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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${\rm \ddot{o}}$dingerradial 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 weinvestigate 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 inthe color-singlet QCD factorization formula (CSQCDFF), i.e., the two $S$-wavestates, $|(b\bar{s})[1^1S_0] \rangle$ (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 complexprocesses, the "improved trace technology" is adopted to deal with the decaychannels at the amplitudes. Moreover, various differential distributions anduncertainties of the concerned processes are analyzed carefully. By adding theuncertainties caused by the ${b}$ and ${s}$-quark masses in quadrature, weobtain $\Gamma{(t\to |(b\bar{s})[n]\rangle+W^{+}s)}=14.19^{+4.36}_{-3.20}$~MeV. At the LHC with the luminosity ${\calL}\propto 10^{34}cm^{-2}s^{-1}$ and the center-of-mass energy $\sqrt{S}=14$TeV, sizable $|(b\bar{s})[n]\rangle$ or $|(\bar{b}s)[n]\rangle$ meson eventscan 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.

${\bar{B}^{0}_{s}}$ and its excited meson production via top quark decays at the LHC