Numerical Study of Two-Phonon Sidebands in the Optical Absorption of Free Polarons in the Strong-Coupling Limit

Using the theory and the analytical results of Kartheuser, Evrard, and Devreese (KED), we have calculated numerically the two-phonon sideband in the optical absorption coefficient of free polarons. Therefore, integrals of the type ${∫}({d{{→}}{k}}{{{{→}}{k}}²}){∫}({d{{{→}}{k}}^{{'}}}{{{{→}}{k}}^{{'}2}}){{{→}}{k}}^{{'}2j}{cos}²ʲ{{θ}}^{{'}}{{{→}}{k}}²ⁱ{cos}²ⁱ{θ}{δ}({({{→}}{k}+{{{→}}{k}}^{{'}})}²{-}{{ε}}²){e}^{{-}x{{{→}}{k}}²{-}y{{{→}}{k}}^{{'}2}}$ as obtained by KED, are treated analytically and reduced to a finite sum of single integrals for integer $i$ and $j>~0$, for all real $x$, $y>0$ and for any real ${ε}$. Our numerical results, combined with the zero-phonon line and the one-phonon sideband obtained by KED, confirm the mechanism for the absorption of light by free polarons proposed by KED.