• Bethe ansatz
• Lieb-Liniger model
• transverse Ising model
• entanglement
• quantum quench

This thesis focuses on the study of the properties of non equilibrium condensed matter many body quantum systems, a subject which is becoming more and more interesting. This is because, parallel to the theoretical analysis of simplified low dimensional models, experimental techniques that allow to observe the quantum dynamics in a more or less controlled way are being developed, such as non linear transport, ultrafast pump probe and ultracold atoms.
Among the possible ways to perturb a system from an equilibrium configuration, I have focused on the most schematic protocol, the so called quantum quench, i.e. the instantaneous change of a parameter governing the static Hamiltonian of the system. This has the consequence of bringing the system in an initially out of equilibrium configuration; from this time on, we let the system evolve isolated from the rest of the environment according to this time independent Hamiltonian.
With this kind of setup, some of the interesting questions that can be addressed concern the long time dynamics of the system, in particular finding an ensemble formulation for the equilibrium distribution function locally reached by the system at long times after the quench. Such an assessment is particularly useful because it allows to make predictions about any local correlation function in the stationary state without the need to follow the actual dynamics. I have investigated this issue in a variety of exactly solvable models, both free and interacting, in lattice systems and in the continuum, starting from several different initial conditions. As a particularly interesting class of initial conditions, I focused on highly excited states of the pre-quenched Hamiltonian, as opposed to the usually studied ground states.
In the first part of the thesis, I considered the dynamical and stationary behaviour of
observables in the transverse field Ising model after a quench of the magnetic field. Choosing an excited state as the initial state yields the same statistical distribution as from initial ground states, namely the Generalized Gibbs distribution: this is an extension of the classical statistical Gibbs ensemble which takes into account all the mutually commuting integrals of motion of the integrable model. On the other hand, the initial excitation profile has consequences on the power law's exponent with which the equilibrium value is reached in some observables.
In the second part of the thesis, I considered the relaxation to a stationary state that takes place in two more experimentally relevant processes. More specifically, in the first place I considered the interplay between a quantum quench of the external potential on half the system's size in a system of relativistic fermions in one dimension and the well-known Klein tunneling. I showed how the phenomenology of Klein tunneling can be obtained with this specific quench and how it can be rigorously derived from the steady state reached in the long time limit, which turns out to be a non-equilibrium-steady-state. In the second place, I have developed a schematic representation of the out of equilibrium dynamics that takes place in one of the most celebrated quench experiment, the so called Quantum Newton’s Cradle experiment. In this experiment, an ultracold bosonic gas initially put in a superposition of opposite momentum states is let evolved under repulsive contact interactions and was shown not to thermalize even after thousands of collisions. The system, which is a close realization of the Lieb-Liniger model, is interacting and requires a non trivial analysis in terms of Bethe ansatz techniques. The usual path to construct the stationary state, which in free-like systems implies the explicit computation of all the integrals of motion, is difficult to apply in a truly interacting model. We thus resort to use a newly developed technique which allows to identify, by means of a variational method, the single representative state important for the long time evolution of the system and then to obtain the stationary momentum distribution function.