  The formation of novel two-dimensional electron gas (2DEG) with high mobility
in metal/amorphous interfaces has motivated an ongoing debate regarding the
formation and novel characteristics of these 2DEGs. Here we report an optical
study, based on infrared spectroscopic ellipsometry, of nonmagnetic metal and
amorphous semiconducting oxide (Cu/Bi$_2$O$_3$) interfaces that confirms the
formation of a 2DEG with spin orbit coupling (SOC). The 2DEG optical response
was simulated with a uniaxial diagonal dielectric tensor within a sub-nanometer
thin layer, where its $x$ and $z$ components lineshapes resolved in both
free-electron and peak-like contributions, resulted very similar to theoretical
predictions [M. Xie et al., Phys. Rev. B $\bf{89}$, 245417 (2014)] of a two
dimensional electron gas confined in the normal direction of a perovskite
interface. In particular, the small but finite conducting character of the $z$
component provides a unambiguous signature of the presence of the 2DEG in the
Cu/Bi$_2$O$_3$ system. Although the original constituent materials do not
possess spin-orbit coupling (SOC), the resulting interfacial hybridization of
such states induce electronic asymmetric wave functions. This work demonstrates
the detection of 2DEG in amorphous crystals allowing to study its challenging
interfacial phenomena such as SOC and interface-bulk coupling, overcoming an
experimental impediment which has hold back for decades important advancements
for the understanding of 2DEGs in amorphous materials.
