Introduction -- Spin-1/2 ladders -- Methods -- Static properties and NMR relaxation rate -- Dynamical correlations of a spin ladder -- Conclusions and perspectives.

This thesis shows how a combination of analytic and numerical techniques, such as a time dependent and finite temperature Density Matrix Renormalization Group (DMRG) technique, can be used to obtain the physical properties of low dimensional quantum magnets with an unprecedented level of accuracy. A comparison between the theory and experiment then enables these systems to be used as quantum simulators; for example, to test various generic properties of low dimensional systems such as Luttinger liquid physics, the paradigm of one dimensional interacting quantum systems. Application of these techniques to a material made of weakly coupled ladders (BPCB) allowed the first quantitative test of Luttinger liquids. In addition, other physical quantities (magnetization, specific heat etc.), and more remarkably the spin-spin correlations – directly measurable in neutron scattering experiments – were in excellent agreement with the observed quantities. We thus now have tools to quantitatively assess the dynamics for this class of quantum systems.