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Journal of Applied Physics

Analytic potential in a linear radio-frequency quadrupole trap with cylindrical electrodes

Ruthann K. Melbourne, John D. Prestage, Lute Maleki

Published Mar 1, 1991• 7 citations
Publisher

Abstract

We have analyzed a linear radio-frequency ion trap of finite length employing a four-sectored hollow cylinder enclosed between two end caps. Solutions to Laplace’s equation determine the effective potential inside the trap due to the rf trapping field, the static confining potential from dc-biased end caps, and the potential arising when a dc bias is present on the cylinder sectors. The finite length of a linear trap results in field distortion at the trap ends due to fringing effects. We have demonstrated that near the ends, the effective potential arising from the rf fields acts to propel ions out of the trap. We have further shown that the addition of a dc bias on two neighboring cylinder sectors generates an inhomogeneous field in the trap which produces a force on the ions along the trap’s long axis in a direction dependent on the sign of the bias. An exact analytic determination of the potential can be used to determine the profile of the trapped ion cloud.

DOI
10.1063/1.348635

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We have analyzed a linear radio-frequency ion trap of finite length employing a four-sectored hollow cylinder enclosed between two end caps. Solutions to Laplace’s equation determine the effective potential inside the trap due to the rf trapping field, the static confining potential from dc-biased end caps, and the potential arising when a dc bias is present on the cylinder sectors. The finite length of a linear trap results in field distortion at the trap ends due to fringing effects. We have demonstrated that near the ends, the effective potential arising from the rf fields acts to propel ions out of the trap. We have further shown that the addition of a dc bias on two neighboring cylinder sectors generates an inhomogeneous field in the trap which produces a force on the ions along the trap’s long axis in a direction dependent on the sign of the bias. An exact analytic determination of the potential can be used to determine the profile of the trapped ion cloud.
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 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 analyze the possibility of cooling ions with a single laser beam, due to the coupling between the three components of their motion induced by the Coulomb interaction. For this purpose, we numerically study the dynamics of ion clouds of up to 140 particles, trapped in a linear quadrupole potential and cooled with a laser beam propagating in the radial plane. We use molecular dynamics simulations and model the laser cooling by a stochastic process. For each component of the motion, we systematically study the dependence of the temperature on the anisotropy of the trapping potential. Results obtained using the full radio-frequency (rf) potential are compared to those of the corresponding pseudopotential. In the rf case, the rotation symmetry of the potential has to be broken to keep ions inside the trap. Then, as for the pseudopotential case, we show that the efficiency of the Coulomb coupling to thermalize the components of motion depends on the geometrical configuration of the cloud. Coulomb coupling appears to be not efficient when the ions organize as a line or a pancake and the three components of motion reach the same temperature only if the cloud extends in three dimensions.
The design of surface-electrode ion traps is studied analytically. The classical motion of a single ion in such a trap is described by coupled Mathieu equations. An analytic boundary-value solution for the Laplace equation in three dimensions with a piecewise-constant Dirichlet boundary condition is described. This solution is used to model the electrostatic potential field generated by a series of electrodes lying in a single plane. The model is applied to the problem of designing surface-electrode ion traps, including calculating important trap design parameters such as the center position of the trapped ion, trap depth, and stability parameters. The model is used to determine optimized dimensions for trap electrodes in various cases.
We report on development of a miniature linear rf ion trap and preliminary efforts to trap and laser-cool {sup 24}Mg{sup +} ions. This trap has a radial dimension of 50 {mu}m, one tenth the size of our previous trap. The rms confinement of a trapped ion scales as R{sup -2}, where R is the trap radius, so tight confinement of ions is possible at a modest rf voltage. Although the three-dimensional trapping potential for our trap cannot be calculated analytically, it can be described in terms of dimensionless parameters, measured by studying macroscopic charged particles in a much larger model of the trap. We expect that with this trap it will be possible to achieve secular frequencies high enough for resolved-sideband cooling of light ions using allowed dipole transitions. This will make it much easier to perform experiments on the quantum motion of trapped ions. This trap uses a two-wire design rather than the more common four-wire one, so the trap electrodes occlude a much smaller solid angle than either a four-wire linear trap or the more common quadrupole trap. Thus, it will be possible to couple optical devices, such as high-finesse cavities, much more efficiently to trapped ions.
The resonant frequency of a helical resonator, that efficiently delivers high voltage rf to an electrodynamic ion-trap, depends on capacitive and inductive loads. Estimation of these loads for the parts that are attached to the resonator are thus necessary for designing a resonator operating at a desired frequency. On the other hand, the trapped ion experiences a voltage induced by the capacitance resulting from the electrodes. This results to excess micromotion to the ion. Here, we estimate the capacitive and resistive loads arising from different parts of ourtrap.