Ph.D. Thesis - Physics

Figure 7-8: Crystal structures for 120 ions with ωˆx = ωˆy = 2π × 150 s −1 . Left: 2-D
projection of the 3-D ion crystal that results for a vertical frequency of ωz = 2π × 500 s −1 .
Right: 2-D ion crystal structure for ωˆz = 2π × 650 s −1 , greater than the critical frequency
for crystal planarity (according to Eq. 7.1) of ωˆz = 2π × 608 s −1 .
7.2 Micromotion scaling and its effect on quantum simula-
tion
7.2.1 Scaling of the micromotion amplitude
We now turn to the calculation of the micromotion amplitude for the ions in the elliptical
trap, as well as the effect that it has on quantum simulation. The micromotion amplitude
Aµ along ˆx is given to first order by Aµ = 1
2 qˆx∆x, where qˆx is the Mathieu parameter
along the direction ˆx and ∆x is the displacement of the ion from the rf null along ˆx. Similar
relations apply for the other directions. Given a numerical solution for the trapping potential
along with the ion geometry calculation presented in the last section, we can calculate this
numerically as well.
The scaling of the micromotion amplitude for two ions is straightforward. Suppose two
ions align along the ˆy axis of an elliptical trap. Then, by balancing the Coulomb and
trapping forces, the ion-ion distance is
d =
e 2 c
2πǫ0mω 2 ˆy
1/3
, (7.2)
and the displacement from the rf null is just ∆y = d/2. Since the frequency ωˆy scales as
1/r0, the basic scaling law for micromotion is A ∝ r 2/3
0 , therefore it decreases as the overall
size of the trap is reduced. This is one motivation for decreasing the overall size of the trap.
The micromotion amplitudes of ions in larger 2-D crystals may be estimated from the
ratio of displacements of ions given above in the ion crystal structure calculations. In-
teractions between neighboring ions are impacted only by the relative motion of the ions
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