Approach to chaos in ultracold atomic and molecular physics: Statistics of near-threshold bound states for Li+CaH and Li+CaF

We calculate near-threshold bound states and Feshbach resonance positions for atom–rigid-rotor models of the highly anisotropic systems Li+CaH and Li+CaF. We perform statistical analysis on the resonance positions to compare with the predictions of random matrix theory. For Li+CaH with total angular momentum $J=0$ we find fully chaotic behavior in both the nearest-neighbor spacing distribution and the level number variance. However, for $J>0$ we find different behavior due to the presence of a nearly conserved quantum number. Li+CaF $\left(J=0\right)$ also shows apparently reduced levels of chaotic behavior despite its stronger effective coupling. This may indicate the development of another good quantum number relating to a bending motion of the complex. However, continuously varying the rotational constant over a wide range shows unexpected structure in the degree of chaotic behavior, including a dramatic reduction around the rotational constant of CaF. This demonstrates the complexity of the relationship between coupling and chaotic behavior.

Figure 1

The ab initio ${}^3{A}^{\prime }$ interaction potential for (a) Li+CaH and (b) Li+CaF. Contours are labeled in ${\mathrm{cm}}^{-1}$; $\theta =0^{\circ }$ corresponds to the Li-H(F)-Ca geometry.

Figure 2

Statistical analysis of calculated levels of Li+CaH ($J=0$ and $0.7\le \lambda \le 1.6$). From top to bottom: level positions in $\lambda$; staircase function, with fitted smooth function $\xi \left(X\right)$ in blue; histogram of the NNS distribution, with the Poisson (green dotted), Wigner-Dyson (red dashed) and fitted Brody (black) distributions; level number variance, with Poisson (green dotted) and GOE (red dashed) predictions.

Figure 3

Near-threshold bound states as a function of $\lambda$ for Li+CaH $\left(J=0\right)$. The bottom panel shows an expanded view with a state approaching threshold, crossed by several steeper states. The dashed line has been added to follow this state through the avoided crossings and help guide the eye.

Figure 4

Nearest-neighbor spacing distributions for Li+CaH: $J=1 \left(0.5\le \lambda \le 1.5\right),J=3 \left(0.8\le \lambda \le 1.4\right)$, and $J=8$ ($0.8\le \lambda \le 1.3$). Lines are as in Fig. 2.

Figure 5

The NNS distributions for Li+CaH ($J=1$ and $0.5\le \lambda \le 1.5$) using the helicity-decoupling approximation for $K=0,\left|K\right|=1$, and $K=0$ and $\left|K\right|=1$. Lines are as in Fig. 2; the magenta dash-dotted line in the bottom panel is the predicted NNS distribution for two overlapping GOEs.

Figure 6

Near-threshold bound states as a function of $\lambda$ for Li+CaH $\left(J=1\right)$: using the helicity-decoupling approximation (top) (red, $K=0$; green, $\left|K\right|=1$) and the full calculation (bottom).

Figure 7

Calculated statistics for Li+CaF ($J=0$ and $0.8\le \lambda \le 1.4$): NNS distribution and level number variance. Lines are as in previous figures.

Figure 8

Near-threshold bound states as a function of $\lambda$ for Li+CaF $\left(J=0\right)$.

Figure 9

Rotational constant dependence of the fitted Brody parameter for Li+CaH $\left(J=0\right)$. Labeled points are (a) 0.05, (b) 0.25, (c) 1.0, (d) 12.5, (e) 21, and (f) $50 {\mathrm{cm}}^{-1}$; $b_{\mathrm{CaF}}$ and $b_{\mathrm{CaH}}$ are also indicated.

Figure 10

Examples of the NNS distribution (left) for values of rotational constant highlighted in Fig. 9 and the corresponding level number variance (right). Lines are as in previous figures.