Ph.D. Thesis - Physics

Figure 8-1: Schematic of the experimental setup. The two ions are trapped in different
potential wells, set by the segmented dc electrodes. Distances are approximately to scale
for our experiments; actual values are presented below.
is the time required for the two ions to completely exchange motional states. Normally, the
1/d 3 scaling of the interaction rate ωex means that this coupling quickly becomes negligible
as the ions are moved apart. Here, we will focus on the case where free-space coupling may
be ignored, and the only observable coupling is due to the wire.
Fig. 8-1 depicts a schematic of the experimental setup. A single wire is positioned close
to two trapped ions, which are confined in a linear surface-electrode ion trap and are set
far enough apart so that coupling due to their shared normal mode is negligible. The
creation of two different trap regions is possible by adjusting the segmented dc electrodes;
fine adjustment of these voltages can also aid in setting the secular frequencies of the two
ions to be equal, a resonance condition which is important for optimal coupling.
In what follows we will follow two approaches to describe the dynamics of this system.
In the first, we solve the dynamics directly from electrostatic calculations. In the second, we
use an effective circuit model, a useful approach when studying the effects of decoherence in
the next section. This approach was followed by Heinzen and Wineland in a paper that first
treated the coupling of ions over a wire, among other related situations [HW90]. The new
contribution here is the solution of the dynamics directly from electrostatics, presented in
Sec. 8.2.1 which was undertaken mostly by the author’s collaborator Nikos Daniilidis, and
provides a more rigorous justification of the circuit model for our experimental situation.
8.2.1 Electrostatic solution
In order to solve for the behavior of the above system, we need to make some simplifying
(but reasonable) assumptions. We consider a wire of radius a and length L, positioned a
height H above an infinite conducting plane, and parallel to this plane. Treating this plane
(which is, in reality, the trap) as infinite in extent and a continuous conductor is the first
assumption. The two ions are situated at heights h1 and h2 (h1, h2 < H) above the trap,
and are some distance d apart. We assume also that h1, h2, H ≪ d < L. The ions are
treated as point charges, a very reasonable assumption. We depict this situation in Fig. 8-2.
Considering both ions to have a charge ec, and the wire to have zero net charge, it can
be shown using Green’s function techniques that the electrostatic potential of the wire with
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