Rydberg Molecules for Ion-Atom Scattering in the Ultracold Regime

We propose a novel experimental method to extend the investigation of ion-atom collisions from the so far studied cold, essentially classical regime to the ultracold, quantum regime. The key aspect of this method is the use of Rydberg molecules to initialize the ultracold ion-atom scattering event. We exemplify the proposed method with the lithium ion-atom system, for which we present simulations of how the initial Rydberg molecule wave function, freed by photoionization, evolves in the presence of the ion-atom scattering potential. We predict bounds for the ion-atom scattering length from ab initio calculations of the interaction potential. We demonstrate that, in the predicted bounds, the scattering length can be experimentally determined from the velocity of the scattered wave packet in the case of ${{}^6\mathrm{Li}}^{+}\text{−}{}^6\mathrm{Li}$ and from the molecular ion fraction in the case of ${{}^7\mathrm{Li}}^{+}\text{−}{}^7\mathrm{Li}$. The proposed method to utilize Rydberg molecules for ultracold ion-atom scattering, here particularized for the lithium ion-atom system, is readily applicable to other ion-atom systems as well.

Figure 1

In the past few years, the ion-atom interaction could be studied down to the millikelvin regime for various ion-atom combinations by the use of hybrid traps, with the ion held in a Paul trap (inset). However, the ultracold, quantum scattering regime could not be reached yet (see color-coded lines for the respective $S$-wave scattering limits $E_0$). We propose an experimental method to enter the ultracold ion-atom scattering regime using Rydberg molecules. We demonstrate this method with the lithium ion-atom system which features an early onset of the $S$-wave scattering regime due to its small reduced mass.

Figure 2

Using Rydberg molecules to initialize an ultracold ion-atom scattering event, exemplified with lithium. The Rydberg molecule wave function of interest, more precisely, $R^2{|{\tilde{\mathrm{\Psi }}}_{A\mathsf{A}}(R)|}^2$, is shown in (a) for both isotopes. It is bound in the homonuclear $\mathrm{Li}(30s)\text{−}\mathrm{Li}(2s)$ potential ${\tilde{V}}_{A\mathsf{A}}(R)$. It is in its spherically symmetric rovibrational ground state ($\tilde{v}=\tilde{J}=0$), which can be experimentally addressed, evident from (b), as the rotational constant is larger than the Rydberg molecule decay linewidth ${\gamma }_{A\mathsf{A}}$. Freed by photoionization, the initial Rydberg molecule wave function evolves in the spherically symmetric ion-atom interaction potential ${\tilde{V}}_{I\mathsf{A}}(R)$, with $R^{\star }$ denoting its characteristic radius. Because of angular momentum conservation, only $S$-wave scattering occurs ($\tilde{J}=0$) despite the components above $E_0$ in the Rydberg molecule energy spectrum; see (c). The overlap between ${\tilde{\mathrm{\Psi }}}_{A\mathsf{A}}$ and the last bound molecular ion wave function ${\tilde{\mathrm{\Psi }}}_{I\mathsf{A}}$ [see (a)] determines the bound fraction in the scattered wave packet.

Figure 3

Ultracold lithium ion-atom scattering processes, initialized with Rydberg molecules, for different scattering lengths $\mathcal{A}$. For $\mathcal{A}>0$ [demonstrated in (a) for ${{}^6\mathrm{Li}}^{+}\text{−}{}^6\mathrm{Li}$ scattering with $\mathcal{A}=+R_6^{\star }$], the scattered wave packet splits into a free, expanding shell and a bound shell, indicating molecular ion formation. For $\mathcal{A}<0$ [exemplified in (b) for $\mathcal{A}=-R_6^{\star }$], the entire scattered wave packet is free. In (a) and (b), the time evolution of the scattered wave packet $R^2{|{\mathrm{\Psi }}_{I\mathsf{A}}(R,t)|}^2$ is shown from $t=0$ [${\mathrm{\Psi }}_{I\mathsf{A}}(t=0)={\tilde{\mathrm{\Psi }}}_{A\mathsf{A}}$, unfilled curve, vertical axis applies] to $1\mu \mathrm{s}$ in steps of 50 ns (top to bottom, curves shifted by $-3.25\times {10}^{-6}\mathrm{a}.\mathrm{u}.$ each). The insets demonstrate the $S$-wave character of the scattered wave packet [$R$ extension of $7\times {10}^3\mathrm{a}.\mathrm{u}.$, same color scale in (a) and (b)]. For $\mathcal{A}<0$ ($\mathcal{A}>0$), the shell expansion velocity $\zeta$ (the bound fraction $b$) is a sensitive quantity to determine the scattering length; see (c) and (d), where also the scattering lengths ${\mathcal{A}}_{6,7}$ from our ab initio calculations are indicated (vertical lines with the shaded areas marking the bounds).