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arXiv:1206.4006 · New Journal of Physics

Modes of oscillation in radiofrequency Paul traps

Haggai Landa, Michael Drewsen, Benni Reznik, Alex Retzker

Published Sep 13, 2012• 65 citations
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Abstract

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.

DOI
10.1088/1367-2630/14/9/093023

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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.
The quadrupole linear Paul trap is one of the key instruments in building highly stable atomic clocks. However, a frequency reference based on a single trapped ion is limited in stability due to the time needed for the interrogation cycle which cannot be further shortened. A promising strategy is the utilization of multiple trapped ions. The ions of the same kind then repulse each other with the Coulomb force, which is countered by the ponderomotive force of the time depended field in the trap. A few ions form a chain along the axis of a linear Paul trap. Adding more ions (a few tens or hundreds) gives rise to Coulomb crystals. We created an efficient simulation code which calculates the motion of such collections of ions in quasi-static radiofrequency fields of real linear quadrupole traps (including the micromotion). We include a model for laser cooling of the ions. The simulation tool can be used to study the formation and the dynamics of Coulomb crystals under conditions corresponding to various 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.
Ion Coulomb crystals form when trapped ions are cooled below a certain critical temperature. In radiofrequency (RF) traps, such crystals are usually stationary (i.e. non-rotating) with shapes similar to those of cold, homogeneously charged liquids when the number of ions is sufficient for three-dimensional structures to appear. ‘Dynamically stable’ structures can, however, also be created either by directly applying a torque to an initially stationary crystal or by choosing trap parameters just within the stability region where stationary crystals will melt due to parametric resonances between the RF trap field and ion plasma modes.
Abstract. While the linear radiofrequency trap finds various applications in high-precision spec-troscopy and quantum information, its higher-order cousin, the linear multipole trap, is almost exclu-sively employed in physical chemistry. Recently, first experiments have shown interesting features by laser-cooling multipole-trapped ion clouds. Multipole traps show a flatter potential in their cen-tre and therefore a modified density distribution compared to quadrupole traps. Micromotion is an important issue and will certainly influence the dynamics of crystallized ion structures. Our experi-ment tends to investigate possible crystallization processes in the multipole. In a more general way, we are interested in the study of the dynamics and thermodynamics of large ion clouds in traps of different geometry.
We study the probability distribution of an atomic ion being laser-cooled in a radio-frequency Paul trap, in terms of Hamiltonian action-angle coordinates appropriate for the time-dependent potential. Close to the effective minimum in many Paul traps, the periodically-driven potential is approximately quadratic in the coordinates. We find that despite the microscopic nonequilibrium forces, a stationary thermal-like exponential distribution in the action is reached after coarse-graining the angles, with a mean action corresponding to the Doppler cooling limit. However, residual electric fields can push the ion from the origin of the quadrupole potential and set it into a large-amplitude oscillation. Above a threshold amplitude of such excess micromotion, the action distribution for motion expanded about the driven oscillation broadens and becomes distinctly nonthermal. We find that by a proper choice of the laser detuning the distribution can be made exponential again, with a mean action close to the Doppler cooling limit. This is important for quantum information processing and other applications, and in particular the derived approach can be immediately applied to crystals of trapped ions in planar configurations, where the driven motion of ions is unavoidable.