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Physical Review A

Prediction of an alignment transition region of two-ion crystals in a Paul trap

M. G. Moore, R. Blümel

Published Dec 1, 1994• 12 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.50.r4453

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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.
With the help of detailed molecular-dynamics calculations we prove the existence of an instability region I embedded in the single-ion Mathieu stability region M of Paul-trap control parameters a and q. Two-ion crystals do not exist in I. A two-ion crystal created outside of I can be dragged into I by an adiabatic change of a and q. A deterministic order-chaos transition occurs at the point of entry into I. This type of melting transition does not depend on the presence of laser noise and occurs even in the absence of collisions with rest-gas atoms. The prediction can be checked with existing experimental 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.
Micromotion of ions in Paul traps has several adverse effects, including alterations of atomic transition line shapes, significant second-order Doppler shifts in high-accuracy studies, and limited confinement time in the absence of cooling. The ac electric field that causes the micromotion may also induce significant Stark shifts in atomic transitions. We describe three methods of detecting micromotion. The first relies on the change of the average ion position as the trap potentials are changed. The second monitors the amplitude of the sidebands of a narrow atomic transition, caused by the first-order Doppler shift due to the micromotion. The last technique detects the Doppler shift induced modulation of the fluorescence rate of a broad atomic transition. We discuss the detection sensitivity of each method to Doppler and Stark shifts, and show experimental results using the last technique.