• dynamical localization
• kicked Bose-Hubbard
• thermalization

In classical physics the emergence of statistical mechanics is quite well understood in terms of chaotic behaviour in phase space and the ergodic hypothesis. It seems instead much more complex to understand how the unitary dynamics of a complex isolated quantum system can bring an arbitrary initial state to thermalization at long enough times. Many progresses have been made in the last few decades, partly prompted by the new experimental platforms of ultra-cold atoms in optical lattices and trapped ions, which allow for the simulation of isolated systems. Some questions however remain open.

In particular, a problem which is attracting an increasing attention is thermalization of local observables in periodically-driven (also termed Floquet) systems. From a foundational perspective, Floquet thermalization is extremely peculiar already in classical systems and even more in the quantum case. In fact, being energy not conserved, the thermal ensemble in this case must yield every state with the same probability independently from its energy, such as a T=\infty canonical ensemble. Thus thermalization is strictly connected to energy absorption from the driving. Moreover, in order to have thermalization, it is very important the structure of the eigenstates of the dynamics (Floquet states) which must be strongly entangled and locally equivalent to the T=\infty thermal ensemble (Eigenstate thermalization). Furthermore quantum Floquet systems can support new exciting phases of matter, such as time crystals, which do not have a counterpart in static systems. Therefore it is extremely important to understand when thermalization occurs and when it is suppressed, in order to stabilize these peculiar phases.

In parallel one may wonder what is the role of quantum mechanics in thermalization. Is the thermal or non-thermal behaviour dominated by the underlying semi-classical dynamics, or may new quantum phenomena emerge, completely altering the classical picture? Along this line, an interesting single-particle phenomenon, known as dynamical localization, was discovered in the 80s in the kicked rotor model (a periodically-driven system). In this model quantum coherence totally hinders thermalization, even when the underlying classical dynamics is ergodic. This phenomenon can in fact be understood as an Anderson localization in energy space, via a suitable mapping.

In more recent years, a debate has been opened regarding whether dynamical localization can be present also in many-body systems or if, increasing the system size, dynamical localization will eventually be broken and ergodicity recovered. So far, various works have focused on different version of coupled kicked rotors. However, the debate is far from being settled for two main reasons. First of all, until the very recent paper (Notarnicola et al., 2018) in which a new mapping is proposed, the connection between dynamical localization and Anderson localization was not clear for more than one rotor. Secondly, numerical studies of coupled kicked rotors are extremely limited (so far a maximum of $3$ coupled rotors has been considered), due to the huge Hilbert space of the model.

In the present thesis I try to overcome the second difficulty by considering, instead, the kicked Bose-Hubbard model with number conservation, which in fact allows for numerical studies up to a quite large system size. The kicked Bose-Hubbard model appears to have much in common with kicked rotors models. Indeed, I will show both analytically and numerically that dynamical localization can appear also in a Bose-Hubbard dimer (a chain with only L=2 sites) with a finite number N of bosons in it.

I will then move to analyse how localization properties change when the number of interacting bosons increases.
One way of of going towards this many-body scenario is to keep the number of sites L fixed and to increase N. In this respect I will discuss in great detail the dimer and I will show, both analytically and numerically, that as N increases the model gradually recovers ergodicity.

Afterwards, I will study localization properties as a function of L aiming to understand the behaviour of the model as L and N grow, while N/L is kept fixed.
For this purpose, analytical methods do not provide us with a useful insight any more. Our study then heavily relies on the numerics. I identify a few useful indicators of dynamical localization, apart from the energy absorption suppression. I then employ finite-size numerical simulations (exact diagonalization, Krylov technique, time-evolving block decimation) to evaluate the considered indicators and understand if dynamical localization can survive as L increases. I will show in this way that there seems to be a parameter region where dynamical localization can persist up to the maximum L which we are able to access numerically, suggesting the possibility of dynamical localization also in the thermodynamic limit.
I furthermore stress that dynamical localization in the model under study is completely due to quantum effects, as in the kicked rotor model. Indeed, I also simulate the semi-classical dynamics of the model and find a very different behaviour showing diffusion and T=\infty thermalization.

Finally, I also study the kicked Bose-Hubbard model when quenched disorder is added. I argue that the system may undergo a transition to a many-body localized phase. Indeed, I numerically show that, at a finite system size and for strong enough disorder, the entanglement entropy in the chain logarithmically increases in time.