Periodic Reporting for period 1 - RydIons (Coherent manipulation of cold trapped ions in Rydberg states) Reporting period: 2018-04-01 to 2020-03-31 Summary of the context and overall objectives of the project 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. Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far Key results and advances achieved over the course of the project:(1) The project started by Rydberg excitation of trapped calcium ions in a linear segmented radiofrequency (RF) ion trap. Rydberg states of calcium ions were excited from metastable D states in a single-step excitation using a VUV laser system at 123 nm. This nontrivial radiation source was used for studying a certain Rydberg P state, see figure (1).(2) Over the course of the project, we have developed another experimental setup in which Rydberg excitation was carried out using two high-power commercial UV laser systems. This setup became fully operational as an essential progress in line with the proposed research. (3) The two-step Rydberg excitation scheme was used for an elaborate investigation of nS and nD Rydberg series. These results were used to precisely determine the value for the second ionisation energy of calcium ions with significantly improved uncertainties, see figure (2). (4) We performed a theoretical study for implementing fast entangling operations using electric pulses act on trapped Rydberg ions. In this scheme, a fast electric pulse generates a state-dependent force and leads to accumulation of a phase which can be precisely controlled in our experiment. These calculations have established a powerful technique to realise fast quantum gates using trapped Rydberg ions with applications in quantum information processing. Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far) Further results and potential impacts are considered as follows:(1) Further measurements and technical improvements will be performed using the newly built setup to enable coherent manipulation of Rydberg states. The setup has been also prepared for microwave spectroscopy, and microwave-dressed Rydberg states will be characterised. (2) This project shows great potential for investigating many-body quantum systems. One interesting direction is to use Rydberg excitation for generating the state-dependent force that can lead to structural phase transitions. (3) The experimental work is expected to reach the stage that can be used for realising a Rydberg quantum simulator in which dipolar interactions between ions can be used to mimic energy exchange in linear chains.(4) The theoretical calculation performed has certainly open doors for implementing fast quantum gates. We are determined to take advantage of our advance setup to realise such gates and to improve the record by optimal control of our experimental parameters.