The last years have witnessed dramatic progress in experimental control and refinement of quantum simulations based on ultracold atoms. New detection techniques and preparation techniques such as the quantum gas microscope now give direct access to the nonequilibrium dynamics of interacting quantum many-body systems.

I will discuss two recent theoretical approaches for investigating dissipative dynamics and the quasiparticle excitation structure of strongly correlated bosons in optical lattices:

1) We present a generalized quasi-particle theory for bosonic lattice systems, which naturally contains all relevant collective modes, including the Higgs amplitude in the strongly correlated superfluid. It provides a systematic framework for efficiently calculating observables beyond the Gutzwiller approximation and for including external perturbations, as well as higher order decay and interactions in terms of quasi-particle operators. It also allows for the construction of an alternative path integral approach in terms of quasi-particle coherent states.

2) In recent years, controlled dissipation has proven to be a useful tool for probing of quantum systems in the ultracold setup. We consider the dynamics of bosons induced by a dissipative local defect, in the superfluid and supersolid phases, by solving the master equation using the Gutzwiller approximation. We find that in the superfluid phase repulsive nearest neighbour interactions can lead to enhanced dissipation processes. On the other hand, effective loss rates are strongly suppressed deep in the supersolid phase, where repulsive nearest neighbour interactions play a dominant role. In the limit of strong dissipation we recover the quantum Zeno effect.