  We use first-principles methods to study doped strong ferroelectrics (taking
BaTiO$_3$ as a prototype). Here we find a strong coupling between itinerant
electrons and soft polar phonons in doped BaTiO$_3$, contrary to
Anderson/Blount's weakly coupled electron mechanism for "ferroelectric-like
metals". As a consequence, across a polar-to-centrosymmetric phase transition
in doped BaTiO$_3$, the total electron-phonon coupling is increased to about
0.6 around the critical concentration, which is sufficient to induce
phonon-mediated superconductivity of about 2 K. Lowering the crystal symmetry
of doped BaTiO$_3$ by imposing epitaxial strain can further increase the
superconducting temperature via a sizable coupling between itinerant electrons
and acoustic phonons. Our work demonstrates a viable approach to modulating
electron-phonon coupling and inducing phonon-mediated superconductivity in
doped strong ferroelectrics and potentially in polar metals. Our results also
show that the weakly coupled electron mechanism for "ferroelectric-like metals"
is not necessarily present in doped strong ferroelectrics.
