Ph.D. Thesis - Physics

Ei = V
2H
(H 2 − h 2 i ) ln 2H−a
a
. (8.7)
The details of this derivation are found in Ref. [DLC + 09b]. We now invoke a result of
Shockley [Sho38] to find that the charge induced in the wire by a single ion may be written
as
qind = − ec
α ln
H + h
. (8.8)
H − h
Noting that the current I in the wire is proportional to qind ˙z, the above result enables
us to write the mechanical equation of motion as an electrodynamic one involving currents
and voltages:
dI 1
Ui = Li +
dt Ci
Idt, (8.9)
where the effective inductance Li and capacitance Ci of a single trapped ion are given as
and
and the geometric factor ξi given by
ξi =
Li = 1
ξ 2 i
mH 2
e 2 c
(8.10)
Ci = 1
ω2 i Li
, (8.11)
2H 2
α(H 2 − h 2 i
) . (8.12)
These formulas enable us to answer a very interesting question: what are the effective
inductance and capacitance of a single trapped ion? The answer is that the inductance is
very large, while the capacitance is very small. Let us assume some feasible values for this
experiment: H = 200 µm, h1,2 = 150 µm, L = 10 mm, a = 12.5 µm, and ω/(2π) = 1 MHz.
For these values, L1,2 = 3.7 ×10 4 H and C1,2 = 6.9 ×10 −19 F. This explains why trapped
ions are such excellent resonators; the usual Q factor for an electrical circuit is given by
Q = L/R 2 Ci, and is on the order of 10 11 for our trapped ions.
The motional coupling rate ωex in terms of these quantities is given by
ωex = 1
, (8.13)
2ωLC
where C is the capacitance of the wire to ground, as detailed in Ref. [HW90]. We may now
calculate the expected coupling rate. Using the above parameters and a wire capacitance
of C = 2 × 10 −15 F, it is ωex = 10 3 s −1 .
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