  Superconductivity forms out of the condensation of Cooper pairs of electrons.
The mechanism by which Cooper pairs are created in non-conventional
superconductors is often elusive because experimental signatures that connect a
specific pairing mechanism to the properties of superconducting state are rare.
The recently discovered superconducting oxide-insulator/KTaO$_3$ interface may
offer clues about its origins. Here we observe distinct dependences of the
superconducting transition temperature Tc on carrier density n$_{2D}$ for
electron gases formed at KTaO$_3$ (111), (001) and (110) interfaces. For the
KTaO$_3$ (111) interface, a remarkable linear dependence of Tc on n$_{2D}$ is
observed over a range of nearly one order of magnitude. Further, our study of
the dependence of superconductivity on gate electric fields reveals the role of
the interface in mediating superconductivity, which also allows for a
reversible electric switching of superconductivity at T = 2 K. We found that
the extreme sensitivity of superconductivity to crystallographic orientation
can be explained by Cooper pairing via inter-orbital interactions induced by
the inversion-breaking transverse optical (TO1) phonons and quantum
confinement. This mechanism is also consistent with the dependence of Tc on
n$_{2D}$ at the KTaO$_3$ (111) interface. Our study may shed light on the
pairing mechanism in other superconducting quantum-paraelectrics.
