Ph.D. Thesis - Physics

Figure 7-4: Scaling of the three secular frequencies in an elliptical trap as a function of
ion height (trap scale). The 1/r0 scaling is assumed. Frequencies are angular, in units of
2π s −1 . Blue corresponds to ωˆy, green to ωˆx, and red to ωˆz.
We note, with the addition of ions, that the crystal transitions from an almost perfectly
2-D shape to a 3-D shape whose extent along ˆz is still considerably less than that along
ˆx and ˆy. We would now like to study how the extent into the vertical direction changes
with ion number. To impart a sense of this, the order of magnitude of vertical extent of the
crystal is plotted vs. ion number in Fig. 7-7.
The results of this calculation are interesting: for small ion numbers, the crystal is
almost perfectly planar, in that the extent along ˆz is orders of magnitude less than that
along ˆx and ˆy, which is essentially zero. At 26 ions, the crystal transitions into a 3-D
shape, for which the ˆz extent is about a tenth of the others. It alternates between this
and the planar shape as additional ions are added, until above 30 ions it remains in the
approximately planar 3-D state.
Another important note is that this elliptical trap was made with a higher eccentricity
than is probably required for breaking the degeneracy. For testing purposes, we decided
to err on the side of caution. With this eccentricity, however, the crystal is less regular
than it would be with a higher degree of symmetry. This situation has been studied by
authors including those of Ref. [BKGH08]. In fact, it may be possible to create more-
regular planar crystals for larger number of ions by increasing the ˆz motional frequency
through the addition of a dc quadrupole field. In Ref. [BH09], it is noted that the condition
for planar crystals in an ion trap is given by the following condition:
ω2 ˆz
ω2 ˆx
70N
>
π3 156
1/2
, (7.1)