Ph.D. Thesis - Physics

Figure 8-3: Equivalent circuit of two ions, each with inductance Li and capacitance Ci,
defined as in the text. They are coupled by a wire with ohmic resistance R and a capacitance
of C to ground. The currents Ii are determined by the velocity of each ion, according to
Eq. 8.8.
circuit model to be of great utility. In Fig. 8-3, we have drawn a schematic of the equivalent
circuit that describes two ions and the wire.
The primary types of decoherence we will treat here are dissipation (Joule heating of
the wire), Johnson noise, and anomalous heating.
8.3.1 Dissipation
To treat dissipation, we need to calculate the current in the wire due to the motion of the
ions. Again invoking Eq. 8.8, this current is given by
I = ec ˙zξ
H ≈ ecξ ω/m
. (8.16)
H
For the above parameter set, this current amounts to 0.1 fA, for a wire resistance of 0.6 Ω;
thus 2 ×10 5 s would be required to dissipate only one quantum of vibrational energy. Given
the very small current and the I 2 dependence of the heating law, this is not surprising.
Therefore, the dissipation of the induced currents will not pose a problem.
8.3.2 Electric field noise
Johnson noise and anomalous heating are two manifestations of the uncontrolled fluctuations
in electric field (electric field noise) that acts upon an ion. These processes can result in the
ion gaining kinetic energy in an uncontrolled way, and therefore decohering the motional
state of the ion.
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