Dynamics and stability of Bose-Einstein condensates with attractive $1/r$ interaction

The time-dependent extended Gross-Pitaevskii equation for Bose-Einstein condensates with attractive $1/r$ interaction is investigated with both a variational approach and numerically exact calculations. We show that these condensates exhibit signatures known from the nonlinear dynamics of autonomous Hamiltonian systems. The two stationary solutions created in a tangent bifurcation at a critical value of the scattering length are identified as elliptical and hyperbolical fixed points, corresponding to stable and unstable stationary states of the condensate. The stable stationary state is surrounded by elliptical islands, corresponding to condensates periodically oscillating in time, whereas condensate wave functions in the unstable region undergo a collapse within finite time. For negative scattering lengths below the tangent bifurcation no stationary solutions exist, i.e., the condensate is always unstable and collapses.

Figure 1

(Color online) Chemical potentials of the ground state $\left({\epsilon }_{+}\right)$ and the nodeless excited state $\left({\epsilon }_{-}\right)$ in the variational calculation. They emerge in a tangent bifurcation at a critical value of the scaled scattering length $a_{\text{cr}}=-3\pi /8=-1.1780$ [12].

Figure 2

(Color online) Eigenvalues of the linearization of the extended Gross-Pitaevskii system (3) for the two stationary solutions emerging at the tangent bifurcation. Both the variational (var) and the numerically exact (num) solutions are shown. The two eigenvalues ${\lambda }_{+}$ of the stationary ground state (cf. Ref. [3]) are purely imaginary demonstrating that the state is an elliptic fixed point. In contrast to this finding, the two eigenvalues ${\lambda }_{-}$ of the nodeless excited state are purely real. One is positive, the other is negative. The state is an hyperbolic fixed point. Vanishing real or imaginary parts are not shown.

Figure 3

(Color online) Phase portraits of the dynamics obtained from time-dependent trajectories of the complex function $A\left(t\right)$. (a) For $a=-1$ the two stationary solutions appear as fixed points. (b) They coalesce for $a=-1.18$. (c) For $a=-1.3$, below the bifurcation point, there are no stationary solutions.

Figure 4

(Color online) Periodically oscillating condensate for the scattering length $a=-1$ and the initial condition $A_i\left(0\right)=0.3$.

Figure 5

(Color online) Collapse of the unstable eigenstate. (a) Scaled scattering length $a=-1$ and initial condition close to the hyperbolic fixed point at $A_r=0$, $A_i=0.3787$. The collapse after finite time is clearly visible in the inset, but depends sensitively on the initial conditions (see text). (b) Collapse of the nonstationary state at $a=-1.3$ with initial condition $A_i\left(0\right)=\frac{1}{6a}+\frac{\pi }{8a^2}=0.10416$.

Figure 6

(Color online) “Potential” part of the mean-field energy for different scattering lengths. This potential part of the mean field energy (28) agrees with the complete mean-field energy for stationary states of Ref. [1] (see text). The motion of the Gaussian wave function is interpreted as the one-dimensional motion of a classical particle in the potential $V\left(q\right)$.

Figure 7

(Color online) Phase portrait of the mean-field energy formulated in the canonical variables $q$ and $p$ for $a=-0.8$.

Figure 8

(Color online) (a) Width of a slightly perturbed stationary state ($a=-0.85$, $f=1.001$) as a function of time. (b) Wave functions of the state for selected times marked by circles in (a). The collapse of the condensate is obvious from both plots.

Figure 9

(Color online) Evolution of the unstable “stationary” state $\left(f=1\right)$ at $a=-1.0$. (a) Due to numerical deviations an oscillation of the width starts for times larger than $t\approx 25$. (b) Corresponding wave functions for the times marked in (a).

Figure 10

(Color online) (a) The width of the condensate, initially in the vicinity of the unstable fixed point with $f=0.99$ and $a=-0.85$ increases linearly with time for $t\gtrsim 65$. (b) Wave functions of the same state for different times marked by circles in (a). The wave functions at $t=240$ and $t=400$ seem to be sharper than the wave function at $t=20$. Note that a larger step size on the grid in position space than in Figs. 8, 9 was used. Therefore there is a visible distance between $r=0$ and the first point on the grid.

Figure 11

(Color online) Double logarithmic plot of the wave functions presented in Fig. 10b. The long range tail occurring with increasing propagation times and leading to a monotonically increasing width in Fig. 10a becomes visible.

Figure 12

(Color online) (a) Quasiperiodically oscillating condensate at a scaled scattering length of $a=-0.85$ and $f=1.01$. (b) An initially quasi periodically oscillating condensate at $a=-0.85$, $f=1.25$ turns into expanding dynamics after long time.