  Superconductivity has been observed in doped SrTiO$_3$ at charge-carrier
densities below 10$^{18}$ cm$^{-3}$, where the density of states at the Fermi
level of the itinerant electrons is several orders of magnitude lower than that
of conventional metals. In terms of the Bardeen-Cooper-Schrieffer description,
this implies the existence of an extraordinarily strong interaction driving the
formation of Cooper pairs, potentially comparable in order of magnitude to that
in some high Tc superconductors. Under suitable conditions the interaction
might remain effective at densities approaching metallic densities, leading to
the possibility of pair formation at elevated temperatures. Here we investigate
the pressure dependence of the resistivity and superconducting transition
temperature, Tc, of SrTiO$_3$ at a carrier density near to optimal doping. Our
experiments show that Tc collapses rapidly with pressure and hence with
increasing frequency of the soft transverse-optical phonon mode connected to
the ferroelectric quantum critical point. We show that the superconductivity
phase diagram can be understood in terms of the coupling of electrons via two
hybrid longitudinal polar modes, based on a model of dipolar fluctuations of
the charge carrier-ion system. In particular, we predict that for carrier
densities above the order of 10$^{18}$ cm$^{-3}$, Tc can be strongly enhanced
on approaching the ferroelectric quantum critical point, as seen in our
measurements of SrTiO$_3$ and as found in many electrically conducting magnetic
analogues. However below this density we predict the reverse behaviour, namely
that Tc is suppressed on approaching the ferroelectric quantum critical point.
Our model is also relevant to superconductivity found in gated ferroelectric
quantum critical systems such as KTaO$_3$ and can guide searches for new
superconductors in a diversity of materials.
