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.

In this work, we present a method for measuring the motional state of a two-level system coupled to a harmonic oscillator. Our technique uses ultranarrowband composite pulses on the blue sideband transition to scan through the populations of the different motional states. Our approach does not assume any previous knowledge of the motional state distribution and is easily implemented. It is applicable both inside and outside of the Lamb-Dicke regime. For higher phonon numbers especially, the composite pulse sequence can be used as a filter for measuring phonon number ranges. We demonstrate this measurement technique using a single trapped ion and show good detection results with the numerically evaluated pulse sequence.

Trapped Rydberg ions are a novel approach to quantum information processing [1, 2]. Qubit rotations in the ion's lower lying electronic states are combined with entanglement operations that take advantage of strong Rydberg interactions [3]. In our experimental setup we excite trapped strontium ions from the metastable 4D state to the Rydberg manifold using a two photon excitation process. Certain properties of the ion become more prominent for highly excited states. One example is the polarizabilty which reacts to the surrounding electric field of the trapping electrodes. While effects due to the polarizability are negligible for lower lying states, they become more prominent for highly-excited states. This change leads to an altered trapping potential for the high lying Rydberg states, as shown in Fig. 1 [4]. For previous experiments those shifts have always been compensated for to perform, for example, sub-microsecond entangling gates between trapped ions [5]. However, this trapping field displacement can also be used to coherently control the ions' motion.

We present a single-shot method to measure motional states in the number basis. The technique can be applied to systems with at least three nondegenerate energy levels which can be coupled to a linear quantum harmonic oscillator. The method relies on probing an Autler-Townes splitting that arises when a phonon-number changing transition is strongly coupled. We demonstrate the method using a single trapped ion and show that it may be used in a nondemolition fashion to prepare phonon number states. We also show how the Autler-Townes splitting can be used to measure phonon number distributions.

Trapped ion systems are one of the leading technologies for the development of various novel quantum devices, including computers, simulators and enhanced sensors. Ion traps offer a high level of control in terms of both the electronic and motional states of individual quantum systems. This enables information to be efficiently encoded and transferred, often via interaction with highly controlled laser fields. In particular, the use of collective motional excitation to generate entanglement is a principle that underlies the majority of gate operations that have thus far been implemented in trapped ion systems. Another method, originally devised to circumvent the need for collective motional excitation in entangling operations, is to use Rydberg states which can be engineered to produce strong, direct and state-dependent interactions between ions. Due to their sensitivity to electric fields, trapped Rydberg ions have also opened the possibility for the development of entirely new gate schemes and for use in quantum metrology. In this work we employ the high level of motional control available to trapped ion systems in order to perform three experiments, involving quantum simulation, computation and metrology respectively. In the first we make use of a single ion and couple its electronic and motional states to simulate sub- and superradiant emission for an atom confined in a 2D cavity. The results obtained showed that technical limitations in our experimental system currently inhibit the formation of such states. In the second experiment we introduce a novel gate scheme for quantum computation using trapped Rydberg ions, initial results demonstrating the feasibility of the gate scheme are presented and future plans for this work are summarised. For the final experiment we investigate the effects of an electric quadrupole field interacting with a single ion to second and higher orders. We observe resonance shifts for the Rydberg excitation and the presence of sidebands due to the interaction when performing spectral scans. These observations show good agreement when compared with theoretical models. Finally a new technique for performing coherent spectroscopy of Ryberg states is demonstrated.

We minimize the stray electric field in a linear Paul trap quickly and accurately, by applying interferometry pulse sequences to a trapped ion optical qubit. The interferometry sequences are sensitive to the change of ion equilibrium position when the trap stiffness is changed, and we use this to determine the stray electric field. The simplest pulse sequence is a two-pulse Ramsey sequence, and longer sequences with multiple pulses offer a higher precision. The methods allow the stray field strength to be minimized beyond state-of-the-art levels. Using a sequence of nine pulses we reduce the 2D stray field strength to (10.5 +/- 0.8) mV m(-1) in 11 s measurement time. The pulse sequences are easy to implement and automate, and they are robust against laser detuning and pulse area errors. We use interferometry sequences with different lengths and precisions to measure the stray field with an uncertainty below the standard quantum limit. This marks a real-world case in which quantum metrology offers a significant enhancement. Also, we minimize micromotion in 2D using a single probe laser, by using an interferometry method together with the resolved sideband method; this is useful for experiments with restricted optical access. Furthermore, a technique presented in this work is related to quantum protocols for synchronizing clocks; we demonstrate these protocols here.

The response of matter to fields underlies the physical sciences, from particle physics to astrophysics, and from chemistry to biophysics. We observe an atom's response to an electric quadrupole field to second- and higher orders; this arises from the atom's electric quadrupole polarizability and hyperpolarizabilities. We probe a single atomic ion which is excited to Rydberg states and confined in the electric fields of a Paul trap. The quadrupolar trapping fields cause atomic energy level shifts and give rise to spectral sidebands. The observed effects are described well by theory calculations.