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International Journal of Modern Physics Conference Series

VALIDITY OF THE BOGOLIUBOV APPROXIMATION FOR THE BOSE-EINSTEIN CONDENSATION IN A TRAP

Hiroshi Ezawa, Keiji Watanabe, Koichi Nakamura

Published Jan 1, 2012• 1 citation
Publisher

Abstract

In treating system of bosons localized in a trapping potential, having a macroscopic number N0 of them condensing at the lowest single-particle level v0, Bogoliubov approximation is to replace the creation/annihilation operators [Formula: see text] of the state v0 by [Formula: see text]. We show that this approximation is justified if the inter-particle potential is repulsive in the sense specified. In fact, we show, by using [Formula: see text], that [Formula: see text] is effectively of the order [Formula: see text] under the condition stated.

DOI
10.1142/s2010194512008033

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In treating system of bosons localized in a trapping potential, having a macroscopic number N0 of them condensing at the lowest single-particle level v0, Bogoliubov approximation is to replace the creation/annihilation operators [Formula: see text] of the state v0 by [Formula: see text]. We show that this approximation is justified if the inter-particle potential is repulsive in the sense specified. In fact, we show, by using [Formula: see text], that [Formula: see text] is effectively of the order [Formula: see text] under the condition stated.
We evaluate the zero temperature properties and Bose-Einstein condensation (BEC) of hard sphere bosons in a harmonic trap using Quantum Monte Carlo methods. The chief purpose of this work is to go beyond the dilute limit, to test the limits of the Gross-Pitaevskii equation and related mean field approximations and to explore the role of interactions in determining the zero temperature properties of the trapped Bose gas. The density is increased by adjusting both the number of trapped bosons, N , and the s-wave scattering length, a, to study systems from the highly dilute region corresponding to early experiments in trapped BEC [2, 3] up to liquid He densities and beyond. Rather than relying on the perturbative Bogoliubov approximation (which assumes a large condensate fraction) to describe the condensate, we use the one-body density matrix (OBDM) formulation of BEC so that the properties of the condensate for systems with arbitrarily large interactions may be studied. In this formulation of BEC, condensate properties are obtained by diagonalizing the OBDM and obtaining the corresponding single particle “natural orbitals” and their occupation numbers for the system. The condensate wave function and condensate fraction are then obtained from the single particle orbital(s) with macroscopic occupation (N0 >> 1). Within this framework, we calculate the effects of interactions and increased density on the ground state energy, the density profile, the momentum distribution, the condensate fraction and condensate “wave-function” and several other properties.
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The Hohenberg theorem on the absence of Bose-Einstein condensation (BEC) in homogeneous systems of space dimensions D< 2D<~2D< 2 is based on a well-known Bogoliubov inequality. Applied to an assembly trapped in a harmonic potential we show that the Bogoliubov inequality rules out BEC in all dimensions at finite temperatures. However, this conflicting result with both theory and experiment disappears when the effect of the order parameter is properly taken into account in the boson-field commutation relations. For a hard-sphere Bose gas the theory is consistent with the expansion of the condensate when a positive scattering length is increased, as well as the collapse of the condensate when the sign of the scattering length is reversed and it reaches a minimum critical value.
The particle distribution in a Bose condensate under the trapping potential and its time evolution after switching off the trapping potential suddenly are calculated. We investigate the problem from the viewpoint of quantum field theory,using a model of self-interacting neutral boson field. Within the approximation of retainng the most dominant term in the Hamiltonian after applying the Bogoliubov replacement, we can calculate analytically the particle distribution as a function of space and time coordinates.