  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
$S\rightarrow\infty$ 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.
