Signatures of the electron-phonon interaction in the far-infrared

Motivated by a recent optical experiment on ${\mathrm{Ba}}_{1-x}{\mathrm{K}}_x\mathrm{Bi}{\mathrm{O}}_3$, we calculate the far-infrared conductivity in the normal and superconducting states of an electron-phonon driven superconductor, and note the signatures of the electron-phonon interaction. We find that two methods in particular are effective ways to determine the electron-phonon coupling strength, both of which are applicable in the normal state. One relies on a sum rule, and requires independent knowledge of the plasma frequency, the other relies on the Drude-like width, and was used by Puchkov et al. in their infrared measurements. We note quantitative corrections to this latter method. However, we support their conclusion that the electron-phonon interaction in ${\mathrm{Ba}}_{1-x}{\mathrm{K}}_x\mathrm{Bi}{\mathrm{O}}_3$ is too weak to account for superconductivity. A third method relies on observing shifts of structure in the superconducting state. However, we find that for systems with intermediate-coupling strength, such structure can be very difficult to detect, especially if the system is moderately dirty.