Simple model for interactions and corrections to the Gross-Pitaevskii equation

One of the assumptions leading to the Gross-Pitaevskii equation (GPE) is that the interaction between atom pairs can be written effectively as a $\delta$ function so that the interaction range of the particles is assumed to vanish. A simple model that takes into account the extension of the interparticle potential is introduced. The correction to the GPE predictions for the energy of a condensate confined by a harmonic trap in the Thomas-Fermi regime is estimated. Although it is found to be small, we believe that in some situations it can be measured using its dependance on the frequency of the confining trap. Due to the simplicity of the model, it may have a wide range of applications.

Figure 1

Ground state of rubidium atoms $m=87\left(\mathrm{amu}\right)$ for various interaction parameters $g$ marked in the figure. The results for $g=0.3\left(\mathrm{Hz}\mathrm{m}\right)$ are colored while the ones for stronger or weaker nonlinearities are marked by light gray. If $g$ is too small, the TF approximation is not valid and there is a big difference between the theory and the numerical results. (a) Ground-state wave function of the GPE calculated with the help of (37) (solid line) and the Thomas-Fermi approximation (31) or (32) (dashed line) for $\omega =1\left(\mathrm{kHz}\right)$. (b) Solid line: Calculated density deviations [using (37)] due to nonzero van der Waals radius $\rho \left[l=5\left(\mathrm{nm}\right)\right]-{\rho }_0\left(l=0\right)$, $\omega =1\left(\mathrm{kHz}\right)$. Dashed green line: simple approximation (B3) for the deviations. Here $d=0.3\left(\mu \mathrm{m}\right)$ [for $g=0.3\left(\mathrm{Hz}\mathrm{m}\right)]$, as can be found from (20) and (34). Note that the deviations between the TF results and the GP ones as well as between $\rho$ and ${\rho }_0$ are largest in a region of size $d$ around $x=R$. The deviations for $g=7.5\times {10}^{-4}\left(\mathrm{Hz}\mathrm{m}\right)$ are divided by 10 in order to make the figure clear. (c) Density deviations at the center $x=0$ of the trap $\Delta \rho ={\rho }_0\left(l=0\right)-\rho \left[l=5\left(\mathrm{nm}\right)\right]$ as a function of the trapping frequency $\omega$. The solid line is the prediction (35) and the stars are the numerically calculated values using (37). The scale is logarithmic.

Figure 2

Energy of the ground state of rubidium BEC as a function of the trap frequency $\omega$ for various interaction parameters $g$ marked in the figure. The results for $g=0.3\left(\mathrm{Hz}\mathrm{m}\right)$ are colored while the ones for stronger or weaker nonlinearities are marked by light gray. (a) Energy for $l=0$. solid line: the prediction (27); stars: the calculated values integrating (37). (b) The correction to the energy $\Delta E\left(l\right)=E\left[l=5\left(\mathrm{nm}\right)\right]-E\left(l=0\right)$. Solid red line: the prediction (48); stars: the calculated values integrating (37).