  In this work we evaluate the masses of the $|(b\bar{s})[n]\rangle$ or
$|(\bar{b}s)[n]\rangle$ quarkonium ($\bar{B}^{0}_{s}$ or ${B}^{0}_{s}$ meson)
under the B.T. potential, and the values of the Schr${\rm \ddot{o}}$dinger
radial wave function at the origin of the $|(b\bar{s})[n]\rangle$ or
$|(\bar{b}s)[n]\rangle$ quarkonium within the five potential models. Then we
investigate a systematic study on the production of the $|(b\bar{s})[n]\rangle$
or $|(\bar{b}s)[n]\rangle$ quarkonium via top quark or antitop quark decays in
the color-singlet QCD factorization formula (CSQCDFF), i.e., the two $S$-wave
states, $|(b\bar{s})[1^1S_0] \rangle$ (or $|(\bar{b}s)[1^1S_0] \rangle$) and
$|(b\bar{s})[1^3S_1] \rangle$ (or $|(\bar{b}s)[1^3S_1] \rangle$), and its four
$P$-wave excited states, $|(b\bar{s})[1^1P_1] \rangle$ (or $|(\bar{b}s)[1^1P_1]
\rangle$) and $|(b\bar{s})[1^3P_J] \rangle$ (or $|(\bar{b}s)[1^3P_J] \rangle$)
(with $J =[0, 1, 2]$). For deriving compact analytical results for complex
processes, the "improved trace technology" is adopted to deal with the decay
channels at the amplitudes. Moreover, various differential distributions and
uncertainties of the concerned processes are analyzed carefully. By adding the
uncertainties caused by the ${b}$ and ${s}$-quark masses in quadrature, we
obtain $\Gamma{(t\to |(b\bar{s})[n]\rangle
+W^{+}s)}=14.19^{+4.36}_{-3.20}$~MeV. At the LHC with the luminosity ${\cal
L}\propto 10^{34}cm^{-2}s^{-1}$ and the center-of-mass energy $\sqrt{S}=14$
TeV, sizable $|(b\bar{s})[n]\rangle$ or $|(\bar{b}s)[n]\rangle$ meson events
can be produced through ${t}$-quark or ${\bar{t}}$-quark decays; i.e., about
$1.3~\times10^6$ ${\bar{B}^0_s}$ or ${B^0_s}$ events per year can be obtained.
