Ph.D. Thesis - Physics

Figure 9-9: Linear fit of the ωˆy (green) and ωˆx (red) secular frequencies to 1/D 2 , where D
is the ion-wire distance.
Why might this 1/D 2 dependence appear? Our hypothesis is as follows. The rf voltage
on the wire arises from a capacitive coupling from the rf electrodes to the wire. The electric
field seen by the ion falls off as 1/D, from the elementary problem in electrostatics (assuming
the wire to be long). In addition, the induced voltage on the wire scales as 1/D. This may be
understood intuitively by considering the wire-trap system to be a parallel plate capacitor
with one plate (the wire) much smaller than the other. Since the secular frequencies are
directly proportional to the magnitude of the electric field acting on the ion, this produces
a 1/D 2 scaling.
Heating rates
We now present the heating rate measurements. These were done using the Doppler re-
cooling method. At each wire position, the time-dependent fluorescence was recorded 200
times and added up to produce the data presented here. Fig. 9-10 shows an example plot
of this data and the fit to it. The initial energy can be inferred from this plot, according to
the method presented in Ref. [WEL + 07].
Such plots were done for each data set, and the resulting heating rate measurements are
presented in Fig. 9-11 and Fig. 9-12.
Although we do not see a clear trend as a function of the ion-wire distance, this is
actually a positive result. After all, the ion is closer to the trap than to the wire, and
the surface area of the trap is much greater. We would expect the wire to need to get
significantly closer to the ion than the trap is before heating effects due to the wire are
noticeable. This bodes well for getting the wire close to the ions without inducing excess
heating. Indeed, in the future, cryogenic cooling should suppress this baseline heating that
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