Quantum fluctuations and Gross-Pitaevskii theory

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.