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Dynamical screening is a key property of charged many-particle systems. Itstheoretical description is based on the $GW$ approximation that is extensivelyapplied for ground-state and equilibrium situations but also for systems drivenout of equilibrium. The main limitation of the $GW$ approximation is theneglect of strong electronic correlation effects that are important in manymaterials as well as in dense plasmas. Here we derive the dynamically screenedladder (DSL) approximation that selfconsistently includes, in addition to the$GW$ diagrams, also particle--particle and particle--hole $T$-matrix diagrams.The derivation is based on reduced-density-operator theory and the result isequivalent to the recently presented G1--G2 scheme [Schl\"unzen \textit{etal.}, Phys. Rev. Lett. \textbf{124}, 076601 (2020); Joost \textit{et al.},Phys. Rev. B \textbf{101}, 245101 (2020)]. We perform extensive time-dependentDSL simulations for finite Hubbard clusters and present tests against exactresults that confirm excellent accuracy as well as total energy conservation ofthe approximation. At strong coupling and for long simulation durations,instabilities are observed. These problems are solved by enforcing contractionconsistency and applying a purification approach.

The dynamically screened ladder approximation: Simultaneous treatment of strong electronic correlations and dynamical screening out of equilibrium