## 3 Discussion  
We have analyzed the process of vibrational quenching of an initially excited homonuclear molecular ion due to collisions with the background gas. The effective couplings between the molecular states turn out to decay with a power law in the same way as the polarization potential, but with different coupling coefficients depending on the overlap of the molecular wave functions. The long-range nature of the process makes it possible to rely on distorted wave Born approximation to determine the distribution of the collision products. This approximate quantum treatment agrees qualitatively with previous estimates based on classical trajectory calculations.  
The total inelastic rate coefficient for vibrational relaxation of the weakly bound molecular ions is given by the Langevin formula \(\mathcal{K}_{L}=2\pi\hbar R_{\rm mol}^{\ast}/\mu^{\star}\), with \(R_{\rm mol}^{\ast}\) being the characteristic length of the molecular ion-atom pair, irrespective of the initial vibrational state. Assuming the case of Rb atoms, \(\mathcal{K}_{L}\approx 2.1\times 10^{-9}\) cm\({}^{3}\)/s, which for high atomic densities \(n\approx 10^{15}\) cm\({}^{-3}\), typical for a Bose-Einstein condensate, results in lifetimes of the order of microseconds. In order to estimate the role of such secondary relaxation processes after an initial three-body recombination event for experiments,  
Figure 4: K matrix elements calculated within the distorted wave Born approximation (DWBA) for vanishing collision energy as a function of the energy difference between the channels for the entrance channel scattering length \(a=R^{\ast}\) and initial kinetic energy \(E=10^{-4}E^{\star}\).  
Figure 5: (**a**) Distribution of product states for vanishing collision energy after the inelastic collision for the 5 th vibrational state (\(v=5\)). (**b**) Same, but for the most weakly bound state (\(v=1\)).  
we consider a typical kinetic energy release of \(\approx\)\(h\times(10\,...\,100)\) MHz associated with the formation of a weakly bound molecular ion. Evidently, this yields a mean free path of the molecular ion in the order of few micrometers between successive collision events, significantly smaller than the typical spatial extent of a trapped condensate. This means that in a dense gas secondary collisions after the production of the molecular ion can be highly important and can be detected, e.g., by monitoring the arrival time of the molecule at the detector. For comparison, in the case of neutral weakly bound Rb\({}_{2}\) molecule the expected rate coefficient is 50 times smaller.  
In our model we neglected the role of possible complex formation. Once all three particles involved come close together, the resulting complex state may be long-lived and provide high density of resonance states. This would presumably impact the result of the collision by redistributing the particles along all possible product states in a statistical way. Extension of our model to involve three-body resonances and complex formation will be the subject of future work.  
Conceptualization, K.J. and F.M.; methodology, K.J.; calculations, K.J.; writing, K.J. and F.M. All authors have read and agreed to the published version of the manuscript.  
Funding:F.M. acknowledges support from Deutsche Forschungsgemeinschaft [Project No. PF 381/17-1, part of the SPP 1929 (GiRyd)], funding by the Carl-Zeiss foundation, and is indebted to the Baden-Wurttemberg-Stiftung for the financial support by the Eliteprogramm for Postdocs.  
The authors declare no conflict of interest.