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Physical Review Letters

Dynamics of charged particles in a Paul radio-frequency quadrupole trap

John D. Prestage, Ann Katherine Williams, Lute Maleki, M. Jahed Djomehri, E. Harabetian

Published Jun 10, 1991• 65 citations
Publisher

Abstract

We have carried out a molecular-dynamics simulation of hundreds of ions confined in a Paul trap. The simulation includes the trapped particles' micromotion and interparticle Coulomb interactions. We have implemented a random walk in velocity to bring the secular motion to a given temperature which we have numerically measured. When the coupling \ensuremath{\Gamma} is large the ions form concentric shells which undergo a quadrupole oscillation at the rf frequency, while the ions within a shell form a 2D hexagonal lattice. Ion clouds at 5 mK show no rf heating for ${\mathit{q}}_{\mathit{z}}$\ensuremath{\lesssim}0.6, whereas rapid heating is seen for ${\mathit{q}}_{\mathit{z}}$=0.8.

DOI
10.1103/physrevlett.66.2964

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We have carried out a molecular-dynamics simulation of hundreds of ions confined in a Paul trap. The simulation includes the trapped particles' micromotion and interparticle Coulomb interactions. We have implemented a random walk in velocity to bring the secular motion to a given temperature which we have numerically measured. When the coupling \ensuremath{\Gamma} is large the ions form concentric shells which undergo a quadrupole oscillation at the rf frequency, while the ions within a shell form a 2D hexagonal lattice. Ion clouds at 5 mK show no rf heating for qz{\mathit{q}}_{\mathit{z}}qz​\ensuremath{\lesssim}0.6, whereas rapid heating is seen for qz{\mathit{q}}_{\mathit{z}}qz​=0.8.
We discuss a radio-frequency (rf) ion trap, the rotating-radio-frequency (rotating-rf) trap, in which the motion of a charged particle is described by trigonometric functions rather than the usual Mathieu functions of a normal rf trap. In the rotating-rf trap, a rotating quadrupole electric field confines charged particles, whereas in a normal rf trap, an oscillating quadrupole electric field does. Ion motion in a rotating-rf trap is a superposition of two nondegenerate circular secular motions and two corresponding circular micromotions. The cases of applying a uniform dc magnetic field or a quadrupole dc electric field in addition to the rotating-rf field are also discussed. Confinement in a rotating-rf trap can be tighter than in a normal linear rf trap with the same experimental parameter values.
Molecular dynamics simulations of the ion clouds stored in a linear quadrupole rf ion trap are performed at low temperatures (0.1 mK--15 K). The introduction of periodic boundary conditions allowed us to study the translationally uniform ion cloud geometry. The time evolution of the kinetic energy of the ion cloud is monitored during the simulation and the increase in the kinetic energy of the cloud (rf heating) is determined; its dependence on temperature, trapping voltage, and the number of ions is studied. The dependence of rf heating rate on temperature shows that rf heating is undetectable until the temperature reaches approximately 0.5 K; it then increases rapidly until it reaches a maximum at around 2 K. It starts to decrease slowly at higher temperatures. The rf heating rate is shown to increase very sharply with the amplitude of the trapping voltage. The dependence of the rf heating rate on the number of ions shows the influence of the ion crystal shell structure at low temperatures, and has a simple linear dependence at higher temperatures.
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 discuss a radio-frequency (rf) ion trap, the rotating-radio-frequency (rotating-rf) trap, in which the motion of a charged particle is described by trigonometric functions rather than the usual Mathieu functions of a normal rf trap. In the rotating-rf trap, a rotating quadrupole electric field confines charged particles, whereas in a normal rf trap, an oscillating quadrupole electric field does. Ion motion in a rotating-rf trap is a superposition of two nondegenerate circular secular motions and two corresponding circular micromotions. The cases of applying a uniform dc magnetic field or a quadrupole dc electric field in addition to the rotating-rf field are also discussed. Confinement in a rotating-rf trap can be tighter than in a normal linear rf trap with the same experimental parameter values.
We investigate the energy dynamics of noncrystallized (melted) ions, confined in a Paul trap. The nonperiodic Coulomb interaction experienced by melted ions forms a medium for nonconservative energy transfer from the radio-frequency (rf) field to the ions, a process known as rf heating. We study rf heating by analyzing numerical simulations of noncrystallized ion motion in Paul trap potentials, in which the energy of the ions' secular motion changes at discrete intervals, corresponding to ion-ion collisions. The analysis of these collisions is used as a basis to derive a simplified model of rf heating energy dynamics, from which we conclude that the rf heating rate is predominantly dependent on the rf field strength. We confirm the predictability of the model experimentally: Two trapped 40Ca+^{40}\mathrm{Ca}^{+}40Ca+ ions are deterministically driven to melt, and their fluorescence rate is used to infer the ions' energy. Under these experimental conditions, we investigate which Doppler cooling parameters are required for efficient recrystallization, and compare our results to simulations.
The accurate evaluation of the second-order Doppler frequency shift (SODFS) of trapped ions in a linear Paul trap has been studied with experiments and molecular dynamics (MD) simulations. The motion of trapped ions in the trap has three contributions, and we focus on the ion excess micromotion, which is rarely discussed when evaluating the SODFS of ions. Based on the hypothesis that the ion density is uniformly distributed in the radial direction, we propose a model to accurately evaluate the total SODFS for an ion ensemble with temperature less than 100 mK, which consists of three contributions of ion motion. The effectiveness of the model has been verified by MD simulation, and the experimental results of the excess micromotion of trapped ions are consistent with the theory. We believe that our model offers advantages in accurately evaluating the SODFS for the ion trap, especially those of laser-cooled ion microwave clocks based on large ion clouds.