Quantum Corrections to the Ground State of a Trapped Bose-Einstein Condensate

TL;DR

This paper derives local quantum correction terms to the Gross-Pitaevskii equation for a trapped Bose-Einstein condensate, highlighting the dominant quantum fluctuations and the limitations of gradient expansion.

Contribution

It introduces a second-order gradient expansion to include quantum fluctuation effects in the mean-field description of Bose-Einstein condensates.

Findings

Quantum fluctuations dominate corrections at wavelengths of order 1/rom ho a

Derived local correction terms to the Gross-Pitaevskii equation

Gradient expansion breaks down at fourth order

Abstract

In the mean-field approximation, the number density \rho(r) for the ground state of a Bose-Einstein condensate trapped by an external potential V(r) satisfies a classical field equation called the Gross-Pitaevskii equation. We show that quantum corrections to \rho are dominated by quantum fluctuations with wavelengths of order 1/\sqrt{\rho a}, where a is the S-wave scattering length. By expanding the equations for the Hartree-Fock approximation to second order in the gradient expansion, we derive local correction terms to the Gross-Pitaevskii equation that take into account the dominant effects of quantum fluctuations. We also show that the gradient expansion for the density breaks down at fourth order.