Universal Rate Constants for Reactive Collisions of Ultracold Molecules

A simple quantum-defect model gives analytic expressions for the complex scattering length and threshold collision rates of ultracold molecules. If the probability of reaction in the short-range part of the collision is high, the model gives universal rate constants for $s$- and $p$-wave collisions that are independent of short-range dynamics. This model explains the magnitudes of the recently measured rate constants for collisions of two ultracold ${}^{40}\mathrm{K}{}^{87}\mathrm{Rb}$ molecules, or an ultracold ${}^{40}\mathrm{K}$ atom with the ${}^{40}\mathrm{K}{}^{87}\mathrm{Rb}$ molecule [S. Ospelkaus et al., Science 327, 853 (2010)].

Figure 1

Real and imaginary parts of ${\tilde{a}}_{\ell =0}/\overline{a}$ (left panel) and $({\tilde{a}}_{\ell =1}/\overline{a})(k\overline{a}{)}^{-2}$ (right panel) as $k\to 0$ for different values of the loss parameter $y$ (set of circles) as the phase parameter $s=a/\overline{a}$, color coded according to the value of $\arctan (s)/\pi$, varies over its full range of $-\infty <a<+\infty$. The blue-red (dark gray-medium gray) boundary occurs where $U_{\ell }(r)$ has a bound state at $E=0$.

Figure 2

Loss rate constant ${\mathcal{K}}^{\mathrm{ls}}$ of ${}^{40}\mathrm{K}{}^{87}\mathrm{Rb}$ molecules versus temperature. Experimental data [4] for a single component gas of molecules in spin states $|F=-4,M_F=\frac{1}{2}⟩$ (blue squares), $|F=-4,M_F=\frac{1}{2}⟩$ (red triangles), and for $50/50$ mixture of these two spin states (green dots), are compared with prediction of MQDT in the universal regime ($y=1$, dashed lines), and in the nonuniversal regime ($y<1$, solid lines). In the latter case the values of $y$, and $s$ parameters for $\ell =0$ and $\ell =1$ are determined by fitting to the experimental data.