  Atomically precise superlattices involving transition metal oxides provide a
unique opportunity to engineer correlated electron physics using strain
(modulated by choice of substate) and quantum confinement (controlled by layer
thickness). Here we use the combination of density functional theory and
dynamical mean field theory (DFT+DMFT) to study Ni E$_g$ $d$-orbital
polarization in strained LaNiO$_3$/LaAlO$_3$ superlattices consisting of four
layers of nominally metallic NiO$_2$ and four layers of insulating AlO$_2$
separated by LaO layers. The layer-resolved orbital polarization is calculated
as a function of strain and analyzed in terms of structural, quantum
confinement, and correlation effects. The effect of strain is determined from
the dependence of the results on the Ni-O bond-length ratio and the octahedral
rotation angles; quantum confinement is studied by comparison to bulk
calculations with similar degrees of strain; correlation effects are inferred
by varying interaction parameters within our DFT+DMFT calculations. The
calculated dependence of orbital polarization on strain in superlattices is
qualitatively consistent with recent X-ray absorption spectroscopy and resonant
reflectometry data. However, interesting differences of detail are found
between theory and experiment. Under tensile strain, the two inequivalent Ni
ions display orbital polarization similar to that calculated for strained bulk
LaNiO$_3$ and observed in experiment. Compressive strain produces a larger
dependence of orbital polarization on Ni position and even the inner Ni layer
exhibits orbital polarization different from that calculated for strained bulk
LaNiO$_3$.
