Trapped Rydberg ions are a novel platform for quantum information processing. This approach combines the advanced quantum control of trapped ions and the strong dipolar interaction of Rydberg atoms. In this thesis, a strong dipole-dipole interaction has been demonstrated and a sub-microsecond entangling gate has been implemented in a cold two-ion crystal. After minimizing the polarizability of the Rydberg state by microwave dressing and understanding the effect of the quadrupole radio-frequency trap,the entangling gate has been applied in a warm 12-ion crystal.

Trapped ion systems are at the forefront of the development of various forms of quantum technology. Continuing to improve and establish new devices and techniques for the control of trapped ions is a vital element of ongoing research. In this thesis, a range of experiments which aim to expand the quantum toolkit of trapped ion systems are presented. These results primarily focus on the control and detection of bound motional states of a single trapped ⁸⁸Sr+ ion for the purposes of quantum simulation and computation. We demonstrate how the interference between motional modes can reveal an interesting new interpretation of the mechanism behind light-matter interaction and introduce two separate techniques for the detection of motional states, based on the Autler-Townes effect and the use of composite pulses respectively. Additionally, we introduce a novel method to perform micromotion compensation and build upon previous works studying the effects of trapping electric fields on a single trapped Rydberg ion, observing the second-order quadrupolar response of the ion with a highly precise sensitivity.