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Quasiperiodic Regime

  Blümel claims that, for immediately below the cut-off for region C, there exists a nonheating quasiperiodic region of phase space. That is, the root-mean-square displacements from the trap center which are immediately on the ``dense'' side of the heating regime constitute non-heating configurations. This assertion is important because it implies that there is a nonheating, neutral ``buffer'' zone between the crystalline phase (discussed below) and the chaotic phase.[4] Blümel states that the range of values which form this quasiperiodic regime decreases as Mathieu parameter q increases for a given a. However, although the width of the non-heating quasiperiodic regime is compressed as the configuration approaches the Mathieu instability, Blümel conjectures that in stability region A ``there is always a small phase-space region around the crystalline solution in which no heating occurs.''[4] This statement contradicts conclusions drawn by Hoffnagle, et al.: Hoffnagle's research group claimed the existence of reproducible order chaos transitions (displaying hysteresis) as a function of the Mathieu parameters a, q. The research group claimed, for a=0, a well-marked order chaos transition at and a chaos order transition at . Hoffnagle's work contradicts Blümel's theory on two points: ( a) Hoffnagle, et al., claim order chaos transitions as a function of the Mathieu parameters, whereas Blümel's criterion for predicting ordered and chaotic states is the amount of energy in the system (which can be gauged by ); ( b) Hoffnagle claims the existence of an order chaos transition at a=0, , which is well below the Mathieu instability of , while Blümel asserts the stability of few-ion crystals in Mathieu regime A--- even in the absence of damping--- right up to the Mathieu instability.gif The stability of crystals will be further investigated after a discussion of the crystalline phase.

The configuration of file cloud5A.trp depicts 5 Mg ions in this non-heating ``quasiperiodic'' regime. The term ``quasiperiodic'' may be somewhat of a misnomer--- in this closely-coupled cloud phase,gif non-linear effects are strong. As shown in Fig. gif, Fourier transforms of position for the non-heating cloud do not yield the sharp, noiseless peaks of truly quasiperiodic behavior, which is composed of discrete frequencies related by an irrational ratio (Figs. gif, gif, gif). Nevertheless, Fig. gif is dominated by certain frequency ranges, so one might call this behavior ``pseudoperiodic.''


next up previous contents
Next: Crystalline Phase Up: Behaviors Observed for Previous: Chaotic Regime



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