  Recent infrared measurements of phonon peaks in gated bilayer graphene reveal
two striking signatures of electron-phonon interaction: an asymmetric Fano
lineshape and a giant variation of the peak intensity as a function of the
applied gate voltage. In this Letter we provide a unified theoretical framework
which accounts for both these effects and unveils the occurrence of a switching
mechanism between the symmetric ($E_g$) and anti-symmetric ($E_u$) phonon mode
as dominant channel in the optical response. A complete phase diagram of the
optical phonon response is also presented, as a function of both the charge
density and the bandgap.
