Behavior of a Bose-Einstein condensate containing a large number of atoms interacting through a finite-range interatomic interaction

We investigate Bose-Einstein condensates (BEC) containing a large number of bosonic atoms interacting via a finite-range semirealistic interatomic interaction. Ground state properties for an increasing number of atoms in the condensate have been calculated using a modification of the potential harmonic expansion method (including a short range correlation function in the expansion basis) to solve the many-body Schrödinger equation. An improved numerical algorithm for the calculation of the potential matrix elements permits us to have up to 14 000 atoms in the condensate. Although our approach is approximate and justified for dilute condensates, our results agree well with available diffusion Monte Carlo results for the same case. The ground state energies also agree well with those by the Gross-Pitaevskii equation method for up to 100 particles in the trap and become gradually larger than the latter (up to 5% for 14 000 atoms). The difference is attributed to the effects of finite range interatomic interaction and two-body correlations. Our approach presents a clear physical picture of the condensate, being computationally economical at the same time.

Figure 1

Comparison of ground state energy per particle by PHEM and GP for ${}^{87}\mathrm{Rb}$ using $V_0{\mathrm{sech}}^2(r_{ij}∕r_0)$ potential. Note the logarithmic scale for the number of particles in the condensate.

Figure 2

Plot of wave function ${\zeta }_0\left(r\right)$ (vertical peaks) and the lowest eigenpotential ${\omega }_0\left(r\right)$ as a function of hyper-radius for 20, 500, and 1000 atoms of ${}^{87}\mathrm{Rb}$. The actual value of ${\omega }_0\left(r\right)$ can be obtained by multiplying the vertical scale by 4000.0. The peak of the wave function occurs at the position of the minimum of lowest eigenpotential.

Figure 3

Plot of $⟨V_{\mathit{\text{trap}}}⟩$, $⟨V⟩$, and $⟨T⟩$ per particle as a function of the number of particles $A$ for ${}^{87}\mathrm{Rb}$ $(a_s=0.00433\mathrm{o.u.})$. Note that the calculated points are joined by lines to guide the eye.

Figure 4

Comparison of ground state energy per particle by PHEM and GP equation for ${}^{23}\mathrm{Na}$ using $V_0{\mathrm{sech}}^2(r_{ij}∕r_0)$ potential.

Figure 5

Plot of $⟨V_{\mathit{\text{trap}}}⟩$, $⟨V⟩$, and $⟨T⟩$ per particle as a function of the number of particles $A$ for ${}^{23}\mathrm{Na}$ $(a_s=0.00284\mathrm{o.u.})$. Note that the calculated points are joined by lines to guide the eye.