  We have conducted a detailed high-pressure (HP) investigation on $Eu-$doped
$BaTiO_3$ using angle-resolved x-ray diffraction, Raman spectroscopy,
dielectric permittivity and dc resistance measurements. The x-ray diffraction
data analysis shows a pressure-induced structural phase transition from the
ambient tetragonal to the mixed cubic and tetragonal phase above 1.4 GPa. The
tetragonality of the sample due to the internal deformation of the $TiO_6$
octahedra caused by charge difference from Eu doping cannot be lifted upon by
pressure. Softening, weakening, and disappearance of low-frequency Raman modes
indicate ferroelectric tetragonal to the paraelectric cubic phase transition.
But the pressure-induced increase in the intensity of [E(LO), A1(LO)] and the
octahedral breathing modes indicate the local structural inhomogeneity remains
in the crystal and is responsible for spontaneous polarization in the sample.
Low-frequency electronic scattering response suggests the pressure-induced
carrier delocalization, leading to a semi-metallic state in the system. Our HP
dielectric constant and dc resistance data can be explained by the presence of
pressure-induced localized clusters of microscopic ferroelectric ordering. Our
results suggest HP phase coexistence leads to a ferroelectric-like
semi-metallic state in $Eu-$doped $BaTiO_3$ under the extreme quantum limit.
