Interband Kadanoff-Baym equations

(Two-time semiconductor Bloch equations)

Nonequilibrium Greens functions approach to optical excitation and relaxation in semiconductors
Selfconsistent treatment of electron number dynamics, optical response and time-dependent band renormalization
Starting point: Kadanoff-Baym equations for the two-time Greens functions generalized to multi-band systems (corresponding to the two indices)
For an explanation of the two time indices, click here
Band-diagonal functions correspond to electron populations in the respective band, off-diagonal functions describe the probability of transitions between two bands, in analogy to two-level atoms (and the corresponding density matrix).

E ist the classical electric field - laser field plus field induced in the semiconductor (the solution of Maxwell's equations), d is the dipole moment
Omega denotes the renormalized Rabi frequency - the electric field modified by the Hartree-Fock mean field due to the Coulomb interaction (V - Coulomb potential) between the laser pulse excited electrons and holes.

The sigmas are the electron selfenergies.
Note the non-local character of the integrals (time integration, "memory" effect)

The sigmas are generalized scattering rates due to correlation effects between electrons and electrons, phonons, impurities etc.
Here we consider, as an example, scattering due to collisions between electrons and holes, including band-off-diagonal effects (scattering with excitons)
V_s is the statically screened Coulomb potential, pi - the longitudinal polarization ("bubble")