  High-pressure x-ray absorption spectroscopy was performed at the Ir $L_3$ and
$L_2$ absorption edges of Sr$_3$Ir$_2$O$_7$. The branching ratio of white line
intensities continuously decreases with pressure, reflecting a reduction in the
angular part of the expectation value of the spin-orbit coupling operator,
$\left\langle {\bf L} \cdot {\bf S} \right\rangle$. Up to the high-pressure
structural transition at 53 GPa, this behavior can be explained within a
single-ion model, where pressure increases the strength of the cubic crystal
field, which suppresses the spin-orbit induced hybridization of $J_{\text{eff}}
= 3/2$ and $e_g$ levels. We observe a further reduction of the branching ratio
above the structural transition, which cannot be explained within a single-ion
model of spin-orbit coupling and cubic crystal fields. This change in
$\left\langle {\bf L} \cdot {\bf S} \right\rangle$ in the high-pressure,
metallic phase of Sr$_3$Ir$_2$O$_7$ could arise from non-cubic crystal fields
or a bandwidth-driven hybridization of $J_{\text{eff}}=1/2,\,3/2$ states, and
suggests that the electronic ground state significantly deviates from the
$J_{\text{eff}}=1/2$ limit.

Strongly Correlated Electrons