## 2 Results  
### System  
Let us consider a single ion of mass \(m\) moving in a dilute gas of neutral atoms of the same species. Throughout this work we will focus on the case of Rb atoms which are experimentally most relevant, but our result can also be applied to other cases. The ion motion is assumed to be unconstrained, meaning that the characteristic length scale of the ion trap (if present at all) is larger than the ion-atom interaction range as well as the typical interparticle distance. Such a situation can be realized, e.g., by utilizing optical ion traps or by ionization of an ultracold atom directly from the gas. Once produced, the ion will undergo collisions with the surrounding atoms. The ion-atom interaction at long range is governed by the \(-C_{4}/r^{4}\) term (leading induction coefficient in the multipole expansion) [1], with \(C_{4}=\frac{1}{2}q^{2}\alpha\). Here, \(q\) is the ionic charge, \(\alpha\) the atomic polarizability, and \(r\) denotes the ion-atom distance. One can define the length scale \(R^{\star}\) and energy scale \(E^{\star}\) associated with this potential as  
\[R^{\star}=(2\mu C_{4}/\hbar^{2})^{(1/2)}\,\ E^{\star}=\hbar^{2}/2\mu(R^{\star})^{2} \tag{1}\]  
with \(\mu=m/2\) denoting the reduced mass of the ion-atom system. In addition to two-body elastic collisions and charge transfer processes which determine the ion mobility [22], three-body recombination is expected to occur, leading to association of molecular ions [2]. The classical three-body cross section for this process has been obtained in reference [15] and is described with a universal formula \(\sigma_{3}(E)=\frac{8\pi^{2}}{15}2^{5/4}(R^{\star})^{5}(E/E^{\star})^{-5/4}\). Although little is known about the product state distribution after the recombination process, experiments and theory developed for neutral Rb atoms interacting via van-der-Waals forces have led to the conclusion that the resulting molecules are mostly weakly bound [3]. Here, rather than in the recombination, we are interested in the dynamics of the molecular ion after it has been produced. Note that it is also technically possible to associate molecular ions using a Raman process without relying on three-body recombination, which should give access to better control over the initial state [23]. Atom-atom-ion three-body recombination has been observed in references [2; 24] for trapped Rb\({}^{+}\) and Ba\({}^{+}\) ions immersed in a Rb gas prepared at typical densities of \(\approx\)\(1\times 10^{12}\) cm\({}^{-3}\). For the trapped ions, the typical collision energies are in the few to tens of mK regime, set by the trap-induced ion micromotion. For these conditions, recombination timescales on the order of a second have been observed. Note that for higher atomic densities typical of a Bose-Einstein condensate (\(\approx\)\(1\times 10^{15}\) cm\({}^{-3}\)), the recombination time can be expected to reach even the microsecond timescale, highlighting the much larger cross section for ion-neutral-neutral recombination as compared to the recombination of three neutral atoms. The homonuclear molecular ion does not posses a dipole moment, and thus its long-range interaction with the surrounding atoms is also described by the \(1/r^{4}\) potential. The complete interaction potential surface consists of interactions among three particles, the ion (labeled as \(1\) in the equation below) and two atoms (labeled \(2\) and \(3\))  
\[V(\mathbf{r}_{12})+V(\mathbf{r}_{13})+V(\mathbf{r}_{23})\,, \tag{2}\]  
where \(V(\mathbf{r}_{12})\) and \(V(\mathbf{r}_{13})\) are the ion-atom potentials and \(V(\mathbf{r}_{23})\) describes the van-der-Waals interaction between neutral atoms. For Rb, we have \(R^{\star}=5028\,a_{0}\) with \(a_{0}\) being the Bohr radius, while the characteristic van-der-Waals length for Rb is only about \(165\,a_{0}\). The characteristic energy \(E^{\star}\) in the case of Rb is \(h\times 1640\) Hz, or equivalently \(k_{B}\times 78.7\) nK [1].