A time convolution less density matrix approach to the nonlinear optical response of a coupled system-bath complex

Time convolution less density matrix theory (TCL) is a powerful and well established tool to investigate strong system-bath coupling for linear optical spectra. We show that TCL equations can be generalised to the nonlinear optical response up to a chosen order in the optical field. This goal is achieved via an time convolution less perturbation scheme for the reduced density matrices of the electronic system. In our approach, the most important results are the inclusion of a electron-phonon coupling non-diagonal in the electronic states and memory effects of the bath: First, the considered model system is introduced. Second, the time evolution of the statistical operator is expanded with respect to the external optical field. This expansion is the starting point to explain how a TCL theory can treat the response up to in a certain order in the external field. Third, new TCL equations, including bath memory effects, are derived and the problem of information loss in the reduced density matrix is analysed. For this purpose, new dimensions are added to the reduced statistical operator to compensate lack of information in comparison with the full statistical operator. The theory is benchmarked with a two level system and applied to a three level system including non-diagonal phonon coupling. In our analysis of pump-probe experiments, the bath memory is influenced by the system state occupied between pump and probe pulse. In particular, the memory of the bath influences the dephasing process of electronic coherences developing during the time interval between pump and probe pulses.