Ph.D. Thesis - Physics
Ph.D. Thesis - Physics
Ph.D. Thesis - Physics
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Using the expression for ωˆr derived from Eq. 5.3,<br />
x1sd 2 ≈ Ω 2 mr4 1Q2 8πǫ0V 2 . (5.5)<br />
Q1<br />
When Q1 is not equal to Q2, then the confining forces are characterized by different ωˆr<br />
and the two ions have different offsets from equilibrium. The ratio in offsets, if the masses<br />
are comparable (m1 ≈ m2), should be (Q2/Q1) 2 . We observed exactly such an asymmetry<br />
between the offsets of the two ions, where typically Q2/Q1 is between 1 and 5. There may<br />
be additional small asymmetries due to edge effects and the presence of the 4-rod trap as<br />
well as differences in charge. Fig. 5-14 shows the displacement of a pair of ions as the rf<br />
drive frequency is varied, for one experimental run. The spread in charge to mass ratios and<br />
accordingly unknown values of Q and m for each ion (as in Ref. [PLB + 06]) does not permit<br />
us to compare the observed repulsion to a theoretical model. Indeed, if these data were<br />
available, this repulsion experiment would be a very useful way to measure the screening<br />
factor s for a given trap, perhaps prior to trapping atomic ions.<br />
We conclude that ion-ion interaction in a mm-scale lattice trap is observable by the<br />
mutual Coulomb repulsion of macroions. Such an experiment could be used to measure<br />
the screening parameter s for a given trap, if knowledge of the charges and masses of the<br />
individual particles involved were available. Although we have been able to measure the ion-<br />
ion interaction of macroions and fit it to a model (in a certain region of parameter space),<br />
it will be necessary to scale the trap down further in order to observe ion-ion interactions<br />
between the atomic ions that would be used for quantum simulation.<br />
5.6 Scaling laws for the simulated interactions in lattice traps<br />
The lattice trap discussed in this chapter provides a fairly straightforward method for real-<br />
izing a two-dimensional array of trapped ions. The first two steps enumerated in Sec. 4.3.3,<br />
design and testing of the trap, have now been reported. We turn now to evaluation of<br />
the trap design, and ask: how useful could this system be for quantum simulation of two-<br />
dimensional spin models [PC04b]? We will need to calculate both the motional coupling<br />
rate ωex, and the simulated coupling rate J.<br />
5.6.1 Motional coupling rate<br />
We begin with the motional coupling rate, which is the rate at which two coupled ions swap<br />
motional states (provided they have the same secular frequency). Let us review the basic<br />
physics of the system formed by two trapped ions undergoing mutual Coulomb repulsion.<br />
The lowest-order term in the Taylor expansion of the Coulomb potential that contains an<br />
interaction between the ions is<br />
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