When Feynman proposed quantum computing in 1982, he suggested as one application the simulation of quantum systems that are too hard for analytical methods or classical computers. In this collaboration with the experimental group of Immanuel Bloch (LMU and MPQ) a simulator for the study of the relaxation of a non-equilibrium many-body quantum state towards thermal equilibrium was proposed. While theory allowed the validation of this dynamical quantum simulator without a single free fit parameter, the experiment itself could go way beyond the simulation times reached by the best classical methods, allowing the observation oft he build-up of non-trivial quantum correlations and the thermalization process. Ref.: Trotzky, Chen, Flesch, McCulloch, Schollwöck, Eisert, Bloch, Nature Physics 8, 325 (2012) Experiments with ultra-cold atomic gases give access to the non-equilibrium dynamics of many-body systems. Here we study a cloud of interacting fermions that expands in an optical lattice after being released from the harmonic trap (the figure shows a typical contour plot of the particle density profile during such an expansion). We show that such a system, described by the Hubbard model, can expand ballistically in one dimension, allowing us to study the expansion velocity as a function of initial conditions. Our predictions for the expansion velocity could be verified in experiments similar to the one by Schneider et al. (Nature Physics 8, 213- (2012)) that was done for the two-dimensional case. Ref.: Langer, Schuetz, McCulloch, Schollwöck, Heidrich-Meisner (Phys. Rev. A 85, 043618 (2012) [more] (arxiv: 1011.3404) We study the effects of the Coulomb interaction in the one dimensional Kondo lattice model on the phase diagram, the static magnetic susceptibility and electron spin relaxation. We show that onsite Coulomb interaction supports ferromagnetic order and nearest neighbor Coulomb interaction drives, depending on the electron filling, either a paramagnetic or ferromagnetic order. Furthermore we calculate electron quasiparticle life times, which can be related to electron spin relaxation and decoherence times, and explain their dependence on the strength of interactions and the electron filling in order to find the sweet spot of parameters where the relaxation time is maximized. We find that effective exchange processes between the electrons dominate the spin relaxation and decoherence rate.