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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2310.18528 |
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| _version_ | 1866912131303079936 |
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| author | Tomchenko, Maksim |
| author_facet | Tomchenko, Maksim |
| contents | A nonuniform condensate is usually described by the Gross-Pitaevskii (GP) equation, which is derived with the help of the c-number ansatz $\hat{ Ψ}(\mathbf{r},t)=Ψ(\mathbf{r},t)$. Proceeding from a more accurate operator ansatz $\hatΨ(\mathbf{r},t)=\hat{a}_{0}Ψ(\mathbf{r},t) \sqrt{N}$, we find the equation $i\hbar \frac{\partial Ψ(\mathbf{r},t)}{\partial t}=-\frac{\hbar ^{2}}{2m}\frac{\partial ^{2}Ψ(\mathbf{r},t)}{\partial \mathbf{r}^{2}}+\left( 1-\frac{1}{N}\right) 2cΨ(\mathbf{r},t)|Ψ(\mathbf{r},t)|^{2}$ (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 $ρ(x)$ are determined for various values of~$N$, the mean particle density $\barρ$, and the coupling constant $γ=c/\barρ$. The solutions are compared with the exact ones obtained by the Bethe ansatz. The results show that in the weak coupling limit ($N^{-2}\ll γ\lesssim 0.1$), the GP and GP$_{N}$ equations describe the system equally well if $N\gtrsim 100$. For few-boson systems ($N\lesssim 10$) with $γ\lesssim N^{-2}$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_18528 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Nonuniform Bose-Einstein condensate. I. An improvement of the Gross-Pitaevskii method Tomchenko, Maksim Quantum Gases A nonuniform condensate is usually described by the Gross-Pitaevskii (GP) equation, which is derived with the help of the c-number ansatz $\hat{ Ψ}(\mathbf{r},t)=Ψ(\mathbf{r},t)$. Proceeding from a more accurate operator ansatz $\hatΨ(\mathbf{r},t)=\hat{a}_{0}Ψ(\mathbf{r},t) \sqrt{N}$, we find the equation $i\hbar \frac{\partial Ψ(\mathbf{r},t)}{\partial t}=-\frac{\hbar ^{2}}{2m}\frac{\partial ^{2}Ψ(\mathbf{r},t)}{\partial \mathbf{r}^{2}}+\left( 1-\frac{1}{N}\right) 2cΨ(\mathbf{r},t)|Ψ(\mathbf{r},t)|^{2}$ (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 $ρ(x)$ are determined for various values of~$N$, the mean particle density $\barρ$, and the coupling constant $γ=c/\barρ$. The solutions are compared with the exact ones obtained by the Bethe ansatz. The results show that in the weak coupling limit ($N^{-2}\ll γ\lesssim 0.1$), the GP and GP$_{N}$ equations describe the system equally well if $N\gtrsim 100$. For few-boson systems ($N\lesssim 10$) with $γ\lesssim N^{-2}$ 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. |
| title | Nonuniform Bose-Einstein condensate. I. An improvement of the Gross-Pitaevskii method |
| topic | Quantum Gases |
| url | https://arxiv.org/abs/2310.18528 |