Reduced density matrix theory dynamics

In principle, the theory of nonlinear spectroscopy with femtosecond laser pulses is well developed. A comprehensive and up-to-date exposition of nonlinear optical spectroscopy in the femtosecond time domain is provided by the monograph of Mukamel. ° For additional reviews, see Refs. 7 and 11-14. While many theoretical papers have dealt with the analysis or prediction of femtosecond time-resolved spectra, very few of these studies have explicitly addressed the dynamics associated with conical intersections. In the majority of theoretical studies, the description of the chemical dynamics is based on rather simple models of the system that couples to the laser fields, usually a few-level system or a set of harmonic oscillators. In the case of condensed-phase spectroscopy, dissipation is additionally introduced by coupling the system to a thermal bath, either at a phenomenological level or in a more microscopic maimer via reduced density-matrix theory. 

T. Yanai and G. K. L. Chan, Canonical transformation theory for dynamic correlations in multireference problems, in Reduced-Density-Matrix Mechanics With Application to Many-Electron Atoms and Molecules, A Special Volume of Advances in Chemical Physics, Volume 134 (D.A. Mazziotti, ed.), Wiley, Hoboken, NJ, 2007. 

In the DC-biased structures considered here, the dynamics are dominated by electronic states in the conduction band [1]. A simplified version of the theory assumes that the excitation occurs only at zone center. This reduces the problem to an n-level system (where n is approximately equal to the number of wells in the structure), which can be solved using conventional first-order perturbation theory and wave-packet methods. A more advanced version of the theory includes all of the hole states and electron states subsumed by the bandwidth of the excitation laser, as well as the perpendicular k states. In this case, a density-matrix picture must be used, which requires a solution of the time-dependent Liouville equation. Substituting the Hamiltonian into the Liouville equation leads to a modified version of the optical Bloch equations [13,15]. These equations can be solved readily, if the k states are not coupled (i.e., in the absence of Coulomb interactions). 

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