Fig. 1.

Loop diagrams for two-molecule (Eq. 5) X-ray scattering processes with the amplitude-squared form explicitly indicated. The shaded area represents an arbitrary excitation that prepares the system in a superposition state of $|g⟩$ and $|e⟩$ (further explained in SI Appendix, Fig. S1). We denote modes of the X-ray probe pulse with $p$ and $p\prime$, whereas $s$, $s\prime$ represent relevant scattering modes [${\mathbf{k}}_{p^{\left(\prime \right)}}$ has frequency ${\omega }_{p^{\left(\prime \right)}}$ and ${\mathbf{k}}_{s^{\left(\prime \right)}}$ has frequency ${\omega }_{s^{\left(\prime \right)}}$]. Elastic scattering processes come with ${\widehat{\sigma }}_{gg}$ or ${\widehat{\sigma }}_{ee}$ and are denoted by black field arrows. Inelastic processes in which the molecule gains (Stokes) or loses (anti-Stokes) energy to the field come with ${\widehat{\sigma }}_{ge}$ or ${\widehat{\sigma }}_{eg}$ depending whether the action is on the ket or bra and are denoted with red and blue field arrows to indicate the field’s spectral shift due to the particular diagram. Note: In diagram J, the energy order of states $e,e\prime$ is not set. We have depicted the elastic case for specificity. Diagrams I–L identify the corresponding terms in Eq. 11.