Long-lived complexes and chaos in ultracold molecular collisions

Estimates for the lifetime of collision complexes formed during ultracold molecular collisions based on density-of-states arguments are shown to be consistent with similar estimates based on classical trajectory calculations. In the classical version, these collisions are shown to exhibit chaos and their fractal dimension is calculated versus collision energy. From these results, a picture emerges that ultracold collisions are likely classically ergodic, justifying the density-of-states estimates for lifetimes. These results point the way toward using the techniques of classical and quantum chaos to interpret molecular collisions in the ultracold regime.

Figure 1

Schematic of the initial conditions. The lone atom is given an initial velocity $v_{\mathrm{col}}$ corresponding to a collision energy $E_{\mathrm{col}}$. For collisions with an impact parameter, the lone atom is displaced a distance $b$ in the $yz$ plane.

Figure 2

The fraction of collision complexes yet to decay as a function of time at a collision energy of 30 K for Li + Li${}_2$, K + K${}_2$, and Cs + Cs${}_2$. The solid lines show an exponential decay fitted to the data points for each system. For each system over 1000 trajectories were run for random initial $\theta$ between 0 and $\pi /2$ with no impact parameter.

Figure 3

Lifetime as a function of collision energy for the collisions of Li + Li${}_2$, K + K${}_2$, and Cs + Cs${}_2$. Shown are lifetimes computed from explicit dwell times from classical trajectories (dots) and from the RRKM approximation (solid line). The dotted line shows a power-law fit to the classical trajectory data.

Figure 4

Representative trajectories for collisions of Li + Li${}_2$ over a range of collision energies, the different colors labeling different atoms. All calculations were performed without impact parameter for $\theta /\pi =0.25$.

Figure 5

Lifetime as a function of initial $\theta$ for collisions of Li + Li${}_2$ with zero impact parameter at $E_{\mathrm{col}}=450$ K. Different colors correspond to different final basins for the trajectory. The lower panel shows a 500$\times$ magnified region of the upper panel.

Figure 6

$f\left(\delta \right)$ as a function of $\delta$ for collision of Li + Li${}_2$. Shown are three representative collision energies demonstrating the full range of behavior. The corresponding $\alpha$ values are 0.01 at 31 K, 0.24 at 316 K, and 1.00 at 3162 K.

Figure 7

Fractal dimension $d$ as a function of collision energy for collisions of Li + Li${}_2$, both with and without an impact parameter. $d-2$ is plotted for the result, including an impact parameter for comparison. The vertical dotted line is $E_{\mathrm{chaos}}$, as defined in Eq. (7), and the vertical solid line is the atom-atom well depth.

Figure 8

Fractal dimension $d$ as a function of collision energy for collisions of Li + Li${}_2$, K + K${}_2$, and Cs + Cs${}_2$ for collisions without an impact parameter. The vertical dotted line is $E_{\mathrm{chaos}}$ as defined in Eq. (7), and the vertical solid line is the atom-atom well depth for each species.