  There exist two methods to study two-baryon systems in lattice QCD: the
direct method which extracts eigenenergies from the plateaux of the temporal
correlator and the HAL QCD method which extracts observables from the non-local
potential associated with the tempo-spatial correlator. Although the two
methods should give the same results theoretically, qualitatively different
results have been reported. Recently, we pointed out that the separation of the
ground state from the excited states is crucial to obtain sensible results in
the former, while both states provide useful signals in the latter. In this
paper, we identify the contribution of each state in the direct method by
decomposing the two-baryon correlators into the finite-volume eigenmodes
obtained from the HAL QCD method. We consider the $\Xi\Xi$ system in the
$^1$S$_0$ channel at $m_\pi = 0.51$ GeV in 2+1 flavor lattice QCD using the
wall and smeared quark sources. We demonstrate that the "pseudo-plateau" at
early time slices (t = 1~2 fm) from the smeared source in the direct method
indeed originates from the contamination of the excited states, and the true
plateau with the ground state saturation is realized only at t > 5~15 fm
corresponding to the inverse of the lowest excitation energy. We also
demonstrate that the two-baryon operator can be optimized by utilizing the
finite-volume eigenmodes, so that (i) the finite-volume energy spectra from the
HAL QCD method agree with those from the optimized temporal correlator and (ii)
the correct spectra would be accessed in the direct method only if highly
optimized operators are employed. Thus we conclude that the long-standing issue
on the consistency between the L\"uscher's finite volume method and the HAL QCD
method for two baryons is now resolved: They are consistent with each other
quantitatively only if the excited contamination is properly removed in the
former.
