Fully quantum perturbative description of correlated Stokes--anti-Stokes scattering

The process in which Raman scattering produces correlated Stokes and anti-Stokes radiation is known as Stokes--anti-Stokes (SaS) scattering. It has been shown recently that this process can generate entangled photon pairs, making it a promising tool for quantum optical technologies, but a proper quantum theoretical description was lacking. In this paper, a fully quantum derivation of the electric polarization in a medium with vibrational Raman response, with quantized electromagnetic fields, is developed. Using quantum perturbation theory for Heisenberg operators, we find the solution for the material electric polarization and show that a four-wave mixing-like correlated SaS scattering appears in the first order of perturbation and completely characterizes the non-resonant SaS photon pair production. We also discuss how to construct the third-order non-linear optical susceptibility for the SaS scattering from the quantum formalism, and show that it coincides with the one derived for classical fields in stimulated Raman.