Delayed collapses of Bose-Einstein condensates in relation to anti-de Sitter gravity

We numerically investigate spherically symmetric collapses in the Gross–Pitaevskii equation with attractive nonlinearity in a harmonic potential. Even below threshold for direct collapse, the wave function bounces off from the origin and may eventually become singular after a number of oscillations in the trapping potential. This is reminiscent of the evolution of Einstein gravity sourced by a scalar field in anti de Sitter space where collapse corresponds to black-hole formation. We carefully examine the long time evolution of the wave function for continuous families of initial states in order to sharpen out this qualitative coincidence which may bring new insights in both directions. On the one hand, we comment on possible implications for the so-called Bosenova collapses in cold atom Bose–Einstein condensates. On the other hand, Gross–Pitaevskii provides a toy model to study the relevance of either the resonance conditions or the nonlinearity for the problem of anti de Sitter instability.

Figure 1

Examples of delayed collapses in $d=3$. We depict the value of ${\left|\psi (t,r=0)\right|}^2$ for ${\sigma }^2=2/5$ and varying $\epsilon$. In each case, the wave function promptly collapses after hitting the upper limit of the plot. In panel (a), ordered by growing $t_c$, we have ${\epsilon }_1=3.1286$ (green line), ${\epsilon }_2=3.1285$ (brown), ${\epsilon }_3=3.1280$ (purple), ${\epsilon }_4=3.1277$ (magenta), and ${\epsilon }_5=3.1273$ (red). Notice the approximate overlap for long ranges of $t$. In panel (b), ${\epsilon }_6=3.1270$ (blue line) results in collapse after many bounces. We remark that the timescale of both graphs is different.

Figure 2

Time of collapse for a family of initial conditions with ${\sigma }^2=1/2$ in $d=3$. Prompt collapse occurs for $\epsilon >2.891$. The resolution of the peak at the transition $\epsilon \sim 2.889$ exhibits a chaotic structure (in the inset).

Figure 3

Sectors for prompt and delayed collapse in $d=3$. The dashed line signals the transition from initial conditions where $d^{2}V/d{t}^2{|}_{t=0}$ changes from negative (above) to positive (below). All delayed collapses happen with $V\ne 0$.

Figure 4

Doubly logarithmic plot of $C_{nnnn}$ as a function of $n$ for different values of $d$. In the inset, we depict $C_{n,4n,2n,3n}$, showing that they present a similar behavior.

Figure 5

Curve of $t_c$ vs $\epsilon$ for $d=7$ with ${\sigma }^2=\frac{1}{2}$ and $V\left(r\right)=\frac{1}{2}{r}^{\alpha }$. Two nonresonant cases $\alpha =1.9$ and $\alpha =2.1$ present limiting values of $\epsilon$ below which no singularity is reached in computational time. The harmonic resonant case $\alpha =2$ [Eq. (1)] shows a regularly spaced set of steps extending to much lower values of $\epsilon$.

Figure 6

Temporal evolution of the spectrum $E_n={\mu }_n{\left|{\alpha }_n\right|}^2$ for some low-lying modes until collapse. Panel (a) corresponds to an example with $d=3$. Panel (b) corresponds to an example with $d=7$. The changes in the spectrum mostly take place around the time when the wave function periodically peaks at the origin. This periodic energy transfer towards the ultraviolet underlies the stepwise patterns of collapse in Figs. 2 and 5.

Figure 7

A simulation with $d=3,\epsilon =2.79,w=14,{\sigma }^2=1/2$. Panel (a) represents ${\left|\psi \right|}^2$ at the origin. Panels (b) and (c) depict the relative deviations for the norm and Hamiltonian, respectively. Panel (d) compares the second derivative of the variance from the numerical computation to that from the right-hand side of Eq. (4) computed at each step. Finally, panel (e) shows the convergence level along the whole simulation. On a grid of $2^w$ points, $Q_w=\left|\right|{\psi }_{w+1}-{\psi }_w\left|\right|/\left|\right|{\psi }_{w+2}-{\psi }_{w+1}\left|\right|$ where $\left|\right|·\left|\right|$ stands for a $L^2$ norm. The plot shows convergence at fourth order with high accuracy. The tests show that the numerical method is reliable until times very near collapse.