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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 the 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. In addition, in the axial direction, the $g$-atom experience a 1D optical lattice and the $e$-atom is localized by a harmonic potential. In this work, we propose two appropriate tight-binding models, which are applicable for the cases that the odd-wave scattering between the $g$- and $e$-atom is negligible or not, respectively. We further give a microscopic derivation for the inter-atomic interaction parameters of these tight-binding models, by exactly calculating the low-energy inter-atomic scattering amplitude. Our results show that, as one can predict, these interaction parameters can be efficiently controlled by the confinement potentials. We further exam the simple "projection approximation" with which one derives the interaction parameters by directly projecting the 3D Huang-Yang pseudopotential on the ground state of the confinement and the lowest band of the optical lattice. We find that one should be very careful about determining the interaction parameters in the tight-binding models. Furthermore, we calculate the spin-exchanging rate, which dependents on the incident quasi-momentum $k$ of the $g$-atom, for the recent experiment (L. Riegger, {\it et. al.,} Phys. Rev. Lett. {\bf 120}, 143601 (2018)) of $^{173}$Yb atoms in this quasi-(1+0)D system, and study finite-momentum effect in this experiment. Our results show that in this system the finite-momentum effect of the $g$-atom is very significant, and the momentum of the $g$-atoms in this experiment may be pretty high (already in the second Brillouin zone of the optical lattice)