of laser radiation [6] is widely used to describe the ultracold gas. Wherein, validity of the Gross-Pitaevskii equation is shown for the inhomogeneous system, containing a finite number of particles N in an infinite volume V [7]. The consideration of such system does not correspond to the thermodynamic limit transition (N → ∞, V → ∞, N/V = const). At the same time the derivation of the Gross-Pitaevskii equation for an inhomogeneous system in thermodynamic limit is based on the hypothesis of the "anomalous averages" existence (see [8]). As shown in [9][10][11][12][13][14][15] the description of homogeneous systems with BEC using of anomalous averages is dubious.An alternative approach proposed in the present paper is based on applying the conventional diagram technique of the perturbation theory to an equilibrium system in a large but finite volume [16] and on existence of the off-diagonal long-range order (ODLRO) in the one-particle density matrix. Such approach allows the self-consistent consideration of BEC by transferring to the thermodynamic limit [17] (see [18] for more details). On this basis we establish in the present paper two new results: a) the explicit expression for kinetic part of the ground state energy of Bose system with BEC in an external field, which is valid for arbitrary strong interaction between particles, b) the ground state energy of the system under consideration in the Hartree-Fock approximation, which differs from one in the theory, based on anomalous averages.We use ODLRO to describe an inhomogeneous system of bosons with zero spin and mass m, which is in a static exte...