Ph.D. Thesis - Physics

Figure 7-13: Scaling of the J-coupling rates for all three directions as a function of the ion
height in an elliptical trap, for the three concentric square rings wire configuration. The
value for each direction will only hold if the projection of the electron spin is entirely along
that direction; this would give an Ising-type interaction.
Numerically, this is nicely verified. We apply the above scaling law to the secular
frequencies, compute the mean ion-ion distance, and calculate the J coupling as a function
of h, the height of the ion above the trap surface (calculation with respect to r0 would
yield the same scaling). The highest coupling rate plotted here is 700 s −1 , which will be
observable if the dominant decoherence rates are significantly lower than that.
By contrast, the fields due to the rings produce a field gradient at the location of the
ions with the form ∂ B
∂xi
∂Bˆx
∂Bˆy ∂Bˆz
= ∂x ˆx + ∂y ˆy + ∂z ˆz. We expect basically the same scaling law
for the rings as for the triple-Z, due to the above arguments. However, the total interaction
rates are different (and higher, overall). This is plotted in Fig. 7-15.
7.3.2 Discussion
The above results are but a small sample of the interactions that may be created with
magnetic field gradients. Indeed, one appealing thing about this approach is the sheer
variety of forces that may be created. The source of this freedom is the fact that the
wires that create the gradients are separated from the source of the trapping potentials.
Thus, the tight integration of wires and trapping electrodes proposed in Ref. [CW08] is not
required. However, this greater flexibility in the global potentials does not come without a
cost; individually switching interactions between individual pairs of ions is not possible.
The methods presented in the section will apply the same state-dependent force to
every ion in a 2-D crystal in an elliptical trap. These translate into Ising or Heisenberg
Hamiltonians, depending on the experimenter’s choice. There remain two major questions:
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