Trapped Rydberg ions represent a flexible platform for quantum simulation and information processing. They combine a high degree of control over electronic and vibrational degrees of freedom, typical of trapped ion setups, with the possibility to individually excite ions to high-lying Rydberg levels, allowing one to engineer strong interactions between pairs of excited ions. 

In the first part of the talk, I will show how the coupling between Rydberg pair interactions and collective motional modes can be exploited to realise effective long-range multi-body interactions, consisting of two, three, and four-body terms. By focusing on an experimentally feasible quasi one-dimensional setup of Sr Rydberg ions, I will demonstrate that multi-body interactions are enhanced by the emergence of a soft mode associated, e.g., with a structural phase transition. This has a striking impact on many-body electronic states and results, for example, in a three-body anti-blockade effect.

The second part of the presentation will illustrate the capabilities of trapped Rydberg ions in simulating the quantum behaviour in the neighbourhood of a conical intersection. Conical intersections occur at the degeneracy points between two (or more) electronic potential energy surfaces and are a paradigmatic example of the non-adiabatic processes characterising the excited dynamics of large molecules. I will show how, thanks to the interplay between large polarizability and strong dipolar exchange interactions, trapped Rydberg ions make it possible to engineer tunable conical intersections and to study their ensuing dynamics in a highly controllable environment. In particular, I will demonstrate how the presence of a conical intersection affects both the nuclear and electronic dynamics leading, for instance, to the inhibition of the nuclear motion.