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We perform a systematic analysis of the influence of phonon driving on thesuperconducting Holstein model coupled to heat baths by studying both thetransient dynamics and the nonequilibrium steady state (NESS) in the weak andstrong electron-phonon coupling regimes. Our study is based on thenonequilibrium dynamical mean-field theory, and for the NESS we present aFloquet formulation adapted to electron-phonon systems. The analysis of thephonon propagator suggests that the effective attractive interaction can bestrongly enhanced in a parametric resonant regime because of the Floquet sidebands of phonons. While this may be expected to enhance the superconductivity(SC), our fully self-consistent calculations, which include the effects ofheating and nonthermal distributions, show that the parametric phonon drivinggenerically results in a suppression or complete melting of the SC order. Inthe strong coupling regime, the NESS always shows a suppression of the SC gap,the SC order parameter and the superfluid density as a result of the driving,and this tendency is most prominent at the parametric resonance. Using thereal-time nonequilibrium DMFT formalism, we also study the dynamics towards theNESS, which shows that the heating effect dominates the transient dynamics, andSC is weakened by the external modulations, in particular at the parametricresonance. In the weak coupling regime, we find that the SC fluctuations abovethe transition temperature are generally weakened under the driving. Thestrongest suppression occurs again around the parametric resonances because ofthe efficient energy absorption.

Nonequilibrium steady states and transient dynamics of conventional superconductors under phonon driving