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Atoms

Influence of End Cap Structure on the Axial Geometric Parameter of the Linear Paul Trap

Lin Li, Zi Li

Published Feb 5, 2026• 0 citations
Publisher

Abstract

Through finite-element simulation, the axial potential distribution of the ion trap is analyzed. The effects of the central hole diameter of the end cap and the spacing between the two end caps on the axial geometric parameters of the ion trap are investigated. Based on these findings, a set of linear Paul traps is designed by selecting suitable end caps and quadrupoles. Stable trapping of calcium ions (Ca+) is successfully achieved, and these ions are subsequently laser-cooled into ionic Coulomb crystals. In the experiment, secular motion excitation of the Ca+ ion Coulomb crystal is performed, yielding an axial geometric parameter of 0.115(1) for the ion trap. This value aligns well with the simulation result of 0.114(2). The precise determination of the axial geometric parameter provides a solid foundation for subsequent high-precision optical or mass spectrometry measurements.

DOI
10.3390/atoms14020010

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Through finite-element simulation, the axial potential distribution of the ion trap is analyzed. The effects of the central hole diameter of the end cap and the spacing between the two end caps on the axial geometric parameters of the ion trap are investigated. Based on these findings, a set of linear Paul traps is designed by selecting suitable end caps and quadrupoles. Stable trapping of calcium ions (Ca+) is successfully achieved, and these ions are subsequently laser-cooled into ionic Coulomb crystals. In the experiment, secular motion excitation of the Ca+ ion Coulomb crystal is performed, yielding an axial geometric parameter of 0.115(1) for the ion trap. This value aligns well with the simulation result of 0.114(2). The precise determination of the axial geometric parameter provides a solid foundation for subsequent high-precision optical or mass spectrometry measurements.
An effective method for determining the geometric parameters of a linear Paul trap is provided and carried out in this paper. The method is based on exciting the secular motion of laser-cooled Ca+ ions in a linear Paul trap with varied trapping potentials. Taking advantage of the segmented structure of the linear Paul trap, both the axial and the radial geometric parameters are obtained by applying external excitation fields to the trapping electrodes directly. The axial geometric parameter of our linear Paul trap is determined to be 0.374(3), and, for the first time, the radial geometric parameter is determined to be 0.928(2).
In this work, we have studied the dynamics of 7-ions Ca+ Coulomb crystal in a linear quadrupole trap with flat end-cap electrodes. We have introduced geometrical parameters to determine points of phase transitions in the crystal. Then we have numerically calculated a phase diagram for this case and simulated the corresponding Coulomb crystals. It was shown that new one-dimensional radial Coulomb crystals are observed with the application of both high end-cap voltage (more than 1200 V) and a bias voltage on linear electrodes.
In order to demonstrate laser cooling on negatively charged ions, it is necessary to confine the ions in a region free of strong magnetic fields. In the course of this thesis, the pre-existing design of a linear Paul trap was modified and taken into operation. The new design possesses two additional end cap electrodes whose shape provides effective axial confinement without blocking optical access to the trapped particles. The influence of these end cap electrodes on the trapping potential was simulated and compared to experimental results. Measurements with AU- ions show that the trap allows storage of ions for multiple hours. Analysis ofthe storage time suggests that the loading process heats up the trapped ions. The lifetime τHot of the hot ions was determined to be 23(3) min. After thermalization, the lifetime increases to 75(2) min. Measurements of the ion cloud’s radius provide an estimation of the equilibrium temperature T∞ ≈5000 K. The experimentally determined time constant of the cooling process is τcool = 41:6(4) s.
Deviations from both the ideal linear Paul trap (LPT) geometry as well as the ideal quadrupole driving scheme introduce imperfections to ion trapping potentials. We investigate the effects of these imperfections in a LPT operated in a conventional dipole-drive configuration. We demonstrate the trapping of the Li+^++ ions along the axial direction with zero and negative end-cap voltages. This occurs due to the modified axial a−qa-qa−q space resulting from radial-to-axial coupling of the electric field. The dipole drive configuration lifts the degeneracy of the radial trapping potentials, resulting in unequal radial secular frequencies, and this is demonstrated experimentally. The combined effects of dipole drive and trap dimensions are summarized in a two-dimensional map that quantifies deviations from ideal behaviour. Based on this map, we propose a geometric modification that significantly reduces radial-to-axial coupling of the potential.
Coulomb crystals have applications in many areas such as quantum computing and simulation, quantum logic spectroscopy, nonlinear dynamics and chaos, phase transitions, and chemical reaction process. The structure of the Coulomb crystal and the trajectory of each trapped ion are typically determined by the parameters of the trap and the ion species. However, dark ions are often inevitable in experiment, which introduces uncertainty into the desired crystal structures and ion trajectories. Few researches have been conducted to investigate the configuration change of the crystal in the presence of dark ions and the influence of a dark ion on its surrounding ion trajectories in a multi-ion system. In this work, we utilize the molecular dynamics simulation software LAMMPS and the (py)LIon package (modified to adapt the semi-classical theory of laser cooling) for simulating the three-dimensional ion trajectories of Coulomb crystals. The formation process of 40Ca+ Coulomb crystal in a linear trap is simulated. With the micromotion and secular motion trajectories of each ion, we calculate the temperature of Coulomb crystal and the average velocity of specific ions. It is observed that the crystal structure exhibits obvious layering phenomenon when the trapped ions yield a large difference in their charge-to-mass ratio (CMR), however, layering is not obvious with a small difference in the CMR. In addition, we simulate and compare the Coulomb crystal structure formed by pure 40Ca+ ions with that formed by 40Ca+ ions mixed with a small number of dark ions including isotopic ions (44Ca+) and impurity ions (CaH+). Three different cases are investigated, namely the one-dimensional ion string, two-dimensional planar structure and three-dimensional helical structure. The results show that the ions in the neighborhood of a dark ion exhibit around micron-order position change compared with their positions before the dark ion is formed. Such a change can be measured in experiment through microscopic imaging, thereby providing a way to identify the formation of dark ions in Column crystals with a large ion number.
Coulomb crystal yields a wide range of applications including quantum computing and simulation, phase transitions, quantum logic spectroscopy, nonlinear dynamics and chaos, chemical reaction process, etc. The structure of the Coulomb crystal and the trajectory of each trapped ion are typically determined by the parameters of the trap and the ion species. However, dark ions are often inevitable in experiments, which introduces uncertainty to the desired crystal structures and ion trajectories. Few research has been conducted to investigate the configuration change of the crystal in the presence of dark ions and the influence of a dark ion on its surrounding ion trajectories in a multi-ion system. In this paper, we utilize the molecular dynamics simulation software LAMMPS and the (py)Lion package (modified to adapt the semi-classical theory of laser cooling) for the simulation of the 3D ion trajectories of Coulomb crystals. The formation process of 40Ca+ Coulomb crystals in a linear trap is simulated. With the micromotion and secular motion trajectories of each ion, we calculate the temperature of Coulomb crystal and the average velocity of specific ions. It’s observed that the crystal structure exhibits obvious layering phenomenon when the trapped ions yield a large difference in their charge-to-mass ratio (CMR), however, layering is not obvious with a small difference in the CMR. In addition, we simulate and compare the Coulomb crystal structure formed by pure 40Ca+ ions and that formed by 40Ca+ ions mixed with a small number of dark ions including isotopic ions (44Ca+) and impurity ions (CaH+). Three different cases are investigated, namely the 1D ion string, 2D plane structure and 3D helical structure. Results show that ions in the neighborhood of a dark ion exhibit around micron-order position changes compared to their positions before the dark ion is formed. Such a change can be measured in experiment through microscopic imaging, providing a way to identify dark ions in Column crystals with a large number of ions.