Ph.D. Thesis - Physics

HTI = −
i,j=i+1
Ji,jZiZj +
BXi
i
(7.3)
using the approach outlined in Sec. 4.2. Noting that we make use of the vertical modes,
leading to βˆz ≪ 1, the coupling constant Jˆz along ˆz is given by
Jˆz = cˆze 2 cF 2 ˆz
4πǫ0m 2 ω 4 ˆz d3,
(7.4)
where ωˆz is the trap frequency along ˆz, Fˆz is the component of the state-dependent force
along ˆz, and d is the distance between the ions. In this situation, the constant cˆz = −2.
The state-dependent force also adds a magnetic field along the ˆz direction:
µˆαB ′ ˆz = µˆαBˆz + 4Fˆz
3mω2 , (7.5)
ˆz
The above expressions describe the simulated coupling between two two ions; this is ade-
quate for our purposes since in the β ≪ 1 limit, the interactions are approximately nearest-
neighbor. This effective magnetic field is in addition to the transverse magnetic field Bˆx
that appears (as just B) in Eq. 7.3. Note that within this section, we will now drop the
directional superscripts, i.e. F ≡ Fˆz.
With nonzero micromotion, the ion-ion spacing d is no longer constant in time. This
means that the normal mode frequencies ωn also change in time, and as a consequence, J
becomes time dependent. This dependence is apparent by writing
d(t) = d (1 + 2Aµ cos (Ωt + φ)) , (7.6)
where Ω and φ are the rf drive frequency and phase. The micromotion amplitude Aµ
given here is relative to the ion-ion distance d. That is, if A is the actual micromotion
amplitude, Aµ = A/d. The time dependence of J renders the equations of motion difficult
to analytically integrate; therefore, we use numerical methods exclusively.
To begin the simulations, we pick a constant amount of evolution time, equal to approx-
imately 10J, and pick an initial state. We propagate the system forward in time, adjusting
the Hamiltonian at each time step. This requires a time step τ that satisfies τ ≪ 2π/Ω.
We then measure some expectation value 〈M(t, Aµ)〉 at each time step, and compare these
values to that which is obtained in the absence of micromotion. The error between the two
is calculated simply as the difference between 〈M(t, A = 0)〉 and 〈M(t, A = 0)〉 Following
this, we increase the simulation time. Unless otherwise stated, we choose the initial state
|Ψ0〉 = |↑↑〉 (in the z basis) and measure M = Z1 + Z2. The Matlab codes to do the
simulations presented here are included in Appendix A.
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