  Bound states of the neutron-deficient, near-dripline nucleus $^{36}$Ca were
populated in two-neutron removal from the ground state of $^{38}$Ca, a direct
reaction sensitive to the single-particle configurations and couplings of the
removed neutrons in the projectile wave function. Final-state exclusive cross
sections for the formation of $^{36}$Ca and the corresponding longitudinal
momentum distributions, both determined through the combination of particle and
$\gamma$-ray spectroscopy, are compared to predictions combining eikonal
reaction theory and shell-model two-nucleon amplitudes from the USDB, USDC, and
ZBM2 effective interactions. The final-state cross-section ratio
$\sigma(2^+_1)/\sigma(0^+)$ shows particular sensitivity and is approximately
reproduced only with the two-nucleon amplitudes from the ZBM2 effective
interaction that includes proton cross-shell excitations into the $pf$ shell.
Characterizing the proton $pf$-shell occupancy locally and schematically, an
increase of the $sd$-$pf$ shell gap by $250$ keV yields an improved description
of this cross-section ratio and simultaneously enables a reproduction of the
$B(E2;0^+_1\to2^+_1)$ excitation strength of $^{36}$Ca. This highlights an
important aspect if a new shell-model effective interaction for the region was
to be developed on the quest to model the neutron-deficient Ca isotopes and
surrounding nuclei whose structure is impacted by proton cross-shell
excitations.
