Self-consistent calculation of the coupling constant in the Gross-Pitaevskii equation

TL;DR

This paper introduces a self-consistent method to evaluate the coupling constant in the Gross-Pitaevskii equation, accounting for medium effects and trapping potentials without pseudopotential replacement, applicable to low-dimensional and crossover regimes.

Contribution

It presents a novel variational approach based on a simplified Hartree-Fock-Bogoliubov approximation that avoids unphysical divergences and allows analytical estimations in various trapping scenarios.

Findings

Coupling constant renormalization due to medium effects and trapping potentials.

Analytical estimations for low-dimensional and crossover regimes.

Significant contributions of short-range correlations to energy calculations.

Abstract

A method is proposed for a self-consistent evaluation of the coupling constant in the Gross-Pitaevskii equation without involving a pseudopotential replacement. A renormalization of the coupling constant occurs due to medium effects and the trapping potential, e.g. in quasi-1D or quasi-2D systems. It is shown that a simplified version of the Hartree-Fock-Bogoliubov approximation leads to a variational problem for both the condensate and a two-body wave function describing the behaviour of a pair of bosons in the Bose-Einstein condensate. The resulting coupled equations are free of unphysical divergences. Particular cases of this scheme that admit analytical estimations are considered and compared to the literature. In addition to the well-known cases of low-dimensional trapping, cross-over regimes can be studied. The values of the kinetic, interaction, external, and release energies in low dimensions are also evaluated and contributions due to short-range correlations are found to be substantial.