Ph.D. Thesis - Physics

Results
Before presenting the numerical results, let us offer an intuitive view of the effect of micro-
motion on the evolution of the two-ion system. Micromotion is a driven oscillation at the
rf drive frequency Ω. This causes the two ions that are undergoing an effective spin-spin
interaction to have a time-dependent spacing d, as noted above. Since the J-coupling de-
pends upon d approximately as J ∝ 1/d 3 , the effective J-coupling rapidly oscillates during
a simulation. The ions, being driven by the rf, will spend as much time closer to each other
as they spend further apart. However, due to the 1/d 3 dependence of J, the time-averaged
interaction rate will be higher than it would be if the ions were stationary. We also expect
that since Ω ≫ J, the varying J due to micromotion may be treated as a value that is
time-averaged over many periods of 2π/J. Another way of stating this is that the only
term in the Hamiltonian which effects a spin-spin interaction is JZ1Z2, and therefore the
fastest interaction rate is set by J, even if J oscillates at a frequency Ω.
This picture is confirmed by our numerical simulations. As an example, we compute the
evolution of the initial state |↑↑〉 under the Hamiltonian written in Eq. 7.3. The parameters
for this calculation are J = −10 3 s −1 , Ω = 10 6 s −1 , and B = -J. The relative micromo-
tion amplitude was assumed to be Aµ = 0.1, a sensible value for many experiments. We
numerically integrate the equations of motion for the two-spin state both with and without
micromotion, and calculate the time-dependent error as the difference between the Aµ = 0
and Aµ = 0.1 time-dependent expectation values for the observable Z1 + Z2. We find that
over a time of several periods of J, the error quickly grows to a maximum value of 0.5, to
be compared to the maximum possible expectation value of 2. 2 This result is plotted in
Fig. 7-9.
This calculation permits us to calculate the time-averaged J-coupling constant if the ions
are undergoing micromotion; for the above parameters, this is equal to Jav = 1.036×10 3 s −1 .
As expected, this value is slightly higher than the J for Aµ = 0. When the simulation is
performed with Aµ = 0.1 and the calculated value for Jav, the error falls strikingly, by
about three orders of magnitude. This result is also presented in Fig. 7-9.
Although the intuitive picture of the effects of micromotion is confirmed, at least in
theory, there are important further questions. Generally, one may wish to prepare an
arbitrary state of the two ions, then apply the simulated interaction. In addition, position
dependence of the state-dependent force will also alter Jav; we must show that a similar
averaging technique works in this case as well. Further, the applied potentials may have
time-dependence, for example in the event that one wished to adiabatically apply a spin-spin
interaction in order to observe a phase transition.
We now treat each of these situations in turn, beginning with the operations on arbitrary
states. We note that in practice the two-qubit states used will not be truly arbitrary,
2 This scale is arbitrary; to convert into actual angular momentum values for spin-1/2 particles, multipli-
cation by /2 is required.
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