The Gross-Pitaevskii equations and beyond for inhomogeneous condensed bosons

TL;DR

This paper derives the static Gross-Pitaevskii equation from an energy principle, discusses its limitations in describing inhomogeneous Bose gases, and explores an alternative integral equation approach based on the Bogoliubov-de Gennes equations.

Contribution

It provides a simple derivation of the static GP equation, reviews experimental applications, and introduces an integral equation approach to address GP limitations.

Findings

Derivation of static GP equation from energy variational principle

Review of experimental tests of GP predictions

Proposal of an integral equation based on Bogoliubov-de Gennes equations

Abstract

A simple derivation of the static Gross-Pitaevskii (GP) equation is given from an energy variational principle. The result is then generalized heuristically to the time-dependent GP form. With this as background, a number of different experimental areas explored very recently are reviewed, in each case contact being established between the measurements and the predictions of the GP equations. The various limitations of these equations as used on dilute inhomogeneous condensed Boson atomic gases are then summarized, reference also being made to the fact that there is no many-body wave function underlying the GP formulation. This then leads into a discussion of a recently proposed integral equation, derived by taking the Bogoliubov-de Gennes equation as starting point. Some limitations of the static GP differential equation are thereby removed, though it is a matter of further study to determine whether a correlated wave function exists as underpinning for the integral equation formulation.