Excited configurations of hydrogen in the oxyhydride BaTiO$_{3-x}$H$_x$ ($x=0.1-0.5$), which are considered to be involved in its hydrogen transport and exchange processes, were investigated by positive muon spin relaxation ($mu^+$SR) spectroscopy using muonium (Mu) as a pseudoisotope of hydrogen. Muons implanted into the BaTiO$_{3-x}$H$_x$ perovskite lattice were mainly found in two qualitatively different metastable states. One was assigned to a highly mobile interstitial protonic state, which is commonly observed in perovskite oxides. The other was found to form an entangled two spin-$frac{1}{2}$ system with the nuclear spin of an H$^-$ ion at the anion site. The structure of the (H,Mu) complex agrees well with that of a neutralized center containing two H$^-$ ions at a doubly charged oxygen vacancy, which was predicted to form in the SrTiO$_{3-delta}$ perovskite lattice by a computational study [Y. Iwazaki $et$ $al$., APL Materials 2, 012103 (2014)]. Above 100 K, interstitial Mu$^+$ diffusion and retrapping to a deep defect were observed, which could be a rate-limiting step of macroscopic Mu/H transport in the BaTiO$_{3-x}$H$_x$ lattice.
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