Finding new materials with antiferromagnetic (AFM) Kitaev interaction is an urgent issue to broaden and enrich the quantum magnetism research significantly. By carrying out inelastic neutron scattering experiments and subsequent analysis, we conclude that Na$_3$Co$_2$SbO$_6$ and Na$_2$Co$_2$TeO$_6$ are new honeycomb cobalt-based AFM Kitaev systems. The spin-orbit excitons at 20-28~meV in both compounds strongly supports the idea that Co$^{2+}$ ions of both compounds have a spin-orbital entangled $J_rm{eff}=1/2$ state. Furthermore, we found that a generalized Kitaev-Heisenberg Hamiltonian can well describe the spin-wave excitations of both compounds with additional 3rd nearest-neighbor interaction. Our best-fit parameters show large AFM Kitaev terms and off-diagonal symmetric anisotropy terms of a similar magnitude in both compounds. We should stress that our parameters optimized magnetic structures are consistent with the magnetic structures reported from neutron diffraction studies. Moreover, there is also the magnon-damping effect at the higher energy part of the spin waves, as usually observed in other Kitaev magnets. We demonstrate that Na$_3$Co$_2$SbO$_6$ and Na$_2$Co$_2$TeO$_6$ are the first experimental realization of AFM Kitaev magnets based on the systematic studies of the spin waves and analysis.
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