The amplitude of the signal carrier (voltage, current, photons, etc.) must
exceed the amplitude of the noise which arises throughout the transfer
process: Signal / Noise $\gg 1$

In this section we focus on simultaneously satisfying both criteria. The first
criterion is considered in the particular context of $1/f^{\alpha}$ motional
heating. The second criterion is examined for the case of classical thermal
noise. Shot noise and the ($1/f^{\alpha}$) electronic flicker noise which
arises in the induced current during transmission of the quantum state are not
considered here, as it remains uncertain how to estimate shot noise and
($1/f^{\alpha}$) electronic flicker noise in a superconductor. We proceed with
the following three constraints, all of which must be satisfied to transfer
quantum information using a conducting wire.

  1. i)

$t_{\mathrm{deco.}}/t_{\mathrm{ex.}}\geq 10$

  2. ii)

$V_{\mathrm{sig.}}/V_{\mathrm{J.N.}}\geq 10$

  3. iii)

$\left(d\bar{n}/dt\right)\times t_{\mathrm{ex.}}\leq 1$

In point iii), $d\bar{n}/dt$ is the rate at which the average motional mode
number of the ion increases, in quantums per second. For ratio i), presently
the maximum achievable decoherence time $t_{\mathrm{deco.}}$ of motional mode
states is primarily limited by $1/f^{\alpha}$ heating noise. Thus, for the
trap and coupling system ensemble considered here, ratio i) and condition iii)
are related by a relationship between $t_{\mathrm{deco.}}$ and $d\bar{n}/dt$.
The raw motional state lifetime is $T_{1}=1/\left(d\bar{n}/dt\right)$, where
$T_{1}$ is the typical time it takes one quantum of motional energy to enter
the ion. For an ion in the ground state it is generally true that $T_{1}\geq
T^{*}_{2}=t_{\mathrm{deco.}}$, which means $1/\left(d\bar{n}/dt\right)\geq
t_{\mathrm{deco.}}$. Ratio i) can be rewritten as $t_{\mathrm{deco.}}\geq
10\times t_{\mathrm{ex.}}$. Combining these inequalities gives
$1/\left(d\bar{n}/dt\right)\geq t_{\mathrm{deco.}}\geq 10\times
t_{\mathrm{ex.}}$, or $1/\left(d\bar{n}/dt\right)\geq 10\times
t_{\mathrm{ex.}}$. Rearranging gives $\left(d\bar{n}/dt\right)\times
t_{\mathrm{ex.}}\leq 1/10$ , which shows that satisfying ratio i)
automatically places a more stringent bound on criterion iii), than criterion
iii) itself. Therefore, moving forward we will only be concerned with
satisfying ratio i), and assume that satisfying criterion iii) automatically
follows. We note that the above reasoning with inequalities no longer holds if
$T_{1}\leq T^{*}_{2}=t_{\mathrm{deco.}}$, which can occur for example when the
coherence of the overall motional state is embedded in many motional quanta.
This is the case in the experiment in Talukdar  _et al._ (2016), where the
phase coherence in a state with large displacements $\bar{n}\sim 50$ is
$3.9~{}(\pm 0.5)~{}$ms while $T_{1}=1~{}$ms.

###  IV.2 Finding an expression for the heating rate $dn/dt$ for traps
demonstrating a power-law behavior with increasing temperature

Since $t_{\mathrm{ex.}}$ is known from section II, finding an expression for
criterion i), $t_{\mathrm{deco.}}/t_{\mathrm{ex.}}\geq 10$, reduces to finding
an expression for the decoherence rate $t_{\mathrm{deco.}}$.
$t_{\mathrm{deco.}}$ is presently constrained by $1/f^{\alpha}$ heating, so to
estimate $t_{\mathrm{deco.}}$ one must express $t_{\mathrm{deco.}}$ in terms
of $d\bar{n}/dt$. Often (not always), $t_{\mathrm{deco.}}\leq
T_{1}=1/\left(d\bar{n}/dt\right)$. Here, we are concerned with general order-
of-magnitude estimations so we make the approximation
$t_{\mathrm{deco.}}\approx 1/\left(d\bar{n}/dt\right)$. This is not a bad
approximation; using again the result from Talukdar  _et al._ (2016), at room
temperature and for $f=880~{}$kHz and $d=50~{}\mu$m,
$t_{\mathrm{deco.}}=3.9~{}$ms, while $T_{1}=1~{}$ms. This is consistent with
other observations where the coherence time of a motional superposition state
is $t_{\mathrm{deco.}}=100~{}$ms, comparable to the motional heating time,
$T_{1}=70~{}$ms to $190~{}$ms per phonon in the same system Schmidt-Kaler  _et
al._ (2003). The trend is also supported by measurements in Lucas  _et al._
(2007). We further assume the relationship $t_{\mathrm{deco.}}\approx
1/\left(d\bar{n}/dt\right)$ remains valid at all temperatures. If this is
true, decreasing the heating rate $1/\left(d\bar{n}/dt\right)$ by going to
cryogenic temperatures can increase $t_{\mathrm{deco.}}$ by two orders of
magnitude compared to the room-temperature values given above (from reference
Talukdar  _et al._ (2016)), leading to $t_{\mathrm{deco.}}\sim 0.39~{}$s.
Additionally, the experimental results in reference Sedlacek  _et al._ (2018b)
for the heating rate at $T=4~{}$K, $d\bar{n}/dt\sim 3$ quanta per second,
allow us to estimate
$1/\left(d\bar{n}/dt\right)=T_{1}=t_{\mathrm{deco.}}=0.33~{}$s. Using the
assumption $t_{\mathrm{deco.}}\approx 1/\left(d\bar{n}/dt\right)$, criterion
i) above can be rewritten as
$t_{\mathrm{deco.}}/t_{\mathrm{ex.}}=1/\left(d\bar{n}/dt\times
t_{\mathrm{ex.}}\right)\geq 10$.

The mechanisms underlying $1/f^{\alpha}$ heating are not settled, but its
characteristics are sufficiently established to formulate a crude
phenomenological model that captures the dependence of $1/f^{\alpha}$ heating
on frequency $f$, distance between the ion and the surface $d$, and
temperature $T$, of the trap electrodes. Using Bayesian information criterion
(BIC) values, Sedlacek et al. Sedlacek  _et al._ (2018b) (2018) find that
before treatment with ex situ ion milling (ESIM), the temperature dependence
of the heating rate at $f=1.3~{}$MHz and $d=50~{}\mu$m in a surface trap with
sputtered niobium electrodes on a sapphire substrate is best described by a
power scaling,
$d\bar{n}/dt=B^{{}^{\prime}}\times\left[1+\left(T/T_{\mathrm{p}}\right)^{1.51~{}(\pm
0.04)}\right]$ where $B^{{}^{\prime}}$ is a proportionality constant equal to
a baseline heating rate $\left(d\bar{n}/dt\right)_{0}\equiv B^{{}^{\prime}}$
which persists at $T=0~{}$K, and $T_{\mathrm{p}}$ is the thermal activation
temperature of some currently unknown process. In a separate experiment on a
similar system (sputtered niobium electrodes on sapphire substrate), Sedlacek
et al. Sedlacek  _et al._ (2018a) show that with no ESIM treatment, at
$T=295~{}$K and $d=64~{}\mu$m the change in heating rate with changing trap
frequency appears to follow a power scaling $d\bar{n}/dt=C^{{}^{\prime}}\times
f^{-2.4~{}(\pm 0.2)}$ where $C^{{}^{\prime}}$ is a proportionality constant.
In this same study reference Sedlacek  _et al._ (2018a) (2018) finds that at
$f=850~{}$kHz and $T=295~{}$K the change in heating rate with changing
distance between the ion and the trap electrodes follows a power scaling
$d\bar{n}/dt=D^{{}^{\prime}}\times d^{-4.0~{}(\pm 0.2)}$ where
$D^{{}^{\prime}}$ is a proportionality constant. Although these power scalings
provide important insights, they are insufficient to propose a general
phenomenological expression of the form

$d\bar{n}/dt=\tilde{A}\times\bigg{(}\frac{1}{f^{\tilde{\alpha}}d^{\delta}}\times\left[1+\left(T/T_{\mathrm{p}}\right)^{\beta}\right]\bigg{)}~{},$
(53)

where $\tilde{A}\equiv B^{{}^{\prime}}C^{{}^{\prime}}D^{{}^{\prime}}$ is a
proportionality constant. Expression (53) is only justified if the temperature
exponent $\beta=1.5$ is independent of $f$ and $d$, the frequency exponent
$\tilde{\alpha}=2.4$ is independent of $T$ and $d$, and the distance exponent
$\delta=4.0$ is independent of $f$ and $T$. (To be completely general, $\beta$
must also be independent of $T$, $\tilde{\alpha}$ independent of $f$, and
$\delta$ independent of $d$. However, the empirical observations of power laws
would be difficult if this were not true.) These independences must be true
within the ranges of frequency, distance, and temperature of interest. We now
consider this point in more detail. For the following discussion the reader
may find it useful to refer to table 13 which contains a literature review
with measured values of the exponents $\beta$, $\tilde{\alpha}$, $\delta$ for
$T$, $f$, and $d$ respectively, assuming a power law behavior. Notably, traps
treated with ion milling (e.g. described in Hite  _et al._ (2012)) have been
shown to not follow a power-law scaling with temperature Sedlacek  _et al._
(2018b), though some evidence suggests they may still follow a power-law
scaling with frequency Hite  _et al._ (2012). Measurements on traps treated
with ion milling are not considered in table 13.

Table 5: A non-exhaustive list of measurements of power-law scaling of the heating rate $d\bar{n}/dt$ as a function of frequency, $\propto f^{-\tilde{\alpha}}$, distance, $\propto d^{-\delta}$, and temperature, $\propto\left[1+\left(T/T_{\mathrm{p}}\right)^{\beta}\right]$, for surface or needle traps not treated, or prior to treatment using ion milling bombardment with energetic ions (e.g. $\mathrm{Ar}^{+}$).131313For historical context, surface traps were first demonstrated around the year $2005$ Chiaverini  _et al._ (2005); Wineland  _et al._ (2005); Stick  _et al._ (2006). Measurements of scaling behavior prior to $2005$ were performed using non-planar traps. Columns give experimentally measured values of the relevant exponent. For each case, the values of the two variables which remained fixed during the measurement are indicated in parentheses. E.P. stands for Electro Plated, E.V. stands for Evaporative deposition, Spt. stands for sputtered, and saph. stands for sapphire. Where relevant, the letter ”p.” or ”n.” denotes whether the measurement is for a mode parallel or normal to the surface. A question mark ”?” indicates a value measured repeatedly in the study with wide variations. Reference |  $\tilde{\alpha}$ ($f^{-\tilde{\alpha}}$) |  $\delta$ ($d^{-\delta}$) |  $\beta$ ($T^{\beta}$) |  trap type |  ion
---|---|---|---|---|---
Da An et al. 2019, An  _et al._ (2019) |  $1.97$ & $2.2$ $(\pm 0.13,0.3)$ ($d=170~{}\mu$m & $d=70~{}\mu$m, $T=$ R.T.), p. |  $2.75~{}(\pm 0.09)$ ($f=1~{}$MHz, $T=$ R.T.), n. |  – |  Al-Cu on silica, planar |  ${}^{40}\mathrm{Ca}^{+}$
Sedlacek et al. 2018, Sedlacek  _et al._ (2018b) |  – |  – |  $1.51~{}(\pm 0.04)$ ($f=1.3~{}$MHz, $d=50~{}(\pm 1)~{}\mu$m), $T_{\mathrm{p}}\sim 10~{}$K (Au/Nb), p. |  E.P. Au and Spt. Nb on saph., planar |  ${}^{88}\mathrm{Sr}^{+}$
Sedlacek et al. 2018, Sedlacek  _et al._ (2018a) |  $2.4$ to $2.3~{}(\pm 0.2)$ ($d=64~{}\mu$m, $T=295~{}$K and $T=5~{}$K), p. |  $4.0$, $3.9~{}(\pm 0.2)$, ( $f=850~{}$kHz, $T=295~{}$K and $T=5~{}$K), p. |  – |  $2~{}\mu$m Spt. Nb on saph., planar |  ${}^{88}\mathrm{Sr}^{+}$
Boldin et al. 2018, Boldin  _et al._ (2018) |  $2.13~{}(\pm 0.10)$ ($d=134~{}\mu$m, $T=$ R.T.), p. |  $3.79~{}(\pm 0.12)$ ($f=196~{}$kHz, $T=$ R.T.), p. |  – |  E.P. Au on saph., planar |  ${}^{172}\mathrm{Yb}^{+}$
Talukdar et al. 2016, Talukdar  _et al._ (2016) |  $1.9~{}(\pm 0.2)$ ($d=50~{}\mu$m, $T=$ R.T.) |  – |  – |  Al-Cu ”same as Daniilidis  _et al._ (2014)”, planar |  ${}^{40}\mathrm{Ca}^{+}$
Bruzewicz et al. 2015, Bruzewicz  _et al._ (2015) |  $1.6~{}(\pm 0.1?)$ ($d=50~{}\mu$m, $T=4$ to $295~{}$K) |  – |  $1.59~{}(\pm 0.03)$ ($f=0.6$ to $1.4~{}$MHz, $d=50~{}\mu$m), $T_{\mathrm{p}}\sim 14-24~{}$K |  $2~{}\mu$m Spt. Nb on saph., planar |  ${}^{88}\mathrm{Sr}^{+}$
Daniilidis et al. 2014, Daniilidis  _et al._ (2014) |  Between $2.27$ & $2.50$ $(\pm 0.23)$ ($d=100~{}\mu$m, $T=4$ to $295~{}$K) |  – |  – |  E.V. Ti-Al-Cu on fused quartz, planar |  ${}^{40}\mathrm{Ca}^{+}$
Chiaverini & Sage 2014, Chiaverini and Sage (2014) |  – |  – |  $2.13~{}(\pm 0.05)~{}~{}>70~{}K$ and $0.54~{}(\pm 0.04)~{}~{}<70~{}K$ ($f=1.32~{}$MHz, $d=50~{}\mu$m), $\frac{d\bar{n}}{dt}\propto T^{\beta}$ |  2 $\mu$m Spt. Nb or $500~{}$nm Au on saph., planar |  ${}^{88}\mathrm{Sr}^{+}$
Hite et al. 2012, Hite  _et al._ (2012) |  $2.53~{}(\pm 0.07)$ ($d=40~{}\mu$m, $T=$R.T. ) |  – |  – |  $10~{}\mu$m E.P. Au on crystalline quartz, planar |  ${}^{9}\mathrm{Be}^{+}$
Labaziewicz et al. 2008, Labaziewicz  _et al._ (2008) |  Between $1.7$ & $2.0$ ($\pm 0.3$) ($d=75~{}\mu$m, $T=10$ to $90~{}$K) |  – |  Temperature dependence measured, but scaled freq. w/ $1/f^{2}$ |  Ag-Ti on crystal quartz, E.P. Au sol’n, planar |  ${}^{88}\mathrm{Sr}^{+}$
Epstein et al. 2007, Epstein  _et al._ (2007) |  $2.4~{}(\pm 0.4)$ ($d=40~{}\mu$m, $T=$ R.T.) |  – |  – |  Au on fused quartz, planar |  ${}^{25}\mathrm{Mg}^{+}$
Deslauriers et al. 2006, Deslauriers  _et al._ (2006) |  $1.8~{}(\pm 0.2)$ ($d=103~{}\mu$m, $T=300~{}$K) |  $3.5~{}(\pm 0.1)$ ($f=2.07~{}$MHz, $T=300~{}$K) |  Observed $d\bar{n}/dt$ decrease btwn $T=300~{}$K and $T=150\pm 20~{}$K |  W (tungsten), needle |  ${}^{111}\mathrm{Cd}^{+}$

In table 13, reference Sedlacek  _et al._ (2018a) finds that the frequency
dependence of the heating rate is essentially identical at two different
temperatures, $\tilde{\alpha}=2.4~{}(\pm 0.2)$ at $T=295~{}$K and
$\tilde{\alpha}=2.3~{}(\pm 0.2)$ at $T=5~{}$K ($d=64~{}\mu$m). From this we
guess that the frequency exponent $\tilde{\alpha}$ is the same at all
temperatures in between, and therefore that it is independent of temperature.
We note this temperature-independence is somewhat different from some results
which give $\tilde{\alpha}\sim 1.7$ at temperatures below $75~{}$K Bruzewicz
_et al._ (2015); Labaziewicz  _et al._ (2008), although the room temperature
value in the range of $\sim 1.97~{}(\pm 0.13)$ ($d=170~{}\mu$m) and
$2.2~{}(\pm 0.3)$ ($d=70~{}\mu$m) An  _et al._ (2019) to $2.4~{}(\pm 0.4)$
Epstein  _et al._ (2007) is consistent with other results. Reference Sedlacek
_et al._ (2018a) also observes the distance dependence at two different
temperatures and finds $\delta=4.0~{}(\pm 0.2)$ at $T=295~{}$K and
$\delta=3.9~{}(\pm 0.2)$ at $T=5~{}$K ($f=850~{}$kHz), from which they
conclude that the distance exponent $\delta$ is independent of temperature. In
addition to these direct findings, we propose that the distance exponent
$\delta$ either depends weakly on the frequency $f$, or does not depend on $f$
at all. This is supported by the fact that in a surface trap, reference
Sedlacek  _et al._ (2018a) finds a value $\delta=4.0~{}(\pm 0.2)$ at a
frequency of $f=850~{}$kHz (at $T=295~{}$K), while different experiments with
various ion frequencies (most not on surface traps) show similar values
$\delta\sim 4$, (at $T\sim$ R.T.) Epstein  _et al._ (2007), and reference
Deslauriers  _et al._ (2006) finds a similar value $\delta=3.5~{}(\pm 0.1)$ in
a dedicated distance-dependence experiment, at a frequency $f=2.07~{}$MHz (at
$T=300~{}$K). The value $\delta=3.5~{}(\pm 0.1)$ observed in reference
Deslauriers  _et al._ (2006) differs slightly from $\delta\sim 4$, which could
be related to characteristics of the double-needle trap used in that
experiment. Also, we propose that the frequency exponent $\tilde{\alpha}$ is
either weakly dependent or independent of the distance $d$. This is based on
the measurements of $\tilde{\alpha}=2.4~{}(\pm 0.2)$ at a distance
$d=64~{}\mu$m (at $295~{}$K) Sedlacek  _et al._ (2018a),
$\tilde{\alpha}=2.2~{}(\pm 0.3)$ at a distance $d=70~{}\mu$m An  _et al._
(2019) (at R.T.), and $\tilde{\alpha}=1.97~{}(\pm 0.13)$ at a distance
$d=170~{}\mu$m (at R.T.) An  _et al._ (2019). Turning to the temperature-
dependence exponent, $\beta$ appears independent of $f$ based on a study where
$\beta$ was measured five times at different frequencies. This study gives
values from $\beta=1.5$ to $1.65$ with no clear pattern for $5$ frequencies
from $0.6~{}$MHz to $1.4~{}$MHz, giving a weighted average value of
$\beta=1.59~{}(\pm 0.03)$ Bruzewicz  _et al._ (2015). Finally, to justify the
use of equation (53), $\beta$ must also be independent of $d$. Unfortunately,
we are unaware of any studies where $\beta$ has been measured for multiple
different distances $d$, so we make the experimentally unsupported assumption
that the temperature scaling exponent $\beta$ does not depend on the distance
$d$ between the ion and the trap electrodes. This seems theoretically
plausible; the electric field noise arises from processes within the trap
electrodes, and it is reasonable to imagine that these processes are not
activated by the ion. Thus, the distance at which the ion is positioned should
not dramatically impact the electric field noise, and hence should not affect
how the heating rate of the ion changes with the temperature of the trap
electrodes.

We have now given motivations for why $\tilde{\alpha}$ is independent of $f$,
$T$, $d$, why $\delta$ is independent of $d$, $T$, $f$, and why $\beta$ is
independent of $T$, $f$, $d$. Although results should be interpreted with
appropriate caution, we believe for the case of un-milled trap electrodes and
using the systems in Sedlacek  _et al._ (2018a) and Sedlacek  _et al._
(2018b), it is justified to model the heating rate as

$d\bar{n}/dt=\tilde{A}\times\bigg{(}\frac{1}{f^{\tilde{\alpha}}d^{\delta}}\times\left[1+\left(T/T_{\mathrm{p}}\right)^{\beta}\right]\bigg{)}$
(54)

For our purposes we will use equation (54) with the explicit values
$\tilde{\alpha}=2.4$, $\delta=4.0$, and $\beta=1.51$. These values are chosen
because all three were measured in two studies using essentially the same
experimental system in references Sedlacek  _et al._ (2018a) and Sedlacek  _et
al._ (2018b). Assuming $\beta$, $\tilde{\alpha}$, and $\delta$ were the same
in both studies, we can use these studies to estimate the proportionality
constant $\tilde{A}$. Using a single data point in Sedlacek  _et al._ (2018a)
or Sedlacek  _et al._ (2018b) estimated from the heating rate $d\bar{n}/dt$ at
a given frequency $f$, distance $d$, and temperature $T$, we calculate
$\tilde{A}$ as

$\tilde{A}=\frac{f^{\tilde{\alpha}}d^{\delta}}{1+\left(T/T_{\mathrm{p}}\right)^{\beta}}\times
d\bar{n}/dt~{},$ (55)

which has units of
$\left(\mathrm{Hz}\right)^{\tilde{\alpha}}\cdot\left(\mathrm{m}\right)^{\delta}\cdot\mathrm{quanta}\cdot\mathrm{s}^{-1}$.
Extracted values of $\tilde{A}$ are given in table 7 in appendix I. The
overall mean value of $\tilde{A}$ calculated using only values for niobium
(Nb) traps, is $\tilde{A}=0.012~{}\pm 0.003$. With this value, equation (54)
characterizes the heating rate of an un-milled Nb trap such as used in
Sedlacek  _et al._ (2018a) and Sedlacek  _et al._ (2018b), for the frequency
range $f$ from $470~{}$kHz to $1.2~{}$MHz, the distance range $d$ from
$29~{}\mu$m to $83~{}\mu$m, and the temperature range $T$ from $\sim 3~{}$K to
$300~{}$K.

Next, we use a theoretical result for the transition rate of an ion between
motional states to write the constant $\tilde{A}$ in equation (54) more
explicitly. This allows us to express $\tilde{A}$ as a function of the charge
$q_{\mathrm{c}}$ and mass $m$ of the ion. Since both $q_{\mathrm{c}}$ and $m$
appear in the expression for the time required for two coupled ions to
exchange states, $t_{\mathrm{ex.}}$, the dependence of $\tilde{A}$ on
$q_{\mathrm{c}}$ and $m$ must be included to capture the full dependence of
the ratio $t_{\mathrm{deco.}}/t_{\mathrm{ex.}}$ on $q_{\mathrm{c}}$ and $m$.

An ion in a trap may be subjected to an electric field perturbation which can
be described by a time-dependent Hamiltonian $H^{\prime}\left(t\right)$. Using
first-order time-dependent perturbation theory and without verifying the
result, for an atom in the motional state $|n\rangle$ at time $t=0$, the
average rate of transition to the state $|m\neq n\rangle$ in a time interval
$T_{h}$ is

$\Gamma_{m\leftarrow
n}\equiv\frac{1}{T_{h}}\left|\frac{-i}{\hbar}\int_{t^{\prime}=0}^{t^{\prime}=T_{h}}dt^{\prime}H^{\prime}_{mn}\left(t^{\prime}\right)e^{i\omega_{mn}t^{\prime}}\right|^{2}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$=\frac{1}{\hbar^{2}}\int_{\tau=-\infty}^{\tau=\infty}d\tau
e^{i\omega_{m}\tau}\langle\epsilon\left(t\right)\epsilon\left(t+\tau\right)\rangle\left|\langle
0|q_{\mathrm{c}}x|1\rangle\right|^{2}\equiv\Gamma_{0\rightarrow 1}~{}.$

To quote reference Savard  _et al._ (1997): ”…we have assumed that the
averaging time $T_{h}$ is short compared to the time scale over which the
level populations vary, but large compared to the correlation time of the
fluctuations so that the range of $\tau$ extends formally to $\pm\infty$”. In
addition, $\omega_{m}$ is the angular frequency of the ion in the harmonic
trap potential and $\hbar$ is the reduced Plank constant. The second line
introduces an explicit expression for the perturbation,
$H^{\prime}\left(t\right)=-q_{\mathrm{c}}\epsilon\left(t\right)x$, where
$q_{\mathrm{c}}$ is the charge subjected to a time-dependent perturbative
uniform electric field $\epsilon\left(t\right)$, and $x$ is a displacement
within that field. This Hamiltonian is proposed in Turchette  _et al._ (2000)
for a trapped ion. The second line is also for a specific transition from the
ground state $|0\rangle$ to the first excited state $|1\rangle$ of a quantum
harmonic oscillator. Evaluating the motional matrix element $\left|\langle
0|q_{\mathrm{c}}x|1\rangle\right|^{2}$ by expressing $x$ in terms of creation
and annihilation operators gives

$\Gamma_{0\rightarrow
1}\equiv\frac{q^{2}_{\mathrm{c}}}{4m\hbar\omega_{m}}S_{E}\left(\omega_{m}\right)~{},$
(56)

where $S_{E}\left(\omega\right)\equiv 2\int_{-\infty}^{\infty}d\tau
e^{i\omega\tau}\langle\epsilon\left(t\right)\epsilon\left(t+\tau\right)\rangle$
is defined as the spectral density of electric-field fluctuations. This result
is often quoted in the literature Daniilidis  _et al._ (2014); Bruzewicz  _et
al._ (2015); Brownnutt  _et al._ (2015); Sedlacek  _et al._ (2018b); An  _et
al._ (2019). $\Gamma_{0\rightarrow 1}$ is equal to $d\bar{n}/dt$ for the ion
in its ground state. If the experimentally observed heating in equation (54)
is accurately described by equation (56), then one can draw an association
between the two equations by defining $\tilde{A}\equiv
q^{2}_{\mathrm{c}}A/\left(4mh\right)$, where $A$ is a constant, and factoring
out a factor of $1/f$ from equation (54). Looking at equations (54) and (56),
we can now define the spectral density $S_{E}$ of electric-field fluctuations
as

$S_{E}\equiv\frac{A}{f^{\alpha}d^{\delta}}\times\left[1+\left(T/T_{\mathrm{p}}\right)^{\beta}\right],$
(57)

where $\alpha\equiv\tilde{\alpha}-1$ is often the value reported in the
literature. Combining the empirically motivated expression in (54) with the
theoretical result in (56) gives a more explicit expression for the heating
rate

$d\bar{n}/dt=A\times\frac{q^{2}_{\mathrm{c}}}{4mh}\times\bigg{(}\frac{1}{f^{\tilde{\alpha}}d^{\delta}}\times\left[1+\left(T/T_{\mathrm{p}}\right)^{\beta}\right]\bigg{)}.$
(58)

To summarize the various parameters, $d\bar{n}/dt$ is the heating rate of the
ion in quanta per second, $A$ is a constant, $q_{\mathrm{c}}$ is the charge of
the ion, $m$ is the mass of the ion, $h$ is Plank’s constant, $f$ is the
secular frequency at which the ion oscillates, $d$ is the distance between the
ion and the planar trap surface, $T$ is the temperature of the trap
electrodes, $T_{\mathrm{p}}$ is the thermal activation temperature of some
currently unknown process, and $\tilde{\alpha}$, $\delta$, $\beta$ are
exponents which are fairly or fully independent of the other parameters. Based
on the parameter ranges explored in Sedlacek  _et al._ (2018a, b), expression
(58) is expected to be valid for $5~{}\mathrm{K}<T<295~{}$K,
$0.5~{}\mathrm{MHz}<f<1~{}$MHz, and $30~{}\mu\mathrm{m}<d<100~{}\mu$m. The
constant $A$ can be calculated from the measured mean value of
$\tilde{A}=0.012$, the charge $q_{\mathrm{c}}$ of the ion, and the mass $m$ of
the ion as $A\equiv\tilde{A}\times 4mh/q^{2}_{\mathrm{c}}=1.8\times 10^{-22}$
$\mathrm{kg}^{2}\cdot\mathrm{s}^{-2}\cdot\mathrm{C}^{-2}\cdot\left(\mathrm{Hz}\right)^{\tilde{\alpha}}\cdot\mathrm{m}^{\delta+2}\cdot\mathrm{quanta}$.
Finally, since equation (58) is based on equation (56), it is only strictly
valid for describing the heating rate of an ion in the motional ground state.

It is tempting to devise a similar expression to equation (58) for the case
where the temperature dependence follows an Arrhenius behavior, as observed in
Sedlacek  _et al._ (2018b) and Labaziewicz  _et al._ (2008). However, we
hypothesize that most of the experimental evidence gathered to date is for
traps where the heating rate is not governed by an Arrhenius-type process. For
experiments measuring only $\tilde{\alpha}$ and/or $\delta$ but not $\beta$,
which is to say experiments performed at a constant temperature, we don’t know
for sure whether the trap would have manifested Arrhenius behavior if it were
subjected to different temperatures. Since a power-law temperature dependence
is more common than an Arrhenius dependence when trap electrodes do not
receive special treatment Labaziewicz  _et al._ (2008); Chiaverini and Sage
(2014); Bruzewicz  _et al._ (2015); Sedlacek  _et al._ (2018b), we assume it
as the default. If the heating rate is governed by an Arrhenius behavior, this
indicates a regime where heating could be driven by different mechanisms. It
cannot be taken for granted that key parameters such as $\tilde{\alpha}$,
$\delta$, and $\beta$ (or the Arrhenius-regime equivalent) are independent of
$f$, $d$, and $T$. Presently, there is a paucity of experiments to indicate
such independence. Until a broader range of experiments are performed on
systems where an Arrhenius behavior is explicitly demonstrated, it seems
premature to postulate an empirically-motivated analytical model for heating
rates in Arrhenius-type systems.

To conclude this section, it would be valuable for our understanding of
heating in both power-law type systems and Arrhenius-type systems, to perform
experiments which thoroughly explore planes in the two-dimensional phase space
of ”$\left(d\bar{n}/dt\right)$ as a function of $f$ and $d$”. This would
confirm that $\alpha$ is independent of distance and $\delta$ is independent
of frequency. Reference Bruzewicz  _et al._ (2015) (2015) maps out a full
plane of $dn/dt$ as a function of frequency and temperature. Exploring the
third dimension of the phase space by mapping out full planes of the heating
rate $\left(d\bar{n}/dt\right)$ as a function of $f$ and $d$ for various
temperatures $T$, would also be of great benefit. To our knowledge, a
systematic experimental phase diagram for $\left(d\bar{n}/dt\right)$ as a
function of frequency $f$, distance $d$, and temperature $T$ does not exist.

###  IV.3 Simultaneously satisfying criteria i) and ii) from section IV.1 for
traps demonstrating a power-law behavior with increasing temperature

In section IV.2 we made the approximation $t_{\mathrm{deco.}}\approx
1/\left(d\bar{n}/dt\right)$, and found an explicit expression for
$d\bar{n}/dt$, equation (58). Therefore, we have

$t_{\mathrm{deco.}}\approx\frac{4mhf^{\tilde{\alpha}}d^{\delta}}{Aq^{2}_{\mathrm{c}}\times\left[1+\left(T/T_{\mathrm{p}}\right)^{\beta}\right]}~{}.$
(59)

The expression for the exchange time is $t_{\mathrm{ex.}}=\pi\omega m/\gamma$
Portes  _et al._ (2008)141414This is similar to the expression for a single
Rabi oscillation in a two-level system (see footnote in section II.5)., and we
found an explicit expression for $\gamma$ in section II.4, equation (29). This
led to an explicit expression for $t_{\mathrm{ex.}}$, in section II.5. We are
now ready to put together the expressions for $t_{\mathrm{deco.}}$ and
$t_{\mathrm{ex.}}$ into a single expression for
$t_{\mathrm{deco.}}/t_{\mathrm{ex.}}$. Expressing the capacitances of the
disks and conducting wire in terms of their dimensions,
$C_{a}=8\epsilon_{\mathrm{o}}r$ and
$C_{b}=2\pi\epsilon_{\mathrm{o}}l_{\mathrm{w}}\big{/}\mathrm{ln}\left(l_{\mathrm{w}}/a\right)$,
we find:

$t_{\mathrm{deco.}}/t_{\mathrm{ex.}}=\frac{4mhf^{\tilde{\alpha}}d^{\delta}}{Aq^{2}_{\mathrm{c}}\times\left[1+\left(T/T_{\mathrm{p}}\right)^{\beta}\right]}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\times\frac{1}{\pi\omega
m}\left[\frac{q^{2}_{\mathrm{c}}}{4\pi\epsilon_{\mathrm{o}}}\left(\frac{r}{2r+\frac{2\pi
l_{\mathrm{w}}}{8\mathrm{ln}\left(l_{\mathrm{w}}/a\right)}}\right)\left(\frac{dr^{2}}{\left(d^{2}+r^{2}\right)^{3}}\right)\right]~{}.$

This simplifies to

$t_{\mathrm{deco.}}/t_{\mathrm{ex.}}=\frac{hf^{\tilde{\alpha}-1}d^{\delta+1}}{2A\pi^{3}\epsilon_{\mathrm{o}}\times\left[1+\left(T/T_{\mathrm{p}}\right)^{\beta}\right]}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\times\left[\left(\frac{r}{2r+\frac{2\pi
l_{\mathrm{w}}}{8\mathrm{ln}\left(l_{\mathrm{w}}/a\right)}}\right)\left(\frac{r^{2}}{\left(d^{2}+r^{2}\right)^{3}}\right)\right]~{}.$
(60)

Equation (60) can be used to predict whether or not criterion i) from section
IV.1 is satisfied. It is independent of the mass $m$ and the charge
$q_{\mathrm{c}}$ of the particle.

In equation (60), the exponents $\tilde{\alpha}$, $\delta$, and $\beta$ cannot
presently be controlled, but they can be measured experimentally in a
dedicated system. Similarly, although the process by which trap electrodes
generate radiation is not well understood, for a power-law behavior its
activation temperature $T_{\mathrm{p}}$ can be measured. These parameters are
externally imposed. In addition, equation (60) depends on five factors which
can be controlled experimentally: 1) the frequency $f$ at which the ion
oscillates in the trap, 2) the distance $d$ between the ion and the trap
surface (which determines the optimal disk radius $r$), 3) the temperature $T$
of the trap electrodes, 4) the length $l_{\mathrm{w}}$ of the coupling wire,
and 5) the radius $a$ of the coupling wire. As we are unaware of any
experiments measuring $\tilde{\alpha}\leq 1$ (for general ranges refer to
table 13), it is expected that $t_{\mathrm{deco.}}/t_{\mathrm{ex.}}$ increases
with increasing frequency $f$. Additionally, $\beta>0$ so
$t_{\mathrm{deco.}}/t_{\mathrm{ex.}}$ increases if the trap electrodes are
cooled. If they are cooled to $T=T_{\mathrm{p}}$ the dependence on $\beta$
vanishes, and if $T\ll T_{\mathrm{p}}$ the temperature dependence vanishes
entirely.

We can roughly understand how $t_{\mathrm{deco.}}/t_{\mathrm{ex.}}$ scales
with distance by letting $r=d/\sqrt{2}$, which maximizes the coupling strength
$\gamma$ and hence minimizes $t_{\mathrm{ex.}}$ in the limiting case
$2r\gg\frac{C_{b}}{8\epsilon_{\mathrm{o}}}$, or $C_{a}=C_{c}$ and $C_{b}=0$
(see section II.4). With this, equation (60) becomes

$t_{\mathrm{deco.}}/t_{\mathrm{ex.}}=\frac{h}{27A\pi^{3}\epsilon_{\mathrm{o}}}\times\frac{f^{\tilde{\alpha}-1}d^{\delta-3}}{\left[1+\left(T/T_{\mathrm{p}}\right)^{\beta}\right]}~{}.$
(61)

From equation (61), the dependence on the distance between the ion and the
trap electrodes is not clear. If $\delta>3$, then
$t_{\mathrm{deco.}}/t_{\mathrm{ex.}}$ increases with increasing distance $d$.
This is expected for the majority of the systems in table 13 and in particular
for Sedlacek  _et al._ (2018a) and Sedlacek  _et al._ (2018b), where
$\delta=4$, based on which equations (60) and (61) are derived. However, if
$\delta<3$ as observed in reference An  _et al._ (2019), where $\delta=2.75$,
the ratio $t_{\mathrm{deco.}}/t_{\mathrm{ex.}}$ decreases with increasing
distance $d$. At this point it should be stated that the approximation
$C_{a}=C_{c}\gg C_{b}=0$ is not usually valid, as the wire capacitance $C_{b}$
is typically larger than the total capacitance of the two disks. For a
coupling wire of length $l_{\mathrm{w}}=0.01~{}$m and radius $a=10~{}\mu$m,
for $d=200~{}\mu$m we have $r_{\mathrm{opt.}}\sim d/\sqrt{2}=140~{}\mu$m,
which gives $2\times C_{a}=2\times 8\epsilon_{\mathrm{o}}r=20~{}$femto-Farads
$<C_{b}=2\pi\epsilon_{\mathrm{o}}l_{\mathrm{w}}\big{/}\mathrm{ln}\left(l_{\mathrm{w}}/a\right)=80~{}$femto-
Farads. Equation (61) gives a simplified picture for how
$t_{\mathrm{deco.}}/t_{\mathrm{ex.}}$ responds to changes in $d$ in one
limiting case. In the opposite, more realistic limit where the wire
capacitance is much greater than the disk capacitance, looking at equation
(60) shows that the numerator gains a factor of $r$ (when $C_{b}\gg
C_{a}=C_{c}$, the factor of $2r$ in the denominator becomes negligible
compared to $C_{b}$). The extra factor of $r$ in the numerator corresponds
roughly to a factor of $d$, which means the exponent in the distance scaling
is increased by $1$ compared with equation (61), such that
$t_{\mathrm{deco.}}/t_{\mathrm{ex.}}\propto d^{\delta-2}$. In this limit, even
for the small value $\delta=2.75$ measured in reference An  _et al._ (2019),
the exponent of $d$ is positive and the ratio
$t_{\mathrm{deco.}}/t_{\mathrm{ex.}}$ increases with increasing distance $d$.

We now have an expression for criterion i) in section IV.1. We can also write
an explicit expression for criterion ii), for the ratio of the induced voltage
signal to the Johnson-Nyquist noise voltage,
$V_{\mathrm{sig.}}/V_{\mathrm{J.N.}}$. Using the expressions for
$V_{\mathrm{sig.}}$ and $V_{\mathrm{J.N.}}$ from section III gives

$V_{\mathrm{sig.}}/V_{\mathrm{J.N.}}=\left(\frac{\sigma\pi
a^{2}}{4k_{\mathrm{B}}T\Delta
fl_{\mathrm{w}}}\right)^{1/2}\times\left(\frac{2}{2\times
8\epsilon_{\mathrm{o}}r+\frac{2\pi\epsilon_{\mathrm{o}}l_{\mathrm{w}}}{\mathrm{ln}\left(l_{\mathrm{w}}/a\right)}}\right)$
$\qquad\qquad\qquad\times~{}\frac{2q_{\mathrm{c}}r^{2}}{(r^{2}+d^{2}_{\mathrm{eq.}})^{3/2}}\left(\frac{h}{8\pi^{2}mf}\right)^{1/2}~{}.$
(62)

This can be made more general by writing the resistance of a normal conducting
wire, $\left(l_{\mathrm{w}}/\sigma\pi a^{2}\right)$, simply as $R$.

$V_{\mathrm{sig.}}/V_{\mathrm{J.N.}}=\left(\frac{1}{4k_{\mathrm{B}}T\Delta
fR}\right)^{1/2}\times\left(\frac{2}{2\times
8\epsilon_{\mathrm{o}}r+\frac{2\pi\epsilon_{\mathrm{o}}l_{\mathrm{w}}}{\mathrm{ln}\left(l_{\mathrm{w}}/a\right)}}\right)$
$\qquad\qquad\qquad\times~{}\frac{2q_{\mathrm{c}}r^{2}}{(r^{2}+d^{2}_{\mathrm{eq.}})^{3/2}}\left(\frac{h}{8\pi^{2}mf}\right)^{1/2}~{}.$
(63)

Equation (63) can be used to predict whether or not criterion ii) from section
IV.1 is satisfied for a normal conducting wire or a superconducting wire. It
depends on both the mass $m$ and the charge $q_{\mathrm{c}}$ of the particle.

The aim is to satisfy both criteria i) and ii) simultaneously. Equation (60)
depends on fewer parameters than equation (63), which gives it less
flexibility for adjustment. Therefore, the modus operandi is to first ensure
i) is satisfied, and then if necessary, to use the extra variables in equation
(63) to ensure that ii) is also satisfied. We first examine how equation (60)
depends on the radius $a$ of the coupling wire. Equation (60) can be plotted
by plugging in values for $f$, $d$, $T$, $l_{\mathrm{w}}$, and letting the
radius of the coupling wire $a$ vary from $\geq 20~{}$nm to $0.1\times$ the
radius of the pickup disks, for example $\sim 10~{}\mu$m for $d=100~{}\mu$m.
The lower bound on $a$ is based on the minimum wire width at which
superconductivity persists, while the upper bound on $a$ ensures the validity
of the model of two disks connected by a wire. The goal is for
$t_{\mathrm{deco.}}/t_{\mathrm{ex.}}$ to be as large as possible, so we start
by choosing the optimal values of $f$, $d$, $T$ within the range where
equation (60) is expected to be valid. The maximum frequency is $1~{}$MHz, and
a reasonable minimum temperature is $10~{}$K. We assume $\tilde{\alpha}=2.4$,
$\beta=1.51$, and $\delta=4.0$, which we know to be a consistent combination
for a Niobium trap from Sedlacek  _et al._ (2018a, b). Figure 7($a$) shows a
plot of $t_{\mathrm{deco.}}/t_{\mathrm{ex.}}$ for three different distances
between the ion and the surface, $d=30~{}\mu$m, $d=50~{}\mu$m, $d=100~{}\mu$m.

Figure 7: Ratio $t_{\mathrm{deco.}}/t_{\mathrm{ex.}}$ as a function of wire
radius, for various parameter values. ($a$) shows a plot of
$t_{\mathrm{deco.}}/t_{\mathrm{ex.}}$ as a function of the coupling wire’s
radius $a$ for a set of parameter values for which equation (60) should be
valid, based on experimental observations in Sedlacek  _et al._ (2018a, b).
Even for the best combination, $d=100~{}\mu$m and $f=1~{}$MHz, the criterion
$t_{\mathrm{deco.}}/t_{\mathrm{ex.}}>10$ is not satisfied. ($b$) shows a
similar plot made by assuming equation (60) continues to be valid up to
$d=200~{}\mu$m and $f=5~{}$MHz (extrapolating). For these values, the
criterion $t_{\mathrm{deco.}}/t_{\mathrm{ex.}}>10$ is satisfied. ($c$) shows
that changes in the exponents $\tilde{\alpha}$ and $\delta$ can have a
dramatic effect on the ratio $t_{\mathrm{deco.}}/t_{\mathrm{ex.}}$. This is
illustrated using the least favorable values of $\tilde{\alpha}$ and $\delta$
listed in table 13. All panels are plotted for $\tilde{A}=0.012$ (or
$A=1.8\times 10^{-22}$), $T=10~{}$K, $T_{\mathrm{p}}=10~{}$K and a wire length
$l_{\mathrm{w}}=0.01~{}$m.

The dependence on $a$ is weak within the given range. Additionally, for all
radii $a$ the ratio $t_{\mathrm{deco.}}/t_{\mathrm{ex.}}$ is less than $10$.

To achieve $t_{\mathrm{deco.}}/t_{\mathrm{ex.}}>10$ for $\tilde{\alpha}=2.4$,
$\beta=1.51$, and $\delta=4.0$, we are obligated to let the frequency $f$ and
distance $d$ exceed the known range of validity of equation (60). We assume
the scaling for $d\bar{n}/dt$ as a function of $f$ and $d$ remains the same,
and extrapolate equation (60) to higher frequencies and larger distances.
Figure 7($b$) gives a sense of how far $f$ and $d$ must exceed known limits to
reach $t_{\mathrm{deco.}}/t_{\mathrm{ex.}}>10$ by plotting equation (60) as a
function of $a$ for $f=5~{}$MHz and $d=50~{}\mu$m, $d=100~{}\mu$m,
$d=200~{}\mu$m. The upper two lines in figure 7($b$) suggest that for
$\tilde{\alpha}=2.4$, $\beta=1.51$, and $\delta=4.0$, at $T=10~{}$K, and for
an ion trapped $100~{}\mu$m or $200~{}\mu$m above a Nb surface, criterion i)
is satisfied. This result is extremely sensitive to changes in
$\tilde{\alpha}$ and $\delta$. We can look at what happens for the least
favorable reported values of $\tilde{\alpha}$ and $\delta$ in table 13. The
least favorable value of $\tilde{\alpha}$ is $\tilde{\alpha}=1.5$ (including
the experimental uncertainty of $0.1$). The least favorable value of $\delta$
is $\delta=4.2$. With these, plotting equation (60) as a function of $a$ for
$f=5~{}$MHz and $d=50~{}\mu$m, $100~{}\mu$m, $200~{}\mu$m gives a ratio
$t_{\mathrm{deco.}}/t_{\mathrm{ex.}}$ on the order of $1\times 10^{-5}$
(figure 7($c$)). Achieving $t_{\mathrm{deco.}}/t_{\mathrm{ex.}}\sim 1$ for the
combination $\tilde{\alpha}=1.5$, $\delta=4.2$, $\beta=1.51$ requires both
high frequencies and large distances, for instance the combination $f=1~{}$GHz
and $d=10~{}$cm. Such values are not feasible due to experimental limitations
and make it unrealistic to satisfy criterion ii). Therefore, before
undertaking an experiment to couple ions in separate traps it is important to
characterize the ion trap and verify that $\tilde{\alpha}$ is large enough and
$\delta$ is small enough to guarantee $t_{\mathrm{deco.}}/t_{\mathrm{ex.}}\geq
10$ for the targeted frequency and distance ranges.

Assuming the combination $\tilde{\alpha}=2.4$, $\delta=4.0$, and $\beta=1.51$
from Sedlacek  _et al._ (2018a, b), we now plot equation (60) as a function of
both $f$ and $d$. The three-dimensional surface plot in figure 8($a$) shows
combinations of $f$ and $d$ for which $2\times
t_{\mathrm{deco.}}/t_{\mathrm{ex.}}\geq 20$. (Scaling
$t_{\mathrm{deco.}}/t_{\mathrm{ex.}}$ by a factor of $2$ improves the
visualization.) This can be compared with the three-dimensional surface plot
of $V_{\mathrm{sig.}}/V_{\mathrm{J.N.}}$ in figure 8(b) over the same ranges
of $f$ and $d$, with a wire resistance $R=0.01~{}\Omega$, a frequency
bandwidth $\Delta f=500~{}$Hz, the mass $m_{\mathrm{Be}}$ of a beryllium atom
and charge $q_{\mathrm{c}}=4~{}e$. (The charge and resistance values used are
different from the ones in table II, to make it easier to see the shape of the
plot.) Ideally, a given coordinate in the frequency-distance plane should
yield a value on the vertical axis greater than $20$ in figure 8($a$) and
greater than $10$ in figure 8(b). However, values of $f$ and $d$ which
increase $t_{\mathrm{deco.}}/t_{\mathrm{ex.}}$ tend to decrease
$V_{\mathrm{sig.}}/V_{\mathrm{J.N.}}$ and vice-versa. If figure 8($a$) and
figure 8(b) are superimposed, a compromise between the two criteria is reached
for combinations of $f$ and $d$ that lie between the largest values on each
surface. To visualize the target ranges clearly, we impose the ratio
$t_{\mathrm{deco.}}/t_{\mathrm{ex.}}=10$ which specifies a single contour line
in the $f-d$ plane in figure 8($a$). We also set
$V_{\mathrm{sig.}}/V_{\mathrm{J.N.}}=10$ which specifies a contour line in
figure 8(b). The two contour lines can be placed on a common set of
coordinates, and if they overlap, the region between them yields values of $f$
and $d$ for which both the ratios $t_{\mathrm{deco.}}/t_{\mathrm{ex.}}$ and
$V_{\mathrm{sig.}}/V_{\mathrm{J.N.}}$ are greater than $10$. Rearranging
equations (60) and (63), the functions relating frequency $f$ to ion distance
$d$ for the contour lines $t_{\mathrm{deco.}}/t_{\mathrm{ex.}}=10$ and
$V_{\mathrm{sig.}}/V_{\mathrm{J.N.}}=10$ are:

$f_{t_{\mathrm{deco.}}/t_{\mathrm{ex.}}}=\Bigg{(}10\times\left[\frac{2A\pi^{3}\epsilon_{\mathrm{o}}\times\left[1+\left(T/T_{\mathrm{p}}\right)^{\beta}\right]}{hd^{\delta+1}}\right]$
$\times\left[\left(\frac{2r+\frac{2\pi
l}{8\mathrm{ln}\left(l/a\right)}}{r}\right)\left(\frac{\left(d^{2}+r^{2}\right)^{3}}{r^{2}}\right)\right]\Bigg{)}^{1/\left(\tilde{\alpha}-1\right)}~{},$
(64)

and

$f_{V_{\mathrm{sig.}}/V_{\mathrm{J.N.}}}=\frac{h}{8\pi^{2}m}\Bigg{(}\frac{1}{10}\times\left(\frac{1}{4k_{\mathrm{B}}T\Delta
fR}\right)^{1/2}$ $\times\left(\frac{2}{2\times
8\epsilon_{\mathrm{o}}r+\frac{2\pi\epsilon_{\mathrm{o}}l}{\mathrm{ln}\left(l/a\right)}}\right)\times\frac{2q_{\mathrm{c}}r^{2}}{(r^{2}+d^{2}_{\mathrm{eq.}})^{3/2}}\Bigg{)}^{2}~{}.$
(65)

Figure 8(c) shows a plot of equations (64) and (65) with common vertical and
horizontal axes. The shaded region corresponds to combinations of $f$ and $d$
where the criteria $t_{\mathrm{deco.}}/t_{\mathrm{ex.}}=10$ and
$V_{\mathrm{sig.}}/V_{\mathrm{J.N.}}=10$ are simultaneously satisfied.

The shaded region between the two curves increases rapidly with improved
values of the exponents $\tilde{\alpha}$, $\delta$, and $\beta$. However, as
it is not well understood how to control these exponents, the position of the
curve for $t_{\mathrm{deco.}}/t_{\mathrm{ex.}}=10$ is fairly inflexible.
(Decreasing the temperature below $T_{\mathrm{p}}\sim 10~{}$K provides little
added benefit, and changing the wire radius is of limited use.) On the other
hand, the contour line for $V_{\mathrm{sig.}}/V_{\mathrm{J.N.}}=10$ can be
displaced in the $f-d$ plane by changing the temperature $T$ of the coupling
system, the charge $q_{\mathrm{c}}$ or mass $m$ of the ion, the frequency
range $\Delta f$ over which Johnson-Nyquist noise disturbs the system, or the
resistance $R$ of the coupling wire. For instance, reducing the temperature
from $T=10~{}$K to $T=0~{}$K causes the range of $f$ and $d$ for which both
criteria are simultaneously satisfied to expand to infinity, as
$V_{\mathrm{J.N.}}$ goes to zero. We observe that the right-most parentheses
in equation (63), which depends on the mass $m$ and frequency of oscillation
$f$ of the ion, is the spread (standard deviation of the position) of the
zero-point wavefunction. This spread is $\sim 10~{}$nm for a
${}^{9}\mathrm{Be}^{+}$ ion oscillating at $5~{}$MHz. In section I.2.1 we
chose this as the amplitude of oscillation of the ion, but the amplitude of
oscillation can be increased by active driving.151515Increasing the amplitude
by reducing the trap potential is not a suitable strategy because it leads to
increased $1/f^{\tilde{\alpha}}$ Anomalous heating. Ion shuttling experiments
show that ions can be displaced without causing a loss of motional state
coherence.

Figure 8: Simultaneously satisfying $t_{\mathrm{deco.}}/t_{\mathrm{ex.}}\geq
10$ and $V_{\mathrm{sig.}}/V_{\mathrm{J.N.}}\geq 10$. ($a$) A plot of $2\times
t_{\mathrm{deco.}}/t_{\mathrm{ex.}}$ as a function of the distance $d$ between
the ions and their respective pickup disks and the frequency $f$ at which the
ions oscillate. Equation (60) is evaluated for the following parameters: $h$
is Plank’s constant, $\epsilon_{\mathrm{o}}$ is the vacuum permittivity, the
heating coeff. $A=1.8\times 10^{-22}$, $T=10~{}$K, activation temperature
$T_{\mathrm{p}}=10~{}$K, wire length $l_{\mathrm{w}}=0.01~{}$m, wire radius
$a=10\times 10^{-6}~{}$m, disk radius $r=d/\sqrt{2}$, and the
$1/f^{\tilde{\alpha}}$ heating exponents $\tilde{\alpha}=2.4$, $\delta=4.0$,
and $\beta=1.51$ from Sedlacek  _et al._ (2018a, b). ($b$) A plot of
$V_{\mathrm{sig.}}/V_{\mathrm{J.N.}}$ for the same parameter values as in
($a$), with the following additions: $k_{\mathrm{B}}$ is Boltzmann’s constant,
the coupling system temperature $T=10~{}$K, the motional mode frequency range
$\Delta f=500~{}$Hz, wire resistance $R=0.01~{}\Omega$, charge
$q_{\mathrm{c}}=4\times 1.6\times 10^{-19}~{}$C, and mass
$m_{\mathrm{Be}}=1.5\times 10^{-26}~{}$kg. ($c$) Two contour lines: the blue
contour corresponds to $t_{\mathrm{deco.}}/t_{\mathrm{ex.}}=10$ and the purple
contour is for $V_{\mathrm{sig.}}/V_{\mathrm{J.N.}}=10$. Parameter values are
the same as in ($a$) and ($b$) but with $R=0.0001~{}\Omega$ and charge
$q_{\mathrm{c}}=e$.

Shuttling ions a distance of $280~{}\mu$m over the course of $5$ ion
oscillations (in $3.6~{}\mu$s) has been achieved with motional heating below
$0.1$ quantums Walther  _et al._ (2012). This is equivalent to a displacement
on the order of $50~{}\mu$m in one ion oscillation, suggesting that driving
the ion could be exploited to increase the amplitude of oscillation to well
over $1~{}\mu$m. This would increase $V_{\mathrm{sig.}}$ and hence the ratio
$V_{\mathrm{sig.}}/V_{\mathrm{J.N.}}$ by a factor of $100$, allowing
constraints on the Johnson-Nyquist noise to be relaxed by permitting a higher
resistance $R$ of the coupling wire or a broader range of frequencies $\Delta
f$ to interfere with the signal voltage.

An important step for future experiments is to measure $\tilde{\alpha}$,
$\beta$, $\delta$, $A$, and $T_{\mathrm{p}}$ over the parameter ranges at
which experiments are to be performed. This not only lays the foundation for
using expression (60), but also confirms that expression (60) is valid for the
targeted ranges of $f$, $T$, and $d$. For the example presently considered
based on references Sedlacek  _et al._ (2018a, b), and using figure 8(c), the
target ranges of $f$, $d$, and $T$ are ($f=1~{}$MHz to $f=10~{}$MHz),
($d=50~{}\mu$m to $d=500~{}\mu$m), ($T=5~{}$K to $T=300~{}$K). A technical
point for experiments is that the parameter ranges for $f$, $d$, and $T$
cannot be known before $\tilde{\alpha}$, $\beta$, $\delta$, $A$, and
$T_{\mathrm{p}}$ are measured. This means $\tilde{\alpha}$, $\beta$, $\delta$,
$A$, and $T_{\mathrm{p}}$ must initially be measured over arbitrarily chosen
ranges of $f$, $d$, and $T$. Once initial values are determined, the ranges of
$f$, $d$, and $T$ can be adjusted to ensure criteria i) and ii) are satisfied.
It may be necessary to measure $\tilde{\alpha}$, $\beta$, $\delta$, $A$, and
$T_{\mathrm{p}}$ a second time over the adjusted ranges of $f$, $d$, and $T$
to verify the validity of equation (60) for the adjusted ranges. Ultimately,
before attempting the coupling experiment proposed herein, figure 8(c) should
be replicated using experimentally measured values of $\tilde{\alpha}$,
$\beta$, $\delta$, and $T_{\mathrm{p}}$ for the specific trap that will be
used in experiments.

###  IV.4 Ranges of experimental parameters

The values of many parameters in equations (60) and (63) are bounded by
experimental constraints. We now discuss these constraints, starting with the
most rigid ones and progressing towards more flexible constraints.

The strongest constraints are the $1/f^{\tilde{\alpha}}$ (Anomalous noise)
heating exponents $\delta$, $\tilde{\alpha}$, and $\beta$. These vary
substantially between different trap setups and determine the scaling of the
decoherence time with distance $d$, frequency $f$, and temperature $T$. The
scaling exponents, along with the activation temperature $T_{\mathrm{p}}$, are
presently difficult to control. Next, are the distance $d$, frequency $f$, and
temperature $T$. For the experimentally measured coefficients
$\tilde{\alpha}=2.4$, $\delta=4.0$, and $\beta=1.51$ in Sedlacek  _et al._
(2018a, b), $d$, $f$, and $T$ are constrained by $1/f^{\tilde{\alpha}}$ noise
to roughly $d>50~{}\mu$m, $f>2.5~{}$MHz,161616There is another, theoretical
limit on $f$ which is irrelevant in the regime where $1/f^{\tilde{\alpha}}$
noise is the dominant constraint. The frequency $f=5~{}$MHz gives the spacing
between the harmonic oscillator modes. Since the spacing between sideband
transitions at $5~{}$MHz is $5~{}$MHz, the sideband transitions are separated
from each other by much more than their width $\Delta f$. From a technical
standpoint the frequency can be reduced significantly. To keep the sidebands
resolved, the frequency must not be reduced past the limit where the width
$\Delta f$ of the first-sideband peak overlaps with the second-sideband peak
or the carrier transition. This imposes the condition $f>\Delta f$. and
$T<10~{}$K as shown by the blue curve for
$t_{\mathrm{deco.}}/t_{\mathrm{ex.}}=10$ in figure 8(c) (plotted for
$T=10~{}K$). With $d$ constrained by $1/f^{\tilde{\alpha}}$ heating, the
radius of the pickup disk $r$ is also constrained, as we showed in section
II.4 that the optimal relationship between $d$ and $r$ is
$d_{\mathrm{eq.}}/\sqrt{2}\leq r_{\mathrm{opt.}}\leq d_{\mathrm{eq.}}$.
Collectively, the above observations show that the main constraints for
satisfying the criterion $t_{\mathrm{deco.}}/t_{\mathrm{ex.}}\geq 10$,
described by equation (60), are imposed by $1/f^{\tilde{\alpha}}$ heating due
to the trap electrodes.

Until now, we have used the notation $T$ to describe both the temperature of
the trap electrodes, $T^{\mathrm{trap}}$ and the temperature of the coupling
system, $T^{\mathrm{c.s.}}$. However, in practice these may be different.
$T^{\mathrm{c.s.}}$ determines the impact of Johnson-Nyquist thermal noise and
should be reduced as much as possible. For a superconducting coupling system,
$T^{\mathrm{c.s.}}$ must be below the critical temperature $T_{\mathrm{c}}$ of
the superconductor. The critical temperature of aluminum, which may be
advantageous due to its long coherence length ($\xi\approx 1,600~{}$nm ), is
$T^{\mathrm{Al}}_{\mathrm{c}}\sim 1~{}$K Hansen (2006). The critical
temperature of niobium is $T^{\mathrm{Nb}}_{\mathrm{c}}\approx 9~{}$K.

The range of frequencies $\Delta f$ over which Johnson-Nyquist noise disturbs
the system is determined by the range of frequencies transmitted via the
coupling system. We want to transmit information encoded in motional modes of
an ion, so $\Delta f$ is the range of frequencies that can excite these
motional modes. Two factors influence this range. First, the intrinsic spread
of frequencies which can induce a motional mode transition, known as the
natural linewidth of the transition. Second, the jitter of the center
frequency of the distribution as a whole due to technical noise. A motional
mode transition can occur via a sideband transition of an ion, which involves
a change in its internal state along with a change in its motional state. In
the Lamb-Dicke regime, the spread of frequencies which induce a sideband
transition is smaller than the spread of frequencies which induce a carrier
transition (a change in internal state without an associated change in
motional state). Thus, the natural linewidth of the carrier transition
provides a bound on the natural linewidth of the sideband transition.
Depending on the atomic species and atomic transition, this range can be quite
narrow. To give an example, the natural linewidth of the carrier transition of
the $6S_{1/2}$ to $5D_{3/2}$ electric-quadrupole transition in 138Ba+ is
$13~{}$mHz Kleczewski  _et al._ (2012). This implies the natural linewidth of
its sideband transition is less than $13~{}$mHz.

Technical noise influences the spacing between motional modes. The spacing
between motional modes is given by $\hbar\omega$, the frequency of oscillation
of the ion, which depends on the strength of the ion’s confinement in the
trap. If the strength of confinement changes, for example due to fluctuations
in the voltages on the trap electrodes, the spacing between motional modes
also varies. This results in a shift of the center of the distribution of
frequencies which excite the motional mode transition. If the center frequency
shifts rapidly and by a large amount compared to the natural linewidth of the
motional mode, the range of frequencies which induce motional mode transitions
is effectively broadened. For changes in the motional modes that occur through
sideband transitions, variations in the internal energy levels of the ion also
contribute to jitter of the center frequency of the motional mode transition.
However, this effect tends to be smaller than jitter due to variations in the
trap potential. In practice, all factors which cause the motional mode energy
levels to shift at high enough frequencies broaden the range of $\Delta f$.
Reference Johnson  _et al._ (2016) shows that technical noise on the sideband
transition between two hyperfine states of the 171Yb+ ion can be reduced to
$\Delta f\sim 500~{}$Hz. A further discussion of $\Delta f$ may be found in
appendix J.

Next, we consider the wire resistance $R$. Figure 8(c) is plotted for a
temperature $T=10~{}$K, and a wire of radius $a=10\times 10^{-6}~{}$m, length
$l=1~{}$cm, and resistance $R=0.0001~{}\Omega$. A resistance of
$R=0.0001~{}\Omega$ is difficult to achieve for a normal (non superconducting)
wire of these dimensions. In appendix E, we use the expression $R\approx
l_{\mathrm{w}}/\left(\sigma\pi a^{2}\right)$, where $\sigma$ is the electrical
conductivity, and calculate a resistance $R\sim 0.003~{}\Omega$ for a
cylindrical copper conducting wire with the dimensions above, with a
conductivity $\sigma_{10\mathrm{K}}$ at $T=30~{}$K. Therefore, we recommend
fabricating the coupling wire from a superconducting material to suppress
Johnson-Nyquist noise and help satisfy the constraint
$V_{\mathrm{sig.}}/V_{\mathrm{J.N.}}\geq 10~{}$. In figure 8(c), this equates
to ensuring that the purple curve for $V_{\mathrm{sig.}}/V_{\mathrm{J.N.}}$ is
positioned far enough to the right along the distance axis to allow the
existence of the shaded overlap region. A more detailed discussion of
resistance can be found in appendix E.

We now consider the wire radius $a$. Equations (60) and (63) for
$t_{\mathrm{deco.}}/t_{\mathrm{ex.}}$ and
$V_{\mathrm{sig.}}/V_{\mathrm{J.N.}}$ both depend on $a$ via the wire
capacitance
$C_{\mathrm{w}}=2\pi\epsilon_{\mathrm{o}}l_{\mathrm{w}}\big{/}\mathrm{ln}\left(l_{\mathrm{w}}/a\right)$.
In the case of a normal conducting wire, $V_{\mathrm{sig.}}/V_{\mathrm{J.N.}}$
also depends on $a$ via the wire’s resistance
$R=l_{\mathrm{w}}/\left(\sigma\pi a^{2}\right)$ (see equation (62) for an
expression of $V_{\mathrm{sig.}}/V_{\mathrm{J.N.}}$ with $R$ explicitly
written out). Increasing the cross-sectional area of a non superconducting
cylindrical wire, in other words increasing its radius $a$, decreases the
wire’s resistance and associated noise voltage. Overall, increasing $a$
decreases the ratio $t_{\mathrm{deco.}}/t_{\mathrm{ex.}}$,171717Increasing $a$
increases the capacitance of the coupling wire, $C_{\mathrm{w}}$, which
decreases the coefficient $\zeta$ and hence the coupling strength $\gamma$.
This increases the exchange time $t_{\mathrm{ex.}}$ and decreases the ratio
$t_{\mathrm{deco.}}/t_{\mathrm{ex.}}$. but for a normal conducting wire it
increases $V_{\mathrm{sig.}}/V_{\mathrm{J.N.}}$.181818Due to increasing wire
capacitance, increasing $a$ decreases $V_{\mathrm{sig.}}$ according to
$V_{\mathrm{sig.}}\propto
1/\left(A+\frac{2\pi\epsilon_{\mathrm{o}}l_{\mathrm{w}}}{\mathrm{ln}\left(l_{\mathrm{w}}/a\right)}\right)$,
where $A$ represents the capacitance of the two pickup disks of the coupling
system. For a non superconducting wire, increasing $a$ decreases the thermal
noise voltage fluctuations as $V_{\mathrm{J.N.}}\propto 1/a$. Therefore, the
ratio $V_{\mathrm{sig.}}/V_{\mathrm{J.N.}}$ scales as
$V_{\mathrm{sig.}}/V_{\mathrm{J.N.}}\propto
a/\left(A+\frac{2\pi\epsilon_{\mathrm{o}}l_{\mathrm{w}}}{\mathrm{ln}\left(l_{\mathrm{w}}/a\right)}\right)~{}$.
$V_{\mathrm{sig.}}/V_{\mathrm{J.N.}}$ starts off growing with increasing $a$,
until $a$ approaches $~{}l_{\mathrm{w}}$. For the expression
$C_{\mathrm{w}}=2\pi\epsilon_{\mathrm{o}}l_{\mathrm{w}}\big{/}\mathrm{ln}\left(l_{\mathrm{w}}/a\right)$
to hold, we only consider the regime in which $a\ll l_{\mathrm{w}}$, in other
words where $V_{\mathrm{sig.}}/V_{\mathrm{J.N.}}$ increases with increasing
$a$. ($V_{\mathrm{sig.}}/V_{\mathrm{J.N.}}$ may also increase with $a$ for a
superconducting wire, see appendix E). Since equations (60) and (63) both
depend on the same parameter $a$ but respond to changes in $a$ in opposite
ways, this leads to an optimization problem where, with all other parameters
fixed, some values of $a$ may satisfy the criterion
$t_{\mathrm{deco.}}/t_{\mathrm{ex.}}\geq 10$ but not
$V_{\mathrm{sig.}}/V_{\mathrm{J.N.}}\geq 10$, or vice versa. Although figure 7
shows that the ratio $t_{\mathrm{deco.}}/t_{\mathrm{ex.}}$ only depends weakly
on the wire radius $a$, optimization can be achieved by plotting both ratios
as a function of $a$ and overlaying the two plots on the same graph with a
common vertical axis (not shown here). This allows one to visualize which
range of wire radiuses, if any, simultaneously satisfies $t_{\mathrm{ex.}}\geq
10$ and $V_{\mathrm{sig.}}/V_{\mathrm{J.N.}}\geq 10$, i.e. constraints i) and
ii) in section IV.1. Given that $t_{\mathrm{deco.}}/t_{\mathrm{ex.}}$ only
depends weakly on $a$, we assume it is generally preferable to improve
$V_{\mathrm{sig.}}/V_{\mathrm{J.N.}}$ at the expense of
$t_{\mathrm{deco.}}/t_{\mathrm{ex.}}$ by using larger wire radii and hence
lower resistance.

In addition to the optimization outlined above, the choice of $a$ should take
into account at least four other factors. First, for large wire radiuses, $a$
should remain small compared to the radii of the pickup disks, $a\ll
r=d/\sqrt{2}$. This ensures the overall model of two pickup disks connected by
a coupling wire remains valid. Second, the wire radius must respect $a\ll
l_{\mathrm{w}}$ to avoid an unrealistic divergence of the expression
$C_{\mathrm{w}}=2\pi\epsilon_{\mathrm{o}}l_{\mathrm{w}}\big{/}\mathrm{ln}\left(l_{\mathrm{w}}/a\right)$
for the capacitance of a thin wire (see for example figure 10 and the
associated discussion in appendix K). Third, if one intends to use a
superconducting wire, there is a minimum wire dimension at which
superconductivity can be maintained, around $6~{}$nm according to Bezryadin
_et al._ (2000), (see appendix E). This sets a minimum bound
$5~{}\mathrm{nm}\lesssim a$. Fourth, for small wires the expression
$R_{\mathrm{wire}}=l_{\mathrm{w}}/\left(\sigma\pi a^{2}\right)$ ceases to be
valid. The limit of small wires requires a different expression for the
resistance. The conductance $G$ becomes quantized and the resistance is given
by a form of the Landauer formula, $1/R=G=\frac{e^{2}}{\pi\hbar}\sum_{n}T_{n}$
Landauer (1957), where $e$ is the charge of an electron, $\pi=3.14...$,
$\hbar$ is the reduced Plank’s constant, the sum over $n$ is a sum over the
number of conductive channels in the system, and $T_{n}$ are the transmission
eigenvalues which describe the probability of transmission of an electron
through the input interface and output interface of a conductor.

For the coupling system to be useful, the length $l_{\mathrm{w}}$ of the
coupling wire should be long enough that the coupling system provides an
advantage over direct free-space coupling via the Coulomb interaction. In
other words, $\gamma$, which denotes the coupling strength via the conducting
wire, should be larger than $\gamma_{\mathrm{Coul.}}$, which denotes the
coupling strength via the direct Coulomb interaction. We can set the target
$\gamma/\gamma_{\mathrm{Coul.}}\geq 10~{}$. For proof of concept purposes, it
would also be nice to demonstrate a situation where the Coulomb interaction is
not strong enough to achieve state exchange within the decoherence time of the
ion, $t_{\mathrm{deco.}}/t^{\mathrm{Coul.}}_{\mathrm{ex.}}\leq 1$, but
interaction via the coupling system is strong enough to achieve state exchange
within the ion’s decoherence time, $t_{\mathrm{deco.}}/t_{\mathrm{ex.}}\geq
1$. This would prove that any observed state exchange happens via the wire. To
design such a system, one must know the maximum distance $l_{\mathrm{Coul.}}$
at which state exchange via the Coulomb interaction can occur faster than the
decoherence time. An inequality placing an upper bound on $l_{\mathrm{Coul.}}$
is derived in appendix K, expression (98). For intuition, we consider two
singly-charged ${}^{9}\mathrm{Be}^{+}$ ions with mass $m=1.5\times
10^{-26}~{}$kg , trapped with an axial mode frequency $f=5~{}$MHz , and
displaying a decoherence time $t_{\mathrm{deco.}}=0.32~{}$ seconds.191919The
decoherence time $t_{\mathrm{deco.}}=0.32~{}$ is calculated using equation
(59), evaluated for the parameter values above and a distance
$d_{\mathrm{eq.}}=50\times 10^{-6}~{}$m between the ions and the surface trap
electrodes, a temperature $T=10~{}$K of the trap electrodes, which are
characterized by the activation temperature $T_{\mathrm{p}}=10~{}$K, and a
proportionality constant $A=1.8\times 10^{-22}$, and the exponents
$\tilde{\alpha}=2.4$, $\delta=4.0$, and $\beta=1.51$ from Sedlacek  _et al._
(2018a, b). For comparison with an actual experimental value, reference
Sedlacek  _et al._ (2018b) measures a heating rate of about $10$ quanta/s, and
hence a decoherence time of $t_{\mathrm{deco.}}\sim 0.1~{}$seconds, for ions
trapped with an axial mode frequency $f=1.3~{}$MHz,
$d_{\mathrm{eq.}}=50~{}\mu$m, and $T=10~{}$K. In this system, the maximum
distance $l_{\mathrm{Coul.}}$ at which state exchange can occur via direct
Coulomb interaction is $l_{\mathrm{Coul.}}=0.5~{}$mm. Therefore, if a coupling
system is designed with $l_{\mathrm{wire}}>0.5~{}$mm and motional state
exchange is observed, one can safely conclude the state exchange is occurring
via the coupling system. We have used the length $l_{\mathrm{wire}}=1~{}$cm
for illustrative calculations throughout this study. As shown in figure 10,
appendix K, with this length the criterion $\gamma/\gamma_{\mathrm{Coul.}}\geq
10~{}$ is satisfied. However, to enhance the coupling strength and mitigate
decoherence in the wire, for a proof-of-concept experiment we propose using a
wire $l_{\mathrm{wire}}=1~{}$mm long, even though this length does not satisfy
the target $\gamma/\gamma_{\mathrm{Coul.}}\geq 10~{}$.

Finally, the mass $m$ and charge $q_{\mathrm{c}}$ of the ion are somewhat
flexible constraints. Indeed, equation (60) is independent of both these
parameters. Equation (63) increases with smaller $m$ and larger
$q_{\mathrm{c}}$. To maximize $V_{\mathrm{sig.}}/V_{\mathrm{J.N.}}$ it is
advantageous to use particles with the lowest mass and the highest charge
possible. As it is generally simplest to produce singly-charged ions, here we
mainly consider $q_{\mathrm{c}}=1$ and the mass $m_{\mathrm{Be}}$ of a
beryllium ion. However, by removing additional electrons the charge of a
trapped ion may be increased up to around $q_{\mathrm{c}}=10$ Safronova  _et
al._ (2014). To reduce the mass as much as possible, a single electron qubit
could also be considered, with a mass of $m_{\mathrm{e}}=9\times
10^{-31}~{}$kg $\approx\left(1/10,000\right)\times m_{\mathrm{Be}}$. For such
a small mass it is important to verify the validity of the rotating wave
approximation before applying the results for the exchange time
$t_{\mathrm{ex.}}$ used in this work. The rotating wave approximation holds
when the coupling between two harmonic oscillators is weaker than strength of
the harmonic confinement, meaning in the limit $(\gamma/(m\omega^{2}))<0.1~{}$
Estes  _et al._ (1968). For the value of $\gamma$ in table 2 for $d=50\times
10^{-6}~{}$m, $l_{\mathrm{w}}=0.01~{}$, and using the mass of an electron
$m_{\mathrm{e}}=9.1\times 10^{-31}~{}$kg and a frequency $f=5~{}$ MHz, we find
$(\gamma/(m\omega^{2}))\approx 0.02$ which is not far from the cutoff of
$0.1$. However, trapped electrons typically oscillate at frequencies on the
order of $f=100$ MHz. In this case, $(\gamma/(m\omega^{2}))\approx 4\times
10^{-5}\ll 0.1$ and the rotating wave approximation is verified. Although the
prospect of using electrons is appealing, designing an experiment with
electrons may require special considerations. Electrons are typically trapped
in Penning traps rather than Paul traps, and other additional factors may need
to be considered.

Table 6 provides a summary of the constraints discussed above along with their
degree of flexibility. The values should be used as intuitive guidelines, but
the given boundaries do not guarantee a frequency-distance diagram where both
criteria $t_{\mathrm{deco.}}/t_{\mathrm{ex.}}>10$ and
$V_{\mathrm{sig.}}/V_{\mathrm{J.N.}}>10$ are simultaneously satisfied, as in
figure 8(c). In outlining the constraints, where relevant we assume the
exponents $\tilde{\alpha}=2.4$, $\beta=1.51$, and $\delta=4.0$ for the
parameters $f$, $d$, and $T$ respectively, which are a consistent combination
for a Niobium trap, based on references Sedlacek  _et al._ (2018a, b). Prior
to implementing an experiment, the exponents $\tilde{\alpha}$, $\beta$, and
$\delta$ should be measured for a specific target experimental system, and the
diagram in figure 8(c) should be reproduced for a specified combination of
parameters to check for a successful overlap region. For each parameter in
table 6, a single inequality indicates the direction which increases both the
ratios $t_{\mathrm{deco.}}/t_{\mathrm{ex.}}$ and
$V_{\mathrm{sig.}}/V_{\mathrm{J.N.}}$. For cases where a given parameter
increases one of the ratios but decreases the other, a general range is given.
The Tunability columns give a sense of how much the ratio can be improved by
tuning the indicated variable. Low tunability means if the variable is pushed
to known experimental limits, roughly outlined in the Present limitation
column, the ratio can be improved by a factor $\leq 5$. Moderate means $>5$
but $\leq 10$.

Table 6: Summary of variables and their suggested values and tunability. The
Suggested value column gives a list of bounds based on experimental
feasibility. If all of the bounds in this column are used to evaluate either
$t_{\mathrm{deco.}}/t_{\mathrm{ex.}}$ or
$V_{\mathrm{sig.}}/V_{\mathrm{J.N.}}$, they yield a result on the order of
$10$. When two colored values are shown, the purple bound (top value) is
chosen such that the ratio $t_{\mathrm{deco.}}/t_{\mathrm{ex.}}\approx 10$ ,
and the red bound (bottom value) is chosen such that the ratio
$V_{\mathrm{sig.}}/V_{\mathrm{J.N.}}\approx 10$. Bounds given in black
represent that, where applicable, the same value is used to evaluate
$t_{\mathrm{deco.}}/t_{\mathrm{ex.}}$ and
$V_{\mathrm{sig.}}/V_{\mathrm{J.N.}}$. For the wire radius, the value
$a=10~{}\mu\mathrm{m}$ is used in all cases. The ratio
$t_{\mathrm{deco.}}/t_{\mathrm{ex.}}$ is associated with $5$ free parameters,
while $V_{\mathrm{sig.}}/V_{\mathrm{J.N.}}$ is associated with $9$ free
parameters. Two of the free parameters associated with
$V_{\mathrm{sig.}}/V_{\mathrm{J.N.}}$ have moderate tunability. This indicates
that the ratio of coherence time to exchange time is a less flexible criterion
than the signal to noise ratio for Johnson-Nyquist (thermal) noise.

Description |  Variable |  Suggested value |  Tunability of $t_{\mathrm{deco.}}/t_{\mathrm{ex.}}$ |  Tunability of $V_{\mathrm{sig.}}/V_{\mathrm{J.N.}}$ |  Present limitation
---|---|---|---|---|---
Distance between ion and pickup disk |  $d_{\mathrm{eq.}}$ |  ${\color[rgb]{0.54,0.17,0.89}\definecolor[named]{pgfstrokecolor}{rgb}{0.54,0.17,0.89}\geq 50~{}\mu\mathrm{m}}$,
${\color[rgb]{0.9,0.17,0.31}\definecolor[named]{pgfstrokecolor}{rgb}{0.9,0.17,0.31}\leq 200~{}\mu\mathrm{m}}$ |  low |  low |  $\sim 50~{}\mu$m, ion heating from trap electrodes. a
Radius of pickup disk |  $r$ |  $\approx d_{\mathrm{eq.}}/\sqrt{2}$ |  N.A. |  N.A. |  N.A.
Length of wire |  $l_{\mathrm{w}}$ |  $\leq 1~{}$cm |  low |  low |  $l_{\mathrm{w}}\sim 0.5~{}$mm. b
Motional mode range of frequencies c |  $\Delta f$ |  $\leq 500~{}$Hz |  N.A. |  low |  $\leq 500~{}$Hz, due to stability of the trap Johnson  _et al._ (2016).
Secular frequency |  $f$ |  $\geq 5~{}\mathrm{MHz}$ |  low |  low |  $\sim 2.5$ MHz, heating of motional mode proportional to $1/f^{\tilde{\alpha}}$. a
Charge |  $q_{\mathrm{c}}$ |  $\geq 1$ |  N.A. |  moderate |  Complexity of the experiment needed to ionize ions.
Mass ${}^{d,~{}e}$ |  $m$ |  $\leq m_{{}_{{}^{9}\mathrm{Be}^{+}}}=1.5\times 10^{-26}~{}$kg |  N.A. |  low |  No limit. Must be determined on case by case basis.
Wire radius |  $a$ |  $100~{}\mathrm{nm}\leq a\leq 0.2\times d_{\mathrm{eq.}}/\sqrt{2}$ |  low |  low,
linked
to $R$ |  $\sim 5~{}$nm, minimum radius at which superconductivity persists Bezryadin  _et al._ (2000).
Temperature |  $T$ |  $\leq 10$ degrees Kelvin |  low |  low |  $\sim 5~{}$K , ion trap operation raises temperature above $4~{}$K Chiaverini and Sage (2014).
Resistance f |  $R$ |  $\ll g\left(f,d_{\mathrm{eq.}},w\right)$ AND
$\leq 0.001~{}\Omega$ |  N.A. |  high |  Minimal residual resistance of superconducting wire.

a This effect is sometimes called Anomalous heating Leibfried  _et al._
(2003); Deslauriers  _et al._ (2006). See sections III.3, IV.2 and appendix I
for further discussion.
b At $f=5~{}$MHz , at $l_{\mathrm{w}}\sim 0.5~{}$mm the ratio
$t_{\mathrm{deco.}}/t^{\mathrm{Coul.}}_{\mathrm{ex.}}\sim 1$ for two singly-
charged ${}^{9}\mathrm{Be}^{+}$ ions coupled via direct Coulomb interaction
(see appendix K). Therefore, at $l_{\mathrm{w}}\leq 0.5~{}$mm a coupling wire
is theoretically not necessary.
c The relevant range of frequencies $\Delta f$ is estimated in appendix J.
d Suggested mass cutoff for the case $q_{\mathrm{c}}=1$, $R=0.001~{}\Omega$.
If $q_{\mathrm{c}}>1$ or $R<0.001~{}\Omega$ the mass constraint can be relaxed
and higher masses become more viable.
e Electrons with mass $m_{\mathrm{e}}$ could also be explored, see section
IV.4.
f The function $g\left(f,d_{\mathrm{eq.}},w\right)$ is the resistance at which
the ratio $V_{0}/V_{f}\sim 10$ after induced charge $Q_{\mathrm{transf.}}$ has
drained off of a pickup disk, given the frequency of oscillation $f$ and
assuming the pickup disk is grounded. See section I.3 for discussion. a This
effect is sometimes also called Anomalous heating Leibfried  _et al._ (2003);
Deslauriers  _et al._ (2006). See sections (LABEL:Flickernoise) and IV.2 for
further discussion.
b At $f=5~{}$MHz , at $l_{\mathrm{w}}\sim 0.5~{}$mm the ratio
$t_{\mathrm{deco.}}/t^{\mathrm{Coul.}}_{\mathrm{ex.}}\sim 1$ for two singly-
charged ${}^{9}\mathrm{Be}^{+}$ ions coupled via direct Coulomb interaction
(see appendix K). Therefore, at $l_{\mathrm{w}}\leq 0.5~{}$mm a coupling wire
is theoretically not necessary.
c The relevant range of frequencies $\Delta f$ is estimated in appendix J.
d Suggested mass cutoff for the case $q_{\mathrm{c}}=1$, $R=0.001~{}\Omega$.
If $q_{\mathrm{c}}>1$ or $R<0.001~{}\Omega$ the mass constraint can be relaxed
and higher masses become more viable.
e Electrons with mass $m_{\mathrm{e}}$ could also be explored, see section
IV.4.
f The function $g\left(f,d_{\mathrm{eq.}},w\right)$ is the resistance at which
the ratio $V_{0}/V_{f}\sim 10$ after induced charge $Q_{\mathrm{transf.}}$ has
drained off of a pickup disk, given the frequency of oscillation $f$ and
assuming the pickup disk is grounded. See section I.3 for discussion.

##  V A few considerations for implementing a practical system

In the previous section we considered experimental constraints on the
parameters of the coupling system. Here, we discuss general considerations for
a real implementation. We begin with the question of whether to use a single
coupling wire, or two parallel coupling wires.

To minimize attenuation of the signal during propagation, one study Zurita-
Sánchez and Henkel (2008) has proposed a design using two parallel coupling
wires. For the present system such a transmission line design would not be
helpful. At low frequencies, an electrical signal can propagate along a single
conducting strand of wire with minimal losses. A rule of thumb is that a
transmission line configuration is only necessary if the length of the wire
$l_{\mathrm{w}}$ is greater than $1/4~{}$ of the propagating wavelength
$\lambda$ Tra (a) (a cutoff of $1/10~{}$th of the propagating wavelength is
also used Tra (b)). The trapped ions considered here oscillate with a
frequency $f\sim 10~{}$MHz. For an electromagnetic wave in a conductor, the
velocity factor $v/c$ is on the order of $0.5$, meaning the speed of
propagation $v$ is $\sim~{}0.5~{}\times$ the speed of light. For a frequency
$f\sim 10~{}$MHz, the wavelength is $\lambda=v/f\sim 15~{}$m. Since early
experiments will likely attempt propagation over no more than a few
centimeters, the approximation $\lambda\gg l_{\mathrm{w}}$ is valid.

The material from which the coupling wire is fabricated is also an important
consideration. So far, we have used decoherence of the ions’ motional modes in
the ions as the benchmark by which to gauge whether the exchange time
$t_{\mathrm{ex.}}$ is short enough to successfully transfer quantum
information between two ions. It is also important to consider whether and for
how long quantum information can survive in the coupling system itself. In
superconductors, the coherence length $\xi$ of Cooper pairs—the basic building
blocks of superconductivity—can be over $1,600~{}$nm ($\xi\approx 1,600~{}$nm
for Aluminium at $\sim 1~{}$K Hansen (2006)). A recent experiment successfully
transferred quantum information over a distance of $\sim 3~{}$mm using a
niobium superconducting wire cavity $600~{}$nm wide with a center frequency
$f_{\mathrm{c}}=5.8~{}$GHz and quality factor $Q_{\mathrm{c}}=4,700$ Mi  _et
al._ (2018) (the coherence length for niobium is $\xi_{\mathrm{Nb}}=38~{}$nm).
Additionally, a theoretical study spanning two publications predicts quantum
information can propagate as an electric signal along a normal conducting
lossy electrical transmission line Zurita-Sánchez and Henkel (2006,
2008).202020This result depends on several factors including an estimation of
the decoherence time in the wire, $t^{\mathrm{wire}}_{\mathrm{deco.}}$. Other
approaches in the literature for estimating the decoherence time in the
coupling system can be found in appendix L. The work finds that a wire
manifests properties of a string of harmonic oscillators, and shows that
quantum information can be exchanged via a $1~{}$cm wire connecting two
$100~{}\mu$m disks with a fidelity up to $0.989$ at a temperature of $4~{}$
Kelvin, for the coupling strengths in Zurita-Sánchez and Henkel (2008). The
authors state: in the dispersive regime, conventional conductors, despite
their intrinsic thermal resistive noise, do not necessarily degrade the
coherent dynamics of the confined electrons on the timescale relevant for
typical quantum manipulations. These results suggest that transferring quantum
information using a conducting wire is feasible.

Several factors may hinder propagation of a quantum signal. Among these,
dissipation can lead to decoherence Zurita-Sánchez and Henkel (2008). In a
two-fluid model for superconductivity, a superconductor contains a
superconducting component which carries current without interaction, along
with a normal conducting component Weingarten (1996). The normal conducting
electrons give rise to power dissipation from which a frequency-dependent
surface resistance, or sheet resistance is defined as
$R_{\mathrm{s}}=\frac{1}{2}\left(2\pi
f\right)^{2}\mu^{2}_{0}\lambda_{\mathrm{L}}^{3}\sigma_{1}$, where $\mu_{0}$ is
permeability, $\lambda_{\mathrm{L}}$ is the London penetration depth, and
$\sigma_{1}$ is the normal state conductivity of the normal conducting
electrons Hansen (2006); Weingarten (1996). This expression has been verified
experimentally Hansen (2006). A precise description yields the BCS surface
resistance, $R^{\mathrm{BCS}}_{\mathrm{s}}\propto
f^{2}e^{-\Delta/k_{\mathrm{B}}T}/T$, where $f$ is frequency, $\Delta$ is
defined to be half the Cooper pair binding energy, $k_{\mathrm{B}}$ is
Boltzmann’s constant, $T$ is temperature in Kelvin, and $2\Delta\approx
3.5k_{\mathrm{B}}T_{\mathrm{c}}$, where $T_{\mathrm{c}}$ is the critical
temperature of the superconductor Sup . For niobium with
$T_{\mathrm{c}}=9.3~{}$K, at $T<T_{\mathrm{c}}/2$,
$\Delta/k_{\mathrm{B}}\approx 17.67$ Sup . The BCS sheet resistance in ”Ohms
per square” (not ”per square meter”), has been measured for niobium to be
about $100~{}\mathrm{n}\Omega/\mathrm{square}$ at $T=4.2~{}$K and $1~{}$GHz,
with extrapolation suggesting a decrease to about
$1~{}\mathrm{n}\Omega~{}\mathrm{square}^{-1}$ at $10~{}$MHz Weingarten (1996).
Other electrical noise may also exist in superconductors. In niobium at
$4.2~{}$K, reference Weingarten (1996) measures a ”residual surface
resistance” of $R_{\mathrm{res.}}\approx
1~{}\mathrm{n}\Omega~{}\mathrm{square}^{-1}$. This resistance is shown to be
relatively frequency independent over the range of $\sim 100~{}$MHz to
$100~{}$GHz Weingarten (1996). Additionally, in type II superconductors, there
may be dissipative interactions involving the movement of weakly-pinned
vortices Kogan (2008). Moreover, reference Boogaard  _et al._ (2004) describes
a spreading resistance associated with electrons flowing into or out of normal
conducting reservoirs at either end of a superconducting wire. In some cases
the spreading resistance is projected to be as large as
$\sim~{}10~{}\mathrm{m}\Omega$ Boogaard  _et al._ (2004). Radiation resistance
could also lead to dissipation within the coupling system. Finally, in any
realization, the pickup disks should connect to the coupling wire in a tapered
fashion as indicated in figure 5, to ensure adiabatic flow of electrons
between the disks and the coupling wire, reducing reflections and hence the
possibility of shot noise Li  _et al._ (1990). Appendix E discusses additional
points related to resistance.

Photons emitted by the ion may destroy the superconductivity of the coupling
system, leading to normal resistance and associated dissipation and
decoherence. In a superconductor, superconductivity exists due to electrons in
Cooper pairs which have zero resistance. If these Cooper pairs are broken, the
electrons in the pair become ”normal electrons”. In BCS theory, if a material
is in a superconducting state it is because the thermal energy
$k_{\mathrm{B}}T$ is smaller than the binding energy of the Cooper pairs.
Breaking Cooper pairs requires energy on the scale of the Cooper pair binding
energy, $E_{\mathrm{C.p.}}=2\Delta\sim 1~{}$meV $=1.6\times 10^{-22}~{}$J
Dressel (2013). The energy of a motional mode photon emitted by the ion is
fixed by its motional frequency $f\sim 10\times 10^{6}~{}$Hz, and is
$E^{\mathrm{motional}}_{\mathrm{ion}}=hf\sim 6.6\times 10^{-27}~{}$J where $h$
is Plank’s constant. Thus, motional photons from the trapped ions cannot break
Cooper pairs. However, the frequency of optical photons emitted by the ions or
used to address the internal orbital states of the ions is on the order of
$E^{\mathrm{optical}}_{\mathrm{ion}}=100\times 10^{12}~{}$Hz. The extra seven
orders of magnitude in energy means these are likely to break apart Cooper
pairs. It could be necessary to take precautions to avoid exposing the
superconducting coupling system to visible light or other high-energy photon
sources.

For the layout of the device, the coupling system can be incorporated into a
surface-trap configuration, which lends itself naturally to scaling. It can be
manufactured using standard nanofabrication techniques, for example following
the process we have outlined in Van Horne and Mukherjee (2020b), with
additional steps to make the wire and introduce the coupling system within a
planar trap. In the design outlined in Van Horne and Mukherjee (2020b), the
pickup-disks are positioned on top of two pillars which stick out above the
plane of the surface trap, to decrease the distance between the ions and the
disks and thereby increase the coupling strength which scales strongly with
distance, $\gamma\propto 1/d^{5}$. It should be noted that the configuration
of pickup-disks protruding towards the trapped ions departs from the model
used for the calculation of $\gamma_{\mathrm{pickup-disk}}$ in this work,
since the pickup disks are no longer surrounded by an infinite ground plane.
This results in fewer (zero) induced negative charges around the pickup-disk
when a positively charged particle moves closer to it. Negative charges
surrounding the pickup disk would exert a mutually repulsive force on negative
charges in the disk, reducing the amount of negative charge that accumulates
on the disk. Thus, without this effect the implementation in Van Horne and
Mukherjee (2020b) presumably leads to a stronger coupling strength compared to
the estimates in subsection II.5. If the majority of $1/f^{\tilde{\alpha}}$
(Anomalous) heating comes from effects happening in the plane of the trap
electrodes or from the trap electrodes themselves, positioning the pickup
disks close to the ions will not significantly increase $1/f^{\tilde{\alpha}}$
heating. With an equal heating rate and increased coupling strength, the ratio
$t_{\mathrm{deco.}}/t_{\mathrm{ex.}}$ can be enhanced.

Since the signal-to-noise ratio for thermal noise scales with the charge of
the trapped particles, $V_{\mathrm{sig.}}/V_{\mathrm{J.N.}}\propto
q_{\mathrm{c}}$, using multiply-ionized species up to a maximum of about
$q_{\mathrm{c}}=+10$ Safronova  _et al._ (2014) can increase the ratio
$V_{\mathrm{sig.}}/V_{\mathrm{J.N.}}$ by an order of magnitude. This strategy
should not affect the ratio of coherence time to exchange time,
$t_{\mathrm{deco.}}/t_{\mathrm{ex.}}$, which we have shown to be independent
of the charge (equation (60)). It does, however, significantly reduce the
coherence time $t_{\mathrm{deco.}}$ of the ion (equation (59)), as the
$1/f^{\tilde{\alpha}}$ heating rate $d\bar{n}/dt$ scales with the charge
squared (equation (58)).

##  VI Conclusions and outlook

This work suggests that despite the deleterious effects of
$1/f^{\tilde{\alpha}}$ (Anomalous) heating, it is possible to exchange quantum
information between ion qubits in separate surface traps using a conducting
wire. Moreover, this can be achieved using existing technologies and
materials, and singly-charged ions. However, successful implementation is
subject to a number of engineering requirements.

Based on the observed variability of $1/f^{\tilde{\alpha}}$ noise in published
literature, $1/f^{\tilde{\alpha}}$ noise is predicted to be one of the main
constraints on the proposed scheme. As part of designing an experiment, it is
strongly recommended to measure the exponents $\tilde{\alpha}$, $\delta$, and
$\beta$ which characterize $1/f^{\tilde{\alpha}}$ noise in the setup, to
ensure anomalous heating is low enough that quantum state exchange is
theoretically possible, and $t_{\mathrm{deco.}}/t_{\mathrm{ex.}}\geq 10$.
Working at cryogenic temperatures below $T=10~{}$K is also advised in early
experiments, for two reasons. First, temperatures down to about $10~{}$K have
been shown to suppress $1/f^{\tilde{\alpha}}$ noise, lengthening coherence
time and hence increasing the ratio of coherence time to information exchange
time, $t_{\mathrm{deco.}}/t_{\mathrm{ex.}}$. Second, low temperature favors
superconductivity. The low resistance offered by a superconducting wire
reduces thermal (Johnson-Nyquist) noise in the coupling wire, increasing the
signal-to-noise ratio $V_{\mathrm{sig.}}/V_{\mathrm{J.N.}}$. Using a
superconducting wire is also beneficial to reduce or eliminate shot (Shottky)
noise.

Next, we recommend stabilizing ion traps to around $\Delta f\leq 500~{}$Hz to
minimize the bandwidth over which the ions absorb noise. For the same reason,
it could be desirable to turn off lasers during the transfer of motional
quantum information, to avoid broadening the absorption spectrum $\Delta f$ of
the motional mode transitions. Reducing $\Delta f$ reduces thermal or other
noise which disturbs the system, and helps increase the signal-to-noise ratio.

Regarding the layout of the system, optimization of the design herein leads to
a rule of thumb which relates the distance $d_{\mathrm{eq.}}$ between a
trapped ion and a nearby pickup disk, to the optimal radius
$r_{\mathrm{opt.}}$ of the disk. When the pickup disks’ capacitance is much
larger than the capacitance of the connecting wire, the optimal relationship
is $r_{\mathrm{opt.}}=d_{\mathrm{eq.}}$. When the wire capacitance is much
larger than the disk capacitance,
$r_{\mathrm{opt.}}=d_{\mathrm{eq.}}/\sqrt{2}$. In general,
$d_{\mathrm{eq.}}/\sqrt{2}\leq r_{\mathrm{opt.}}\leq d_{\mathrm{eq.}}$.

Although the technology for interconnecting ion qubits using conducting wires
already has the potential to be viable, over the long run there is room for
progress which could lead to increased robustness and relaxed constraints.
This can happen in several ways. First, if coupling designs such as the one
outlined in Van Horne and Mukherjee (2020b) increase the coupling strength
$\gamma$ without significantly increasing $1/f^{\tilde{\alpha}}$ heating
noise, state exchange times $t_{\mathrm{ex.}}$ could be significantly reduced.
Second, in the future $1/f^{\tilde{\alpha}}$ surface heating noise may become
better understood and ways could be found to diminish it, thereby increasing
coherence times $t_{\mathrm{deco.}}$. For instance, if $1/f^{\tilde{\alpha}}$
heating is activated by processes related to the $\sim$MHz rf trapping
frequencies used in Paul traps, using Penning traps which do not require A.C.
fields could be a way forwards. (Calculations suggest black-body radiation
cannot account for the observed heating rates. The energy which causes
$1/f^{\alpha}$ ion heating must come from somewhere, and it appears to be
radiated from rf trap electrodes (see appendix I.1).) While some of the
derivations herein may need to be revisited for Penning traps, their
adaptation and applicability is likely straight forwards. Both of the advances
above would allow transfer of quantum information at higher temperatures and
over longer distances.

In this work we have focused on results which are immediately applicable for a
planar coupling system, due to the potential for planar systems to be scaled.
Other systems may be less suited to scalability but could facilitate exchange
of quantum information. We hope the analysis and methodology developed herein
will provide a foundation for a rich set of future studies and experiments. On
this note, we wish the reader well in their scientific pursuits.

###### Acknowledgements.

We thank Angelina B. Frank for making figures 2 and 9, and Marco Fellous-
Asiani for verifying the calculation of $Q_{\mathrm{transf.}}$.

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##  Appendix A Charge in an outer ring of a conducting disk; justification of
the ’ring of charge’ approximation

We claim that when charges are placed on a conducting disk, most of the
charges end up in a ”ring” around the edges of the disk. To justify this, one
must demonstrate that at equilibrium, most of the charge is located in an
annular segment of the disk with a width $\xi\ll$ than the radius of the disk.
In this appendix the radius of the disk will be called ”$a$”, following the
notation in McDonald (2003). We define ”$\xi\ll a$” to mean
$\xi\leq\frac{a}{4}$, and ”most of the charge” to mean that at least
$\frac{3}{4}\times Q$ should be within the annular segment covered by $\xi$,
where $Q$ is the total charge on the disk. Following McDonald McDonald (2002),
and Sir Thompson (Lord Kelvin) Thomson (Lord Kelvin), the surface charge
density of a circular disk over most of its surface area (the parts near the
edges of the disk notwithstanding), is given by
$\sigma_{\mathrm{circular~{}disk}}(r)=\frac{Q^{\prime}}{4\pi
a\sqrt{a^{2}-r^{2}}}$, where $a$ is the radius of the disk, $r$ is the
distance away from the center of the disk at which the surface charge density
is evaluated, and $Q^{\prime}$ is a parameter to be determined, (which would
be the total charge $Q$ of the two flat surfaces combined, in the limit that
the height $2b$ of the disk is $2b=0$). A more accurate description of the
surface charge density near a corner is given by:
$\sigma_{s}=\frac{K}{\sqrt[3]{s}}$, where $K$ is a constant and $s$ is
distance away from the corner Jackson (1999) McDonald (2003). In the thin disk
system, the intersection between the top (or bottom) face of the disk, and the
side of the disk, produces a corner which extends as a ring around the center
of the disk. The variable ”$s$” describes moving inwards a distance ”$s$” from
the $90^{\mathrm{o}}$-corner edge of the ring, where $s\leq b$, and $2b$ is
the height of the disk. The same expression $\sigma_{s}=\frac{K}{\sqrt[3]{s}}$
can be used to describe the charge density on the side of the disk, moving
away from the $90^{\mathrm{o}}$-corners, towards the center of the rectangular
strip which wraps around the circumference of the disk. To ensure consistency
there must be continuity of charge density at every point on the top (and
bottom) surface of the disk. Therefore, somewhere on the disk
$\sigma_{\mathrm{circular~{}disk}}(r)=\sigma_{s}(s)$. We can postulate that
this occurs a distance $b$ inwards from the edge of the disk, and then allow
the free parameter $Q^{\prime}$ to vary in such a way as to satisfy the
constraint that the total charge on the disk must sum up to $Q$. Continuity of
charge density imposes: $K\big{/}\sqrt[3]{b}=Q^{\prime}\big{/}4\pi
a\sqrt{a^{2}-(a-b)^{2}}$, or

$K=\frac{Q^{\prime}b^{1/3}}{4\pi a\sqrt{a^{2}-(a-b)^{2}}}$ (66)

The total charge contained in the region ”near the corners” of the disk can be
found by integrating $\sigma_{s}$ a distance $b$ (in other words, half the
disk-height) from the top-corner towards the center of the rectangular strip,
from the bottom-corner towards the center of the rectangular strip, and also
inwards forming an ’outer ring’ of charge on the top and bottom faces of the
disk. This yields $Q_{s}=4\int_{0}^{b}\\!\frac{K}{\sqrt[3]{s}}2\pi a\
ds=\frac{3b}{\sqrt{a^{2}-{(a-b)}^{2}}}Q^{\prime}$, where we have substituted
expression (66) in for $K$. The total charge on the rest of the disk, over two
’inner small disks’ on the top and bottom faces, may be found by integrating
$\sigma_{\mathrm{circular~{}disk}}$ from $r=0$ to $r=\left(a-b\right)$. This
yields $Q_{g}=2\int_{0}^{a-b}\frac{Q^{\prime}}{4\pi a\sqrt{a^{2}-{r}^{2}}}2\pi
rdr=\left(1-\frac{\sqrt{a^{2}-{(a-b)}^{2}}}{a}\right)Q^{\prime}$. The sum of
these two expressions gives the total charge on the disk, $Q=Q_{g}+Q_{s}$,
which provides a relationship between $Q$ and $Q^{\prime}$:

$Q^{\prime}=\frac{Q}{\frac{3b}{\sqrt{a^{2}-{(a-b)}^{2}}}+1-\frac{\sqrt{a^{2}-{(a-b)}^{2}}}{a}}$
(67)

In the limit $a\gg b$, expression (67) reduces to $Q^{\prime}\approx Q~{}$,
which can be understood intuitively. When the disk is thin compared to its
radius, only a small percent of its charge resides on the thin edge
perpendicular to the circular faces of the disk.

We want to find the charge contained in the outer-ring region a distance
$\xi\leq\frac{a}{4}$ away from the corner. In some cases, if the width $2b$ of
the conducting disk is large compared to the radius $a$ of the disk, the
distance $b$ inwards from the side of the disk may be greater than
$\frac{a}{4}$. In this case, it makes sense to only use the expression for the
charge density ”near a corner” to calculate the charge in the outer-ring
$\xi\leq\frac{a}{4}$. Experimentally, the typical thickness of surface-trap
electrodes ranges from $2b=100~{}$nm to $2b=10~{}\mu$m Labaziewicz  _et al._
(2008); Hite  _et al._ (2012); Daniilidis  _et al._ (2014); Sedlacek  _et al._
(2018a), and for what follows we consider a thickness up to $2b=35~{}\mu$m.
From section II.4, for the limiting case where the disk capacitance is much
greater than the wire capacitance, the optimal relationship between the radius
of the disk and the equilibrium height of the suspended ion is $r\sim
d_{\mathrm{eq.}}/\sqrt{2}$ (here, the notation ”$r$” refers to the standard
notation used throughout the rest of the article, outside of this appendix).
Throughout this article we have used the two example values $d=50~{}\mu$m and
$d=200~{}\mu$m, which respectively imply disk radii of
$a=\frac{1}{\sqrt{2}}50~{}\mu\mathrm{m}=35~{}\mu$m, and
$a=\frac{1}{\sqrt{2}}200~{}\mu\mathrm{m}\sim 140~{}\mu$m. We can calculate the
charge in the outer ring for the two extreme aspect ratios: ($2b=35~{}\mu$m,
$a=35~{}\mu$m), and ($2b=100~{}$nm, $a=140~{}\mu$m).

In the first case, the thickness $\xi=a/4$ of the outer ring is equal to
$b/2$. Since here $a/4$ is less than $b$, we only use the expression for the
charge density ”near a corner” to calculate the charge in the outer-ring
$\xi\leq\frac{a}{4}$. We must calculate the charge on the side of the disk, as
well as inwards a distance $b/2$ towards the center of the disk. The charge on
the side of the disk is $Q_{s}/2$. The charge inwards a distance $b/2$ towards
the center of the disk, for the top and bottom faces is
$2\int_{0}^{b/2}\\!\frac{K}{\sqrt[3]{s}}2\pi a\
ds=\frac{3bQ^{\prime}}{2^{5/3}\sqrt{a^{2}-(a-b)^{2}}}$. Adding these together,
the total charge in the region $\xi\leq\frac{a}{4}$ is

$Q^{\xi\leq\frac{a}{4}}_{\mathrm{Edges}}=\frac{3b\left(2^{2/3}+1\right)}{2^{5/3}\sqrt{a^{2}-(a-b)^{2}}}Q^{\prime}$
(68)

Plugging in the relationship between $Q^{\prime}$ and $Q$ from (67), and
evaluating for $b=17.5~{}\mu$m, $a=35~{}\mu$m, yields
$Q^{\xi\leq\frac{a}{4}}_{\mathrm{Edges}}\sim 0.76Q$. Hence, the approximation
of a ring of charge is justified.

Next, we look at the second case, for a thinner disk. Now, $b=\xi=0.05~{}\mu$m
is significantly less than $\frac{a}{4}=35~{}\mu$m, so the amount of charge
contained in the thin sliver defined by $b$ is small compared to the total
charge on the disk. To find the charge in the region
$\xi\leq\frac{a}{4}=35~{}\mu$m we cannot only consider the ”edges” of the
disk. Indeed, since $b/a$ is small and the ratio of charge in the region
$s<b~{}$, to the charge in the rest of the disk, is small, one could consider
using only the expression for the charge density on the circular surface
$\sigma_{\mathrm{circular~{}disk}}(r)=\frac{Q^{\prime}}{4\pi
a\sqrt{a^{2}-r^{2}}}$. In any case, this expression should be valid in the
region $r<\frac{3}{4}a$. To avoid worrying about what happens near the edges
of the disk, since $b<a/4$, we can use a short-cut and calculate the charge at
a distance $\xi\geq\frac{a}{4}$ away from the edges, or in other words, the
charge in the central part of the disk. We use the same integral as used to
calculate $Q_{g}$ (above equation (67)), but integrate from $r=0$ to
$r=\frac{3}{4}a$. Thus,
$Q^{\mathrm{central}}=2\int_{0}^{3a/4}\frac{Q^{\prime}}{4\pi
a\sqrt{a^{2}-{r}^{2}}}2\pi rdr=\left(1-\sqrt{\frac{7}{16}}\right)Q^{\prime}$.
Plugging in the relationship between $Q^{\prime}$ and $Q$ from (67), and
evaluating for $b=50~{}$nm, $a=140~{}\mu$m, yields
$Q^{\mathrm{central}}=0.33Q$. The charge in the outer annular region,
$Q^{\xi\leq\frac{a}{4}}_{\mathrm{Edges}}~{}$, is $Q$ minus the charge in the
central area, in other words
$Q^{\xi\leq\frac{a}{4}}_{\mathrm{Edges}}=Q-0.33Q=0.67Q$. In this case our
criterion for ”most of the charge” ($75\%$) being in the annular segment
covered by $\xi\leq\frac{a}{4}$ is not satisfied, but it is not too far off.
If we calculate $Q^{\mathrm{central}}$ for the region from $r=0$ to $r=a/2$,
we find
$Q^{\mathrm{central}}=\left(1-\sqrt{3/4}\right)Q^{\prime}=0.13~{}Q^{\prime}$
$\approx 0.13~{}Q~{}$, which means over $86\%$ of the charge is contained in
the region $r\geq a/2$. Thus, although the approximation that charge
distributes in a ’ring’ becomes less accurate for thinner disks, the
approximation is still reasonably well justified.

The analysis above can be supplemented with an additional observation. We
showed that more than $50~{}\%$ of the charge lies in the $r\geq a/2$ region,
but this region also accounts for more than $50~{}\%$ of the surface-area of
the disk. Our calculations do not inform us about the charge density
throughout the disk. For this we recall the expression for the charge density,
which in the limit of a thin disk ($Q^{\prime}\approx Q~{}$) is
$\sigma_{\mathrm{circular~{}disk}}(r)=\frac{Q}{4\pi a\sqrt{a^{2}-r^{2}}}~{}$
McDonald (2002); Thomson (Lord Kelvin) . This expression depends on the radial
distance $r$ from the center of the disk, and increases continuously as $r$
approaches $a$. It guarantees the charge density is not, for instance,
homogeneous across the disk. Although we do not know a priori whether
representing the surface-charge distribution on a conducting disk as a ring is
more accurate than representing it as a homogeneous distribution, we can say
by looking at the charge density that the ring of charge model is both
plausible and consistent with known results.

To put numbers to the theoretical results above, we calculate an average
charge density for the two regions, ($r=3a/4$ to $r=a$), and ($r=a/2$ to
$r=a$), using the expression for a thin disk. The area of the first region is
$\pi a^{2}-\pi\left(\frac{3}{4}a\right)^{2}=\pi
a^{2}\left(\frac{7}{16}\right)$. The charge density in this region is then
$\rho^{\xi\leq\frac{a}{4}}=\frac{16}{7}\times 0.67~{}Q/\left(\pi
a^{2}\right)$. The area of the second region is $\pi
a^{2}\left(\frac{12}{16}\right)$. The charge density in this region is
$\rho^{\xi\leq\frac{a}{2}}=\frac{16}{12}\times 0.86~{}Q/\left(\pi
a^{2}\right)$. Since $\frac{16}{7}\times 0.67=1.53$, while
$\frac{16}{12}\times 0.86=1.15$, the charge density nearer to the edge of the
disk is greater than the charge density closer to the center of the disk.

##  Appendix B The distribution of charge on connected conducting self-
capacitors

###  B.1 Two connected metallic self-capacitors

To help guide intuition, a system of two capacitors connected by a wire with
zero capacitance can be mapped to a system of two springs anchored to two
walls on either side, and joined together in the middle. We imagine that the
springs are initially disconnected from one another, and then pulled together,
attached, and left to equilibrate. Therefore, at equilibrium both springs are
somewhat stretched. The conversion of quantities from a system of springs to a
system of electrical components is as follows:

$\vec{F}=\pm
k\Delta\vec{x}\qquad\longleftrightarrow\qquad\vec{V}=\pm\left(\frac{1}{C}\right)\vec{Q}$
(69)

Specifically, $\vec{F}\leftrightarrow\vec{V}$, $k\leftrightarrow\frac{1}{C}$,
and $\Delta\vec{x}\leftrightarrow\vec{Q}$. Here,
$\vec{V}\mathrm{and}~{}\vec{Q}$ are described as vectors to denote that they
can be positive or negative along a one-dimensional number line. The $\pm$
denotes that the direction of the force depends on which side of the spring
one displaces, while keeping in mind that $\Delta\vec{x}$ is always measured
as a positive number (see figure 9). The leftward force due to the leftmost
spring is $\vec{F}_{a}=-(k_{a}\Delta\vec{x}_{a})$, while the right-ward force
due to the rightmost spring is $\vec{F}_{c}=k_{c}\Delta\vec{x}_{c}$, where
$\Delta\vec{x}_{a}$ and $\Delta\vec{x}_{c}$ are both positive numbers measured
along the same $x$-axis. At equilibrium the leftward force and the rightward
force must have the same magnitude and opposite direction, so:
$\vec{F_{a}}=-\vec{F_{c}}$. Hence,
$k_{a}\Delta\vec{x_{a}}=k_{c}\Delta\vec{x_{c}}$, which translates to
$\frac{1}{C_{a}}\vec{Q}_{a}=\frac{1}{C_{c}}\vec{Q}_{c}$. Thus, $C_{a}=C_{c}$
implies $\vec{Q}_{a}=\vec{Q}_{c}$.

Figure 9: Two connected springs. (Top) Two springs attached to walls are
disconnected and in a relaxed position. (Bottom) The two springs are brought
together, attached, and then allowed to relax to the equilibrium state shown.
The notations $\vec{F_{1}}$ and $\Delta\vec{x}_{1}$ in the figure correspond
to $\vec{F_{a}}$ and $\Delta\vec{x}_{a}$ in the text, and similarly for
$\vec{F_{2}}$.

###  B.2 Three connected metallic self-capacitors

The same reasoning is followed as in section B.1, but now with an additional
third spring placed between spring $a$ and spring $c$, called spring $b$. This
leads to the following two equations:

$k_{a}\Delta\vec{x_{a}}=k_{\mathrm{eq_{1}}}\left(\Delta\vec{x_{b}}+\Delta\vec{x_{c}}\right)$
(70)
$k_{c}\Delta\vec{x_{c}}=k_{\mathrm{eq_{2}}}\left(\Delta\vec{x_{a}}+\Delta\vec{x_{b}}\right)$
(71)

The notation $k_{\mathrm{eq_{1}}}$ refers to the equivalent spring constant of
the two springs $k_{b}$ and $k_{c}$ in series, and
$\left(\Delta\vec{x_{b}}+\Delta\vec{x_{c}}\right)$ is the total distance by
which this equivalent spring is stretched.212121The subscript in
$k_{\mathrm{eq_{1}}}$ stands for ”equivalent”, not to be confused with the
subscript in $d_{\mathrm{eq.}}$ in other parts of the manuscript, which stands
for ”equilibrium”. Solving (71) for $\Delta\vec{x_{b}}$ gives
$\Delta\vec{x_{b}}=\frac{k_{c}\Delta\vec{x_{c}}}{k_{\mathrm{eq2}}}-\Delta\vec{x_{a}}$,
which can be inserted into equation (70). Re-arranging (70) then leads to:

$\Delta\vec{x_{c}}=\Delta\vec{x_{a}}\left(1+\frac{k_{a}}{k_{\mathrm{eq1}}}\right)\left(\frac{1}{\frac{k_{c}}{k_{\mathrm{eq2}}}+1}\right)~{}.$
(72)

Writing out $k_{\mathrm{eq1}}=\frac{k_{b}k_{c}}{k_{b}+k_{c}}$,
$k_{\mathrm{eq2}}=\frac{k_{a}k_{b}}{k_{a}+k_{b}}$ and substituting
$\frac{1}{C_{i}}$ with $i=a,b,c$ for $k_{i}$, and $Q_{i}$ for
$\Delta\vec{x_{i}}$ gives (after simplification):

$Q_{c}=Q_{a}\left(1+\frac{C_{c}+C_{b}}{C_{a}}\right)\left(\frac{1}{1+\frac{C_{b}+C_{a}}{C_{c}}}\right)$
(73)

which further reduces to $Q_{c}=Q_{a}\frac{C_{c}}{C_{a}}$. If $C_{a}=C_{c}$,
then $Q_{a}=Q_{c}$, the same as was concluded in section B.1. Similarly, a
relationship can be derived between $\Delta\vec{x_{c}}$ and
$\Delta\vec{x_{b}}$,

$\Delta\vec{x_{c}}=\Delta\vec{x_{b}}\left(1+\frac{k_{a}}{k_{\mathrm{eq1}}}\right)\left(\frac{1}{\left(\frac{k_{a}}{k_{\mathrm{eq1}}}\right)\frac{k_{c}}{k_{\mathrm{eq2}}}-1}\right)~{}.$
(74)

This leads to the relationship between $Q_{c}$ and $Q_{b}$:

$\displaystyle Q_{c}=~{}$ $\displaystyle
Q_{b}\left(1+\frac{C_{c}+C_{b}}{C_{a}}\right)$ (75)
$\displaystyle~{}~{}~{}~{}~{}~{}\times~{}\left(\frac{1}{\left(1+\frac{C_{b}}{C_{c}})\right)\left(1+\frac{C_{b}}{C_{a}}\right)-1}\right)~{},$

which further reduces to $Q_{c}=Q_{b}\frac{C_{c}}{C_{b}}$.

###  B.3 Expressing the charge $Q_{c}$ in terms of $Q_{\mathrm{transf.}}$

Section B.2 provides expressions for $Q_{c}$ in terms of $Q_{a}$, and for
$Q_{c}$ in terms of $Q_{b}$. However, the final intent is to find the charge
deficit (or surplus) that results on disk2, when ion#1 moves closer (or
farther away from) disk1. What is known is the amount of charge
$Q_{\mathrm{transf.}}$ which would flow onto disk1, if it were connected to
ground. This provides the total energy brought to disk1 by the movement of
ion#1, $E=\frac{Q^{2}_{\mathrm{transf.}}}{2\left(C_{a}+C_{b}+C_{c}\right)}$,
where $C_{a}=C_{\mathrm{disk1}}$, $C_{b}=C_{\mathrm{wire}}$, and
$C_{c}=C_{\mathrm{disk2}}$. What remains unknown is how this energy will
distribute among $C_{a}$, $C_{b}$, and $C_{c}$, when the system adjusts to the
new potential. (Presumably this adjustment occurs adiabatically, since the
resistance $R_{\mathrm{wire}}$ and inductance $L_{\mathrm{wire}}$ is low
enough for the charge to flow between the capacitors much more quickly than
the period of one oscillation. See subsection I.3, and appendices E and H).
Specifically, the aim is to determine the amount of charge which ends up on
$C_{c}$, which determines the strength of the resulting force on ion#2 when
ion#1 moves. From the results of section B.2, it is now possible to express
$Q_{c}$ in terms of $Q_{\mathrm{transf.}}$. With
$C_{\mathrm{tot.}}\equiv\left(C_{a}+C_{b}+C_{c}\right)~{}$,

$E=\frac{Q^{2}_{\mathrm{transf.}}}{2C_{\mathrm{tot.}}}=\frac{Q^{2}_{a}}{2C_{a}}+\frac{Q^{2}_{b}}{2C_{b}}+\frac{Q^{2}_{c}}{2C_{c}}~{}.$
(76)

Using (73) and (75), and the notation $\Lambda$ and $\Upsilon$ for the
portions which depend on capacitance in (73) and (75), respectively (so that,
e.g. $Q_{c}=Q_{a}\Lambda$) this becomes

$\frac{Q^{2}_{\mathrm{transf.}}}{2C_{\mathrm{tot.}}}=\frac{Q^{2}_{c}}{2}\left(\frac{1}{C_{a}\left(\Lambda^{2}\right)}+\frac{1}{C_{b}\left(\Upsilon^{2}\right)}+\frac{1}{C_{c}}\right)~{},$

which re-arranges to

$Q_{c}=\frac{1}{\sqrt{\left(\frac{1}{C_{a}\left(\Lambda^{2}\right)}+\frac{1}{C_{b}\left(\Upsilon^{2}\right)}+\frac{1}{C_{c}}\right)C_{\mathrm{tot.}}}}Q_{\mathrm{transf.}}~{},$
(77)

and further simplifies to:
$Q_{c}=\left(C_{c}~{}/~{}\left(C_{a}+C_{b}+C_{c}\right)\right)Q_{\mathrm{transf.}}$.
This can be summarized succinctly as $Q_{c}=\zeta Q_{\mathrm{transf.}}$. In
the limiting case $C_{a}=C_{c}$ and $C_{b}=0$, this reduces to
$Q_{c}=\frac{1}{2}Q_{\mathrm{transf.}}$. Therefore, equation (77) taken
together with equation (73) shows that in this limiting case, the charge
deficit (or surplus) on $C_{a}$ is equal to the charge deficit (or surplus) on
$C_{c}$ and the amount of charge deficit (surplus) on each is
$Q_{a}=Q_{c}=\frac{1}{2}Q_{\mathrm{transf}}$. Note that the absolute amount of
charge on the pickup-disks-and-wire system never changes; the system is never
in contact with an external source (or sink) of charges. If it is assumed that
initially the net charge is zero, then it remains permanently zero.

###  B.4 Expressing the charge $Q_{a}$ in terms of $Q_{\mathrm{transf.}}$

It is also possible to express the charge induced on disk1, $Q_{a}$, in terms
of $Q_{\mathrm{transf.}}$. We start with the expression for the total energy
induced on the coupling system, as in equation (76):

$E=\frac{Q^{2}_{\mathrm{transf.}}}{2C_{\mathrm{tot.}}}=\frac{Q^{2}_{a}}{2C_{a}}+\frac{Q^{2}_{b}}{2C_{b}}+\frac{Q^{2}_{c}}{2C_{c}}~{}.$

This can be rewritten by substituting the relationships derived in section
B.2, $Q_{b}=Q_{c}\frac{C_{b}}{C_{c}}$ and $Q_{c}=Q_{a}\frac{C_{c}}{C_{a}}$, as

$\frac{Q^{2}_{\mathrm{transf.}}}{2C_{\mathrm{tot.}}}=\frac{Q^{2}_{a}}{2C_{a}}+\frac{\left(Q_{c}\frac{C_{b}}{C_{c}}\right)^{2}}{2C_{b}}+\frac{\left(Q_{a}\frac{C_{c}}{C_{a}}\right)^{2}}{2C_{c}}~{}.$

Substituting in $Q_{c}=Q_{a}\frac{C_{c}}{C_{a}}$ again, and simplifying, this
becomes

$\frac{Q^{2}_{\mathrm{transf.}}}{2C_{\mathrm{tot.}}}=\frac{Q^{2}_{a}}{2C_{a}}+\frac{Q^{2}_{a}C_{b}}{2C^{2}_{a}}+\frac{Q^{2}_{a}C_{c}}{2C^{2}_{a}}~{}.$

Multiplying the first term by $C_{a}/C_{a}$ and then pulling out the factor of
$Q^{2}_{a}/2C^{2}_{a}$ which is common to all three terms leads to

$\frac{Q^{2}_{\mathrm{transf.}}}{2C_{\mathrm{tot.}}}=\frac{Q^{2}_{a}}{2C^{2}_{a}}\left(C_{a}+C_{b}+C_{c}\right)~{}.$

Rearranging gives

$\frac{Q^{2}_{\mathrm{transf.}}C^{2}_{a}}{\left(C_{a}+C_{b}+C_{c}\right)^{2}}=Q^{2}_{a}~{}.$

Finally, after taking the square root this becomes

$Q_{a}=\left(\frac{C_{a}}{C_{a}+C_{b}+C_{c}}\right)Q_{\mathrm{transf.}}=\eta~{}Q_{\mathrm{transf.}}~{},$
(78)

where we have defined the coefficient $\eta\equiv
C_{a}/\left(C_{a}+C_{b}+C_{c}\right)$. Comparing the coefficient $\eta$ from
equation (78) with the coefficient $\zeta$ defined below equation (77), we see
that $\eta$ and $\zeta$ are identical in the case of a symmetrical coupling
system where $C_{a}=C_{c}$.

##  Appendix C Neglecting the ring of induced charge in the ground plane

In section II.3 we observed that the movement of ion#1 induces a ring of
charge on disk2, and this ring of charge induces a second ring of opposite
charge in the infinite conducting plane surrounding disk2. The purpose of this
appendix is to argue that this second ring can be neglected when calculating
the force on ion#2 due to the movement of ion#1. Hence, the second ring of
induced charge can be neglected in the calculation of the coupling strength
$\gamma$.

Suppose a tube is drawn around the ring of charge induced on the disk (called
the ”first” or ”smaller-radius” ring), forming a torus. By Gauss’s law the
total electric field flux through this tubular surface is equivalent to the
total net charge contained inside of it. If one small segment of the tube is
considered, it can be approximated as a cylindrical surface surrounding a
cylindrical rod. Let ”$a$” be the radius of this cylindrical surface (the
distance from the core of the tube, to the surface of the tube). Let $\theta$
be the angle which wraps around the core of the tube. Because we are looking
at a small ”cylindrical” segment, we can make the approximation that the
outward flux away from the core of the ring is homogeneous in all directions.
(This assumption is valid if the gap between the disk and the infinite plane
is large compared to the height and radius of the disk. If the gap is small,
induced charges outside of the disk can cause charges near the edge of the
disk to distribute unevenly.) Since the angle $\theta$ through which flux
lands on the second ring is small compared to $2\pi$, the fraction of the
total charge on the first ring which is induced on the outer conducting plane
must necessarily be small. If it were considered, it would result in a slight
reduction in the coupling between ion#1 and ion#2.

Note that mathematically, even the slightest tilt of the infinite plane in
which disk2 is etched will result in the full charge $-Q_{\mathrm{ring}}$
being induced on the infinite plane. However, in an infinite plane which is
only slightly tilted, the majority of the induced charges are induced at a
location in the plane which is far away from the inner ring, and from ion#2.
The contribution of these charges to the force on ion#2 is negligible.

##  Appendix D Optimizing the coupling strength $\gamma$ in the limit of large
wire capacitance

The problem to solve is $\frac{\partial}{\partial r}\gamma\left(r\right)=0$ in
the limit $C_{\mathrm{b}}\gg 2C_{\mathrm{disk}}$. Explicitly,

$\frac{\partial}{\partial
r}\left[\frac{q^{2}_{\mathrm{c}}}{4\pi\epsilon_{\mathrm{o}}}\left(\frac{r}{2r+\frac{C_{b}}{8\epsilon_{\mathrm{o}}}}\right)\left(\frac{dr^{2}}{\left(d^{2}+r^{2}\right)^{3}}\right)\right]=0~{}.$

In the limit $C_{\mathrm{b}}\gg 2\times 8\epsilon_{\mathrm{o}}r$, this becomes

$\frac{\partial}{\partial
r}\left(\frac{dr^{3}}{\left(d^{2}+r^{2}\right)^{3}}\right)=0~{}.$

This reduces to