• Floquet
• complex dynamics
• spectral statistics

In this thesis, we consider complex dynamical evolutions which could be realized experimentally with ultracold atoms in optical lattices.
The first part of the thesis deals with the quantum kicked rotor Hamiltonian, a paradigmatic model of quantum chaos.
We analyze the model in a deeply quantum regime where the dynamical localization in the momentum space manifests itself in the fractal geometry of a conductance-like quantity called survival probability.
We show that the fractal fluctuations of the survival probability can be measured in experimental frameworks with ensembles of either cold or ultracold atoms.
The second part deals with ultracold atoms in a static optical lattice, tilted by an external constant force.
We develop a numerical approach which is capable to treat large system sizes for a single band Bose-Hubbard model.
This allows us to considerably extend the previous studies of the full quantum spectrum of the system.
We derive a reasonable extension of the single-band model to two bands.
This complicates the problem from the numerical point of view, but opens the route to study the impact of interband tunneling to the ``horizontal' quantum transport along the lattice.