Ph.D. Thesis - Physics

in Ref. [LGA + 08] in a 75 µm trap (5 quanta/s), it does seem reasonable to be able to
observe the coupling without interference from heating, provided that the trap and the wire
are sufficiently cooled. Even with a simulated coupling rate of J = 100 s −1 , spin models of
a modest size may be simulated. Both of these estimates, however, depend upon internal
state coherence times being on the same order as or greater than the motional heating time.
8.4 Experimental questions
We have now calculated, or otherwise justified, the expected coupling rates and decoherence
rates for our system. However, these rates are based on a specific model, which will not
hold perfectly in the lab. Can we anticipate some effects that may require adjustment to
the model? Also, what are the steps that should be done before attempting a wire-mediated
coupling experiment? There are three main questions that drive the experimental work of
the next chapter:
1. What are the constraints on the resistance and capacitance between the wire and
ground?
2. How do the rf confining fields affect the potential on the wire?
3. How do the heating rates scale as a function of ion-wire distance?
The first question will guide the setup of the experiment, while the second and third
will require experimental measurements. We discuss each briefly.
8.4.1 DC and RF paths from the wire to ground
Intuitively, paths to ground, whether rf or dc, seem likely to reduce the coupling rate by
allowing charge to escape from the “system” to the “environment.” Here, we will briefly
justify why this is the case, and put it into a more quantitative form, in order to figure out
exactly how isolated the wire must be from ground.
We first consider the wire’s resistance to ground. The wire basically is an RC circuit,
with a very large resistance and very small capacitance. The figure of merit is the time
constant that characterizes the leakage rate of charge on the wire to ground. Clearly, a
fast time constant will result in the shorting of the current moving between the two ions
to ground, reducing the coupling rate. The necessary condition will be τleak = RC ≫ tex.
For a capacitance of a few femtofarads, we find that R = 10 13 Ω will provide a leakage
time greater than 1 s. Current leakage, however, is not the only consideration. Note that a
changing total charge on the wire will randomly change the force exerted on the ions, leading
to decoherence of their motional states. Therefore, whatever the total charge on the wire
is, we require it to remain constant during an experiment. Satisfying the above condition
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