Variational methods with coupled Gaussian functions for Bose-Einstein condensates with long-range interactions. II. Applications

Bose-Einstein condensates with an attractive $1/r$ interaction and with dipole-dipole interaction are investigated in the framework of the Gaussian variational ansatz introduced by S. Rau, J. Main, and G. Wunner [Phys. Rev. A 82, 023610 (2010)]. We demonstrate that the method of coupled Gaussian wave packets is a full-fledged alternative to direct numerical solutions of the Gross-Pitaevskii equation, or even superior in that coupled Gaussians are capable of producing both stable and unstable states of the Gross-Pitaevskii equation and thus of giving access to yet unexplored regions of the space of solutions of the Gross-Pitaevskii equation. As an alternative to numerical solutions of the Bogoliubov–de Gennes equations, the stability of the stationary condensate wave functions is investigated by analyzing the stability properties of the dynamical equations of motion for the Gaussian variational parameters in the local vicinity of the stationary fixed points. For blood-cell-shaped dipolar condensates it is shown that on the route to collapse the condensate passes through a pitchfork bifurcation, where the ground state itself turns unstable, before it finally vanishes in a tangent bifurcation.

Figure 1

(a) Mean-field energy $E_{\mathrm{mf}}$ for self-trapped condensates with attractive $1/r$ interaction as a function of the scattering length obtained by using up to five coupled Gaussian wave packets ($N=1,\dots ,5$, respectively) in comparison with the result of the accurate numerical solution of the stationary GPE (exact). (b) Same as (a) but for the chemical potential $\mu$.

Figure 2

Wave function $\psi (r)$ of (a) the stable ground state and (b) the excited state at scattering length $a_{\mathrm{sc}}=-1$ close to the exact numerical bifurcation. The calculations with coupled Gaussians converge rapidly to the exact numerical wave function (triangles).

Figure 3

Same as Fig. 2 but at scattering length $a_{\mathrm{sc}}=-0.6$.

Figure 4

The constituents of the five-Gaussian ansatz (Nos. $1,\dots ,5$) plotted separately. The superposition of the five Gaussians (coupled), agrees excellently with the exact numerical solution (exact) at scattering length $a_{\mathrm{sc}}=-1$.

Figure 5

First pair of eigenvalues as a function of the scattering length for an increasing number of coupled Gaussians and for the exact numerical calculation. (a) The two lowest eigenvalues of the stable ground state are purely imaginary. (b) The unstable excited state with a pair of purely real eigenvalues. Vanishing real or imaginary parts are not shown. The insets illustrate the rapid convergence of the variationally computed eigenvalues to the exact solutions of the Bogoliubov equations.

Figure 6

(a) Second pair of eigenvalues as a function of the scattering length for varying number of coupled Gaussians and for the exact numerical calculation for the stable ground state and the unstable excited state. Numerically exact eigenvalues (triangles) are only available for the stable ground state. All eigenvalues are purely imaginary; vanishing real parts are not shown. (b) Same as (a) but for the third pair of eigenvalues.

Figure 7

Deviations of the ground-state wave function according to the eigenvectors of the (a) first, (b) second, and (c) third pair of eigenvalues $\pm \lambda$ of the stability analysis at scattering length $a_{\mathrm{sc}}=-0.8$.

Figure 8

(a) Mean-field energy and (b) chemical potential as a function of the scattering length, variational solution (v) using two Gaussians compared with an exact full-numerical lattice calculation (n1). For two Gaussians $(N=2)$, the ground state (v1) and the excited state (v2) emerge in a tangent bifurcation at $a_{\mathrm{sc}}=-0.034\hspace{0.5em}202$. The unstable branch of the chemical potential in (b) has a maximum at ${\mu }_{max}\approx 170\hspace{0.5em}000$ (not shown) and then vanishes as a nearly vertical line at $a_{\mathrm{sc}}=1/6$.

Figure 9

Eigenvalues of the Jacobian for the variational solution with one Gaussian as a function of the scattering length $a_{\mathrm{sc}}$. (a) The ground state is stable, all eigenvalues are purely imaginary; (b) the excited state is unstable since there are real parts of eigenvalues, emerging in a tangent bifurcation at $a_{\mathrm{sc}}^{\mathrm{cr}}=-0.037\hspace{0.5em}891\hspace{0.5em}7$. Eigenvalues which do not reach $\lambda =0$ at $a_{\mathrm{sc}}^{\mathrm{cr}}$ match with the corresponding eigenvalues of the stable and unstable state, respectively.

Figure 10

Same as Fig. 9 but for the variational calculation with two coupled Gaussians: (a) the stable ground state and (b) the unstable excited state. Additional eigenvalues are revealed. The tangent bifurcation is shifted to $a_{\mathrm{sc}}^{\mathrm{cr}}=-0.034\hspace{0.5em}202$. Eigenvalues which do not reach $\lambda =0$ at $a_{\mathrm{sc}}^{\mathrm{cr}}$ match with the corresponding eigenvalues of the stable and unstable state, respectively.

Figure 11

(a) Convergence of the mean-field energy (to the solid line) with increasing number of coupled Gaussian wave functions for $a_{\mathrm{sc}}=0$. The mean-field energy for four coupled functions lies already energetically lower than the numerical value of the lattice computation (dashed line). (b) The converged wave function $\psi$ as a function of the transverse coordinate $\varrho$ at different $z$ coordinates. The variational solution and the numerical lattice solution are denoted $v$ and $n$, respectively.

Figure 12

(a) Overview of the mean-field energy for the bifurcation scenario revealed in the variational calculation. For comparison, some values of the numerical lattice solution for the ground state are presented (see text for more details). The stability eigenvalues in the regions around the critical scattering lengths $a_{\mathrm{sc}}^{\mathrm{cr},\mathrm{t}}=-0.012\hspace{0.5em}24$ and $a_{\mathrm{sc}}^{\mathrm{cr},\mathrm{p}}=-0.003\hspace{0.5em}59$ marked by the black frames in (a) are shown in (b) and (c), respectively. In (b) I and II denote the two merging branches. Only the eigenvalues involved in the bifurcation are shown (for analysis and interpretation, see text). Eigenvalues of the Jacobian as a function of the scattering length in (c) indicate the stability change of the ground state in a pitchfork bifurcation. The unstable state for $a_{\mathrm{sc}}<a_{\mathrm{sc}}^{\mathrm{cr},\mathrm{p}}=-0.003\hspace{0.5em}59$ turns into the stable ground state for $a_{\mathrm{sc}}>a_{\mathrm{sc}}^{\mathrm{cr},\mathrm{p}}$ in a pitchfork bifurcation. Subfigure (c) only shows the lowest eigenvalues, those involved in the stability change.

Figure 13

Contour plot of the mean-field energy (a) for $a_{\mathrm{sc}}=-0.0036$ closely below, and (b) for $a_{\mathrm{sc}}=-0.00358$ closely above the pitchfork bifurcation. The eigenvectors corresponding to the eigenvalues in Fig. 12c linearize the vicinity of the fixed point (${\delta }_1$ and ${\delta }_2$, arbitrary units).

Figure 14

Time evolution of the perturbed particle density $\left|\psi \right|{}^2$ of the unstable stationary solution as a function of $x$ and $y$ for $z=0$ for (a) $t=0.000\hspace{0.5em}1$, (b) $t=0.005$, and (c) $t=0.006$.