Polarization-resolved broadband absorption ($W_x=\pi |d_{21,x}{|}^2{\rho }_x({\omega }_0)/({ϵ}_0{\hslash }^2)$)

Weak incoherent frequency components contribute probabilities. The long-time sinc-squared kernel gives the displayed stimulated absorption rate for a fixed polarization and spectral energy density per angular frequency. Isotropic unpolarized radiation instead contributes $|{\mathbf{d}}_{21}{|}^2/3$. Converting to density per ordinary frequency changes the coefficient by $2\pi$.