Pitchfork bifurcations in blood-cell-shaped dipolar Bose-Einstein condensates

We demonstrate that the method of coupled Gaussian wave packets is a full-fledged alternative to direct numerical solutions of the Gross-Pitaevskii equation of condensates with electromagnetically induced attractive $1/r$ interaction or with dipole-dipole interaction. Moreover, Gaussian wave packets are superior in that they are capable of producing both stable and unstable stationary solutions and thus of giving access to yet unexplored regions of the space of solutions of the Gross-Pitaevskii equation. We apply the method to clarify the theoretical nature of the collapse mechanism of blood-cell-shaped dipolar condensates: 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

Chemical potential $\mu$ for self-trapped condensates with attractive $1/r$ interaction as a function of the scaled scattering length $N^{2}a$ obtained by using up to five Gaussian wave packets in comparison with the result of the exact numerical solution of the stationary Gross-Pitaevskii equation. Note that all forms yield a tangential bifurcation diagram, with a stable (upper) and an unstable (lower) branch. The inclusion of three Gaussians already well reproduces the exact numerical result.

Figure 2

(a) Convergence of the mean-field energy with increasing number of coupled Gaussian wave packets (squares) and comparison with the value obtained by a lattice calculation with grid size $128\times 512$ (dashed line), which lies energetically higher than the exact converged variational solution (solid line). (b) Comparison of the variational wave function for six coupled Gaussians (solid curves) with values of the numerical one (triangles) at different $z$ coordinates. Both solutions show a biconcave-shaped condensate. The figures are for (particle number scaled) trap frequencies $N^2{\gamma }_z=25200$ and $N^2{\gamma }_{\rho }=3600$, and scattering length $a=0$.

Figure 3

(a) Mean-field energy of a dipolar condensate for (particle number scaled) trap frequencies $N^2{\gamma }_z=25\hspace{0.5em}200$ and $N^2{\gamma }_{\rho }=3600$ as a function of the scattering length. In the variational calculation with one Gaussian a stable ground state (${\mathrm{g}}^{N=1}$) and an unstable excited state (${\mathrm{e}}^{N=1}$) emerge in a tangent bifurcation. Using coupled Gaussians two unstable states emerge (labeled ${\mathrm{u}}^{\mathrm{coupled}}$), of which the lower one turns into a stable ground state (${\mathrm{g}}^{\mathrm{coupled}}$) in a pitchfork bifurcation. (b) Stability eigenvalues $\lambda$ of the pitchfork bifurcation point for calculations with six coupled Gaussians, scattering length in rectangle marked in (a). Real and imaginary parts of two selected eigenvalues of the Jacobian (7) as a function of the scattering length. For $a<a_{\mathrm{cr}}^{\mathrm{p}}=-0.00359$ the solution is unstable with one pair of real eigenvalues. At $a_{\mathrm{cr}}^{\mathrm{p}}$ the real eigenvalues vanish in a pitchfork bifurcation and a stable ground state forms with purely imaginary eigenvalues. Only those eigenvalues involved in the stability change are shown.

Figure 4

Contour plot of the mean-field energy with the eigenvectors corresponding to the eigenvalues of Fig. 3b linearizing the vicinity of the fixed point (${\delta }_1,{\delta }_2$ in arbitrary units). The figure shows $a=-0.0036$ close below the pitchfork bifurcation point, showing three fixed points: two stable and one hyperbolic.