The non-equilibrium dynamics of isolated quantum systems represent a theoretical and experimental challenge raising many fundamental questions with applications to different fields of modern physics. In these proceedings, we briefly review some of the recent findings on the subject, with particular emphasis to the existence of stationary expectation values of local observables and to their statistical mechanics description. It turns out that the appropriate statistical ensemble describing these asymptotic values depends on whether the Hamiltonian governing the time evolution is integrable or not.
The recent advance of techniques in controlling ultra-cold gases in optical lattice provides a ideal experimental platform to study quantum many-body physics. The almost isolated and highly tunable systems created in these experimental setups are perfect for studying coherent time evolution of closed quantum many-body systems. In this thesis I present an analytical approach on the time evolution of the anisotropic Heisenberg spin chain (XXZ model), which is a fundamental model but has rich dynamics. This approach is developed through generalizing the Yudson contour representation. With this approach I obtain the exact time-dependent wave functions after a quantum quench in an integral form for generic initial states and for any anisotropy ∆. I reformulate the integrals and show their physical interpretation as propagating magnons and their bound states, which were predicted by the “string” solutions of Bethe Ansatz equations. To apply the new approach to specific problems, I calculate the quench dynamics of various observables from some particular initial states: starting from a local N ́eel state I calculate the time evolution of the antiferromagnetic order parameter–staggered magnetization; starting from a state with consecutive flipped spins I calculate the evolution of the local magnetization, and predict the evolution of the induced spin currents. These results and predictions can be confronted with recently experiments in ultra-cold gases in optical lattices, and are expected to contribute to the understanding of non-equilibrium physics of isolated many-body systems.