Quantum corrections to the ground state of a trapped Bose-Einstein condensate

In the mean-field approximation, the number density $\rho \left(\mathbf{r}\right)$ for the ground state of a Bose-Einstein condensate trapped by an external potential $V\left(\mathbf{r}\right)$ 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 for $\rho$ that take into account the dominant effects of quantum fluctuations. We also show that the gradient expansions for the density and for the condensate break down at fourth order and at second order, respectively.