  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.
