The interface between the insulators LaAlO$_3$ and SrTiO$_3$ accommodates a two-dimensional electron liquid (2DEL) -- a high mobility electron system exhibiting superconductivity as well as indications of magnetism and correlations. While this flagship oxide heterostructure shows promise for electronics applications, the origin and microscopic properties of the 2DEL remain unclear. The uncertainty remains in part because the electronic structures of such nanoscale buried interfaces are difficult to probe, and is compounded by the variable presence of oxygen vacancies and coexistence of both localized and delocalized charges. These various complications have precluded decisive tests of intrinsic electronic and orbital reconstruction at this interface. Here we overcome prior difficulties by developing an interface analysis based on the inherently interface-sensitive resonant x-ray reflectometry. We discover a high charge density of 0.5 electrons per interfacial unit cell for samples above the critical LaAlO$_3$ thickness, and extract the depth dependence of both the orbital and electronic reconstructions near the buried interface. We find that the majority of the reconstruction phenomena are confined to within 2 unit cells of the interface, and we quantify how oxygen vacancies significantly affect the electronic system. Our results provide strong support for the existence of polarity induced electronic reconstruction, clearly separating its effects from those of oxygen vacancies.
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