Ph.D. Thesis - Physics

are in close agreement in the case of small coupling constants (γ/ mω 2 ≪ 1). A more
recent solution has shown that complete exchange of motional states can only occur on
resonance (ω1 = ω2) and for specific initial motional states [PRDB08].
The exchange rate, also known as the coupling rate ωex, is given by
ωex
2π
γ
= , (8.5)
πωm
where γ and α were defined above and ω = ω1 = ω2 (the resonant case). Incidentally, it is
also equal to the classical expression!
Referring to Eq. 8.3, the above formula for ωex allows us to evaluate the sensitivity of
the coupling rate to the various parameters of our model system. For instance, the length
of the wire and ion-wire distances enter as 1/ [L(H − h1)(H − h2)], a strong dependence,
while the dependence on the wire radius a is only logarithmic (it is contained in α). Overall,
we see that a shrinking of the entire system size leads to increased coupling, as the ions
induce more charge when closer to the wire; also, a shorter wire length leads to a higher
overall charge density on the wire. The inverse dependence on the secular frequency ω is an
expected feature of coupled harmonic oscillators; we have seen similar physics at work in
the lattice traps of Ch. 5. Physically, tighter confinement (higher ω) reduces the effective
“dipole moment” of each ion, which scales as 1/ √ ω.
8.2.2 Circuit model solution
The above model leads to a physical picture of the physics of our system. In this section, we
explore a different approach, which is based on a circuit model of the system, in which each
component is treated as a lumped element: an inductor, capacitor, or resistor. This can
have two key advantages: one is that the above physical model is based on assumptions that
are not strictly true, but are good approximations. To take an example, the capacitance
between the wire and ground will differ from that computed for an infinite ground plane,
especially since the length of the wire is not much less than the width of the trap in our
experiment. A circuit model allows one to quickly plug in more reasonable estimates of
this capacitance. The second, and more compelling, reason to use a circuit model is that it
makes the treatment of decoherence simpler. This is because the main decoherence sources
are electrical in nature; dissipation of currents in the wire and Johnson noise heating are
the two prime examples. This approach was presented in Refs. [WD75] and [HW90], and
we follow it here.
We begin with the equations of motion for the ions:
ec
m Ei = ¨zi + ω 2 i zi, (8.6)
where the electric field Ei is due to the voltage in the wire induced by the other ion. We
wish to write this field in terms of quantities in the above model, and the result is
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