Ph.D. Thesis - Physics

Johnson noise
We begin our discussion of Johnson noise by writing down the Johnson noise heating power
PJ:
PJ = kBT∆ν, (8.17)
where, as usual, kB is Boltzmann’s constant, T is the temperature, and ∆ν is the frequency
bandwidth in which the ion accepts the power. The latter is inversely related to the Q-factor
of the ion’s motion, mentioned above. To calculate the time over which one quantum of
vibrational energy is absorbed from the wire by the ion, we use the formulas Ev = hν and
Q = ν/∆ν, and arrive at
τ −1 = PJ
Ev
= kBT∆ν
hν
kBT
= . (8.18)
hQ
An alternative, but equivalent expression for the Q parameter (c.f. Sec. 8.2.2) may be
derived from the dissipated power Pd = I 2 ℜ(Z), where I is the current due to a single ion
and ℜ(Z) is the real part of the impedance. We find that
to be
Q = Eion
Pd/ν = m ˙z2 ν
I2 mνH2
=
ℜ(Z) e2 cξ2 . (8.19)
ℜ(Z)
Putting it all together, we calculate the time constant for the absorption of one quantum
τ −1 = kBTe 2 cξ 2 ℜ(Z)
hνmH 2 . (8.20)
Taking ℜ(Z) = 0.6 Ω at T = 298 K, and using the same parameters given above, the
heating time due to Johnson noise is τ = 0.1 s/quantum. The corresponding rate 1/τ
is thus significantly smaller than the motional coupling rate (O(10 3 s)) calculated above.
However, the heating rates are not expected to be dominated by Johnson noise, especially
at room temperature.
Anomalous heating
Anomalous heating is a motional heating of ions with a poorly-understood origin. It scales
as roughly D −4 , where D is the distance from the equilibrium position of the ion to the
nearest trap electrode. This scaling law implies that as ion traps become smaller, the
noise level can quickly lead to too much decoherence of the ions’ motional state. Even
a very low heating rate in a room temperature trap, such as observed in Ref. [SCR + 06],
will add a quantum of energy to the ion’s motional state in, on average, 200 µs. However,
as we have reported elsewhere in this thesis, cryogenic cooling can greatly mitigate this
heating. Taking our estimate of tex = 1 ms, along with the best-case heating rate reported
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