Ph.D. Thesis - Physics

the rf null in such a trap exists only at the center of the trap, which is the minimum
of the potential in three dimensions. Thus, only one ion may be confined without excess
micromotion. One of the principal questions discussed in this chapter is the effect, in theory,
that this micromotion has on an example quantum simulation. The structure of large 2-D
crystals, as well as the impact of rf heating and background gas collisions, was discussed in
Ref. [BKGH08]. However, the question of micromotion had not previously been adequately
addressed.
We also construct and demonstrate a test elliptical trap, to verify a portion of the above
theoretical work. Since it is crucial to reduce motional decoherence (anomalous heating)
in such a trap, we construct a new closed-cycle cryostat apparatus for the testing of these
traps, in order to provide an electrical noise environment in which progress toward quantum
simulation can be made. We then test our predictions of the crystal structure and motional
frequencies of ions in this trap. We come to the conclusion that quantum simulations with
at least a few ions should be possible in this trap.
The chapter is organized as follows. In Sec. 7.1, we introduce the basic model of the
elliptical trap, and present calculations of the structure of ion crystals therein. In Sec. 7.2,
we discuss the effects of micromotion on the fidelity of a quantum simulation, by numeri-
cally simulating the quantum dynamics with a time-dependent potential resulting from the
micromotion. The expected coupling rates due to laser pushing forces are also calculated.
In Sec. 7.3, we study the origin of the state-dependent forces, looking at optical forces, and
also at magnetic field gradients. In Sec. 7.4, we discuss the closed-cycle cryostat appara-
tus used for testing the elliptical traps. We put our theoretical predictions to the test in
Sec. 7.5, measuring the motional frequencies and structure of crystals in the elliptical test
trap. Finally, we evaluate the suitability of the elliptical trap design for quantum simulation
in Sec. 7.6, and then conclude in Sec. 7.7.
7.1 Elliptical ion trap theory
In this chapter, we explore the idea of using elliptical ion traps to perform quantum simu-
lation in 2-D. Elliptical traps were proposed and demonstrated by DeVoe [DeV98] with the
aim of producing a miniature linear ion trap with relatively simple fabrication and fairly
favorable amounts of micromotion. Here, we study instead 2-D ion crystals in such a trap.
To obtain an ordered lattice of ions in a single plane, it is desirable to have a trap
with approximate cylindrical symmetry. However, in such traps (the ring trap included),
the two “radial” vibrational modes are degenerate; hence, there is no preferred axis along
which the ion crystal may align. This undesirable condition can be rectified by introducing
an asymmetry in these two directions. In such a trap, ions will align in a 2-D array until
a critical ion number is reached, at which point the crystal may minimize its energy by
transitioning to a 3-D shape. This is the primary motivation for making the trap elliptical.
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