  Recent improvements in the energy resolution of resonant inelastic x-ray
scattering experiments (RIXS) at the Cu-L$_3$ edge have enabled the study of
lattice, spin, and charge excitations. Here, we report on the detection of a
low intensity signal at 140meV, twice the energy of the bond-stretching (BS)
phonon mode, in the cuprate superconductor
$\textrm{Bi}_2\textrm{Sr}_2\textrm{Ca}\textrm{Cu}_2\textrm{O}_{8+x}$ (Bi-2212).
Ultra-high resolution polarimetric RIXS measurements allow us to resolve the
outgoing polarization of the signal and identify this feature as a two-phonon
excitation. Further, we study the connection between the two-phonon mode and
the BS one-phonon mode by constructing a joint density of states toy model that
reproduces the key features of the data.
