  The ultrafast optical manipulation of ordered phases in strongly correlated
materials is a topic of significant theoretical, experimental, and
technological interest. Inspired by a recent experiment on light-induced
superconductivity in fullerenes [Mitrano et al., Nature 530, 2016], we develop
a comprehensive theory of light-induced superconductivity in driven
electron-phonon systems with lattice nonlinearities. In analogy with the
operation of parametric amplifiers, we show how the interplay between the
external drive and lattice nonlinearities lead to significantly enhanced
effective electron-phonon couplings. We provide a detailed and unbiased study
of the nonequilibrium dynamics of the driven system using the real-time Green's
function technique. To this end, we develop a Floquet generalization of the
Migdal-Eliashberg theory and derive a numerically tractable set of quantum
Floquet-Boltzmann kinetic equations for the coupled electron-phonon system. We
study the role of parametric phonon generation and electronic heating in
destroying the transient superconducting state. Finally, we predict the
transient formation of electronic Floquet bands in time- and angle-resolved
photo-emission spectroscopy experiments as a consequence of the proposed
mechanism.
