Ph.D. Thesis - Physics

the vibrational temperature is low. We will make this assumption here; we will either as-
sume that cooling can be efficiently performed in the lattice-style trap (as in Ch. 5), or that
the otherwise requisite anisotropy of the trap will suffice to render this term small (as in
Ch. 7).
The coupling rate between ions due to the Coulomb interaction, per unit cycle, is char-
acterized by a dimensionless parameter βˆα, defined as
βˆα =
e 2 c
8πǫ0mω 2 ˆα d3.
When βˆα ≪ 1, the J-coupling rate may be approximated as
(4.37)
Jˆα = cαe2 cF 2 ˆα
4πǫ0m2ω4 , (4.38)
ˆα
d3
where cˆα is a constant of order unity that depends on the trap and laser geometry and
d is the average ion-ion distance. For traps in which ions are confined in separate wells,
βˆα ≪ 1 for all ˆα, and this formula holds. This is not necessarily the case when the ions are
confined in the same well. As an example, suppose two ions are confined a distance d apart
in a linear ion trap, and that the line segment connecting them lies along ˆz. In this case,
βˆz = 0.25, and the coupling rate Jˆz is given by
Jˆz = F 2 ˆz
4mω 2 ˆz
2βˆz
. (4.39)
1 + 2βˆz
For a very good discussion of this protocol as applied to two ions, we refer the reader to
the thesis of Ziliang Lin [Lin08]. The first experimental realization of this protocol, using
two ion-qubits, was published in Ref. [FSG + 08].
4.3 Ion trap design for quantum simulation
Performing quantum simulations of phenomena such as spin frustration requires a 2-D
array of trapped ions with a lattice structure that is similar to the structure of the target
system. This requirement is unique to analog, as opposed to digital, quantum simulation.
Creating such a 2-D array, however, is quite nontrivial. Here we present in detail some of
the challenges associated with trap design, and then outline our methods for solving them
that will comprise the remainder of this part of the thesis.
4.3.1 Challenges for trap design
Recall from Sec. 1.4 that there are three main challenges to quantum simulation: deco-
herence, precision limitations, and scalability. In this part of the thesis, we focus on the
scaling up of analog quantum simulation in 2-D. Despite this focus on only one of the above
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