Attractive Bose gas in two dimensions: An analytical study of its fragmentation and collapse

An attractive Bose-Einstein condensate in two spatial dimensions is expected to collapse for supercritical values of the interaction strength. Moreover, it is known that for nonzero quanta of angular momentum and infinitesimal attraction the gas prefers to fragment and distribute its angular momentum over different orbitals. In this work we examine the two-dimensional trapped Bose gas for finite values of attraction and describe the ground state in connection to its angular momentum by theoretical methods that go beyond the standard Gross-Pitaevskii theory. By applying the best-mean-field approach over a variational ansatz whose accuracy has been checked numerically, we derive analytical relations for the energy, the fragmentation of the ground states, and the critical (for collapse) value of the attraction strength as a function of the total angular momentum $L$.

Figure 1

The LLL is the optimal basis for a given nonzero AM $L$. The left panel shows the total energy per particle $\epsilon$ for $\lambda =5$ and $L/N=0.6$ as a function of each of the six (relative) occupations ${\alpha }_i$ of the non-LLL, while all the rest are kept to zero. Any variation of these occupations increases the total energy of the system. In the right panel we plot, for comparison, the dependence of the energy, for the same total AM, on the occupations of the LLL with $m=2$ (orange, lower line) and $m=3$ (black, upper line). A clear minimum can be seen at a nonzero value of $\alpha$. All calculations are done at the optimal values of $\sigma$ for $N=6000$ particles.

Figure 2

Distributions of the occupation numbers (upper panel) and density $\rho \left(r\right)$, i.e., the diagonal of the single-particle reduced density matrix (lower panel) for three different states, all with $\mathcal{L}=1$ and $N=12$. The blue (solid, thick) line corresponds to the ground state of Eq. (13), the red (dashed) line to the ground state found in Ref. [6], and the green (dotted, fade) to the vortex state [see Eq. (22)].

Figure 3

Occupation numbers of the ground state for different values of $\mathcal{L}$, as found within the BMF theory (blue solid) and as given by Wilkin et al. [6] (red dashed). The agreement is good for an approximate value of $\mathcal{L}\lesssim 0.5$. Above this value the two distributions take on completely different appearances, even though the energies of the configurations are almost equal. For instance, for a weak interaction $\lambda =0.01$ the energy difference of the two is $\Delta E\sim {10}^{-4}$. The number of particles here is $N=12$.