Over the past three decades, many efforts have been invested in investigating various quantum systems that can precisely be controlled and form a reliable platform for quantum simulation and computing. Advancements in ultra-cold atoms and ions have revealed their major advantages and make them one of the most promising candidates. Two pillars are important for this success; unprecedented control over the external and internal degrees of freedom of individual ions in traps and the ability to precisely manipulate collective behaviour mediated via long-range interactions between them. The former has been achieved with the highest fidelities in trapped ions and the latter has found the most successful realisations in ultra-cold atoms in highly-excited electronic states, called Rydberg states. In the proposed research, we envisioned a quantum simulator based on Rydberg enhanced interactions between trapped ions, in which these two remarkable properties are combined. The objectives include coherent spectroscopy of Rydberg states of singly-charged atoms confined in an ion trap. Rydberg states are excited using vacuum ultra-violet (VUV) or ultra-violet (UV) laser systems in single- or two-photon excitation processes. Such excitation allows for quantum state-dependent interactions between ions. A remarkable feature of the system is that external electric or magnetic forces yield state-dependent effects that can be measured using spectroscopy techniques. This offers a powerful tool for generating entanglement. Today quantum computation with trapped ions has largely focused on scalability as a major challenge and thus the quest is to implement much faster quantum gate operations. Rydberg trapped ions show great potential to fill this gap and to enable exploration of many-body quantum systems in a precise fashion. This research is part of the worldwide endeavour that aims at exploring quantum effects which can be used to improve or to revolutionise current technologies in simulation, computing and sensing.
