Ph.D. Thesis - Physics

8.2.3 Simulated coupling rates
We now address how the calculated motional coupling rates above may be translated into
a simulated coupling rate. As in Part II, we restrict our discussion to the simulated J-
coupling rate in a spin model simulation using the methods outlined in Ref. [PC04b]. We
will assume that a state-dependent force of magnitude F is applied to each of two ions,
although they may reside in different trap regions.
The J-coupling rate may be calculated by separating the motional coupling rate (parametrized
by β) from the part of J that is due to the state-dependent force. Recalling the results of
Sec. 4.2.2 and taking the β ≪ 1 limit, we find that J for a given direction may be written
as
J =
2cβF 2
mω 2
(8.14)
where ω is the secular frequency along a given direction, m is the ion’s mass, and c is a
constant of order unity that depends on the direction of the state-dependent force. Here,
we set c = 1 since we are mainly concerned with an order of magnitude for the coupling
rate. Recalling that β is the motional coupling rate per secular vibrational period, we write
it as
β = ωex
. (8.15)
ω
For the parameters used in Sec. 8.2.2, we find that β ≈ 10 −3 . Taking our value from
Ch. 5 of F = 2.7 × 10 −21 N, and using the mass of 40 Ca + , we find that J ≈ 100 s −1 .
Although this seems like a poor figure, we note that the same scaling laws that apply to
the lattice ion traps (Ch. 5) do not apply to the present situation. For instance, it may be
possible to decrease the secular frequency, even with a small ion-wire distance. Part of the
reason for this becomes apparent when we discover the effect that the wire has on the trap
potentials (Ch. 9): it may be easier with wire-mediated coupling to preserve trap depth
while lowering the secular frequency.
Naturally, there are other possible avenues to increasing J, such as applying a stronger
force. For now, we regard the simulated coupling as observable in principle, under favorable
but realistic decoherence rates.
8.3 Decoherence
We now turn to estimating the rates of decoherence in the above system. The decoherence
processes are important to understand for the obvious reason: they may tell us if coherent
information transfer is possible over a wire at all, even if the coupling rate seems adequate,
and may also help us to answer questions such as the requisite temperature and resistivity
of the wire. Because these decoherence sources are electrical in nature, we will find the
195