Ph.D. Thesis - Physics

Figure 7-9: Calculation results for a simulated Ising model for two ion-qubits with a spatially
and temporally constant force. Left: The trajectory of the observable 〈Z〉 ≡ 〈Z1 + Z2〉 as a
function of time in the absence of micromotion is plotted on top. Below this, the trajectory
with a micromotion amplitude of Aµ = 0.1 is plotted. The difference between the two is
plotted on the bottom. Without micromotion, J = 10 3 s −1 . Right: The same plots as on
the left, but with the correct average value Jav = 1.0305 ×10 3 s −1 used for the Aµ = 0 case.
When this is done, the error drops by three orders of magnitude.
but will consist of the tensor product of two arbitrary single-qubit states. Entanglement
between the qubits will not form until after the application of the J-coupling, but the
set of arbitrary two-qubit states states subsumes this special case. We follow the above
approach, but with random, complex amplitudes for each basis state (normalized to 1). We
find that for random states, the error is reduced using the same average coupling as before
(Jav = 1.036 × 10 3 s −1 ). The average error during the course of each simulation, regardless
of the initial state, is 1.42×10 −5 .
The effects of a position-dependent force, i.e. F(x), may be considered by applying a
linear gradient at the position of each ion. The maximal gradient likely to occur in an
experiment is that due to an optical standing wave. We take, for the sake of argument,
400 nm to be the lower-wavelength limit of such a wave. In considering the most extreme
cases, we will also assume a maximum relative micromotion amplitude of Aµ = 0.1. The
reason for this is as follows. Suppose that two ions are on opposite sides of the rf null; then
their displacement from the rf null is equal to d/2, and the micromotion amplitude of each
is A = q q d
2∆x = 2
2 , yielding Aµ ≈ 0.08 for q = 0.3.
We proceed by computing the Jav for a set of force gradients, then confirming that
the error indeed vanishes. A set of gradients Fr from 0 to 1, relative to the micromotion
amplitude, as used. This means that for Fr = 0, the force is constant in space, while for
Fr = 1, the force falls to zero when the ion is located at the position x = A. As expected,
the uncorrected error due to micromotion rises as the force is allowed to vary in space, but
nevertheless these errors may be corrected by using Jav rather than J, and are quenched
by three orders of magnitude. Fig. 7-10 contains plots of Jav and the uncorrected error as
a function of the relative force gradient.
164