if the final state consists of an atom and a deep dimer. However, in the three-fermion system relaxation can also occur into into shallow dimers. For example, in a scattering process of an atom $i$ and a $jk$-dimer, relaxation can proceed into the $ij$-dimer provided the binding energy of the $ij$-dimer is larger than that of the $jk$-dimer. Therefore, the total rate $\beta_i$ is the sum of all relaxation rates into atoms plus shallow dimers and atoms plus deep dimers $$\beta_i=
\sum_{j\neq i} \beta^{\rm sh}_{i\to j}+ \beta^{\rm deep}_{i},
\label{beta:tot}$$ where the index $i$ implies atom $i$ plus $jk$-dimer in the initial state and the $j$ implies atom $j$ plus $ik$-dimer in the final state. For brevity, we refer to $\beta^{\rm sh}_{i\to j}$ as the rate into the shallow dimer or the rate into the (final) $ik$-dimer and to $\beta^{\rm
  deep}_{i}$ as the rate into the deep dimers.

The relaxation rate can be calculated from the T-matrix element for atom-dimer scattering ${\cal T}^{\rm AD}_{ij}(k,p;E)$ where $k$ and $p$ are the relative wave numbers of the atom and dimer in initial and final state, respectively, and $E$ is the total energy. By using the [*optical theorem*]{}, we can calculate the total rate 