Ph.D. Thesis - Physics

Figure 7-7: Scaling of the order of magnitude of the vertical extent of the ion crystal as
a function of the number of ions. Trapping parameters are those specified in the text. A
transition from an almost-perfectly planar crystal to an approximately planar crystal is
seen.
where ωˆx = ωˆy and N is the ion number. The fourth-root dependence on N implies that
modest increases in ωˆz will lead to planar crystals for larger numbers of ions.
As an example, we calculate the crystal structure for 120 ions. According to Eq. 7.1,
and for a radial frequency of ωˆx = ωˆy = 2π × 150 s −1 , the condition for a planar crystal is
ωˆz > 2π × 608 s −1 . For a frequency of ωˆz = 2π × 500 s −1 , according to our algorithm, the
crystal is planar, while for ωˆz > 2π × 608 s −1 it is not. We plot the 2-D projections of the
crystal structure in Fig. 7-8. Near the center of the crystal, the ions form a nice regular
hexagonal lattice.
While it may be possible to produce the above secular frequencies using a top plate
voltage (as in Ch. 6) together with the center electrode, numerical simulations have shown
that it is not possible to create such a trapping region using only the electrodes present on
the surface-electrode elliptical trap. We calculate that the maximum number of ions for
a planar crystal in such a trap is 35, which would still be a very interesting situation for
quantum simulation. In addition, it may be possible to implement quantum simulations
using ion crystals that are only approximately planar, such as those depicted in Fig. 7-6.
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