Nonuniform Bose-Einstein condensate. I. An improvement of the Gross-Pitaevskii method

A nonuniform condensate is usually described by the Gross-Pitaevskii (GP) equation, which is derived with the help of the c-number ansatz $\hat{ \Psi}(\mathbf{r},t)=\Psi (\mathbf{r},t)$. Proceeding from a more accurate operator ansatz $\hat{\Psi}(\mathbf{r},t)=\hat{a}_{0}\Psi (\mathbf{r},t) \sqrt{N}$, we find the equation $i\hbar \frac{\partial \Psi (\mathbf{r},t)}{\partial t}=-\frac{\hbar ^{2}}{2m}\frac{\partial ^{2}\Psi (\mathbf{r},t)}{\partial \mathbf{r}^{2}}+\left( 1-\frac{1}{N}\right) 2c\Psi (\mathbf{r},t)|\Psi(\mathbf{r},t)|^{2}$ (the GP$_{N}$ equation). It differs from the GP equation by the factor $(1-1/N)$, where $N$ is the number of Bose particles. We compare the accuracy of the GP and GP$_{N}$ equations by analyzing the ground state of a one-dimensional system of point bosons with repulsive interaction ($c>0$) and zero boundary conditions. Both equations are solved numerically, and the system energy $E$ and the particle density profile $\rho (x)$ are determined for various values of~$N$, the mean particle density $\bar{\rho}$, and the coupling constant $\gamma =c/\bar{\rho}$. The solutions are compared with the exact ones obtained by the Bethe ansatz. The results show that in the weak coupling limit ($N^{-2}\ll \gamma \lesssim 0.1$), the GP and GP$_{N}$ equations describe the system equally well if $N\gtrsim 100$. For few-boson systems ($N\lesssim 10$) with $\gamma \lesssim N^{-2}$ the solutions of the GP$_{N}$ equation are in excellent agreement with the exact ones. That is, the multiplier $(1-1/N)$ allows one to describe few-boson systems with high accuracy. This means that it is reasonable to extend the notion of Bose-Einstein condensation to few-particle systems.