Ph.D. Thesis - Physics

of the scale being tested, and traps of other scales. These properties include the motional
frequencies, which are important for calculating β and J, as well as quantities of practical
importance such as the trap depth and ion position relative to the trapping electrodes.
Results from a trap of one size can typically be scaled to smaller or larger sizes.
Testing
After theoretical calculations of a given trap are done, they must be confirmed by experiment
in order to compute the coupling rates and thereby evaluate the trap design. Testing the
traps consists of fabricating a trap that is representative of one of the above paradigms,
mounting it in a suitable vacuum vessel, and then measuring certain properties of the trap.
For both paradigms, the motional frequencies are measured. In the case of an array of ion
traps, the lattice geometry is determined by the fabrication of the trap electrodes. However,
for ions which form a crystal within the same trap volume, the ion crystal geometry is
determined by the trapping potentials, and verifying that this crystal matches theoretical
predictions is also important.
Evaluation
Equipped with the ion crystal structure and motional frequencies, the coupling rates may
be calculated. A primary goal of this part of the thesis is to determine, for each paradigm,
how the coupling rates scale with the trap size (which is typically defined as the distance
from the trap center to the nearest electrode). In Ch. 5, we treat the problem of an array
of Paul traps, while in Ch. 7 we study ion crystals within one example of a Paul trap that
creates a 2-D array of ions: a surface-electrode elliptical ion trap.
Included in the evaluation of a trap design is an estimation of the relevant decoherence
rates. Decoherence of internal states depends a great deal on the choice of qubit states,
the ambient fields, fluctuations in the control potentials, and other effects. Our primary
concern is with motional decoherence rates. In Ch. 7, the quenching of these rates at
cryogenic temperatures motivates the construction and use of a 4 K cryostat for ion trap
testing. Studying internal and external decoherence is beyond the scope of this thesis, but
we cite relevant results from other research efforts where appropriate. Our primary concern
is to insure that the coupling rates we determine are much higher than the decoherence
rates reported for similar ion traps.
4.4 Summary
We have now explained how the interaction of trapped ions with laser radiation can be
used to implement quantum control over both the internal and motional states of the ions.
Preparation of the ions’ motional state is done effectively by first Doppler cooling the ions
into a crystalline state near the Doppler limit, then further cooling them (if necessary) close
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