For the description of dynamical effects in quantum mechanical systems on ultrashort time scales, memory effects play an important role. Following a brief motivation I will start to describe how to determine the bath correlation function using a numerical decomposition of the spectral density. After this I will derive a non-Markovian formalism from the time-nonlocal Nakajima-Zwanzig identity. Introducing auxiliary density matrices one arrives at the non-Markovian theory first derived by Meier and Tannor [J. Chem. Phys. 111, 3365 (1999)]. In contrast to this, in a second step a time-local approach will be derived based on the Tokuyama-Mori identity. For a time-independent Hamiltonian one arrives at a generalized Redfield equation while for a time-dependent Hamiltonian one introduces equations of motion for auxiliary operators. So the latter approach is quite similar to the non-local formalism. For the example of a damped harmonic oscillator the derived non-Markovian theories are compared among each other, to the Markovian limit neglecting memory effects and time dependencies, and to exact path integral calculations. Good agreement between the exact calculations and the non-Markovian results is obtained. Some of the non-Markovian theories mentioned above treat the time dependence in the system Hamiltonians non-perturbatively. Therefore these methods can be used for the simulation of experiments with arbitrary large laser fields.
Some details of the talk can be found in J. Chem. Phys. 121, 2505 (2004).