Ph.D. Thesis - Physics

should suffice. Note that the above resistance is also consistent with the requirement due
to Johnson noise presented in Sec. 8.2.2.
Next, we consider how small the capacitance actually needs to be. Intuitively, a ca-
pacitance at rf is the same as a short, and therefore the current moving between the ions
will be coupled to ground and the overall coupling between ions will be weakened. This
is mathematically borne out by Eq. 8.13, which contains the capacitance between the wire
and ground in the denominator. As noted in Sec. 8.2.2, the needed wire capacitance C may
be calculated from a target motional coupling rate, which we take to be ωex = 10 3 s −1 .
Taking the inductance of each ion to be the calculated value of 3.7×10 4 H and the motional
frequencies to be ω/ (2π) = 1 MHz, we arrive at a capacitance of 3 × 10 −15 F.
We see that the wire must be electrically floating to a very high degree. This affects
the choices of materials, as well as the expected potentials on the wire. For instance, a
very good insulator must be used to mount the wire near the ions. This, though, leads in
turn to the possibility of large amounts of stray charge existing on that piece. It is also
certainly true that some unknown charge will reside on the wire as well. This affects the
compensation voltages, and also raises the possibility of yet another decoherence source: the
random gain or loss of electrons on the wire during an experiment. This is not a problem
as long as the resistance is very, very high.
8.4.2 Potentials on the wire due to the rf trapping fields
Unknown, slowly-varying charges on the wire can lead to changing compensation voltages,
as well as (possibly) to decoherence. However, the charges in the wire are also influenced
by the rf trapping fields. Our model above does not include this effect, but we can expect
that the wire will be polarized by these electric fields, in such a way that the fields tend to
be cancelled near the wire. In this sense, we can think of the wire as an rf ground, despite
our best efforts to isolate it from ground. This is due to the effect that the wire will have
on the curvature of the electric field lines.
An important experiment will be to measure how exactly the secular frequencies vary
as a function of the ion-wire distance. As we have seen, increasing secular frequencies are
bad for the coupling rate. How do the changing frequencies then affect the scaling of the
coupling rate as a function of ion-wire distance? If they adversely affect ωex too much,
perhaps then we can reduce the rf trapping voltages in order to keep the ωi low. When we
do this, is the trap still stable? How does the trap depth change?
8.4.3 Heating rates vs. ion-wire distance
It has been generally observed that anomalous heating tends to have a D −4 dependence,
where D is the distance from the ion to the nearest trap electrode. This is a general trend
with a lot of scatter, however. The authors of Ref. [DOS + 06] did a systematic study using
two rf electrodes with a variable distance from the ion, and found that the exponent is more
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