Quantum fluctuations and Gross-Pitaevskii theory

TL;DR

This paper uses the Gross-Pitaevskii equation to analyze quantum fluctuations and response functions in Bose-Einstein condensates, confirming Bogoliubov theory predictions and extending to beyond mean field effects.

Contribution

It demonstrates that Gross-Pitaevskii theory reproduces Bogoliubov results and extends to include quantum depletion and beyond mean field corrections.

Findings

Gross-Pitaevskii reproduces Bogoliubov predictions in uniform systems

Quantum depletion and infrared divergence are confirmed

Extension to non-uniform systems and beyond mean field effects discussed

Abstract

Using the linearized version of the time dependent Gross-Pitaevskii equation we calculate the dynamic response of a Bose-Einstein condensed gas to periodic density and particle perturbations. The zero temperature limit of the fluctuation-dissipation theorem is used to evaluate the corresponding quantum fluctuations induced by the elementary excitations in the ground state. In uniform conditions the predictions of Bogoliubov theory, including the infrared divergency of the particle distribution function and the quantum depletion of the condensate, are exactly reproduced by Gross-Pitaevskii theory. Results are also given for the crossed particle-density response function and the extension of the formalism to non uniform systems is discussed. The generalization of the Gross-Pitaevskii equation to include beyond mean field effects is finally considered and an explicit result for the chemical potential is found, in agreement with the prediction of Lee-Huang-Yang theory.