Motivated by the experiment by Budden et al. [Nat. Phys. 17, 611 (2021)], who observed signatures of long-lived photoinduced superconductivity, we develop an accurate analytical and computational approach to nonequilibrium superconductivity following a quench. We consider the BCS-Holstein model, which includes both integrable local electron-electron interactions and integrability-breaking electron-phonon coupling. We develop the Keldysh-Eliashberg theory on the Kadanoff-Baym contour, which enables us to describe nonequilibrium dynamics of the superconductor. We consider a quench in interactions, which results in a dynamic transition from the initial superconducting state to a normal thermal state in the end of the evolution. It is shown that the dynamics contain two stages: The early-time integrable behavior, involving coherent oscillations of the superconducting order parameter, crosses over to the late-time ergodic dynamics exhibiting a thermal decay into an equilibrium state. In the former regime, our computational approach both reproduces exact analytical results on the integrable dynamics of the order parameter (obtained for the zero-temperature initial state) and generalizes those to the case of an initial thermal state. The method also succeeds in describing both integrable-to-thermalizing crossover and the late-time thermal decay, which is shown to be consistent with the time-dependent Ginzburg-Landau theory (with the exponential decay time dependent on the density of quasiparticle excitations). We observe the electron distribution function approaching the Fermi-Dirac thermal distribution at final stages. The details of two-time nonequilibrium dynamics depend on the density of quasiparticles in the initial state and the integrability-breaking parameters, which under certain conditions may result in a long-lived transient superconductivity consistent with experiment.

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• Electron-phonon coupling
• Superconductivity
• Superconductors
• Nonequilibrium Green's function