Fragmented Metastable States Exist in an Attractive Bose-Einstein Condensate for Atom Numbers Well above the Critical Number of the Gross-Pitaevskii Theory

It is well known that attractive condensates do not posses a stable ground state in three dimensions. The widely used Gross-Pitaevskii theory predicts the existence of metastable states up to some critical number $N_{\mathrm{cr}}^{\mathrm{GP}}$ of atoms. It is demonstrated here that fragmented metastable states exist for atom numbers well above $N_{\mathrm{cr}}^{\mathrm{GP}}$. The fragments are strongly overlapping in space. The results are obtained and analyzed analytically as well as numerically. The implications are discussed.

Figure 1

Energies per particle $E/N$ as a function of the scaling parameter ${\sigma }_0$. Shown are the GP energies for $N<N_{\mathrm{cr}}^{\mathrm{GP}}$ and $N>N_{\mathrm{cr}}^{\mathrm{GP}}$ (in black) and the energy of the fragmented BEC for $N>N_{\mathrm{cr}}^{\mathrm{GP}}$ [upper solid (red) curve]; see the text for more details. $N$ is indicated on the respective curves. The relative fragmentation is $\frac{n_1}{N}=0.15$. The scenario analyzed corresponds to ${}^{85}\mathrm{Rb}$ atoms in an isotropic harmonic trap of frequency $\omega =2\pi ·15\mathrm{Hz}$ and $s$-wave scattering length of $a_s=-7.14a_0$, where $a_0$ is the Bohr radius, which are experimentally accessible [2]. For these parameters $N_{\mathrm{cr}}^{\mathrm{GP}}=5000$. The energy is expressed in units of $\hslash \omega$ and length in units of $\frac{1}{\beta }=\sqrt{\frac{\hslash }{m\omega }}$. Transforming to dimensionless units in which $\hslash =m=\omega =1$, the corresponding interaction strength is ${\lambda }_0=4\pi a_s\beta =-0.001684$.

Figure 2

The energy per particle $E({\sigma }_0,{\sigma }_1)/N$ as a function of ${\sigma }_0$ and ${\sigma }_1$ [see Eq. (6)] compared to the respective GP energy [see Eq. (2)]. $N=6000$. The quantities used are the same as in Fig. 1. For these quantities the minimum of $E$ is at ${\sigma }_0=0.816$ and ${\sigma }_1=0.811$.

Figure 3

The critical number $N_{\mathrm{cr}}$ obtained with $E({\sigma }_0,{\sigma }_1)$ of Eq. (6) [solid (black) dots] and with the analytic result of Eq. (7) [dashed (red) curve] as a function of the relative fragmentation $\frac{n_1}{N}$. The quantities used are the same as in Fig. 1. $N_{\mathrm{cr}}^{\mathrm{GP}}$ is indicated by an arrow.

Figure 4

The radial orbitals ${\phi }_0(r)$ and ${\phi }_1(r)$ obtained fully numerically by minimizing the energy functional (4) for $N=1.7N_{\mathrm{cr}}^{\mathrm{GP}}$ [dashed (black) curve]. The results are compared with those of the “Gaussian model” using Eq. (6) [solid (red) curve]. Here, ${\sigma }_0=0.797$ and ${\sigma }_1=0.788$ at the minimum of the energy in Eq. (6). The relative fragmentation is $\frac{n_1}{N}=0.25$. The other quantities used are the same as in Fig. 1. The inset shows the density $\rho (\overrightarrow{r})$ of the fragmented BEC [dashed (black) curve]. Shown for comparison is the density obtained by putting all the bosons in ${\phi }_0(r)$, thus mimicking the shape of a typical GP density [solid (blue) curve].