An unified theory of quantised electrons, phonons and photons out-of-equilibrium: a simplified {\em ab-initio} approach based on the Generalised Baym-Kadanoff ansatz

We present a full ab-inito description of the coupled out-of-equilibrium dynamics of photons, phonons, and electrons. In the present approach the quantised nature of the electromagnetic field as well as of the nuclear oscillations is fully taken into account. The result is a set of integro-differential equations, written on the Keldysh contour, for the Green's functions of electrons, phonons, and photons where the different kind of interactions are merged together. We then concentrate on the electronic dynamics in order to reduce the problem to a computationally feasible approach. By using the Generalised Baym-Kadanoff ansatz and the Completed Collision approximation we introduce a series of efficient but controllable approximations. In this way we reduce all equations to a set of decoupled equations for the density matrix that describe all kind of static and dynamical correlations. The final result is a coherent, general, and inclusive scheme to calculate several physical quantities: carrier dynamics, transient photo-absorption and light-emission. All of which include, at the same time, electron-electron, electron-phonon, and electron-photon interaction. We further discuss how all these observables can be easily calculated within the present scheme using a fully atomistic ab-initio approach.