Physical Significance of Results

This simulation provides a test of whether an ion will remain within a Paul trap. The dynamics of a single ion within a Paul trap are governed by what is known as the Mathieu differential equation [3]:

where and are dimensionless parameters defined by, in the case of the Paul trap:

and is the static applied potential, is the amplitude of the oscillating potential, is the mass of the ion, is the angular frequency of the oscillating potential, is , and , are trap dimensions (as discussed in section 1).

The Mathieu equation has both stable and unstable solutions as a function of and , corresponding to motions which are bounded (trapped ions) and unbounded (escaping ions) [3]. The regions of stability correspond to the regions A and B of Figure 5. When the parameters and are sufficiently small and the system is in region A of the stability plot, stable periodic motion can be represented by the first approximation of the solution to the Mathieu differential equation [13]. The first-order perturbative mathematical solution predicts that the motion of an ion in region A will be described by the superposition of a quickly oscillating ``micromotion'' on a slowly oscillating ``secular'' motion [3]. Figure 6 shows plots of Paul trap simulations operating in this region. Fourier analysis of the frequencies of the secular motion and micromotion of Paul trap simulations in region A of the Mathieu stability plot agree with theoretical results derived from the Mathieu differential equation, Eq. 10 [3]. Also in agreement with theory is the fact that the ions' orbits are quasiperiodic in Mathieu region A- in other words, the oscillations on the and axes involve frequencies (the secular and micromotions) which are not expressible as integer multiples of each other, so the plot reveals a shaded region rather than a closed curve.

As predicted by the Mathieu stability plot, simulations with parameters and resulted in unbounded motion (the ions escaped). Even when kinetic energy was reduced by initializing the simulation at a very cool temperature- K- the ions escaped.

In the stable domain for which and are not sufficiently small (region B of Figure 5, perturbative solutions are not valid [3]. As of 1989, all ion-trapping experiments operated in region A of the stability plot [3]. Thus, region B is an interesting case because it is relatively unexplored, both theoretically and experimentally. Figure 7 shows plots from a three-ion Paul trap simulation operating in region B. Fourier analysis of an undamped ion in region B (see Figure 7) shows that the axial and radial motions are described by a discrete sum of frequencies, despite the presence of a non-linear Coulomb interaction. Thus, the motion of the simulated ion is quasiperiodic, as opposed to chaotic.

Another form of analysis which is useful with regard to this simulation is the radial distribution function, , which expresses the probability of finding an ion at a distance from another ion. The radial distribution function for a crystal shows discrete peaks from which the crystal's structure can be determined. Figure 8 shows for a four-ion simulation in region B of a Paul trap: the ions are contained to within m of each other, and the figure shows definite peaks. However, whether or not these peaks imply an ordered structure has not yet been determined.