  The magnetic excitation spectrum in the bilayer iridate Sr$_3$Ir$_2$O$_7$ has
been investigated using high-resolution resonant inelastic x-ray scattering
(RIXS) performed at the iridium L$_3$ edge and theoretical techniques. A study
of the systematic dependence of the RIXS spectrum on the orientation of the
wavevector transfer, $\mathbf{Q}$, with respect to the iridium-oxide bilayer
has revealed that the magnon dispersion is comprised of two branches well
separated in energy and gapped across the entire Brillouin zone. Our results
contrast with those of an earlier study which reported the existence of a
single dominant branch. While these earlier results were interpreted as two
overlapping modes within a spin-wave model of weakly coupled iridium-oxide
planes, our results are more reminiscent of those expected for a system of
weakly coupled dimers. In this latter approach the lower and higher energy
modes find a natural explanation as those corresponding to transverse and
longitudinal fluctuations, respectively. We have therefore developed a
bond-operator theory which describes the magnetic dispersion in
Sr$_3$Ir$_2$O$_7$ in terms of quantum dimer excitations. In our model
dimerisation is produced by the leading Heisenberg exchange, $J_c$, which
couples iridium ions in adjacent planes of the bilayer. The Hamiltonian also
includes in plane exchange, $J$, as well as further neighbour couplings and
relevant anisotropies. The bond-operator theory provides an excellent account
of the dispersion of both modes, while the measured $\mathbf{Q}$ dependence of
the RIXS intensities is in reasonable qualitative accord with the spin-spin
correlation function calculated from the theory. We discuss our results in the
context of the quantum criticality of bilayer dimer systems in the presence of
anisotropic interactions derived from strong spin-orbit coupling.
