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International Journal of Modern Physics B

VARIATIONAL DESCRIPTION OF WEAKLY INTERACTING BOSE GASES IN 3 DIMENSIONS

Ferran Mazzanti, M. Saarela, V. Apaja

Published Dec 15, 2006• 0 citations
Publisher

Abstract

Static and dynamic properties of a weakly interacting Bose gas of Hard Spheres in three dimensions are studied in the framework of the Correlated Basis Functions (CBF) approximation. Results are compared with explicit expressions for the same quantities derived within the Bogoliubov model. Despite the good agreement in the energy of the groundstate and the excited states, other quantities such as the dynamic structure function present important differences that become more significant when the density is raised.

DOI
10.1142/s0217979206036107

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Static and dynamic properties of a weakly interacting Bose gas of Hard Spheres in three dimensions are studied in the framework of the Correlated Basis Functions (CBF) approximation. Results are compared with explicit expressions for the same quantities derived within the Bogoliubov model. Despite the good agreement in the energy of the groundstate and the excited states, other quantities such as the dynamic structure function present important differences that become more significant when the density is raised.
The energy and structure of dilute hard- and soft-sphere Bose gases are systematically studied in the framework of several many-body approaches, such as the variational correlated theory, the Bogoliubov model, and the uniform limit approximation, valid in the weak-interaction regime. When possible, the results are compared with the exact diffusion Monte Carlo ones. Jastrow-type correlation provides a good description of the systems, both hard- and soft-spheres, if the hypernetted chain energy functional is freely minimized and the resulting Euler equation is solved. The study of the soft-sphere potentials confirms the appearance of a dependence of the energy on the shape of the potential at gas paremeter values of $x\ensuremath{\sim}0.001.$ For quantities other than the energy, such as the radial distribution functions and the momentum distributions, the dependence appears at any value of x. The occurrence of a maximum in the radial distribution function, in the momentum distribution, and in the excitation spectrum is a natural effect of the correlations when x increases. The asymptotic behaviors of the functions characterizing the structure of the systems are also investigated. The uniform limit approach is very easy to implement and provides a good description of the soft-sphere gas. Its reliability improves when the interaction weakens.
The energy and structure of dilute gases of hard spheres in three dimensions is discussed, together with some aspects of the corresponding 2D systems. A variational approach in the framework of the Hypernetted Chain Equations (HNC) is used starting from a Jastrow wavefunction that is optimized to produce the best two–body correlation factor with the appropriate long range. Relevant quantities describing static properties of the system are studied as a function of the gas parameter x = ρa d where ρ, a and d are the density, s–wave scattering length of the potential and dimensionality of the space, respectively. The occurrence of a maximum in the radial distribution function and in the momentum distribution is a natural effect of the correlations when x increases. Some aspects of the asymptotic behavior of the functions characterizing the structure of the systems are also investigated. PACS numbers: 03.75.Hh, 05.30.Jp, 67.40.Db. 1.
We obtain ab-initio estimations of the dynamic structure factor, S ( q ,ω), of Bose gases at zero temperature. More precisely, we use the Genetic Inversion via Falsification of Theories (GIFT) algorithm to perform analytic continuations of imaginary time correlation functions computed via an exact Path Integral projector method. Using the hard-sphere potential to model the two-body interactions between the atoms, we compute S ( q ,ω) changing the gas parameter from the dilute regime ( na 3 = 10 −4 ) up to the density corresponding to superfluid 4 He at equilibrium ( na 3 = 0.2138). With increasing density, we observe the emergence of a broad multiphonon contribution accompanying the quasiparticle peak and a crossover of the dispersion of elementary excitations from a Bogoliubov-like spectrum to a phonon-maxon- roton curve. Apart from the low wave vector region, for na 3 = 0.2138 the energy-momentum dispersion relation and the static density response function, χ( q ), turns out to be in good agreement with the superfluid 4 He experimental data at equilibrium density.
By using a full many-body approach, we calculate the excitation energy, the effective mass, and the density profile of soliton states in a three-dimensional Bose gas of hard spheres at zero temperature. The many-body wave function used to describe the soliton contains a one-body term, derived from the solution of the Gross-Pitaevskii equation, and a two-body Jastrow term, which accounts for the repulsive correlations between atoms. We optimize the parameters in the many-body wave function via a variational Monte Carlo procedure, calculating the grand-canonical energy and the canonical momentum of the system in a moving reference frame where the soliton is stationary. As the density of the gas is increased, significant deviations from the mean-field predictions are found for the excitation energy and the density profile of both dark and gray solitons. In particular, the soliton effective mass m∗{m}^{*}m∗ and the mass $m\mathrm{\ensuremath{\Delta}}N$ of missing particles in the region of the density depression are smaller than the result from the Gross-Pitaevskii equation, their ratio, however, being well reproduced by this theory up to large values of the gas parameter. We also calculate the profile of the condensate density around the soliton notch, finding good agreement with the prediction of the local-density approximation.
We study the ground state of a system of Bose hard spheres trapped in an isotropic harmonic potential to investigate the effect of the interatomic correlations and the accuracy of the Gross-Pitaevskii equation. We compare a local-density approximation, based on the energy functional derived from the low-density expansion of the energy of the uniform hard-sphere gas, and a correlated wave-function approach, which explicitly introduces the correlations induced by the potential. Both higher-order terms in the low-density expansion, beyond Gross-Pitaevskii, and explicit dynamical correlations have effects of the order of percent when the number of trapped particles becomes similar to that attained in recent experiments $(N\ensuremath{\sim}{10}^{7}).$