We consider the excitation of Higgs modes via the modulation of the Bardeen-Cooper-Schrieffer coupling within the Migdal-Eliashberg-Keldysh theory of time-dependent superconductivity. Despite the presence of phonons, which break integrability, we observe Higgs amplitude oscillations reminiscent of the integrable case. The dynamics of quasiparticles follows from the effective Bogolyubov–de Gennes (BdG) equations, which represent a Floquet problem for the Bogolyubov quasiparticles. We find that when the Floquet-Bogolyubov bands overlap, the homogeneous solution formally leads to a negative superfluid density, which is no longer proportional to the amplitude of the order parameter. This result indicates an instability, which we explore using spatially resolved BdG equations. Spontaneous appearance of spatial inhomogeneities in the order parameter is observed and they first occur when the superfluid density becomes unphysical. We conclude that the homogeneous solution to time-dependent superconductivity is generally unstable and breaks up into a complicated spatial landscape via an avalanche of topological excitations.

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• Superconductivity
• Superfluid density
• Nonequilibrium systems
• Bogoliubov-de Gennes equations
• Eliashberg theory
• Nonequilibrium Green's function