  The essential building blocks of cuprate superconductors are two-dimensional
CuO$_2$ sheets interspersed with charge reservoir layers. In bilayer cuprates,
two closely spaced CuO$_2$ sheets are separated by a larger distance from the
subsequent pair in the next unit cell. In contrast to single-layer cuprates,
prior theoretical work on bilayer systems has predicted two distinct acoustic
plasmon bands for a given out-of-plane momentum transfer. Here we report random
phase approximation (RPA) calculations for bilayer systems which corroborate
the existence of two distinct plasmons bands. We find that the intensity of the
lower-energy band is negligibly small, whereas the higher-energy band carries
significant spectral weight. We also present resonant inelastic x-ray
scattering (RIXS) experiments at the O $K$-edge on the bilayer cuprate
Y$_{0.85}$Ca$_{0.15}$Ba$_2$Cu$_3$O$_7$ (Ca-YBCO), which show only one
dispersive plasmon branch, in agreement with the RPA calculations. In addition,
the RPA results indicate that the dispersion of the higher-energy plasmon band
in Ca-YBCO is not strictly acoustic, but exhibits a substantial energy gap of
approximately 250 meV at the two-dimensional Brillouin zone center.
