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The Pseudopotential As a Tool for Describing Ion Crystal Morphology

Varun Ursekar

Published Jan 1, 2017• 0 citations

Abstract

Describing the structure of ion Coulomb crystals in periodically-driven systems, such as cylindrical and linear Paul traps, is important for applications in quantum information processing, quantum simulation, spectroscopy, and frequency standard determination. The pseudopotential, a time-independent effective potential obtained by averaging the explicitly time-dependent trapping potential, is often used for this specific purpose, among other uses in the general study of trapped ion dynamics. This thesis examines the strengths and weaknesses of the pseudopotential approximation as a tool for describing few-ion crystal configurations in a Paul trap. Numerical evidence for the failure of the standard pseudopotential commonly found in the literature in predicting crystal alignment effects in such systems is provided. A method for deriving an improved pseudopotential for a general set of coupled differential equations is presented and applied to the cylindrical and linear traps, providing analytical evidence for these "exotic'' crystal alignment effects. The limitations of the improved pseudopotential in explaining crystal instability in certain regions of trap parameter space and in terms of scalability to many-ion systems are discussed.

DOI: 10.14418/wes01.1.1322

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•Jan 1, 2017•0 citations

Describing the structure of ion Coulomb crystals in periodically-driven systems, such as cylindrical and linear Paul traps, is important for applications in quantum information processing, quantum simulation, spectroscopy, and frequency standard determination. The pseudopotential, a time-independent effective potential obtained by averaging the explicitly time-dependent trapping potential, is often used for this specific purpose, among other uses in the general study of trapped ion dynamics. This thesis examines the strengths and weaknesses of the pseudopotential approximation as a tool for describing few-ion crystal configurations in a Paul trap. Numerical evidence for the failure of the standard pseudopotential commonly found in the literature in predicting crystal alignment effects in such systems is provided. A method for deriving an improved pseudopotential for a general set of coupled differential equations is presented and applied to the cylindrical and linear traps, providing analytical evidence for these "exotic'' crystal alignment effects. The limitations of the improved pseudopotential in explaining crystal instability in certain regions of trap parameter space and in terms of scalability to many-ion systems are discussed.

Contrary to the results of standard pseudopotential analysis we predict the existence of an alignment transition region in the parameter space of the Paul trap in which the orientation of two-ion crystals with average angular momentum zero is different from x-y or z alignment. This prediction is based on an improved pseudopotential derived by keeping terms up to second order in the micromotion amplitudes. Our analytical predictions are supported by numerical evidence and can be checked experimentally with existing setups.

We experimentally study two-dimensional (2D) Coulomb crystals in the "radial-2D" phase of a linear Paul trap. This phase is identified by a 2D ion lattice aligned entirely with the radial plane and is created by imposing a large ratio of axial to radial trapping potentials. Using arrays of up to 19 ^{171}Yb^{+} ions, we demonstrate that the structural phase boundaries of such crystals are well described by the pseudopotential approximation, despite the time-dependent ion positions driven by intrinsic micromotion. We further observe that micromotion-induced heating of the radial-2D crystal is confined to the radial plane. Finally, we verify that the transverse motional modes, which are used in most ion-trap quantum simulation schemes, are well-predictable numerically and remain decoupled and cold in this geometry. Our results establish radial-2D ion crystals as a robust experimental platform for realizing a variety of theoretical proposals in quantum simulation and computation.

The accurate characterization of eigenmodes and eigenfrequencies of two-dimensional ion crystals provides the foundation for the use of such structures for quantum simulation purposes. We present a combined experimental and theoretical study of two-dimensional ion crystals. We demonstrate that standard pseudopotential theory accurately predicts the positions of the ions and the location of structural transitions between different crystal configurations. However, pseudopotential theory is insufficient to determine eigenfrequencies of the two-dimensional ion crystals accurately but shows significant deviations from the experimental data obtained from resolved sideband spectroscopy. Agreement at the level of 2.5×10(-3) is found with the full time-dependent Coulomb theory using the Floquet-Lyapunov approach and the effect is understood from the dynamics of two-dimensional ion crystals in the Paul trap. The results represent initial steps towards an exploitation of these structures for quantum simulation schemes.

We examine the time-dependent dynamics of ion crystals in radiofrequency traps. The problem of stable trapping of general three-dimensional crystals is considered and the validity of the pseudopotential approximation is discussed. We analytically derive the micromotion amplitude of the ions, rigorously proving well-known experimental observations. We use a recently proposed method to find the modes that diagonalize the linearized time-dependent dynamical problem. This allows one to obtain explicitly the ('Floquet–Lyapunov') transformation to coordinates of decoupled linear oscillators. We demonstrate the utility of the method by analyzing the modes of a small 'peculiar' crystal in a linear Paul trap. The calculations can be readily generalized to multispecies ion crystals in general multipole traps, and time-dependent quantum wavefunctions of ion oscillations in such traps can be obtained.

Molecular dynamics simulations of ion Coulomb crystals in linear Paul traps allow for the extraction of properties such as ion temperature and ion counts enabling the investigation of ion chemistry in the ultra-cold regime. These simulations reproduce experimental CCD images using density plots of simulated ions allowing for comparisons to be drawn between simulated and experimental results and providing information inaccessible to the researcher, such as the locations of clouds of non- uorescing species. In this thesis, a method for creating these simulated CCD images is presented, as well as the demonstration of a fourth order integrator for producing more accurate calculations of the ion energy. The simulation of a single frequency Paul trap is presented and the ability to match simulations with experimental parameters as inputs to experimental results is demonstrated in order to show that the simulations that have been constructed can be used to analyze experimental data and work as a predictive lab tool for future explorations of ultra-cold chemistry.