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Applied Physics B

Stability of two-ion crystals in the Paul trap: A comparison of exact and pseudopotential calculations

John A. Hoffnagle, Richard G. Brewer

Published Jan 1, 1995• 15 citations
Publisher
DOI
10.1007/bf01135851

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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.
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.
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.