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Mathieu Regime

When 10 ions were placed in a high-energy configuration in region B, a single ion's trajectory was composed of a discrete number of Fourier components (Fig. gif). The frequency peaks were not as well-defined as those of the single ion in region B (Figs. gif,gif): the ``dips'' between adjacent peaks are neither as lowgif nor as smoothgif for the 10-ion simulation as for the single-ion simulation. Nevertheless, the frequency peaks of the 10-ion and 1-ion transforms are in excellent agreement, varying by to absolute discrepancy for axial oscillation and to discrepancy for radial oscillations (Tab. gif). Thus, the ion motions for the high-energy 10-ion simulation are virtually uncoupled, are well-approximated by motion of a single ion in stability region B, and demonstrate the features of the Mathieu regime (previously discussed for region A in Sec. gif) in stability region B. However, the 10-ion simulation demonstrated that, even in the Mathieu regime, interactions are occasionally significant when Coulomb repulsion arising from a near-collision ``kicks'' a particle from one trajectory to another. (Note the segmented trajectory of a single ion, Fig. gif).

In the Mathieu regime, the classical result of equipartition of energy between kinetic and potential components was demonstrated for a 10-ion simulation in stability region B (Fig. gif).

  
Figure: Equipartition of energy for 10 Paul-trapped magnesium ions in stability region B: the running averages (evaluated over s) of total, potential, and kinetic energies are displayed. The total energy of about eV is partitioned equally among potential and kinetic components, indicating a temperature of approximately 1660 K. source file: paul10B.trp.

  
Table: Frequency analysis of x-motion and axial motion of 1 and 10 Paul-trapped ions in stability region B. The agreement in frequencies illustrates that the 10 ions are in the Mathieu (non-interacting) regime. source files: paul10B.trp, paul1B.trp

 
Figure: Axial position and x-position vs. time for one of 10 ions Paul trapped in region B of Mathieu stability. source file: paul10B.trp.

  
Figure: Fourier transforms (2048 data points) of x-position ( left) and axial position ( right) of a Paul-trapped ion in stability region B. Peak x-frequencies (MHz): , , , , , , . Peak axial frequencies (MHz): , , , , , , , , , . source file: paul10B.trp.

  
Figure: x-position vs. axial position for one of 10 Paul-trapped ions in region B of Mathieu stability. source file: paul10B.trp.



next up previous contents
Next: Chaotic Regime Up: Behaviors Simulated for Previous: Behaviors Simulated for



Wolfgang Christian
Fri May 12 10:36:01 EDT 1995