Ph.D. Thesis - Physics

Figure 7-2: Calculated q parameters and trap depths for a series of rf drive voltages and
an rf frequency of 3.5 MHz. The q parameters are blue, cyan, and green for x, y, and z
respectively, with qˆz > qˆx > qˆy. The trap depths are plotted in red and measured in eV.
The trap is stable along all three directions and also deep enough to trap for this parameter
set.
and secular frequencies, along with the ion crystal structure in the trap.
7.1.1 Secular frequencies and trap depth
The secular frequencies and trap depth are calculated numerically, using the CPO numerical
modeling software (as in Chs. 5 and 6), for a given set of trap dimensions and rf signal. We
present this calculation here for Uraniborg 2. The trap geometry that was an input to CPO
was generated by parametrically drawing the ellipses using a Matlab program. The rf drive
frequency was Ω/(2π) = 3.5 MHz, which is suitable for a trap of this size. In Fig. 7-2 and
Fig. 7-3, we plot the simulation results. In Fig. 7-2, we plot the trap depth and Mathieu
q parameters as a function of rf drive voltages, and in Fig. 7-3, we plot the three secular
frequencies as a function of the same. All dc voltages were assumed to be zero.
It is only necessary to compute the secular frequencies for a single trap size, since scaling
to smaller traps may be done by changing the unit size in the electrostatic computation.
We expect that the secular frequencies will scale as 1/r0, where r0 is a characteristic length
of the trap. In practice, this quantity is numerically calculated and does not correspond to
some specific distance. The height of a single trapped ion above the plane of the electrodes
does, however, obey the same scaling law. We plot in Fig. 7-4 the secular frequencies as a
function of the ion height. A caveat is that even if the dimensions of the electrodes remain
the same, there will be some minimum spacing between electrodes that is given by the
fabrication process used. For microfabricated traps, this is typically a few microns. For
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