Periodic driving of a coherent quantum many body system and relaxation to the Floquet diagonal ensemble

The coherent dynamics of many body quantum system is nowadays an experimental reality: by means of the cold atoms in optical lattices, many Hamiltonians and time-dependent perturbations can be engineered. In this Thesis we discuss what happens in these systems when a periodic perturbation is applied. Thanks to Floquet theory, we can see that -- if the Floquet spectrum obeys certain continuity conditions possible in the thermodynamic limit-- dephasing among Floquet quasi-energies makes local observables relax to a periodic steady regime described by an effective density matrix: the Floquet diagonal ensemble (FDE). By means of numerical examples on the Quantum Ising Chain and the Lipkin model, we discuss the properties of the FDE focusing on the difference among ergodic and regular quantum dynamics and on how this reflects on the thermal properties ($T=\infty$) of the asymptotic condition. We verify thermalization in the classically ergodic Lipkin model and we demonstrate that this effect is induced by the Floquet states being delocalized and obeying Eigenstate Thermalization Hypothesis.We discuss also, in the Ising chain case, the work probability distribution, whose asymptotic condition is not described by the form (Generalized Gibbs Ensemble) that FDE acquires for local obserbvables because of integrability. Dephasing makes some correlations invisible in the local observables, but they are still present in the system. We consider also the linear response limit: when the amplitude of the perturbation is vanishingly small, the Floquet diagonal ensemble is not sufficient to describe the asymptotic condition given by LRT. For every small but finite amplitude, there are quasi-degeneracies in the Floquet spectrum giving rise to pre-relaxation to the condition predicted by Linear Response; these phenomena are strictly related to energy absorption and boundedness of the spectrum.