  Here we present a quantum-statistical formulation of third-order polarization
P(3)(t), which is induced in a sample by a sequence of incident external
fields, and serves as a source of an emitted radiation field detected as a
signal in pump-probe (PP) experiments. Our treatment is based on the
perturbation expansion of the non-equilibrium density matrix for calculation of
multi-time correlation functions, and the corresponding response function, at
finite temperature. As a model for our study, the high-temperature
superconductor MgB2 has been selected. Knowledge of the electronic structure of
the studied system, and of the corresponding Eliashberg function that
represents pertinent electron-phonon (EP) interactions, enabled us to
distinguish non-equilibrium processes running over different time-periods in a
sequence of interactions with laser pulses on a microscopic level. We have also
derived temperature-dependent relaxation dynamics as a function of delay time
between the pump and probe pulses. For the studied model system of MgB2, we
have shown that an abrupt increase of the relaxation time at Tc, as detected by
experiments, is the direct consequence of sudden changes in the character of EP
coupling in transition from an adiabatic to a stabilized superconducting
anti-adiabatic state, as it predicts the anti-adiabatic theory of
electron-vibration interactions. The BCS model, which preserves the adiabatic
character of EP coupling also below the critical temperature of MgB2, is
basically unable to reflect the enormous sudden increase of the relaxation
time. Based on diagrammatic perturbation theory, differences in the optical
pump-optical probe and the optical pump-terahertz probe settings of MgB2 PP
relaxation dynamics are discussed.
