18 the vibrational quantum number at a collision energy of 10 $_{−}$9 K [96]. Quenching processes are more efficient than the elastic scattering for both high and low vibra- tional levels. Similar results have been found for $^{6}$Li + $^{6}$Li$_{2}$ ( v = 0 − 9 , j = 1) [96] system composed of fermionic atoms. Quenching rate coefficients show a slight de- crease when the molecule is in its highest vibrational state. This is because the overlap of the wavefunctions of highly excited diatomic molecules with low-lying vibrational levels is very small leading to small values for the interaction potential coupling matrix elements between the initial and final states [115]. These results do not directly apply to ultracold molecules composed of fermionic atoms created near a Feshbach resonance. In these experiments, the quenching processes are suppressed because the atom - atom scattering length is tuned to large and positive values leading to efficient Pauli blocking mechanism, as explained by Petrov et al. [116]. In the theoretical study of Li + Li$_{2}$ collisions, the atom - atom scattering length is small and negative and no suppression of the quenching processes is found for the molecule in the last vibrational state. Thus, the sign and magnitude of the Li − Li scattering length play a crucial role in the mechanism that suppresses quenching collisions. Ultracold diatomic molecules produced in photoas- sociation or Feshbach resonance methods are usually created in excited vibrational states. Theoretical studies involving highly vibrationally excited molecules are challenging due to the large number of energetically open reaction channels present in the quantum calcu- lation. This puts severe restriction on the calculations for vibrationally excited molecules at low temperatures. Ultracold quantum dynamics calculations for collisions of highly vibrationally excited molecules have been reported in 2007 by Qu´ em´ ener et al. [96] for the Li + Li$_{2}$ system. A three-atom problem is generally described by two different kinds of asymptotic channels. The first kind are the single-continuum states (SCSs). They correspond to configurations where the triatomic system dissociates asymptotically into an atom and a diatomic molecule. The second kind are the double-continuum states (DCSs). They correspond to configurations where the triatomic system dissociates asymptotically into three separated atoms. Since highly vibrationally excited molecular states lie close to and below the triatomic dissociation limit, they are also coupled with the DCSs which lie above the dissociation limit. As a consequence, DCSs have to be included in quantum simulations of atom - diatom systems involving highly vibrationally excited diatomic molecules [96]. This dramatically increases the size and complexity of the quantum dynamics problem. 3. Relaxation of vibrationally excited alkali metal dimers The quenching rates displayed in Fig. 21 show an irreg- ular dependence on the vibrational state of the molecule. This has already been seen for H + H$_{2}$ collisions [28, 117]. In contrast, experimental measurements of quenching rate coefficients for Cs + Cs$_{2}$ collisions [106, 107] do not show any dependence on the vibrational state of the molecule. The differences between these systems can be explained based on the following considerations. First, the theoretical study applies to spin-polarized atom - dimer alkali-metal systems whereas it is not the case for the experiment. A full theoretical treatment should involve the electronic and nuclear spins of the al- kali metal atoms as well as couplings between electronic surfaces of different spins. This is beyond the scope of quantum dynamics calculations at present and will in- volve significant new code development and massive com- putational efforts. Second, the dynamics of the two systems are differ- ent. The lithium system is lighter and has a more at- tractive three-body term than the cesium system [105]. The triatomic adiabatic potential energy curves are well separated for a light system such as Li$_{3}$ and very dense for a heavy system such as Cs$_{3}$. The density of states strongly influences the nature of vibrational relaxation. The effect of the density of states has been illustrated in calculations for the Li + Li$_{2}$ system which exclude the three-body term. Removing the three-body term makes the energy levels more sparse. This results in a more reg- ular and monotonic dependence of the rate coefficients on the vibrational levels v = 3 − 9 as illustrated in Figure 21. This conclusion is in agreement with a previous work of Bodo et al. [117] for the H + H$_{2}$ system. When they Figure 21 shows the dependence of the rate coeffi- cients for $^{7}$Li + $^{7}$Li$_{2}$ ( v = 0 − 10 , j = 0) collisions on FIG. 21: Dependence of the elastic and quenching rate coef- ficients on the vibrational excitation of the molecule for $^{7}$Li + $^{7}$Li 2 ( v = 0 − 10 , j = 0) scattering at a collision energy of 10 − 9 K. The bold line corresponds to the full calculation and the thin line corresponds to the calculation without the three-body term of the PES. Adapted with permission from Qu´ em´ ener et al. [96]. 0 1 2 3 4 5 6 7 8 9 10 initial vibrational number v 10 -14 10 -13 10 -12 10 -11 10 -10 10 -9 Rate coefficient (cm 3 .s -1 ) Full calculation V$_{3}$ = 0 $^{7}$Li + $^{7}$Li$_{2}$(v,j=0) elastic quenching