Unified theory of quantized electrons, phonons, and photons out of equilibrium: A simplified ab initio approach based on the generalized Baym-Kadanoff ansatz

We present a full ab initio description of the coupled out-of-equilibrium dynamics of photons, phonons, and electrons. In the present approach, the quantized nature of the electromagnetic field as well as of the nuclear oscillations is fully taken into account. The result is a set of integrodifferential equations, written on the Keldysh contour, for the Green's functions of electrons, phonons, and photons where the different kinds 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 generalized 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 kinds of static and dynamical correlations. The final result is a coherent, general, and inclusive scheme to calculate several physical quantities: carrier dynamics, transient photoabsorption, and light emission, all of which include, at the same time, electron-electron, electron-phonon, and electron-photon interactions. We further discuss how all these observables can be easily calculated within the present scheme using a fully atomistic ab initio approach.

Figure 1

Schematic representation of the different processes induced by the interaction of a material with a short and intense laser pulse. The action of the laser creates electron-hole pairs [process (a)]. These pairs interact through the screened Coulomb interaction (zigzag line) creating transient excitonic states. After the photoexcitation, the carriers undergo repeated collisions with other carriers and with the lattice [process (b)]. During these processes, phonons and electron-hole pairs are emitted and/or absorbed. Finally, after picoseconds to microseconds, the excited carriers eventually relax to the ground-state emitting photons [process (c)]. ${\omega }_{\gamma }$ and ${\omega }_{\gamma }^{\prime }$ are photon energies, ${\omega }_{\mathbf{q}\lambda }$ is a phonon energy, and ${\omega }_{\mathrm{e}-\mathrm{h}}$ represents electron-hole pair energy.

Figure 2

The Keldysh contour. Any time index runs on the contour which defines a natural time ordering that only in the upper branch is equivalent to the ordering in the standard real-time axis.

Figure 3

Schematic representation of the mathematical procedure followed to get the Dyson equation within the framework of the Hedin's equations. The initial derivative is split in longitudinal (left) and transverse (right) contributions.

Figure 4

Diagrammatic representation of the Hedin's equations. The bullet represents a $\mathrm{\Pi }$ differential operator, defined in Eq. (51).

Figure 5

Schematic representation of the different formulations of Hedin's equations. (a) Longitudinal case. This is well known and widely used in the literature in its equilibrium and out-of-equilibrium versions. When the interaction with the quantized electromagnetic field is switched on, a new pentagon must be added [case (b)] where the corners are the photon propagator and the transverse electronic polarization and vertex function. The dashed lines correspond to the generalized $GW$ approximation where all vertex functions are neglected.

Figure 6

The generalized $GW$ approximation for the three kinds of interactions: e-e, e-p, and e-$\gamma$. The wiggled propagator in the e-e case represents a statically screened interaction, as explained in Ref. [38].

Figure 7

Graphical representation of Eq. (121). A single line represents the density matrix $\rho$, while the double line is its adjoint $\overline{\rho }$. The wiggled line, in the spirit of Fig. 6, represents a generic interaction propagator.

Figure 8

Schematic representation of the different links between the terms of Eq. (123) and the most elemental physical processes occurring in a typical pump-and-probe experiment: the photoexcitation (a), the relaxation via e-e scattering (b), the relaxation and dissipation via e-p scattering (c), and the final, slow, radiative recombination (d). It is crucial to note that the use of ${\mathrm{\Sigma }}_{\text{COHSEX}}$ allows to include excitonic effects (caused by the electron-hole attraction) in all the processes.

Figure 9

Diagrammatic contributions to the dynamical kernel $K^{\mathrm{dyn}}$ resulting from the application of the functional derivative to $S^{\mathrm{dyn}}$, within the $GW$ approximation.