A fully quantum mechanical description of the precessional damping of Pt/Co bilayer is presented in the framework of the Keldysh Green function approach using {it ab initio} electronic structure calculations. In contrast to previous calculations of classical Gilbert damping ($alpha_{GD}$), we demonstrate that $alpha_{GD}$ in the quantum case does not diverge in the ballistic regime due to the finite size of the total spin, $S$. In the limit of $Srightarrowinfty$ we show that the formalism recovers the torque correlation expression for $alpha_{GD}$ which we decompose into spin-pumping and spin-orbital torque correlation contributions. The formalism is generalized to take into account a self consistently determined dephasing mechanism which preserves the conservation laws and allows the investigation of the effect of disorder. The dependence of $alpha_{GD}$ on Pt thickness and disorder strength is calculated and the spin diffusion length of Pt and spin mixing conductance of the bilayer are determined and compared with experiments.
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