Dynamics of repeatedly driven closed systems
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This thesis covers my work in the field of closed, repeatedly driven, Hamiltonian systems. These systems do not exchange particles with the surrounding environment and their time-evolution is described by Hamilton's equations of motion (in the classical framework) or the Schroedinger equation (in the quantum framework). Their interaction with the environment is encoded into the time-dependence of the system's Hamiltonian. Chapter 1 is an "Overview" in which the status of the field, my contributions and future prospective are outlined. Chapters 2 to 4 provide the theoretical background which is used in Chapters 5 to 7 to derive some original results. These results show that in Hamiltonian systems, after many driving events, universal properties emerge. In particular, using the framework of the linear Boltzmann equation, I have studied the dynamics of a mobile, light impurity in a gas of heavy particles. The impurity's kinetic energy increases and, in the long time limit, approaches a non-thermal asymptotic distribution. The significance of this work is to show explicitly the emergence of a non-thermal distribution in a closed, driven system. Moreover, using the work-fluctuation theorems, I have studied the character of the energy distribution of a generic isolated system driven according a generic protocol. Both thermal and non-thermal distributions can be realized for the same system by changing the characteristics of the driving protocol. These two different regimes are separated by a dynamical phase transition. Finally, I have used the Floquet Theory and the Magnus Expansion to analyze the behavior of a generic interacting system which is driven periodically in time. For fast driving the system is unable to absorb energy and remains localized in the low energy part of the Hilbert space while for slow driving the system absorbs energy and, in the long time limit, it is delocalized in the entire Hilbert space. These two qualitatively different behaviors are separated by a many-body localization transition which is related to the break down of the Magnus expansion at the critical value of the driving frequency.