Figure 1

(a)–(c) The position density ${\left|\psi \right|}^2$ at $t=1999$ for $\beta =1.89\times {10}^{-4}$ with different values of $u$: (a) $u=-0.2586$, (b) $u=0.1552$, and (c) $u=0.50$. Note that increasing repulsive interaction strength can lead to localization. (d) Values of ${\sigma }_x{\sigma }_p$ for different values of $u$ and $\beta$. Bright regions indicate delocalized behavior (large ${\sigma }_x{\sigma }_p$), and dark regions indicate localized behavior (small ${\sigma }_x{\sigma }_p$). The solid lines (blue) are theoretical predictions separating regions of poor initial confinement and strong initial confinement.Reuse & Permissions

Figure 2

Contours of $〈H〉$ (black/dotted curves) for various values of interaction strength $u$: (a) $u=-0.20$, (b) $u=-0.02$, (c) $u=0.10$, and (d) $u=0.5$ with $\beta =2\times {10}^{-4}$. Separatrices are shown in bold, separating free-streaming and confined trajectories. In red/solid curves are numerically calculated trajectories under the influence of an external potential of the form in Eq. (4). The trajectories consist of short-time (fast oscillations) and long-time behavior.Reuse & Permissions

Figure 3

The Husimi functions at (a), (b) $t=0$ and at (c), (d) $t=1999$ under the influence of the GP equation. In (a) and (b), the classical separatrix from the early-time model is plotted in bold. In (a), $u=0.5$, $\beta =2\times {10}^{-4}$, and the initial condition is well contained inside the classical separatrix found with the early-time model. In (b), $u=0.1$, $\beta =2\times {10}^{-4}$, however, the initial condition extends well beyond the classical separatrix. (c) demonstrates that the initially well-contained wave packet continues to remain localized at late times, while (d) demonstrates that the poorly contained wave packet has spread in the potential (color scale narrowed to show detail). [In (c) we have applied a small shift $\Delta \varphi =0.31$ to recenter the Husimi function on $\varphi =0$.]Reuse & Permissions