Radially symmetric nonlinear states of harmonically trapped Bose-Einstein condensates

Starting from the spectrum of the radially symmetric quantum harmonic oscillator in two dimensions, we create a large set of nonlinear solutions. The relevant three principal branches, with $n_r=0,1$, and 2 radial nodes, respectively, are systematically continued as a function of the chemical potential and their linear stability is analyzed in detail, in the absence as well as in the presence of topological charge $m$, i.e., vorticity. It is found that for repulsive interatomic interactions only the ground state is linearly stable throughout the parameter range examined. Furthermore, this is true for topological charges $m=0$ or 1; solutions with higher topological charge can be unstable even in that case. All higher excited states are found to be unstable in a wide parametric regime. However, for the focusing (attractive) case the ground state with $n_r=0$ and $m=0$ can only be stable for a sufficiently low number of atoms. Once again, excited states are found to be generically unstable. For unstable profiles, the dynamical evolution of the corresponding branches is also followed to monitor the temporal development of the instability.

Figure 1

Linear stability analysis along the first four branches: ground state (no radial nodes) and 1st–3rd excited states ($n_r=1,2$, and 3 radial nodes) depicted in the curves from left to right in each panel. The top, middle, and bottom panels display $\sigma$ times the norm of the solution as a function of the chemical potential for $m=0$ (no topological charge), $m=1$ (singly charged), and $m=2$ (doubly charged) solutions. Each point on the branch is depicted by crosses for stable solutions and diamonds for unstable solutions. The areas of stability for $m=0,1$ appear only for $n_r=0$ and only for small powers for $\sigma =-1$ and all powers of the ground state for $\sigma =1$. In the case of $m=2$, the ground state with $n_r=0$ is only linearly stable for stronger nonlinearity per norm $NU$ in the case of $\sigma =1$.

Figure 2

(Color online) Linear stability eigenvalues for chargeless $(m=0)$ solutions. The left column of plots corresponds to the real part of the primary eigenvalue for $q=0,1,2,3$, and 4 for the ground state with $n_r=0$ (top panel) and first and second excited states with $n_r=1$ and 2 (middle and bottom panels, respectively) for $\sigma =-1$; while the right column of plots corresponds to the case of $\sigma =+1$. Note that the entire branch for the ground state for $\sigma =+1$ is stable, as shown in the upper right panel.

Figure 3

(Color online) Same as Fig. 2 for singly charged $(m=1)$ solutions.

Figure 4

(Color online) Same as Fig. 2 for doubly charged $(m=2)$ solutions. Notice that in this case the $n_r=0$ branch can be unstable even for the repulsive interactions, i.e., defocusing nonlinearity as shown in the top right panel.

Figure 5

(Color online) Data for the ground state $(n_r=0)$ for $m=0$ for $\sigma =-1$ (left panels) and $\sigma =+1$ (right panels). The top panels show the profile of the solution and the middle panels show the corresponding eigenvalues for $q=0,\dots ,50$ in Eq. (6) on the complex plane $({\lambda }_r,{\lambda }_i)$ of eigenvalues $\lambda ={\lambda }_r+i{\lambda }_i$; see Table for the correspondence between $q$ values and the different symbols used to depict the eigenvalues. The bottom panels depict the time evolution of the solution in harmonic oscillator units after an initial random perturbation of ${10}^{-2}$. Solutions in this figure correspond to $\mu =-2$ for $\sigma =-1$ and norm $N\mid U\mid =100$ for $\sigma =+1$. For all other figures we use $\mu =-0.5$ when $\sigma =-1$.

Figure 6

(Color online) Same as Fig. 5 $(m=0)$ for the first excited state $(n_r=1)$.

Figure 7

(Color online) Same as Fig. 5 $(m=0)$ for the second excited state $(n_r=2)$.

Figure 8

(Color online) Same as Fig. 5 for $m=1$.

Figure 9

(Color online) Same as Fig. 8 $(m=1)$ for the first excited state $(n_r=1)$.

Figure 10

(Color online) Same as Fig. 8 $(m=1)$ for the second excited state $(n_r=2)$.

Figure 11

(Color online) Same as Fig. 5 for $m=2$.

Figure 12

(Color online) Same as Fig. 11 $(m=2)$ for the first excited state $(n_r=1)$.

Figure 13

(Color online) Same as Fig. 11 $(m=2)$ for the second excited state $(n_r=2)$.

Figure 14

(Color online) Three-dimensional isodensity contour plots. Top plot: isodensity contour for the first excited state $n_r=1$ for $\sigma =+1$, $m=2$, i.e., the solution corresponding to the right panel in Fig. 12. One can observe is the rotation of the unstable $q=4$ mode due to the intrinsic vorticity $(m=2)$ of the solution. Bottom plot: isodensity contour for the ground state $n_r=0$ for $\sigma =-1$, $m=1$, i.e., the solution corresponding to the left panel in Fig. 8. The rotation of the $q=3$ mode is quickly arrested and the three spikes become stationary. The contours correspond to surfaces with density equal to half of the maximum density.

Figure 15

Recreated plot of Fig. 1(b) in Ref. 29 using spectral methods. The figure shows the unstable part of the main eigenvalue as a function of the nonlinear strength $N\times U$ for a doubly quantized vortex $(m=2)$.