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
