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Acta Physica Sinica

Simulation of Coulomb crystal structure and motion trajectory of calcium ions in linear ion trap

Hong-Shuo Zhang, Yongzhuang Zhou, Yong Shen, Hongxin Zou

Published Sep 29, 2022• 1 citation
Publisher

Abstract

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.

DOI
10.7498/aps.72.20221674

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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.
We describe a simple and fast method for simulating observed images of ion Coulomb crystals. In doing so, cold elastic collisions between Coulomb crystals and virtual very light atoms are implemented in a molecular dynamics (MD) simulation code. Such an approach reproduces the observed images of Coulomb crystals by obtaining density plots of the statistics of existence of each ion. The simple method has the advantage of short computing time in comparison with previous calculation methods. As a demonstration of the simulation, the formation of a planar Coulomb crystal with a small number of ions has been investigated in detail in a linear ion trap both experimentally and by simulation. However, also large Coulomb crystals including up to 1400 ions have been photographed and simulated to extract the secular temperature and the number of ions. For medium-sized crystals, a comparison between experiments and calculations has been performed. Moreover, an MD simulation of the sympathetic cooling of small molecular ions was performed in order to test the possibility of extracting the temperature and the number of refrigerated molecular ions from crystal images of laser-cooled ions. Such information is basic to studying ultracold ion-molecule reactions using ion Coulomb crystals including sympathetically cooled molecular ions.
A laser-cooling experiment with Ca+{\mathrm{Ca}}^{+}Ca+ ions trapped in a linear rf octupole ion trap is presented. The phase transition of the laser-cooled Ca+{\mathrm{Ca}}^{+}Ca+ ions from the cloud to the crystal state is observed by an abrupt dip of the laser-induced fluorescence spectrum and indicates that mK temperatures are obtained. We have also performed molecular dynamics simulations under various conditions to confirm this property by deducing axially symmetric structures of Coulomb crystals and by evaluating the translational temperatures of the laser-cooled ions. The simulation results show that for small numbers of ions novel ring-shaped crystals are produced. As the number of ions is increased, cylindrical layers in the ring crystal are sequentially formed. For more than 100 ions, also hexagonal and spiral structures emerge in parts of the large-size ion crystal, which has a length on the order of millimeters for the present geometrical arrangement and voltages. An advantage of the linear rf octupole trap is its large almost-field-free region in the middle of the trap, where the micromotion amplitude is small for trapped ions. These results demonstrate that such a multipole trap has attractive features for quantum computing and ultracold ion-atom collision studies.
This thesis describes the study of a range of ion-molecule reactions at very low collision energies using a newly developed experimental technique which involves the reaction of velocity-selected beams of translationally cold neutral molecules with very low kinetic energy ion ensembles. These studies have been enabled by the construction of a new apparatus for trapping and laser-cooling gas phase atomic ions (40Ca⁺). The laser-cooling process results in the formation of ordered, low kinetic energy, lattice-like ion structures, also known as "Coulomb crystals". The properties of single and multicomponent Coulomb crystals (which may also involve molecular ions), and their manipulation via modulation of the applied fields, are explored experimentally and with the use of molecular dynamics simulations. Variations in the laser-cooling parameters are shown to result in different steady-state populations of the electronic states of 40Ca⁺ involved with the laser cooling cycle, and these are modelled within an appropriate theoretical framework. The imaging of 40Ca⁺ fluorescence as a function of time allows the study of various ion-molecule reactions at collision energies around 300 K, with single ion sensitivity. These reaction studies are extended to low-temperature (collision energies close to 1 K), by combination of the ion trap apparatus with a bent quadrupole guide velocity-selector. Ion-molecule collision energies are shown to be variable over a short range through a change in the quadrupole guide voltage, or the ion trapping parameters; the effect of these modulations on the rate constant is explored for Ca⁺ + CH₃F. Bimolecular rate constants for the reactions of 40Ca⁺ with CH₃F, CH₂F₂ and CH₃Cl have been determined for a range of 40Ca⁺ state populations, allowing resolution of the global rate contributions from the ground and combined excited states. These results are analysed in the context of capture theories and ab initio electronic structure calculations. In each case, suppression of the ground state rate constant is explained by the presence of either a submerged or real barrier on the ground state potential surface. Rates of reaction from the combined excited states are generally found to be in line with capture theories, and in some cases variation is found between the high and low collision energy regimes. Molecular product ions generated in these experiments have been shown to be sympathetically-cooled into the crystal structure, and subsequently identified through resonance-excitation mass spectrometry. Molecular ions were also produced by multiphoton laser ionisation of a thermal background gas of OCS molecules. An ion-molecule reaction involving a molecular ion, that of charge transfer between OCS⁺ and ND₃, has been studied at a collision energy near 1 K for the first time using sympathetically-cooled OCS⁺ and velocity-selected ND₃. These experiments illustrate the generality of the techniques described herein, and should lead to many possibilities for future studies.
Laser cooled ions in a Penning trap can be isolated from the environment by placing them in vacuum and only interacting with them through optical and RF fields. The number of trapped particles can be varied from a single ion up to thousands. Confinement is provided by a static homogeneous magnetic field and a quadrupole electric potential. In the natural frame of the ions, this appears as a 3D simple harmonic potential. Therefore three dimensional structures can be formed in the absence of any additional RF field which may lead to heating as is the case with RF traps. There are 3N different motional modes for N particles. I present an analysis of the motion of a single particle showing that the energy levels for all three modes are equally spaced. I also describe the interaction between a trapped two level atom and an optical field. During my time in the lab the laser and computer control of the experiment has been significantly improved. In addition, an existing trap was modified to provide greater optical access and fluorescence collection. This allowed the vibrational levels superimposed on the internal states of a single 40Ca+ ion to be resolved via a narrow linewidth, electric quadrupole transition. This is the first observation of magnetron and modified cyclotron sidebands on an optical transition. When more than one calcium ion is laser cooled, and their temperature reduced below 5mK, they form a Coulomb crystal. The locations of the ions minimise the total potential energy which is comprised of the Coulomb repulsion and trap potential. The fluorescence collection optics have been arranged to resolve individual ions in these crystals. Information about the motion of the ions is deduced by comparing photos from the experiment to numerical simulations. Previously, only two ions have ever been aligned along the magnetic field in a Penning trap. I present strings of up to 29 particles and suggest the only limitation, apart from the electrode structure, is the overlap of the laser beams with the ions.
Laser-cooled atomic ions form ordered structures in radiofrequency ion traps and in Penning traps. Here we demonstrate in a Penning trap the creation and manipulation of a wide variety of ion Coulomb crystals formed from small numbers of ions. The configuration can be changed from a linear string, through intermediate geometries, to a planar structure. The transition from a linear string to a zigzag geometry is observed for the first time in a Penning trap. The conformations of the crystals are set by the applied trap potential and the laser parameters, and agree with simulations. These simulations indicate that the rotation frequency of a small crystal is mainly determined by the laser parameters, independent of the number of ions and the axial confinement strength. This system has potential applications for quantum simulation, quantum information processing and tests of fundamental physics models from quantum field theory to cosmology.