  Converting charge current into spin current is one of the main mechanisms
exploited in spintronics. One prominent example is the Edelstein effect, namely
the generation of a magnetization in response to an external electric field,
which can be realized in systems with lack of inversion symmetry. If a system
has electrons with an orbital angular momentum character, an orbital
magnetization can be generated by the applied electric field giving rise to the
so-called orbital Edelstein effect. Oxide heterostructures are the ideal
platform for these effects due to the strong spin-orbit coupling and the lack
of inversion symmetries. Beyond a gate-tunable spin Edelstein effect, we
predict an orbital Edelstein effect an order of magnitude larger then the spin
one at the (111) LaAlO$_3$/SrTiO$_3$ interface. We model the material as a
bilayer of $t_{2g}$ orbitals using a tight-binding approach, while transport
properties are obtained in the Boltzmann approach. We give an effective model
at low filling which explains the non-trivial behaviour of the Edelstein
response, showing that the hybridization between the electronic bands crucially
impacts the Edelstein susceptibility.
