  With infrared ellipsometry we studied the response of the confined electrons
in a $\gamma$-Al$_2$O$_3$/SrTiO$_3$ heterostructures in which they originate
predominantly from oxygen vacancies. From the analysis of a so-called Berreman
mode, that develops near the highest longitudinal optical phonon mode of
SrTiO$_3$, we derive the sheet carrier density, ${N_{s}}$, the mobility, $\mu$,
and also the depth profile of the carrier concentration. Notably, we find that
${N_{s}}$ and the shape of the depth profile are similar as in
LaAlO$_3$/SrTiO$_3$ heterostructures for which the itinerant carriers are
believed to arise from a polar discontinuity. The main differences concern the
higher mobility and a relatively stronger confinement of the electrons in
$\gamma$-Al$_2$O$_3$/SrTiO$_3$
