  Hubbard excitons are bound states of doublons and holes that can be
experimentally probed both in real materials, such as cuprates, and in cold
atom quantum simulators. Here we compare properties of a Hubbard exciton to
those of a pair of distinguishable dopants in the $t-J$ model and show how
insights into pair properties can be obtained through excitonic spectra. In
particular, we perform large-scale numerical simulations of spectral functions
and optical conductivities and obtain remarkable agreement between Hubbard
excitons and pairs of distinguishable dopants. The latter can be decomposed
into symmetric (bosonic) and anti-symmetric (fermionic) sectors of
indistinguishable dopants, thus enabling a detailed understanding of different
features observed in the excitonic spectra through comparison with a
semi-analytical geometric string theory approach. We further compare
theoretically computed exciton spectra in a single band Fermi-Hubbard model to
resonant inelastic X-ray scattering (RIXS) studies of the parent insulating
cuprate materials. We find remarkable agreement between the two spectra in both
energy and momentum dependence. Our analysis suggests that multiple long-lived
ro-vibrational exciton resonances have been observed in RIXS spectra.
Experimentally, these features are known to persist up to optimal doping. The
comparison we provide between semi-analytical theory, large-scale numerics, and
experimental data thus provides an explanation of the RIXS measurements and
provides new insight into the nature of pairing in cuprates.
