In this work we discuss the relaxation properties of a quantum one dimensional gas. The interest in studying this kind of systems has recently grown due to the development of innovative experimental techniques that made possible the confinement of particles in quasi one-dimensional optical lattices weakly interacting with the environment. These properties can be explored using a quantum quench: we prepare the system in the ground state of a given hamiltonian, then we suddenly change a parameter and let it evolve unitarly. There are mainly two types of quantum quenches: local and global
We consider a global quantum quench in a confined one dimensional bosonic gas. We show that relaxation occurs in slightly different manner than in the periodic case: the stationary correlation function "feels'' the boundaries also in the thermodynamic limit. Furthermore we find a compact expression for the time dependent density profile and for the fermionic correlation function. Both functions describe the non equilibrium behavior of the system. The solutions of the confined problem present difficulties which were absent in the periodic case. These have been overcome by some ingenious approximations which become exact in the thermodynamical limit, thus providing the analytical solution to the problem. In the course of the computation numerical analysis is often used as a support.
We found out that the long-time state of the confined system is translationally invariant (we demonstrated that non translationally invariant corrections are finite-size effects), in particular the stationary density profile is the same as in the homogeneous case, as naively expected. But the effects of the confinement are visible both in the stationary two point correlation function and in the non trivial evolution of the density profile
