Periodically driven many-body quantum systems : Quantum Ratchets, Topological States and the Floquet-Boltzmann Equation

Controlling and manipulating complex many-body quantum systems will be a key ingredient for the development of next-generation technologies. While the realisation of a universal quantum machine is still out of reach, in recent years experimental systems of ultracold atoms have already evolved into a vivid field of research for quantum simulation. Crucially, such systems even allow for the successful quantum engineering of targeted many-body systems by means of coherent periodic driving. The essential properties of these Floquet systems encompass two main aspects: fast driving facilitates the simulation of effective static systems, and interactions lead to unique heating effects as energy is only conserved modulo the driving frequency. Within this thesis we theoretically study both of these aspects in respective model systems. In part I of this thesis, we investigate the dynamics of excitations of a bosonic Mott insulator in a designed one-dimensional Floquet system. Here, periodic driving in combination with breaking all mirror symmetries of the system can induce directed motion of particles. In the limit of small excitation densities, the effectively non-interacting quantum ratchet determines the motion of holes and doublons in the Mott insulator and can in fact be used to manipulate the dynamics of such. This little quantum machine can also be used to drive particles against an external force, where transport is possible but requires the fulfilment of a commensurability condition for long times. In part II, we discuss the role of interactions for periodically driven systems by means of a Floquet version of the Boltzmann equation. Starting from the Keldysh approach, we develop this semiclassical formalism based on a clear separation of time scales. The result is a description of the dynamics and the scattering of Floquet quasiparticles in such systems. Here, the property of discrete energy violation is naturally encoded in our formalism predicting the heating of interacting Floquet systems to infinite temperatures in the long-time limit. As a first application of this approach, we investigate a cold atom setup realising the Haldane model by means of periodic shaking. While homogeneous systems heat up globally, a confining potential evokes thermoelectric transport effects resulting from spatially dependent heating characteristics. Moreover, we show that the interplay of intrinsic heating, macroscopic diffusion and non-trivial topological properties of the Haldane model lead to an anomalous Floquet-Nernst effect, which describes anomalous particle transport as the result of developing temperature gradients. In part III, we elaborate on the quantum simulator aspect of ultracold atoms by providing a theoretical framework for a possible simulation of a topological edge state in a one-dimensional optical lattice. In this case, the one-dimensional Dirac equation with spatially varying mass is important, which captures the topological properties of a corresponding system of the BDI symmetry class. We analytically discuss such system and investigate the role of mean-field interaction effects. We also identify the emergence of dynamical instabilities in a realisation with bosonic atoms