Out-of-equilibrium dynamics of the Bose-Hubbard model in the strong coupling regime

Experimental advances have made ultracold atoms in optical lattices a favourable setting to study out-of-equilibrium phenomena and attracted considerable attention in recent years. These systems are highly versatile in that experimental parameters can be tuned over a wide range of values in real time. This facilitates the study of quantum quenches, in which parameters in the corresponding Hamiltonian are varied in time faster than the system can respond adiabatically. Such protocols open the door to a rich range of many-body physics and have been studied intensely both theoretically and experimentally. The Bose-Hubbard model (BHM) has been shown to describe interacting ultracold bosons in an optical lattice, allowing the opportunity for experiments to probe the out-of-equilibrium dynamics of the model. The BHM is a particularly convenient context for studying quantum quenches as it displays a quantum phase transition between superfluid and Mott-insulator phases. In this thesis, we develop a strong-coupling approach that allows the study of correlations in space and time in both the superfluid and Mott-insulating phases of the BHM. Specifically, we obtain a two-particle irreducible (2PI) effective action within the contour-time formalism that allows for a description of both equilibrium and out-of-equilibrium phenomena. We derive equations of motion for both the superfluid order parameter and the single-particle many-body Green's functions. First, we assess the accuracy of our formalism by studying the equilibrium solution for the homogeneous BHM and comparing our results to existing strong-coupling methods as well as to exact methods where possible. We then consider homogeneous systems that are initially thermalized in the Mott phase, and which are then subjected to quenches. We solve numerically the equations of motion for this scenario and calculate the single-particle density matrix. We demonstrate a Lieb-Robinson-like maximal propagation velocity for the spreading of single-particle correlations in one, two, and three dimensions. We study the dependence of the maximal propagation velocity on the quench protocol, chemical potential, temperature, and dimensionality. We compare our results to exact methods, existing strong-coupling approaches, and experiments where possible. Lastly, we extend our strong-coupling approach to the disordered BHM and derive equations of motion for the disorder-averaged single-particle Green's function. We discuss how these equations of motion can be used to study the phase stability of many-body localization in the disordered BHM for dimensions higher than one.