  Properties of the baryon-baryon interactions in the strangeness $S=-2$ sector
of chiral effective field theory at the next-to-leading order (NLO) level are
explored by calculating $\Xi$ single-particle potentials in symmetric nuclear
matter. The results are transformed to the $\Xi$ potential in finite nuclei by
a local-density approximation with convolution by a Gaussian form factor to
simulate finite-range effects. The $\Xi$ potential is repulsive in a central
region, and attractive in a surface area when the $\Xi$ energy is low. The
attractive pocket can lower the $\Xi^-$ $s$ and $p$ atomic states. The obtained
binding energies in $^{12}$C and $^{14}$N are found to be conformable with
those found in emulsion experiments at Japan's National Laboratory for High
Energy Physics (KEK). $K^+$ spectra of $(K^-, K^+)$ $\Xi$ production inclusive
processes on $^9$Be and $^{12}$C are also evaluated, using a semi-classical
distorted wave method. The absolute values of the cross section are properly
reproduced for $^9$Be, but the peak locates at a lower energy position than
that of the experimental data. The calculated spectrum of $^{12}$C should be
compared with the forthcoming result from the new experiments recently carried
out at KEK with better resolution than before. The comparison would be valuable
to improve the understanding of the $\Xi N$ interaction, the parametrization of
which has still large uncertainties.
