Tight-Binding Kondo Model and Spin-Exchange Collision Rate of Alkaline-Earth Atoms in a Mixed-Dimensional Optical Lattice

We study the two-body problem of ultracold fermionic alkaline-earth (like) atoms in the electronic $^1$S$_0$ state ($g$-state) and $^3$P$_0$ state ($e$-state) which are confined in a quasi-one-dimensional (quasi-1D) tube simultaneously, where in the axial direction the $g$-atom experiences a 1D optical lattice and the $e$-atom is localized by a harmonic potential. Due to the nuclear-spin exchange interaction between the $g$- and $e$-atom, one can use such a quasi-(1+0)D system to realize the Kondo effect. We suggest two tight-binding models for this system, for the cases that the odd-wave scattering between the $g$- and $e$-atom is negligible or not, respectively. Moreover, we give a microscopic derivation for the inter-atomic interaction parameters of these models, by explicitly calculating the quasi-(1+0)D low-energy scattering amplitude of the $g$- and $e$-atom in this system and matching this exact result with the ones given by tight-binding models. We illustrate our results for the experimental systems of ultracold $^{173}$Yb and $^{171}$Yb atoms and show the control effect of the confinement potentials on these model parameters. Furthermore, the validity of the simple ``projection approximation" is examined. In this approximation, one derives the interaction parameters