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

Abstract

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 $\hat{\Psi }(\mathbf{r},t)={\hat{a}}_0\Psi (\mathbf{r},t)/\sqrt{N}$, where $N$ is the number of Bose particles, we find the equation $i\hslash \frac{\partial \Psi (\mathbf{r},t)}{\partial t}=-\frac{{\hslash }^2}{2m}\frac{{\partial }^2\Psi (\mathbf{r},t)}{\partial {\mathbf{r}}^2}+\left(1-\frac{1}{N}\right)2c\Psi (\mathbf{r},t)|\Psi (\mathbf{r},t){|}^2$, which we call the GP${}_N$ equation. It differs from the GP equation by the factor $\left(1-1/N\right)$. We compare the accuracy of the GP and GP${}_N$ equations by analysing 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 the mean particle density $\bar{\rho }=1$, different values of $N$ and of the coupling constant $\gamma =c/\bar{\rho }$. The solutions are compared with the exact ones obtained by the Bethe ansatz. The results show that the GP and GP${}_N$ equations equally well describe the many-particle system ($N\gtrsim 100$) being in the weak coupling regime ($N^{-2}\ll \gamma \lesssim 0.1$). But for the few-boson system ($N\lesssim 10$) with $\gamma \lesssim N^{-2}$ the solutions of the GP${}_N$ equation are in much better agreement with the exact ones. Thus, the multiplier $\left(1-1/N\right)$ 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.

19 pages, 7 figures v2: small changes in sections 1 and 2, references are added; v3: minor changes, accepted version; additionally, a mathematical error in the Conclusion, which also penetrated into the published version, has been corrected ($N$ has been replaced with $\infty$)

References in corpus (2)

• Quantum dark solitons in ultracold one-dimensional Bose and Fermi gases
• Nonuniform Bose-Einstein condensate. II. Doubly coherent states

Cited by in corpus (2)

• Nonuniform Bose-Einstein condensate. II. Doubly coherent states
• Why a Bose-Einstein condensate cannot exist in a system of interacting bosons at ultrahigh temperatures