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arXiv:1109.2160 · Journal of Applied Physics

Stability analysis of ion motion in asymmetric planar ion traps

Fayaz A. Shaikh, Arkadaş Özakın

Published Oct 1, 2012• 20 citations

Abstract

Motivated by recent developments in ion trap design and fabrication, we investigate the stability of the motion of an ion in asymmetrical, planar versions of the linear Paul trap. The equations of motion of an ion in such a trap are generally coupled due to a nonzero relative angle θ between the principal axes of RF and DC fields, invalidating the assumptions behind the standard stability analysis for symmetric Paul traps. Using numerical methods, we obtain stability diagrams for the coupled system for various values of θ, generalizing the standard q-a stability diagrams. We then use multi-scale perturbation theory to obtain approximate formulas for the boundaries of the primary stability region and check our formulas against results from numerical analysis. Our results show that while the primary stability region is quite robust to changes in θ, a secondary stability region is highly variable, joining the primary stability region at the special case of θ=45°, which results in a significantly enlarged stability region for this particular angle. We conclude that while the standard stability diagrams for classical, symmetric Paul traps are not entirely accurate for asymmetric surface traps (or for other types of traps with a relative angle between the RF and DC axes), they are “safe” in the sense that operating conditions deemed stable according to standard stability plots are in fact stable for asymmetric traps, as well. By ignoring the coupling in the equations, one only underestimates the size of the primary stability region.

DOI: 10.1063/1.4752404

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In this paper, we present and discuss stability diagrams for Paul traps driven by two ac voltages. In contrast to a typical Paul trap, here we suggest a secondary ac voltage whose frequency is twice the frequency of the primary one. The ratio between their amplitudes can be used to expand the region of stability and to access different states of motion of trapped ions. This provides a further mechanism to trap, cool, and manipulate single ions and also to improve the experimental framework where ion clouds and crystals can be prepared and controlled. Such approach opens the possibility of designing more sophisticated trapping architectures, leading to a wide variety of applications on ion trap research and mass analysis techniques.

We study the quantum stability of the dynamics of ions in a Paul trap. We revisit previous results that showed that quantum trajectories did not have the same region of stability as their classical counterpart, contrary to what is obtained from a Floquet analysis of the motion in the periodic trapping field. Using numerical simulations of the full wave-packet dynamics, we confirm that the classical trapping criterion is fully applicable to quantum motion, when considering both the expectation value of the position of the wave packet and its width.

Under specific conditions, a rotating saddle potential can confine the motion of a particle on its surface. This time-varying hyperbolic potential shares key characteristics with the RF-electric-quadrupole ion trap (RF Paul trap), making it a valuable mechanical analog. Previous work has primarily focused on symmetric saddles, characterized by equal curvatures along the trapping and anti-trapping directions. RF Paul traps, however, are asymmetric, because the degeneracy of the transverse motional modes must be lifted for laser cooling and other quantum control. In this paper, we investigate the motion of trapped particles in asymmetric rotating saddles. We demonstrate that even minor asymmetries, including those arising from manufacturing imperfections, can significantly affect particle trajectories and stability. Our analysis includes both theoretical modeling and experimental measurements. We derive the equations of motion for asymmetric saddles and solve them to explore stability and precession effects. Additionally, we present lifetime measurements of particles in saddles with varying degrees of asymmetry to map key features of the a–q stability diagram. This study underscores the importance of incorporating asymmetry into mechanical models of ion traps to better reflect real-world implementations.

We present the single-particle stability diagram for the radial motion in a linear Paul trap in a situation where the applied axial dc potential gives rise to a harmonic defocusing radial potential. Although most linear Paul trap experiments have been conducted in a regime where this approximation is reasonably valid, the effect of the axial confinement on the stability of the radial motion has not previously been analyzed. The defocusing effect in both radial directions leads to a stability diagram different from that of the two-dimensional quadrupole mass filter, and hence points toward new studies of few ion dynamics. Expressions for the effective or pseudopotentials for one and two charged particles are presented and discussed.

The stability regions of an ion in a two-dimensional Paul trap are studied by a quantum-mechanical treatment, which is very important for the experiment when the temperature is fairly low. The result is compared with a classical one published before by Paul [Rev. Mod. Phys. 62, 531 (1990)]. It is shown that their properties are quite similar and their differences are related to the quantum effect, which cannot be included within a classical description. All of these properties are discussed in detail.

Using numerical simulations of the time-dependent Schrödinger equation, we study the full quantum dynamics of the motion of an atomic ion in a linear Paul trap. Such a trap is based on a time-varying, periodic electric field and hence corresponds to a time-dependent potential for the ion, which we model exactly. We compare the center-of-mass motion with that obtained from classical equations of motion, as well as to results based on a time-independent effective potential. We also study the oscillations of the width of the ion’s wave packet, including close to the border between stable (bounded) and unstable (unbounded) trajectories. Our results confirm that the center-of-mass motion always follows the classical trajectory, that the width of the wave packet is bounded for trapping within the stability region, and therefore that the classical trapping criterion is fully applicable to quantum motion.