Non-equilibrium control of quantum systems and their phases

In the last years, one of the major aim of the condensed matter area of interest, at the same rate of fundamental research, is to produce new and substantial breakthrough innovations to be applied in the information technology world. Indeed, semiconductor-based devices have constantly improved their performances thanks to the ability of continually shrink the components inside chips. In this process, however, physical components cannot be reduced in size infinitely. All matter consists of atoms and, at the atomic level, particles behave according to the laws of quantum mechanics. With this respect, the control of quantum systems is becoming fundamental to go beyond the present technology and the engineering of powerful phases of matter, very hard to obtain in standard conditions, is one of the main goals people are trying to achieve. The realization of quantum computation devices, in this sense, strongly depends on these new ideas success. In this respect, researchers have faced the difficulty, both from the theoretical and the experimental points of view, to control the state of a quantum system. Quantum control, i.e. the control of quantum phenomena, is becoming one of the major concerns in condensed matter physics, even if results obtained in the recent past are mainly confined to static systems in equilibrium, due to the difficulty to experimentally manipulate out-of-equilibrium quantum systems and the absence of an efficient general theoretical framework to describe non-equilibrium dynamics. In this thesis, I address these currently open questions by inspecting several different condensed matter models, using various methods to drive the system out of equilibrium and focusing on its dynamical features as well as the properties of its equilibration towards a thermal or, more interestingly, non thermal steady state. I discuss the possibility to manipulate various systems to give rise to peculiar dynamical behaviors and corresponding steady states with properties not attainable in thermal equilibrium, ranging from quantum phase transitions and their dynamical counterparts to superconductivity.