  Understanding and controlling the transport properties of interacting
fermions is a key forefront in quantum physics across a variety of experimental
platforms. Motivated by recent experiments in 1D electron channels written on
the $\mathrm{LaAlO_3}$/$\mathrm{SrTiO_3}$ interface, we analyse how the
presence of different forms of spin-orbit coupling (SOC) can enhance electron
pairing in 1D waveguides. We first show how the intrinsic Rashba SOC felt by
electrons at interfaces such as $\mathrm{LaAlO_3}$/$\mathrm{SrTiO_3}$ can be
reduced when they are confined in 1D. Then, we discuss how SOC can be
engineered, and show using a mean-field Hartree-Fock-Bogoliubov model that SOC
can generate and enhance spin-singlet and triplet electron pairing. Our results
are consistent with two recent sets of experiments [Briggeman et al.,
arXiv:1912.07164; Sci. Adv. 6, eaba6337 (2020)] that are believed to engineer
the forms of SOC investigated in this work, which suggests that metal-oxide
heterostructures constitute attractive platforms to control the collective spin
of electron bound states. However, our findings could also be applied to other
experimental platforms involving spinful fermions with attractive interactions,
such as cold atoms.
