Ph.D. Thesis - Physics

VCoup = e2 c
8πǫ0d 3x1x2,
(5.6)
where d is the mean ion-ion distance and x1 and x2 are, respectively, the displacements of
the first and second ion from their equilibrium positions. Lower-order terms result in either
a shift of the overall potential energy or a shift in the ions’ motional frequencies. Defining
p1 and p2 to be the momentum operators for ions 1 and 2 respectively, the Hamiltonian for
the motional states of the ions is
HV = 1
2m p2 1 + 1
2m p2 2 + 1
2 mω2 x 2 1 + 1
2 mω2 x 2 2 + mg 2 x1x2, (5.7)
where we assume that the ions have the same secular frequency ω and mass m, and have
defined the coupling constant g as g 2 = VCoup/ (mx1x2). This Hamiltonian represents a
coupled Harmonic oscillator; the rate ωex at which energy is exchanged between ions is
given by
ωex = g2
. (5.8)
ω
The factor of ω in the denominator has important implications for the coupling rate
in lattice ion traps. The physics of ion traps demands that for constant trap depth and
stability parameter q, the drive frequency must increase as the inverse of the trap scale.
In a lattice trap, d is directly proportional to the size of the trap, and may be considered
one measure of the trap scale. Therefore, since g 2 ∝ 1/d 3 , ωex ∝ 1/d 2 in a lattice-style ion
trap. In practice, this means that over reasonable length scales for d, ωex in a lattice trap
is much lower than it would be if the ions occupied the same trap region, for instance in a
linear ion trap. We plot this comparison in Fig. 5-15.
5.6.2 Simulated J-coupling rate
So far we have remarked only on the motional coupling rate ωex. A more relevant quantity
is the simulated coupling rate J for quantum simulation of spin models. We first give a brief
review of the spin model simulation scheme of Ref. [PC04b] described in Sec. 4.2.2. This
scheme uses a laser that exerts a state-dependent force on trapped ions that are coupled by
their Coulomb interaction. In the limit in which the Coulomb interaction is small compared
to the trapping potential (which is the case for lattice traps), the coupling rate between the
ions is given by
J =
e 2 cF 2
8πǫ0m 2 d 3 ω 4,
(5.9)
where F is the magnitude of the state-dependent force and the other symbols are as defined
above. For quantum simulation in a lattice trap, the frequency ω in Eq. 5.9 may be ωˆr
or ωˆz. For the sake of argument here, we assume ω = ωˆr. F is assumed to be due to a
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