# Sympathetic cooling schemes for separately trapped ions coupled via image
currents

C<EMAIL_ADDRESS>M. Bohman222Present address: High
Energy Physics Division, Argonne National Laboratory, 9700 S Cass Ave, Lemont,
IL 60439  T. Driscoll  M. Wiesinger  F. Abbass  M. J. Borchert  J. A. Devlin
S. Erlewein  M. Fleck  B. Latacz  R. Moller  A. Mooser  D. Popper  E. Wursten
K. Blaum  Y. Matsuda  C. Ospelkaus  W. Quint  J. Walz  C. Smorra  S. Ulmer

(August 28, 2024)

###### Abstract

Cooling of particles to mK-temperatures is essential for a variety of
experiments with trapped charged particles. However, many species of interest
lack suitable electronic transitions for direct laser cooling. We study
theoretically the remote sympathetic cooling of a single proton with laser-
cooled 9Be+ in a double-Penning-trap system. We investigate three different
cooling schemes and find, based on analytical calculations and numerical
simulations, that two of them are capable of achieving proton temperatures of
about 10 mK with cooling times on the order of 10 s. In contrast, established
methods such as feedback-enhanced resistive cooling with image-current
detectors are limited to about 1 K in $100$ s. Since the studied techniques
are applicable to any trapped charged particle and allow spatial separation
between the target ion and the cooling species, they enable a variety of
precision measurements based on trapped charged particles to be performed at
improved sampling rates and with reduced systematic uncertainties.

##  1 Introduction

The ability to prepare (ultra-)cold particles in a controlled environment
through laser cooling has been a major driving force in the enhanced precision
of recent experiments in the field of atomic, molecular and optical physics
[1]. However, many particle species of interest offer no suitable transitions
for direct laser cooling, so that for these particles sympathetic cooling
techniques have to be employed [2, 3, 4, 5, 6, 7]. Established sympathetic
cooling schemes rely on Coulomb interactions between the target particle and
the laser-cooled ions. In order to obtain a sufficiently high coupling rate,
the two species must either be co-trapped in the same potential well [7, 8] or
trapped in separate potential wells [9, 10] that enable particles distances of
$10-$100\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$. However, trap
designs that enable these length scales and the Coulomb interaction between
the particle species impose large systematic shifts on potential precision
measurements in Penning traps [11].

One coupling technique which is not accompanied by these drawbacks is image-
current coupling. Here, the charged particle species are coupled through image
currents induced in adjacent trap electrodes, allowing a macroscopic
separation of the target species and cooling species. Recently, our group has
published the first experimental results on sympathetically cooling a single
proton in a Penning trap by coupling it to a cloud of laser-cooled 9Be+ ions
located in a separate trap [12]. In this experiment, the coupling between the
ions was mediated by a common superconducting RLC resonator, which is usually
used for particle detection [13]. A temperature reduction of the axial mode
from $17.0\,(2.4)$ K to $2.6\,(2.5)$ K was achieved. Other recent experimental
efforts have demonstrated the coupling between two highly charged ions with a
comparable experimental setup [3] or the coupling of two laser-cooled ions
stored in a Paul trap by a floating room-temperature wire [14].

The recent developments [12] will be one of the cornerstones for the next
generation of precision measurements at low energies. For example, our
collaboration (BASE) probes the fundamental charge, parity and time (CPT)
reversal symmetry by performing ultra-high-precision Penning-trap measurements
on protons and antiprotons [15]. In particular, their charge-to-mass ratio is
compared at a relative precision of $69\text{\times}{10}^{-12}\text{\,}$ [16]
and their magnetic moments are measured with fractional resolutions of
$3\text{\times}{10}^{-10}\text{\,}$ [17] and
$1.5\text{\times}{10}^{-9}\text{\,}$ [18], respectively; all being the most
precise measurements of these quantities to date. However, especially the
magnetic moment measurements are limited by the nonzero particle temperature
of $\gtrsim 1$ K. Other experimental programs that are subject to comparable
limitations include the spectroscopy of highly-charged ions [8, 19], the
cooling of antiprotons and positrons for the synthesis of cold antihydrogen
[5, 20, 21], mass measurements of heavy elements [6, 22], or high-precision
metrology with atomic clocks [7, 24].

In this work, we study image-current coupling mechanisms by means of combined
analytical calculations and numerical simulations. Since the particles and the
RLC resonator constitute a complex stochastic system of three coupled
oscillators, we use the numerical simulations to study the dynamical behavior
and optimize cooling times and temperature limits. The relevant physics is
implemented into the simulation code from first principles and experimentally
established models only. After demonstrating agreement between our simulation
results and existing experimental data, we assess the feasibility and
performance of three different cooling schemes. We find that with two of the
techniques, coupling via a common electrode and off-resonant coupling via a
common RLC circuit, temperatures of $\approx$ 10 mK can be reached with
cooling time constants on the order of $5-20$ s. Compared to the established
method of resistive cooling with an RLC resonator at $4-20$ K [25], this
constitutes a temperature reduction by a factor of $\approx 100$ and a cooling
time reduction by a factor of $\approx 10$. The third technique, on-resonance
coupling with a common RLC resonator, can provide cooling time constants below
1 s but is limited to particle temperatures in the hundred mK-regime with the
current experimental setup.

Although applicable to many systems, we find the image-current cooling
technique to be of immediate relevance to ultra-high precision measurements in
Penning traps and therefore motivate the requirements on the cooling
performance with a future measurement of the proton magnetic moment.

##  2 Magnetic moment measurements in a Penning trap

###  2.1 Penning-trap basics

In a Penning trap, charged particles are trapped radially by a constant
homogeneous magnetic field, in cylindrical coordinates $(\rho,\varphi,z)$
oriented in the axial direction,

$\displaystyle\vec{B}=B_{0}\,\vec{e}_{z}\text{,}$ (1)

and axially by a harmonic electrostatic potential,

$\displaystyle\phi(\rho,z)=C_{2}U_{0}(z^{2}-\rho^{2}/2)\text{,}$ (2)

usually created by a stack of five cylindrical electrodes in a compensated,
orthogonal design [26]. $C_{2}$ is a geometrical parameter defined by the trap
size and $U_{0}$ the electrical potential difference between the endcap and
the central ring elecrode [27]. A simplified schematic is shown in Fig. 1(a).
If higher-order contributions to the magnetic field and axial potential are
neglected, the axial mode in $z$-direction is independent of the other
coordinates and has frequency

$\displaystyle\nu_{\text{z}}=\frac{1}{2\pi}\sqrt{\frac{2qC_{2}U_{0}}{m}},$ (3)

where $q$ and $m$ are the charge and the mass of the particle, respectively.
The radial motional modes split up into a slow and fast mode, referred to as
magnetron ($\nu_{-}$) and modified cyclotron ($\nu_{+})$ modes. At
$B_{0}\approx 2\,$T and $U_{0}\approx 1-10$ V, typical values for the
frequencies of the proton in the trap system used in our experiments are
$\nu_{z}\approx 600$ kHz, $\nu_{-}\approx 7$ kHz and $\nu_{+}\approx 29$ MHz.

The motion of the particle induces image currents in the trap electrodes that
permit a non-destructive measurement of the particle’s motional frequencies.
Fundamental properties of the trapped particle such as its charge-to-mass
ratio and magnetic moment can be derived from these frequency measurements
[28]. The axial mode is most commonly detected [29], for which the magnitude
of the induced image current is

$\displaystyle I_{\text{ind}}=\frac{q}{D}\dot{z}.$ (4)

Here $D$ is a specific length related to the trap size and the chosen pickup
geometry [29, 30] and can be considered an “effective electrode distance”.
$\dot{z}$ is the particle’s axial velocity. Detectors for these currents
consist of ultra low-noise amplifiers connected to superconducting NbTi coils,
in our experiment with inductances $L$ of several mH, which are cooled down to
$4.2\text{\,}\mathrm{K}$ by a liquid helium bath [13, 31]. Due to parasitic
effects, the coil needs to be modelled by an additional capacitance
$C_{\text{res}}$ and a resistance $R$, forming a parallel RLC circuit with
eigenfrequency $\nu_{0}$ as depicted in Fig. 1(a). The quality factor
$Q=R/(2\pi\nu_{0}L)$ of such an axial resonator is typically on the order of
$20\,000\text{\,}$, including limitations caused by the input resistance of
the cryogenic amplifiers and parasitic resistive losses [13, 31]. The thermal
Johnson-Nyquist noise of the parallel resistance,
$\braket{U_{\text{noise}}^{2}}=4k_{B}T_{\text{res}}R\Delta f$ for a frequency
interval $\Delta f$, charges up the whole RLC circuit and a resonance signal
at the circuit’s eigenfrequency is created.

By connecting this resonator to a trap electrode, a trapped charged particle
is exposed to the noisy oscillations of the RLC circuit and the axial energy
dynamically samples a Boltzmann distribution with temperature
$T_{\text{res}}$. The coupling strength between ion and resonator depends
among other parameters on the ion’s frequency detuning relative to the
resonator. On resonance, where the coupling is strongest, the particle
effectively shorts the resonator noise, creating a characteristic dip in the
fast Fourier transform (FFT) spectrum [29]. An example of such a spectrum is
shown in Fig. 3 of Sec. 5.

In the context of our studies, we will refer to energy as the total oscillator
energy, which is, although given in units of $\text{K}\cdot k_{B}$, defined
deterministically for every point in time. For example, for the axial mode of
a particle,
$E_{\text{z}}=\left(qC_{2}U_{0}z^{2}+\frac{1}{2}m\dot{z}^{2}\right)$. In
contrast, we will refer to temperature as the average energy of the oscillator
divided by the Boltzmann constant, i.e. $T_{\text{z}}=\braket{E_{z}}/k_{B}$.

(a) (b)

Figure 1: (a) Simplified schematic of a cylindrical Penning trap. The axial
motion of the ion induces image currents in the trap electrodes, which are
detected by means of a resonant circuit with high quality factor. (b)
Simulated axial frequency data for low and high modified cyclotron
temperatures using the model described in the text. The blue lines indicate
the axial frequency without fluctuations from modified cyclotron transitions.
A spin flip corresponding to a $\approx 170$ mHz frequency shift occurs at
simulation number 20 and 80. The spin flip is easily resolved at a low
modified cyclotron temperature. In contrast, at high $T_{+}$ the superimposed
fluctuations prevent an unambiguous identification of a spin flip. The last
magnetic moment measurement of the proton [17] used $T_{\text{thresh}}=600$
mK.

###  2.2 Magnetic moment measurements and the need for advanced cooling
techniques

Penning-trap based magnetic moment measurements of the proton and antiproton
determine the free cyclotron frequency
$\nu_{\text{c}}=\frac{1}{2\pi}\frac{q}{m}B$ and the spin precession frequency
(Larmor frequency) $\nu_{\text{L}}=\frac{1}{2\pi}\frac{g}{2}\frac{q}{m}B$
simultaneously. The ratio of the two quantities is independent of the magnetic
field and charge-to-mass ratio of the particle and yields the dimensionless
$g$-factor:

$\displaystyle g=2\frac{\nu_{\text{L}}}{\nu_{\text{c}}}.$ (5)

$\nu_{\text{c}}$ is determined using the Brown-Gabrielse invariance theorem
$\nu_{\text{c}}=\sqrt{\nu_{+}^{2}+\nu_{\text{z}}^{2}+\nu_{-}^{2}}$ [27] and
measuring $\nu_{+}$, $\nu_{\text{z}}$, $\nu_{-}$ via image current techniques
and sideband coupling [32] or phase-sensitive detection [33]. However, a
direct measurement of the Larmor frequency is not possible as the spin
orientation cannot be detected by image currents in the trap electrodes.
Instead, spin flips are detected by utilizing the continuous Stern-Gerlach
effect [15]. Here, a strong magnetic inhomogenity is superimposed to the
homogeneous field, $B_{\text{z}}(\rho,z)=B_{0}+B_{2}(z^{2}-\rho^{2}/2)$, where
$B_{2}$ characterizes the strength of the magnetic inhomogeneity. This results
in the axial frequency becoming a function of both the energy of the radial
modes as well as the spin state of the trapped particle. Spin flips can be
induced by applying an external rf drive with frequency close to
$\nu_{\text{L}}$. Measuring the spin-flip probability as a function of the
drive frequency yields a resonance line from which $\nu_{\text{L}}$ can be
extracted [34]. However, due to frequency shifts caused by the nonzero mode
temperatures, the strong magnetic bottle poses limitations to the measurement
precision of $\nu_{\text{c}}$ and $\nu_{\text{L}}$. This limitation can be
overcome by employing multi-trap techniques [35, 36, 37]: The frequency
measurements for the determination of $\nu_{\text{c}}$ and $\nu_{\text{L}}$
are conducted in the precision trap (PT) with a homogeneous magnetic field,
while the spin state is determined in the analysis trap (AT) featuring the
magnetic bottle. In the AT, despite using a strong
$B_{2}=$300\,000\text{\,}\mathrm{T}\mathrm{/}\mathrm{m}^{2}$$, a spin flip of
a single proton corresponds to an axial frequency shift of only $\approx 170$
mHz compared to $\nu_{\text{z}}\approx 600$ kHz for our experimental
parameters. As a result, the axial frequency stability is a key parameter for
high-fidelity identification of the spin-state [36, 38]. We observe that a
major contribution of axial frequency fluctuations can be attributed to
modified cyclotron quantum transitions [36]. The transition rate $\zeta_{+}$
is proportional to the mode energy:

$\displaystyle\zeta_{+}\propto n_{+}\propto E_{+}.$ (6)

Here $n_{+}$ denotes the quantum number of the cyclotron mode with energy
$E_{+}=h\,\nu_{+}(n_{+}+1/2)=k_{B}T_{+}$. The effect of noise-driven cyclotron
transitions on a sequence of axial frequency measurements or simulations is
illustrated in Fig. 1(b) for the two cases of low and high cyclotron
temperatures. At low $T_{+}$, the spin flip is easily resolved from residual
fluctuations. In contrast, at high $T_{+}$ the superimposed fluctuations
prevent a high-fidelity identification of a spin flip. Since the measurement
of $\nu_{+}$ increases the modified cyclotron energy, the preparation of the
cyclotron mode with a subthermal energy must be conducted in each double-trap
measurement cycle.

Our latest magnetic moment measurements employed the technique of resistive
feedback cooling with a cyclotron resonator [39, 40]. When coupled to the
resonator, the cyclotron mode thermalizes with the resonator and samples a
Boltzmann distribution, in the case of the proton measurement with an average
temperature of 3 K [25]. However, at the same time the acceptance threshold
$T_{\text{thresh}}$ for unambiguous spin-state determination was set to
$T_{+}\leq T_{\text{thresh}}=600$ mK, so that a time-consuming stochastic
selection of the cyclotron energy was necessary. In fact, $2/3$ of the
duration of one full experimental cycle of about 90 min were needed for the
preparation of the particle in a state of low cyclotron energy and the
subsequent spin-state detection [17]. As a consequence, the measurement of the
proton $g$-factor is limited by the statistical uncertainty, and the
development of cooling schemes that achieve colder particle temperatures in a
shorter time is crucial for overcoming the current limit in precision.
Ultimately, more powerful cooling methods would not only enable an enhanced
spin state detection fidelity, but also allow to deterministically prepare the
proton with a cold cyclotron mode in much shorter times, which is one of the
cornerstones for achieving $<10^{-10}$ fractional precision in the next
generation of (anti-)proton magnetic moment measurements.

In order to realize this boost in precision experimentally, we develop
technologies and methods to sympathetically cool the proton with a cloud of
laser-cooled 9Be+stored in a separate trap a few cm away, for which we have
recently performed the proof-of-principle measurement [12]. The coupling is
achieved through image currents induced in a common electrode or a common
resonator. Since image-current coupling requires matching an oscillation
frequency of the two species involved and the cyclotron and magnetron
frequencies are, in contrast to the axial frequency, not easily adjustable in
our experiment, we currently apply the sympathetic cooling technique to the
axial mode. The subsequent coupling between the cyclotron mode and axial mode
is usually performed via a quadrupolar radiofrequency drive (sideband
coupling) [32]. However, with this technique the mode temperatures are related
by $T_{+}=\frac{\nu_{+}}{\nu_{\text{z}}}T_{\text{z}}$, in our case
$T_{+}\approx 60\,T_{\text{z}}$ for the proton. In order to achieve a
significantly improved cyclotron mode preparation at the previous threshold of
$T_{+}\leq 600$ mK, cooling techniques that reach axial proton temperatures of
$T_{\text{z}}\lesssim 10$ mK are required. For example, $T_{\text{z}}=10$ mK
axial temperature would enhance the success rate of the cyclotron energy
selection from $\approx 18$% to $\approx 63$%. Furthermore, the temperature
distribution after the $\leq 600$ mK-selection is shifted towards lower
temperatures, leading to an improved spin-state detection fidelity and a
reduction in time for determining the spin state. Since the main objective is
to reduce the experimental cycle time and the cyclotron mode needs to be
prepared for each measurement cycle, the cooling scheme itself should require
less than one minute. Apart from the cooling performance, the technique must
be applicable to both protons and antiprotons and should not induce relative
systematic frequency shifts larger than $1\times 10^{-11}$.

##  3 Sympathetic cooling techniques for a single proton

One technique which satisfies the challenging requirements described in Sec. 1
and Sec. 2.2 is image-current coupling, where the proton is sympathetically
cooled by a cloud of laser-cooled 9Be+stored in a separate trap and the
particles are coupled through image currents. A trap stack consisting of five
traps was developed for our experiment, of which four are used to implement
the coupling schemes and one is used for spin state analysis [12, 41, 42].
From an instrumentation point of view, our apparatus allows two different
types of image-current couplings, namely coupling via a common endcap
electrode or coupling via a common RLC resonator. The experimental
configurations are depicted schematically in Fig. 2 and described in more
detail in Refs. [2, 12, 43].

(a) (b)

Figure 2: (a) Sketch of the experimental setup for the common-endcap coupling.
The common endcap electrode can be modelled as a ideal capacitor. (b) Sketch
of the experimental setup for the common-resonator coupling. The proton and
9Be+ ions are stored in two independent traps which share a common resonator.

When performing the common-endcap coupling (Fig. 2(a)), the proton and
beryllium ions are stored in the coupling trap (CT) and beryllium trap (BT),
respectively. These traps are designed such that they share a common endcap
with which the coupling scheme in the proposals of Refs. [2, 43] can be
realized. The image current induced by the particles charges up the
capacitance of the common-endcap electrode. The resulting voltage generates a
force on the particles, so that a system of two coupled oscillators is
created. By laser cooling the beryllium ions with the 313 nm
${}^{2}\text{S}_{1/2}\rightarrow{{}^{2}\text{P}_{3/2}}$ transition (natural
linewidth: $\approx 2\pi\times$20\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$$),
they act as a heat sink and the proton will be sympathetically cooled as well.
This specific approach has the advantage that no additional source of heating
is present. Alternatively, we can trap the particles in the precision trap
(PT) and the source trap (ST), which are connected by a common RLC resonator,
as depicted in Fig. 2(b). Since the resonator is an oscillator itself, this
configuration represents a system of three coupled oscillators. The advantage
of the RLC resonator is that it amplifies the image current signal by the
$Q$-factor of the resonator, resulting in a fast energy exchange between the
particles. In turn, however, the cooling effect competes with the stochastic
heating by the noise, ultimately limiting the minimal proton temperature. This
resonator-coupling scheme was employed in the first demonstration of
sympathetic cooling through image currents recently conducted by our group
[12]. The particle-to-detector coupling rate, which governs both the heating
and cooling rate, can be reduced by detuning the particles from the resonator
eigenfrequency. The reduction in heating rate allows for colder particle
temperatures and this specific scheme will be numerically investigated in Sec.
6.3.

##  4 Numerical implementation

Next, we describe the numerical implementation of the equations governing our
simulations. The proton and 9Be+ ions induce image currents [29]

$I_{\text{res/com,p}}=\frac{q}{D_{\text{{res/com,p}}}}\dot{z}_{\text{p}}\qquad\text{and}\qquad
I_{\text{res/com,Be}}=\frac{q}{D_{\text{res/com,Be}}}\dot{z}_{\text{Be}}$ (7)

in the resonator or the common endcap electrode, denoted with subscript “res”
and “com”, respectively. In all cases $D_{\text{res/com,p/Be}}$ is a
geometrical parameter defined by the trap geometry and can be considered an
“effective electrode distance”. The common endcap is modelled as an ideal
capacitor $C_{\text{com}}$ charged by the image currents. The potential
$U_{\text{com}}$ due to the accumulated charge leads to a force on the
particles, $F_{\text{com,p/Be}}=\frac{q}{D_{\text{com, p/Be}}}U_{\text{com}}$.
Similarly, the particles are driven and damped by the oscillating voltage of
the resonator $U_{\text{RLC}}$ at a corresponding force
$F_{\text{RLC}}=\frac{q}{D_{\text{res,p/Be}}}U_{\text{RLC}}=\frac{q}{D_{\text{res,p/Be}}}L\,\dot{I}_{\text{L}}$.
The resonator itself is described by a second-order differential equation. In
addition to the ion-induced currents, the resonator is charged by the
effective thermal Johnson-Nyquist noise current $I_{\text{noise}}$. We
describe the resonator in terms of the current through the coil
$I_{\text{L}}$. The full set of coupled differential equations, expressed in
terms of input parameters of the simulation, is then:

$\displaystyle
m_{\text{p}}\,\ddot{z}_{\text{p}}+2\,q_{\text{p}}\,C_{\text{2,p}}\,U_{0,p}\,z_{\text{p}}-\frac{q_{\text{p}}}{D_{\text{res,p}}}L\,\dot{I}_{L}-\frac{q_{\text{p}}}{D_{\text{com,p}}}U_{\text{com}}$
$\displaystyle=0$ (8) $\displaystyle
m_{\text{Be}}\,\ddot{z}_{\text{Be}}+2\,q_{\text{Be}}\,C_{\text{2,Be}}\,U_{\text{0,Be}}\,z_{\text{Be}}-\frac{q_{\text{Be}}}{D_{\text{res,
Be}}}L\,\dot{I}_{L}+\frac{q_{\text{Be}}}{D_{\text{com,Be}}}U_{\text{com}}$
$\displaystyle=0$ (9) $\displaystyle
L\,C_{\text{res}}\,\ddot{I}_{L}+\frac{L}{R}\dot{I}_{L}+I_{L}+I_{\text{noise}}+\frac{q_{\text{p}}}{D_{\text{res,p}}}\dot{z}_{\text{p}}+\frac{q_{\text{Be}}}{D_{\text{res,Be}}}\dot{z}_{\text{Be}}$
$\displaystyle=0$ (10) $\displaystyle
U_{\text{com}}=\frac{1}{C_{\text{com}}}\int\left(\frac{q_{\text{p}}}{D_{\text{com,p}}}\dot{z}_{\text{p}}-\frac{q_{\text{Be}}}{D_{\text{com,
Be}}}\dot{z}_{\text{Be}}\right)\text{dt}.$ (11)

Here, the first two equations describe the axial motion of the particles with
external forces $F_{\text{RLC}}$ and $F_{\text{com}}$ and the latter two the
resonator and common endcap capacitance. The subscripts p and Be denote the
quantities related to the proton and the 9Be+ ion(s), respectively.

We neglect the simulation of the radial modes since residual magnetic and
electric field inhomogeneities which would couple the axial and radial modes
are sufficiently small in our experiment. Furthermore, we neglect Coulomb-
interaction between the beryllium ions since only the common mode of the
beryllium cloud interacts with the resonator, as we will outline in Sec. 6.2.

We solve Eq. (8)$-$(11) numerically with typical time steps of $\Delta t=1-5$
ns compared to $\approx$600\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$
oscillator frequency. The time steps are chosen such that effects which can be
attributed to the finite time step are negligibly small and do not affect the
coupling studies while still ensuring a reasonable computation time. If the
motion of more than one beryllium ion is simulated, either an effective cloud
or an array of individual beryllium ions is calculated, depending on whether
the single particle behaviour is of interest. A key ingredient of our
simulation code is the proper implementation of $I_{\text{noise}}$. Eq. (10)
is a type of Langevin equation where the resistance $R$ determines, according
to the fluctuation-dissipation-theorem, both the dissipation and noise terms.
In this case $I_{\text{noise}}$ obeys a delta-correlated Gaussian probability
distribution with zero mean. The mean squared value of the single-sided power
spectral density is given by
$\left<I_{\text{noise}}^{2}\right>=4k_{B}T_{\text{res}}\Delta f/R$. In the
time domain representation, the bandwidth $\Delta f$ must be substituted by
the Nyquist frequency, which is equal to half of the sampling frequency
$f_{s}=1/\Delta t$ according to the Nyquist–Shannon sampling theorem, yielding
$\Delta f=1/(2\Delta t)$. Furthermore, in order to achieve a delta-correlation
of two subsequent noise samples, each noise value is multiplied with a random
value drawn from a Gaussian distribution with a mean of 0 and a standard
deviation of 1, abbreviated as $\mathcal{N}_{\text{n}}(0,1)$. In total, in
each step $n$ the equation for the noise sample then reads

$I_{\text{noise,n}}=\sqrt{2k_{B}T_{\text{res}}/(R\Delta
t)}\cdot\mathcal{N}_{\text{n}}(0,1).$ (12)

Each equation in Eq. (8)-(11) is solved individually by employing a fourth-
order symplectic integrator [44]. This method conserves a Hamiltonian for
conservative forces and, although the resonator noise damps and excites the
particles, avoids long-term drifts and reproduces energy conservation in the
limit $R\rightarrow\infty$. While the integrator can be applied directly to
the equations of motion of the particles, the integrator for the resonator
equation requires a modification due to the dissipative term. Following the
notation in Ref. [44], one step for the discretized RLC circuit equation then
reads:

$\displaystyle I_{\text{n}}^{(i)}$
$\displaystyle=I_{\text{n}}^{(i-1)}+c_{i}\dot{I}_{\text{n}}^{(i-1)}\Delta t$
(13) $\displaystyle\dot{I}_{\text{n}}^{(i)}$
$\displaystyle=\frac{1}{1+\frac{d_{i}\Delta
t}{RC}}\dot{I}_{\text{n}}^{(i-1)}-\frac{1}{1+\frac{d_{i}\Delta
t}{RC}}\frac{d_{i}\Delta
t}{LC}\left(I_{\text{n}}^{(i-1)}+I_{\text{ext}}\right).$

Here $i\in\\{1,2,3,4\\}$ denotes the sub-steps where $I_{\text{n}}^{(0)}\equiv
I_{\text{n}}$, $\dot{I}_{\text{n}}^{(0)}\equiv\dot{I}_{\text{n}}$,
$I_{\text{n}}^{(4)}\equiv I_{n+1}$ and
$\dot{I}_{\text{n}}^{(4)}=\dot{I}_{\text{n}}^{(3)}\equiv\dot{I}_{n+1}$.
Besides, $I_{\text{ext}}=I_{\text{noise}}+I_{\text{res,p}}+I_{\text{res,Be}}$
is the external current. In order for the algorithm to be symplectic, the
dimensionless coefficients $c_{i}$ and $d_{i}$, which define the length of the
interleaved time steps, must be chosen appropriately. While two out of four
coefficients are negative, i.e. correspond to a step backwards in time, their
sum obeys $\sum c_{i}=\sum d_{i}=1$.

Laser cooling of the 9Be+ ions is modelled by assuming that a narrow linewidth
laser with frequency $f_{\text{L}}$, detuned by $\delta_{\text{L}}$ relative
to the transition frequency $f_{0}=f_{\text{L}}-\delta_{\text{L}}$, interacts
with the 9Be+ ions. In each simulation step in which a 9Be+ ion is in the
ground state it can absorb a photon with probability

$\displaystyle P_{\text{abs}}=B_{12}(\dot{z}_{\text{Be}})\Delta
t=\frac{I_{\text{laser}}}{I_{\text{sat}}}\frac{\Gamma\Delta
t/2}{1+4\left(\frac{2\pi\delta_{\text{L}}+k_{\text{L}}\dot{z}_{\text{Be}}}{\Gamma}\right)^{2}}.$
(14)

$B_{12}$ and $B_{21}$ are the Einstein rate coefficients for absorption and
emission, respectively. Further, $k_{\text{L}}$ is the laser wave vector in
axial direction, $\Gamma$ the natural linewidth of the transition ($\approx
2\pi\times 20$ MHz in our case), $I_{\text{laser}}$ the laser intensity and
$I_{\text{sat}}=\frac{\pi h\Gamma f_{\text{0}}^{3}}{3c^{2}}$ the saturation
intensity [45]. The fraction $\frac{I_{\text{laser}}}{I_{\text{sat}}}$ is a
free parameter and is the simulation equivalent to the laser intensity in the
experiment. If an absorption occurs, the ion’s velocity is reduced by $\hbar
k_{\text{L}}/m_{\text{Be}}$ and it enters the excited internal electronic
state. In the excited state the ion cannot absorb photons but can
spontaneously decay to the electronic ground state with probability
$\Gamma\Delta t$. Upon spontaneous decay, the ion receives an axial velocity
kick, $\hbar k/m_{\text{Be}}\cos(\varphi)$, where $\varphi$ is randomly
distributed from 0 to 2$\pi$ to account for the randomness in the direction of
the spontaneous emission. We note that this is a simplified model as it does
not account for the anisotropy of spontaneous emission in a magnetic field
[46], which is however negligible for the studies in this paper. Additionally,
the ion can emit photons by stimulated emission with probability
$B_{21}(\dot{z}_{\text{Be}})\Delta t=B_{12}(\dot{z}_{\text{Be}})\Delta t$
where the ion’s velocity is increased by $+\hbar k_{\text{L}}/m_{\text{Be}}$.

The above considerations enable us to accurately simulate the dynamics of the
trapped particles and resonator including laser cooling of the beryllium ions
under ideal conditions. However, in practice, several experimental
uncertainties can occur. While long-term axial frequency drifts can easily be
compensated, drifts on the order of the duration of one axial frequency
measurement, typically about 60 s, lead to frequency fluctuations that affect
the energy exchange between the particles. The sources of these fluctuations
are various, for example trapping voltage instabilities and drifts of the
power supplies, finite pressure and temperature stabilities of the liquid
helium bath, drifts due to the Coulomb-interaction between the 9Be+ ions, or
residual inhomogeneities of the electric potential or magnetic field. A
realistic estimate for the axial frequency fluctuation of the proton is
$\sigma(\nu_{z})\approx 50-100$ mHz or
$\sigma(\nu_{z})/\nu_{z}\approx$1\text{\times}{10}^{-7}\text{\,}$$ at 60 s for
our experiment [25]. In the following, we summarize the individual
contributions by introducing an effective dynamical axial frequency variation
by adding a modification $\Delta U$ to the trapping potential, where $\Delta
U$ performs a random walk:

$\Delta U_{n+1}=\Delta U_{\text{n}}+\delta
U\cdot\mathcal{\tilde{N}}_{\text{n}}(0,1).$ (15)

The index $n$ indicates the step number and
$\mathcal{\tilde{N}}_{\text{n}}(0,1)$ is another number drawn from a Gaussian
distribution. $\delta U$ is a scaling factor determining the expectation value
of the standard deviation of the random walk $\sigma(U_{0})$ and is related to
the frequency stability by $\sigma(U_{0})/U_{0}=2\sigma(\nu_{z})/\nu_{z}$. In
order to express $\delta U$ in terms of practical experimental quantities, we
define $\xi_{\text{stab}}$ as the relative voltage stability for frequency
measurements of length 60 s and obtain

$\delta U_{0}=\xi_{\text{stab}}\cdot
U_{0}\cdot\sqrt{\frac{1}{n(t=$60\text{\,}\mathrm{s}$)}}.$ (16)

Here $n(t=$60\text{\,}\mathrm{s}$)$ indicates the number of steps required for
60 s of simulated time. It should be noted that $\Delta U_{\text{n}}$ does not
necessarily correspond to the actual behavior of the power supplies used in
the experiment but is rather only a convenient tool to implement dynamical
frequency fluctuations within one simulation run.

Data preparation and analysis is performed in R [47]. Since typical simulated
time spans are on the order of 30–60 s, time steps of $\Delta t$ = 1–5 ns
imply about $10^{10}$ simulation steps, each with several dozens of
operations. In order to minimize the computation time, the calculation-
intensive part is outsourced to a compiled C++ program via the Rcpp-package
[48]. In addition, the calculations are fully parallelized by the doParallel\-
and foreach-package [49, 50], allowing threads up to the number of virtual
cores to run in parallel.

In summary, we emphasize that the physics in our simulation is implemented
from first principles. We only use experimentally established model
assumptions and no heuristic scaling factors.

##  5 Comparison of simulation results to experiment and theory

Next, we demonstrate that the simulation results agree well with theory and
experiments, in particular those in Ref. [12]. We simulate the 9Be+ ions
individually and consider no additional heating effects besides the one
stemming from the resonator [51]. Fig. 3 shows an FFT spectrum recorded with a
common resonator involving the detector resonance, a proton in the PT detuned
by -35 Hz from the detector’s resonance frequency $\nu_{\text{res}}$ and
several 9Be+ ions with $\nu_{\text{z,Be}}=\nu_{\text{res}}$ in the ST.
Compared to the simulated spectrum in Fig. 3(b), the absolute signal level of
the experimental spectrum in Fig. 3(a) is boosted by several amplifier stages.
The broad resonance with full-width at half maximum (FWHM)
$\gamma_{\text{res}}$ can be matched to the unperturbed lineshape of the
noise-charged parallel RLC-circuit and the two dips stem from the proton and
9Be+ ions. From such a dip spectrum several properties can be derived. First,
the dip position yields the axial eigenfrequency of the particle. Here we
confirm that the simulations produce the dip at the correct frequency at
$\nu_{\text{z}}=1/(2\pi)\sqrt{2qC_{2}U_{0}/m}$. Second, the dip width
$\gamma_{\text{z}}$, defined as full-width at half maximum (FWHM) of the dip
signal, is a measure of the resonator-to-ion coupling strength. Assuming
$\gamma_{\text{z}}\ll\gamma_{\text{res}}$, it can be calculated as

$\gamma_{z}=\frac{1}{2\pi}\frac{R}{m}\frac{q^{2}}{D_{\text{res}}^{2}}N,$ (17)

where $N$ is the number of particles with mass $m$ [29].

(a) (b)

Figure 3:  Example FFT spectra of the voltage drop across the RLC circuit. An
experimental spectrum is shown in (a) and a simulated one with approximately
matched parameters in (b). The broad peak is the resonance of the RLC circuit
and the two dips stem from a proton and several 9Be+ ions stored in
independent traps, but coupled to the same resonator. The proton is detuned
from the detector by about -35 Hz, leading to a dispersive dip. The dip widths
are defined by the trap size, the number of ions in the trap and the ion
masses [29].

The dip widths of the simulated dip spectra match with Eq. (17). Similarly,
the simulated noise resonance has the same FWHM as we expect from theory.
Next, since the resonator is a noisy thermal bath, the particle energies
follow a Boltzmann distribution with the mean temperature defined by the
resonator. We confirm that both the absolute resonator temperature, which is a
free input parameter in our simulations, as well as the Boltzmann
distributions of the single particle energies are correctly reproduced.
Regarding laser cooling, we reproduce the theoretical Doppler limit of
$T_{D}=\frac{\hbar\Gamma}{2k_{B}}\approx 0.5$ mK as minimal temperature of the
9Be+ ions. Furthermore, the scattering rate of photons $\gamma_{s}$ for a
single 9Be+ ion artificially held at a constant velocity matches the
theoretical one [45]:

$\gamma_{s}=\frac{I_{\text{laser}}}{I_{\text{sat}}}\frac{\Gamma/2}{1+\frac{I_{\text{laser}}}{I_{\text{sat}}}+4\left(\frac{2\pi\delta_{\text{L}}+k_{\text{L}}\dot{z}_{\text{Be}}}{\Gamma}\right)^{2}}.$
(18)

While the previously mentioned properties must be met for each oscillator
individually and are straightforward to check, the coupling between them,
especially between the proton and the 9Be+ ions, is more subtle. In order to
reach our goal of $\approx$10\text{\,}\mathrm{m}\mathrm{K}$$ axial proton
temperature, it is crucial to understand the coupling mechanism in detail. In
the following, we will compare the experimental data of Ref. [12] with FFT
spectra of simulation data for three different temperature regions.
Additionally, since the simulations provide us access to the time domain data,
we can demonstrate that our experimental signature is truly caused by a
temperature manipulation.

Ref. [12] employed the coupling scheme with a common RLC resonator. First, the
coupling between the proton and 9Be+ ions was demonstrated above the resonator
temperature by bringing the proton to resonance with excited beryllium ions.
Therefore, the first temperature scale at which we compare simulations against
experiment is at particle temperatures above the resonator temperature,
depicted in Fig. 4. Here we simulate a single proton and a cloud of 10 9Be+
ions. The beryllium ions are excited by a parametric drive with frequency
$2\nu_{\text{z,Be}}$, which is implemented as an additional force
$F_{\text{dr}}(t)=F_{\text{0,dr}}\cos(4\pi\nu_{\text{z,Be}}t)$ in Eq. (9). In
the experiment, special care was taken to ensure that the proton was not
directly affected by the heating drive but only via the coupling to the 9Be+
ions. Since the axial frequencies of both particles are matched, not only the
9Be+ ions but also the proton significantly gains energy, demonstrating
resonator-mediated energy exchange between the particles. Since
$\gamma_{\text{z,p}}<\gamma_{\text{z,Be}}<\gamma_{\text{res}}$, the proton
reaches a temperature higher than the one of the resonator. This results in a
sharp peak above the resonator signal in the FFT spectrum in both the
experimental and simulation FFTs, see Fig. 4(a) and (b), respectively. The
time domain data of the simulations, depicted in Fig. 4(c), support this
explanation. Here, the heating drive is switched on at $t=10$ s. Before that,
the energies of all three oscillators perform a random walk through a
Boltzmann distribution with the same temperature. With the parametric drive
switched on, both the proton and 9Be+ ions gain energy equivalent to several
hundred Kelvin. The average energy per ion is plotted for the beryllium cloud.

Second, the coupling at the equilibrium temperature of the resonator was
demonstrated. To this end, the characteristic response of the coupled three-
oscillator system is recorded, shown in Fig. 5. We employ a single proton and
a single 9Be+ ion, whose frequencies are both detuned 130 Hz from the
resonator eigenfrequency. We then sweep the beryllium frequency over the
proton frequency and display the signal power of the corresponding frequency
spectra as colour-coded columns in a heat map. Both the experimental and
simulated data in Fig. 5(a) and (b) exhibit the characteristic response of a
coupled three-oscillator system. In addition, we plot the energy exchange
between the proton, the 9Be+ ion and the resonator at 130 Hz detuning in the
time domain for the simulated data in Fig. 5(c). To emphasize the coupling
between the proton and 9Be+ ions, noise and dissipation were turned off by
setting $R=\infty$ and $T_{\text{res}}=0$ for this plot. Since the resonator
FWHM $\gamma_{\text{res}}$ is $\approx$30\text{\,}\mathrm{H}\mathrm{z}$$, the
resonator is only weakly excited by the particles. The exact behavior of the
energy exchange in Fig. 5(c) strongly depends on the initial relative particle
phase.

Last, we establish common-resonator coupling between a single proton and 50
laser-cooled 9Be+ ions. The frequency domain data in both experiment and
simulation, Fig. 6(a) and Fig. 6(b) respectively, show a broad dip with
reduced signal-to-noise and a superimposed narrow dip below. The broad dip
corresponds to the beryllium cloud which, due to the laser cooling, cannot
fully compensate the resonator noise current and hence exhibits a reduced
signal strength. The narrow dip corresponds to the proton, which is coupled to
both the resonator and the 9Be+ cloud. In the time-domain plot, Fig. 6(c),
both particles have an initial energy of 10 $\text{K}\cdot k_{B}$. After
turning on the laser at $t=7.5$ s, not only the beryllium ions but also the
proton is significantly cooled within a few seconds. This is exactly the
effect we desire and this specific cooling scheme will be more thoroughly
investigated in section 6.2.

In this section, we have shown that our simulations are able to unambiguously
reproduce the observed experimental signatures [12]. By giving access to the
time domain data, the results provide another layer of understanding. The
laser cooling simulations match the theoretical expectations down to the
Doppler limit, and are therefore proven to be a practical and credible tool to
investigate cooling schemes that achieve particle temperatures in the mK-
regime.

| (a) |
---|---
|  (b) |
---|---
|  (c) |
---|---
Figure 4: Comparison between experiment and simulation for $T_{\text{p}}\gg T_{\text{res}}$. The 9Be+ cloud is heated by a parametric $2\nu_{z}$-drive. The proton is located in a different trap and is heated only due to the coupling to the 9Be+ ions, creating the narrow peak signal in the FFT spectrum. (a) and (b) show the FFT spectrum of the experiment and simulation, respectively. The simulated effect of switching on the heating drive at $t=$10\text{\,}\mathrm{s}$$ (dashed line) is demonstrated in (c). The energy per ion is depicted for the beryllium cloud.  | (a) |
---|---
|  (b) |
---|---
|  (c) |
---|---
Figure 5: The coupling between the proton and 9Be+ is demonstrated at $T_{\text{p}}=T_{\text{res}}$ by sweeping $\nu_{\text{z,Be}}$ over $\nu_{\text{z,p}}$. The experiment data in (a) and simulation data in (b) show the characteristic response of a coupled three-oscillator system. In (c) we show simulated time domain data for the simplified case with no resonator noise. | (a) |
---|---
|  (b) |
---|---
|  (c) |
---|---
Figure 6: Comparison between experiment and simulation for
$T_{\text{p}}<T_{\text{res}}$. Both the experiment data in (a) and the
simulation data in (b) show a common-resonator FFT with a laser-cooled
beryllium cloud (broad dip with reduced signal strength) superimposed by a
single proton (narrow dip). In (c) the cooling laser is switched on at
$t=$7.5\text{\,}\mathrm{s}$$, making the cooling effect on the proton visible.
The energy per ion is depicted for the beryllium cloud.

##  6 Coupling scenarios

Next, we will theoretically investigate the three coupling scenarios whose
experimental setups were described in Sec. 3, namely common-endcap coupling,
resonant coupling via a common RLC circuit, and off-resonant coupling via a
common RLC circuit.

We simulate individual beryllium ions in Sec. 6.2 and effective clouds
otherwise since only in Sec. 6.2 absolute laser intensities are of interest.
The initialization of the particle modes is randomized and has no influence on
the coupling schemes. If not explicitly stated otherwise, we use the following
simulation parameters for the resonator:
$R=$236\text{\times}{10}^{6}\text{\,}\Omega$$,
$L=$3\text{\,}\mathrm{m}\mathrm{H}$$,
$C_{\text{res}}=$21.4\text{\,}\mathrm{p}\mathrm{F}$$, yielding a resonator
Q-factor of $20\,000\text{\,}\mathrm{a}$nd a resonance frequency of
$626.9\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$. Regarding the trap parameters,
we adjust the product $C_{2}U_{0}$ such that the particle oscillates at the
desired frequency. The effective electrode distances are
$D_{\text{res,p}}=$10.0\text{\,}\mathrm{m}\mathrm{m}$$ for the PT,
$D_{\text{res,Be}}=$5.7\text{\,}\mathrm{m}\mathrm{m}$$ for the ST and
$D_{\text{com,p}}=D_{\text{com,Be}}=$3.2\text{\,}\mathrm{m}\mathrm{m}$$ for
the CT and BT (compare Fig. 2) [25]. The common endcap capacitance is
$C_{\text{com}}=$5.5\text{\,}\mathrm{p}\mathrm{F}$$. Additionally, we assume
that the resonator has an effective noise temperature of 10 K, accounting for
the fundamental Johnson-Nyquist noise at 4.2 K (the physical temperature of
the experiment) and the input noise of our cryogenic amplifiers [13, 31].
Finally, a time step of $\Delta t=$4\text{\,}\mathrm{n}\mathrm{s}$$ was used
for all subsequent simulations.

###  6.1 Common-endcap coupling

In order to simulate the common-endcap coupling in our trap system, we detune
the resonator eigenfrequency by $200$ kHz so that the energy exchange is
dominated by the common-endcap coupling. In the experiment, this will be
achieved by adding or subtracting an additional parallel capacitance to the
resonator using a cryogenic switch [42]. This has the advantage that the
particle frequencies can be matched on resonance and no adjustment of the
trapping potentials is necessary when starting the coupling procedure. Since
the resonator terms in Eq. (8) - (11) are then negligible for the ion-ion-
coupling with no laser applied, the system of coupled differential equations
can be solved analytically. In order to derive a general solution we allow for
more than one particle in each cloud by setting $q_{\text{p}}\rightarrow
N_{\text{p}}\,q_{\text{p}}$, $m_{\text{p}}\rightarrow
N_{\text{p}}\,m_{\text{p}}$, $q_{\text{Be}}\rightarrow
N_{\text{Be}}\,q_{\text{Be}}$ and $m_{\text{Be}}\rightarrow
N_{\text{Be}}\,m_{\text{Be}}$ in Eq. (10, 11), where $N_{\text{p,Be}}$ is the
number of protons and beryllium ions, respectively. For the sake of a compact
notation we abbreviate $D_{\text{com,p}}\equiv D_{\text{p}}$ and
$D_{\text{com,Be}}\equiv D_{\text{Be}}$ in the following calculations.
Furthermore, we assume that the particle eigenfrequencies are exactly matched,
i.e.
$\frac{2q_{\text{p}}C_{\text{2,p}}U_{\text{0,p}}}{m_{\text{p}}}=\frac{2q_{\text{Be}}C_{\text{2,Be}}U_{\text{0,Be}}}{m_{\text{Be}}}=\omega_{0}^{2}$.
Solving the coupled differential equation for the proton motion yields

$\displaystyle
z_{\text{p}}(t)=c_{1}e^{i\lambda_{1}t}+c_{2}e^{i\lambda_{2}t}+c_{3}e^{i\lambda_{3}t}+c_{4}e^{i\lambda_{4}t}$
(19)

where $c_{1,2,3,4}$ are free coefficients determined by the initial conditions
and $\lambda_{1,2,3,4}$ are the corresponding eigenfrequencies:

$\displaystyle\lambda_{1,2}$ $\displaystyle=\pm\,\omega_{0}$ (20)
$\displaystyle\lambda_{3,4}$
$\displaystyle=\pm\sqrt{\omega_{0}^{2}-\frac{m_{\text{p}}D_{\text{p}}^{2}N_{\text{Be}}q_{\text{Be}}^{2}+m_{\text{Be}}D_{\text{Be}}^{2}N_{\text{p}}q_{\text{p}}^{2}}{C_{\text{com}}D_{\text{p}}^{2}D_{\text{Be}}^{2}m_{\text{p}}m_{\text{Be}}}}$
(21) $\displaystyle\approx\pm(\omega_{0}-\Omega).$ (22)

Here we abbreviated

$\Omega=\frac{1}{2}\frac{m_{\text{p}}D_{\text{p}}^{2}N_{\text{Be}}q_{\text{Be}}^{2}+m_{\text{Be}}D_{\text{Be}}^{2}N_{\text{p}}q_{\text{p}}^{2}}{\omega_{0}C_{\text{com}}D_{\text{p}}^{2}D_{\text{Be}}^{2}m_{\text{p}}m_{\text{Be}}}.$
(23)

Assuming the initial conditions $z_{\text{p}}=z_{0}$, $\dot{z}_{\text{p}}=0$,
$z_{\text{Be}}=0$ and $\dot{z}_{\text{Be}}=0$, the temporal evolution of the
energy of the proton is

$\displaystyle E_{\text{p}}(t)=$
$\displaystyle\frac{1}{2}\frac{m_{\text{p}}z_{0}^{2}}{(m_{\text{p}}D_{\text{p}}^{2}N_{\text{Be}}q_{\text{Be}}^{2}+m_{\text{Be}}D_{\text{Be}}^{2}N_{\text{p}}q_{\text{p}}^{2})^{2}}\left(\vphantom{\frac{x}{y}}\right.\omega^{2}\left[(m_{\text{p}}D_{\text{p}}^{2}N_{\text{Be}}q_{\text{Be}}^{2})^{2}+\right.$
(24)
$\displaystyle\left.\left.(m_{\text{Be}}D_{\text{Be}}^{2}N_{\text{p}}q_{\text{p}}^{2})^{2}+2m_{\text{p}}m_{\text{Be}}N_{\text{p}}N_{\text{Be}}q_{\text{p}}^{2}q_{\text{Be}}^{2}D_{\text{p}}^{2}D_{\text{Be}}^{2}\cos(\Omega
t)\right]+\mathcal{O}(\Omega\omega)\vphantom{\frac{x}{y}}\right).$

(a) (b)

Figure 7: (a) Simulated energy of a single proton capacitively coupled to a
cloud of 9Be+ ions. In the case of $D_{\text{com,p}}=D_{\text{com,Be}}$ the
energy is exchanged completely only for 9 beryllium ions (upper plot). In
contrast, for 80 beryllium ions the energy exchange is incomplete. (b) Energy
exchange amplitude as a function of relative particle detuning for 80 9Be+
ions. The maximum Rabi oscillation amplitude is restored by detuning the
beryllium ions relative to the proton by $+40$ mHz.

Since $\Omega\ll\omega$, the terms with $\mathcal{O}(\Omega\omega)$ can be
neglegted. From Eq. (24) $\frac{\Omega}{2\pi}$ can be identified as the Rabi
frequency of the coupled proton-beryllium system. The difference between Eq.
(23) and previously reported versions [2, 3] arises from the fact that in this
work, up to Eq. (22) no approximations were used. In order for the energy to
be exchanged completely, i.e. $E_{\text{p}}(t=\pi/\Omega)=0$, the condition

$\frac{N_{\text{p}}q_{\text{p}}^{2}}{m_{\text{p}}D_{\text{p}}^{2}}\overset{!}{=}\frac{N_{\text{Be}}q_{\text{Be}}^{2}}{m_{\text{Be}}D_{\text{Be}}^{2}}$
(25)

must be satisfied. Inserting this condition into Eq. (23) reproduces the
previously reported formula [2, 3] for the Rabi oscillation frequency. Eq.
(25) reflects the simultaneous conservation of energy and conservation of
charge on the common-endcap capacitance within one Rabi cycle. We note that
the studied system of coupled oscillators is symmetric in many experiments and
Eq. (25) is automatically fulfilled. However, we emphasize that if the
condition in Eq. (25) is not met, the energy is exchanged only partially and
with a faster frequency. As demonstrated in the following, this has
considerable implications for the realization of the cooling schemes in the
experiment. Fig. 7(a) shows an exemplary energy exchange between a single
proton and 9 or 80 9Be+ ions, top and bottom plot, respectively. Since
$D_{\text{p}}=D_{\text{Be}}$ and $m_{\text{Be}}\approx 9\,m_{\text{p}}$ in our
case, the energy exchange is complete for 9 beryllium ions. While the Rabi
frequency for 80 beryllium ions is higher, only a fraction of the proton’s
energy is exchanged. This would be a severe limitation of the common-endcap
coupling technique as one would need to choose between a complete energy
exchange ($N_{\text{Be}}=9$) or a fast coupling time ($N_{\text{Be}}\gg 9$).
However, the behavior of the Rabi oscillations when varying the number of
beryllium ions is equivalent to introducing a small relative frequency
detuning between the two species. Consequently, one can reverse the effect of
using $N_{\text{Be}}\gg 9$ by introducing a small relative detuning. The
analytical calculations are laborious, thus we make use of the numerical
simulations here. Fig. 7(b) shows the Rabi oscillation amplitude (peak-to-
peak) for 80 9Be+ ions as a function of relative particle detuning. While for
an exact frequency matching only $\approx 4$$\text{K}\cdot k_{B}$ are
exchanged per Rabi cycle, at +40 mHz relative detuning the full Rabi amplitude
is recovered. In the case of nonzero detunings Eq. (23) does not hold anymore.
However, if the detuning is chosen such that the energy is exchanged
completely, the previously reported $\Omega\propto\sqrt{N_{\text{Be}}}$ is
preserved, i.e.
$\Omega($40\text{\,}\mathrm{m}\mathrm{H}\mathrm{z}$,80\,\text{Be})=\sqrt{\frac{80}{9}}\Omega($0\text{\,}\mathrm{m}\mathrm{H}\mathrm{z}$,9\,\text{Be})$.
The parameters presented here yield
$\Omega($0\text{\,}\mathrm{m}\mathrm{H}\mathrm{z}$,80\,\text{Be})/(2\pi)=$53\text{\,}\mathrm{m}\mathrm{H}\mathrm{z}$$
and
$\Omega($40\text{\,}\mathrm{m}\mathrm{H}\mathrm{z}$,80\,\text{Be})/(2\pi)=$34\text{\,}\mathrm{m}\mathrm{H}\mathrm{z}$$.
Although matching two species to a few ten mHz is experimentally challenging,
it is crucial to account for this in the experimental realization of the
coupling scheme as it impacts the cooling times and the achieved temperature
limits.

(a) (b)

Figure 8: (a) Evolution of the simulated energy gain of the 9Be+ cloud divided
by the time after turning off the laser for the common-endcap coupling. The
depicted data is an average temperature for initial axial frequency mismatches
of $\Delta_{\text{unc}}=$100\text{\,}\mathrm{m}\mathrm{H}\mathrm{z}$$ and a
dynamical stability of $\xi_{\text{stab}}=$1\text{\times}{10}^{-7}\text{\,}$$
as defined in Sec. 16. The maximum energy exchange rate is reached at 6.0 s
for 80 Be+ ions, defining the optimal cycle time for the first laser pulse.
(b) Representative example of the cooling process with a laser pulse applied
every 6.0 s.

The initial proposals [2, 43] envisaged cooling close to the Doppler limit by
coupling the target ion for exactly half a Rabi cycle (one $\pi$-pulse) to the
subsequently laser-cooled 9Be+-cloud. However, Fig. 7(b) shows that a
frequency mismatch of only $\approx 10$ mHz limits the residual proton energy
after one $\pi$-evolution to $\gtrsim 1$$\text{K}\cdot k_{B}$, far higher than
the 10 mK required for our experiment. As a result, we investigate a cooling
scheme that is based on repetitively turning on the cooling laser and
performing several imperfect energy exchanges. Compared to continuous laser
operation, chopping the laser achieves orders of magnitude faster cooling. The
reason for this can be understood in the Rabi oscillation picture: In the
continuous case the 9Be+ cloud is artificially held at its minimal
temperature. The cooling rate for the proton is given by the local slope of
the energy exchange curve, i.e. it vanishes to first order for continuously
cooled 9Be+ ions [3].

(a) (b)

Figure 9:  (a) Simulated proton energy over time when applying the chopped
laser scheme for 80 9Be+ ions, yielding a cooling time constant of
$\tau_{\text{CC}}=$18.6\text{\,}\mathrm{s}$$. The depicted data is an average
temperature for initial axial frequency mismatches of
$\Delta_{\text{unc}}=$100\text{\,}\mathrm{m}\mathrm{H}\mathrm{z}$$ and a
dynamical stability of $\xi_{\text{stab}}=$1\text{\times}{10}^{-7}\text{\,}$$
as defined in Sec. 16. The gray band indicates one standard deviation. (b)
Same as in (a), but with the proton initialized at 1.5 mK. Due to the 200 kHz-
detuning from the resonator effectively no heating sources are present, such
that the ultimate temperature limit is the Doppler temperature of the
beryllium ions of $T_{D}\approx$0.5\text{\,}\mathrm{m}\mathrm{K}$$.

To determine a realistic cooling time constant, we incorporate the
experimental frequency uncertainties in this section and the following ones
twofold: First, we add an additional detuning $\delta\nu$ to the proton
frequency due to non-ideal matching to the 9Be+ frequency. $\delta\nu$ is
drawn uniformly from the interval
$\delta\nu\in[-\Delta_{\text{unc}},+\Delta_{\text{unc}}]$ with
$\Delta_{\text{unc}}=100$ mHz and randomly for every simulation. Second, we
assume a relative voltage stability as defined in Sec. 4 of
$\xi_{\text{stab}}=$1\text{\times}{10}^{-7}\text{\,}$$ for both the proton and
beryllium trapping voltages. In a first step we determine the ideal laser
cycle time for the chopped laser scheme. The 9Be+ ions are initialized at an
energy of $$1\text{\,}\mathrm{m}\mathrm{K}$\cdot k_{B}$ and the proton at an
energy of 10 $\text{K}\cdot k_{B}$. As figure of merit we use the average
temperature gain of the beryllium cloud over time, $\Delta
T_{\text{Be}}/\Delta t$, displayed in Fig. 8(a). By switching on the cooling
laser at the time of maximum $\Delta T_{\text{Be}}/\Delta t$, in this
simulated case at 6.0 s, to first order the fastest cooling time constant is
achieved. We note that more complex laser schemes with varying cycle times
could potentially lead to an even better cooling performance, however, in this
paper we will limit the discussion to schemes with constant laser cycle times.
The on-time of the laser is set to 2% of the laser cycle time. Fig. 8(b) shows
a representative cooling run with the cooling laser turned on every 6.0 s.
Averaging several of such runs with the uncertainties described above allows
us to perform a least-square fit with an exponential function

$T_{\text{p}}=T_{p,0}\,e^{-t/\tau_{\text{CC}}}+T_{\text{p,min}}$ (26)

and extract a cooling time constant $\tau_{\text{CC}}$ as it is done in Fig.
9(a). We obtain $\tau_{\text{CC}}=18.6(1.4)$ s for the simulated case with 80
9Be+ ions. The uncertainty is estimated by fitting the single runs and
calculating the standard deviation of the mean $\tau_{\text{CC}}$.

In addition to the cooling time constant, the other parameter of interest is
the lower temperature limit, determined by the equilibrium of the competing
cooling and heating rate. In case of the common-endcap coupling, the heating
by the resonator noise is strongly suppressed due to the resonator detuning of
several hundred kHz. Additionally, other sources of heating are negligibly
small [51]. We investigate the lower limit that can be achieved in the
simulations by repeating the procedure for the cooling time constant, but
initializing the proton at 1.5 mK, shown in Fig. 9(b). The cooling time
constant is within uncertainty indistinguishable from the one with 10 K
initial proton temperature. More importantly, the lower temperature limit is
dictated by the temperature of the 9Be+ cloud, which itself is cooled to the
Doppler limit of $T_{\text{D}}\approx$0.5\text{\,}\mathrm{m}\mathrm{K}$$.

The last quantity of practical interest is the sensitivity of the cooling
performance to relative axial frequency detunings between the proton and the
9Be+ cloud. The cooling efficiency for a fixed relative detuning depends on
the laser cycle time. In the worst case, even for a rather small relative
detuning of $\approx 10$ mHz no cooling can occur if the laser cycle time is
an integer multiple of the Rabi cycle. The relevant quantity in the experiment
is the precision with which the particle frequencies can be matched. Hence we
vary the frequency matching precision $\Delta_{\text{unc}}$ as defined above
instead of introducing a fixed detuning. Furthermore, since for large detuning
intervals fitting the temperature evolution with Eq. (26) becomes an erroneous
measure of cooling effectiveness, we will use the mean proton temperature at
30 s after initialization instead. Fig. 10(a) depicts the sensitivity of the
proton temperature at 30 s as a function of the detuning uncertainty. The
initial proton energy was 10 $\text{K}\cdot k_{B}$. The proton temperature at
30 s nearly doubles when moving from $100$ mHz to $200$ mHz, indicating that
absolute detunings of $>100$ mHz contribute little to the cooling process,
which is a result of the comparably low Rabi frequency of
$34\text{\,}\mathrm{m}\mathrm{H}\mathrm{z}$. Fig. 10(b) shows the simulated
proton temperature after initializing the proton at 0.5 mK$\cdot k_{B}$ and
letting it evolve for 30 s while applying the chopped laser to the 9Be+ ions.
Since for the common-endcap coupling the resonator is detuned, no heating
process competes with the cooling process. As a result, the lower temperature
limit is not affected and the proton temperature can still reach the Doppler
limit of $T_{\text{D}}\approx$0.5\text{\,}\mathrm{m}\mathrm{K}$$.

(a) (b)

Figure 10:  Simulation of the sensitivity of the common-endcap coupling
approach to relative frequency matching uncertainties between the proton and
80 9Be+ ions. In (a) the proton temperature at 30 s after initialization as a
measure of the cooling time is shown. In (b) the lower temperature limit as a
function of $\Delta_{\text{unc}}$ is displayed for a proton initialized at
$$0.5\text{\,}\mathrm{m}\mathrm{K}$\cdot k_{B}$. Due to the absence of heating
effects in the common-endcap coupling scheme the lower temperature limit for
the proton is not affected by a detuning and remains at the Doppler limit of
$T_{\text{D}}\approx$0.5\text{\,}\mathrm{m}\mathrm{K}$$.

###  6.2 Resonant coupling mediated by an RLC resonator

A sketch of the experimental setup for the resonant coupling scheme is shown
in Fig. 2(b) and the experimental proof of principle was recently demonstrated
in Ref. [12]. This approach is the easiest to observe experimentally as the
proton and the 9Be+ ions are continuously being detected in resonance with the
RLC resonator, which is the standard operation mode of our Penning-trap
experiment.

In the case of the resonant coupling with an RLC circuit, the resonator noise
causes the energy exchange between the proton and 9Be+ ions to be highly
incoherent on a ms-timescale. As a result, a cooling scheme that relies on
single $\pi$-pulses will not be able to consistently achieve the desired 10
mK. Although a chopped laser scheme might offer an improved cooling
performance compared to a continuous laser operation, we will investigate the
continuous case in this work since it has already been experimentally
demonstrated [12].

As an introductive theoretical analysis, we first consider two identical
particles in two separate but identical traps connected to the same resonator.
The motion of these two particles can be described by a symmetric and
antisymmetric mode, where the individual oscillations are in phase or shifted
by a phase of $\pi$, respectively. The individual image currents induced into
the resonator interfere destructively for the antisymmetric mode. As a result,
the resonator only couples to the symmetric mode and only this mode is
thermalized to the detector temperature. The same argument applies to two
identical ions in the same trap. The experimental evidence for this is
provided by the linear scaling of the dip width with the number of ions in the
trap, which arises from substituting $q\rightarrow N_{i}\cdot q$ and
$m\rightarrow N_{i}\cdot m$ in Eq. (17). However, this substitution is only
valid if the particles’ motions are in phase. On the other hand, by
introducing an additional damping constant to one of the particles, energy is
transferred between the symmetric and antisymmetric mode. In our experiment,
the cooling laser constitutes this additional damping constant. The fact that
only the symmetric mode is heated by the resonator and the damping of the 9Be+
ions by the laser continuously transfers energy between all modes implies that
even a weak intensity laser is sufficient to eventually cool all modes except
the symmetric one down to the Doppler limit. In the case of negligible energy
of the non-symmetric modes, the final proton temperature is determined by the
proton’s relative component of the symmetric mode temperature. The relative
components are given by the relative dip widths, $\gamma_{\text{p}}$ and
$\gamma_{\text{Be}}$, respectively:

$\displaystyle T_{\text{res}}=T_{\text{p}}+T_{\text{Be}}$ (27) $\displaystyle
T_{\text{p}}=\frac{\gamma_{\text{p}}}{\gamma_{\text{Be}}}T_{\text{Be}}.$ (28)

These equations result in the $T_{p}\propto 1/N_{\text{Be}}$-scaling initially
proposed in Ref. [43] and measured in Ref. [12]:

$\displaystyle
T_{\text{p}}=\frac{1}{1+\frac{\gamma_{\text{Be}}}{\gamma_{\text{p}}}}T_{\text{res}}\mathchoice{\mathrel{\vbox{
\offinterlineskip\halign{\hfil$#$\cr\displaystyle\propto\cr\kern
2.0pt\cr\displaystyle\sim\cr\kern-2.0pt\cr}}}}{\mathrel{\vbox{
\offinterlineskip\halign{\hfil$#$\cr\textstyle\propto\cr\kern
2.0pt\cr\textstyle\sim\cr\kern-2.0pt\cr}}}}{\mathrel{\vbox{
\offinterlineskip\halign{\hfil$#$\cr\scriptstyle\propto\cr\kern
2.0pt\cr\scriptstyle\sim\cr\kern-2.0pt\cr}}}}{\mathrel{\vbox{
\offinterlineskip\halign{\hfil$#$\cr\scriptscriptstyle\propto\cr\kern
2.0pt\cr\scriptscriptstyle\sim\cr\kern-2.0pt\cr}}}}\frac{1}{N_{\text{Be}}}.$
(37)

However, in this form Eq. (37) is only true for weak laser intensities, i.e.
if the damping rate of the 9Be+ ions due to the laser, $\gamma_{\text{L}}$, is
much smaller than $\gamma_{\text{Be}}$. At higher laser intensities, the
beryllium cloud is decoupled from the resonator. This effect can be accounted
for by a reduction of the effective dip width, expressed by the substitution
$\gamma_{\text{Be}}\rightarrow\tilde{\gamma}_{\text{Be}}=k\,\gamma_{\text{Be}}$
with dimensionless scaling parameter $0\leq k\leq 1$ in Ref. [12].

Our simulation results show that, in contrast to the lower limit of the proton
temperature, a higher laser intensity is beneficial for the cooling time
constant as it increases the rate of energy dissipation. In order to study
these relations quantitatively in terms of experimental parameters, we
simulate the proton and a cloud of 80 continuously laser-cooled 9Be+ ions
tuned on resonance with the detector. Fig. 11 shows the cooling time constant
$\tau_{\text{RC}}$ and the lower temperature limit $T_{\text{p,min}}$ as a
function of laser intensity for two exemplary laser detunings of $-20$ MHz and
$-200$ MHz, depicted in subfigures (a) and (b), respectively. The cooling time
constant is determined by least-squares fits with Eq. (26) for an initial
proton energy of 10 $\text{K}\cdot k_{B}$. The lower temperature limit is
determined by initializing the proton at 0 $\text{K}\cdot k_{B}$ and averaging
the proton energy after it has reached equilibrium.

While a much stronger laser intensity is required for $-200$ MHz laser
detuning, the general behavior of $\tau$ and $T_{\text{p,min}}$ are the same:
The cooling time constant drops with larger laser intensity as the 9Be+ ions
are damped more strongly and reaches values down to a few hundred ms. However,
at such high laser intensities the beryllium ions are decoupled too strongly
from the resonator, leading to a suboptimal proton temperature limit. Instead,
the proton temperature saturates in the region where
$\gamma_{\text{L}}\ll\gamma_{\text{Be}}$. The simulated values in the weak
intensity limit agree well with the theoretical prediction of
$T_{\text{p,min}}=350$ mK by Eq. (37), indicated by the blue dashed line.

According to Eq. (17), the beryllium dip width can be increased not only by
loading more beryllium ions, but also by reducing the effective electrode
distance $D_{\text{res}}$. Since the cooling time constants are a function of
both the laser intensity and $D_{\text{res,Be}}$, we use $T_{\text{p,min}}$ in
the weak intensity limit as a measure of improved cooling performance. In Fig.
12(a), $D_{\text{res,Be}}$ is decreased starting from the currently
manufactured trap with $D_{\text{res,Be}}=$5.7\text{\,}\mathrm{m}\mathrm{m}$$.
The red line is the prediction of Eq. (37) and we observe excellent agreement
between simulation and theory. However, even for a trap with
$D_{\text{res,Be}}=$2.0\text{\,}\mathrm{m}\mathrm{m}$$, which is challenging
to manufacture and difficult to operate with 80 9Be+ ions, the lower
temperature limit is only about 50 mK, a factor of 5 higher than our desired
temperature.

While the resonant coupling scheme with continuous laser operation has a
comparably high temperature limit, it is the most robust technique against
experimental uncertainties. Fig. 12(b) shows the final proton temperature as a
function of the detuning of the 9Be+ cloud. The fact that the minimal
$T_{\text{p,min}}$ is shifted from 0 Hz detuning is caused by the effect of
maximizing the coupling strength between the two particle species by
introducing a relative detuning as described in Sec. 6.1. However, since the
laser is not chopped here, the effect on the cooling performance is
negligible. Since the particles are continuously detected on resonance,
matching and stabilizing them to $\approx 100$ mHz is comparably
straightforward and routinely done in our experiment. As Fig. 12(b) shows, the
increase in $T_{\text{p,min}}$ is negligible for $\approx$ 100 mHz detuning.

(a) (b)

Figure 11: Simulated cooling time constant and lower proton temperature limit
as a function of laser intensity for laser detunings $\delta_{L}=-20$ MHz (a)
and $\delta_{L}=-200$ MHz (b). Although the absolute values of the laser
intensity strongly differ, the scaling of $\tau$ and $T_{\text{p,min}}$ is
very similar, indicating that no global favourable laser detuning exists and
rather the combination of laser intensity and detuning defines the cooling
performance. The blue dashed lines indicate the minimal proton temperature of
350 mK predicted by Eq. (37).

(a) (b)

Figure 12: (a) The minimal proton temperature as a function of the effective
beryllium trap size for 80 9Be+ ions. A smaller trap size increases the width
of the beryllium dip, leading to a lower minimal proton temperature according
to Eq. (37). The simulation data is generated with sufficiently low laser
power. (b) The simulated final proton temperature as a function of the
detuning of the proton from the resonator. The asymmetry arises from the
effect described in Sec. 6.1.

###  6.3 Off-resonant coupling mediated by an RLC resonator

The off-resonant resonator coupling approach combines the ideas of the
previous two cooling schemes by detuning the particle frequencies by
$\delta\nu_{\text{res}}$ from the resonator eigenfrequency, typically several
kHz [3]. Similar to the common-endcap coupling, the particles then perform
Rabi oscillations mediated by a common effective capacitance

$C_{\text{eff}}=\frac{\Im(Z_{\text{RLC}}^{-1})}{\omega}\mathchoice{\mathrel{\vbox{
\offinterlineskip\halign{\hfil$#$\cr\displaystyle\propto\cr\kern
2.0pt\cr\displaystyle\sim\cr\kern-2.0pt\cr}}}}{\mathrel{\vbox{
\offinterlineskip\halign{\hfil$#$\cr\textstyle\propto\cr\kern
2.0pt\cr\textstyle\sim\cr\kern-2.0pt\cr}}}}{\mathrel{\vbox{
\offinterlineskip\halign{\hfil$#$\cr\scriptstyle\propto\cr\kern
2.0pt\cr\scriptstyle\sim\cr\kern-2.0pt\cr}}}}{\mathrel{\vbox{
\offinterlineskip\halign{\hfil$#$\cr\scriptscriptstyle\propto\cr\kern
2.0pt\cr\scriptscriptstyle\sim\cr\kern-2.0pt\cr}}}}\delta\nu_{\text{res}}$
(38)

where $Z_{\text{RLC}}$ is the impedance of the RLC detection circuit. Combined
with Eq. (23) this implies that the Rabi frequency decreases approximately
linear with the resonator detuning. With this scheme, the application of
$\pi$-pulses is not only inhibited by the required frequency matching
precision, but also by the residual heating due to the resonator noise. As a
result, we employ a chopped laser scheme again.

In order to pinpoint the optimal laser cycle time and cooling time constant,
we follow the same procedure as for the common-endcap coupling for different
resonator detunings. However, due to the residual particle heating by the
resonator noise, the lower temperature limit is higher than the Doppler limit
and is a function of $\delta\nu_{\text{res}}$. Moreover, the incoherence
introduced by the noise causes the ideal laser cycle time for the cooling time
constant to become suboptimal for reaching the lowest final temperatures. Fig.
13(a) shows the simulated proton temperature as a function of laser cycle time
at 8 kHz resonator detuning. The proton was initialized at 50 mK and the
system was evolved for 30 s to allow it to come into equilibrium. The
displayed temperature is calculated from the mean energy of an additional 30
s-interval afterwards. While the laser cycle time for the cooling time
constant is optimal at 3.5 s, about a factor of 2 in the lower temperature
limit is gained by reducing it to 1.0 s. Equivalent scans were repeated for
each resonator detuning in the following data.

The simulated performance of the off-resonant coupling scheme in terms of
cooling time constants and lower temperature limits is summarized in Fig.
13(b). The cooling time constants rise roughly linearly with the detuning, as
expected from Eq. (23) and Eq. (38). The lower temperature limits saturate at
slightly below 10 mK, which results from the two opposing effects of the
decline of resonator noise and the reduction of the Rabi frequency.

(a) (b)

Figure 13: (a) The simulated proton temperature limit as a function of laser
cycle time for 8 kHz resonator detuning. Due to the resonator noise the ideal
laser cycle time shifts from 3.5 s for the cooling time constant to 1.0 s for
the minimal temperature. The behavior is similar for all resonator detunings.
(b) Simulated cooling time constant and temperature limit as derived from (a)
as a function of resonator detuning.

We investigate the effect of the frequency matching uncertainty for 8 kHz and
20 kHz resonator detuning in the same manner as for the common-endcap
coupling. Fig. 14(a) shows the temperature at 30 s after initializing the
proton at an energy of 10 $\text{K}\cdot k_{B}$. If the particle frequency
matching precision $\Delta_{\text{unc}}$ is known experimentally, one can
optimize the cooling time by adjusting the laser cycle time accordingly. For
example, if the frequency matching precision is worse than the $100$ mHz
assumed in this work, one can decrease the laser cycle time to account for the
faster (but incomplete) Rabi oscillations. However, this requires
$\Delta_{\text{unc}}$ to be smaller than or at least comparable to the Rabi
frequency. This condition is, in contrast to the common-endcap coupling,
fulfilled for the coupling scheme investigated in this section. The procedure
in context of Fig. 8(a) was repeated for every frequency matching precision
for the blue and green data points labelled “corr.” in Fig. 14(a), leading to
a significantly better cooling performance compared to the uncorrected ones.
Especially in the case of 8 kHz the sensitivity of the proton temperature to
$\Delta_{\text{unc}}$ is significantly smaller compared to the common-endcap
coupling. Additionally, Fig. 14(b) shows the equilibrium proton temperature as
a function of the frequency matching uncertainty. An uncertainty of 1 Hz
causes the temperature to be roughly a factor of 2 higher than for 100 mHz for
both 8 kHz and 20 kHz resonator detuning.

(a) (b)

Figure 14: The simulated effect of the frequency matching uncertainty
$\Delta_{\text{unc}}$ for the off-resonant coupling scheme for resonator
detunings of 8 kHz and 20 kHz. The data points labelled “uncorr.”
(uncorrected) were generated with the ideal laser cycle time for
$\Delta_{\text{unc}}=100$ mHz. The data points labelled “corr.” (for
corrected) were generated with the ideal laser cycle time for each value of
$\Delta_{\text{unc}}$. (a) The proton temperature at 30 s as a measure of
cooling effectiveness. The initial energy was 10 $\text{K}\cdot k_{B}$. (b)
The lower limit of the proton temperature with the same uncertainties as in
(a).

##  7 Conclusion and Outlook

We have theoretically studied the sympathetic cooling of the axial mode of a
single proton with laser-cooled 9Be+ ions. A summary of the results is given
in Tab. 1. Of the three studied cooling techniques, the common-endcap coupling
scheme has a comparably long cooling time constant, but still outperforms the
established method of resistive cooling. Due to the absence of resonator
heating, the proton temperature is only limited by the Doppler limit of the
beryllium ions. However, the method is experimentally challenging since the
particles cannot be detected during the coupling process. Additionally, the
low Rabi frequency of $\approx 30$ mHz makes it the most sensitive scheme to
relative frequency mismatches between the particles. In contrast, the RLC
resonator mediated resonant coupling scheme can be tuned to have cooling time
constants in the $100$ ms-regime. It requires no additional experimental
instrumentation to conduct since all three oscillators are being detected on
resonance, which is the standard operation mode of our Penning trap. However,
it is limited to $\approx 350$ mK axial proton temperature with the current
experimental setup. Nevertheless, it will certainly be used as a pre-cooling
mechanism or as an alternative to feedback cooling. The off-resonant RLC
resonator coupling technique combines both approaches and yields a fast
cooling time constant while capable of achieving $\approx 10$ mK proton
temperature, which suffices as cooling scheme to prepare cold (anti-)protons
for $g$-factor measurements [2, 18, 52]. Furthermore, it is more robust than
the common-endcap coupling against relative frequency mismatches. Its main
advantage is that the ratio between cooling time constant and lower
temperature limit can be adjusted to the needs of the particular application.

Method | $\tau(s)$ |  $T_{\text{p,min}}$ (K) | Sensitivity to detuning
---|---|---|---
Common-endcap coupling | 20 s | 0.5 mK | off-resonance, $\approx 200$ mHz
Resonant RLC coupling | 0.1 – 30 s | 350 mK | on-resonance, $\approx 1$ Hz
Off-resonant RLC coupling | 4 – 10 s | 10 – 30 mK | off-resonance, $\approx 200$ – $500$ mHz
Table 1: Summary of the three different coupling schemes in terms of the
cooling time constant $\tau$ and the proton temperature limit
$T_{\text{p,min}}$. The sensitivity to detuning is defined as the frequency
matching precision where the cooling performance is a factor of 2 worse than
the reference at 100 mHz.

Overall, the studies presented here establish the cooling methods for the next
generation of precision measurements at BASE. While the simulations already
supported the proof-of-principle measurement in Ref. [12], they allow a
quantitative assessment of the performance of future techniques and an
identification of their experimental challenges, which would otherwise require
dedicated measurement campaigns. The next step is to apply these schemes
experimentally and validate the conclusions regarding temperature limits and
cooling times drawn in this work. For this purpose an improved experimental
setup is currently being commissioned [42]. It extends the current version by
a dedicated trap for temperature measurements with $\approx 1$ mK resolution
and a cryogenic switch to detune the resonator for the application of the
chopped laser schemes.

Once cooling to 10 mK is routinely achieved experimentally, the cooling
techniques described here will allow magnetic moment measurements to be
conducted at improved sampling rates and reduced systematic shifts and will
thus have considerable impact on future tests of CPT invariance with protons
and antiprotons. In addition, the two-trap sympathetic cooling techniques are
currently being developed for tests of quantum electrodynamics with highly
charged ions [19] or the determination of the magnetic moment of 3He2+ [53].

Since the techniques are practically independent of the target ion and the
species can be macroscopically separated, applications beyond these specific
cases are conceivable. For example, high-precision mass measurements of stable
ions [54] or radionuclides with moderate lifetimes at online facilities [55,
56], studies with molecular ions in radiofrequency traps [57] as well as
trapped-ion quantum computing [58] could benefit from the techniques presented
in this paper. Moreover, sub-Doppler cooling techniques such as sideband
cooling [59, 60] may be employed in the future to reach even lower
temperatures.

## Acknowledgments

This paper comprises parts of the PhD thesis work of C. Will at the Ruprecht
Karl University of Heidelberg.

We thank the ALPHATRAP-group at the Max-Planck-Institute for Nuclear Physics
for helpful discussions regarding the cooling method presented here and
acknowledge similar developments toward cooling highly charged ions in the
ALPHATRAP collaboration.

We acknowledge financial support from the Max-Planck-Society, RIKEN Chief
Scientist Program, RIKEN Pioneering Project Funding, the RIKEN JRA Program,
the Helmholtz-Gemeinschaft, the DFG through SFB 1227 ‘DQ-mat’, the European
Union (Marie Sklodowska-Curie grant agreement number 721559), the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation programme (Grant agree- ment Nos. 832848 - FunI and 852818 - STEP),
the DAAD RISE program and the Max-Planck–RIKEN–PTB Center for Time, Constants
and Fundamental Symmetries.

## Competing interests

The authors declare no competing interests.

## References

  * [1] M. S. Safronova, D. Budker, D. DeMille, D. F. J. Kimball, A. Derevianko, and C. W. Clark, “Search for new physics with atoms and molecules,” Rev. Mod. Phys., vol. 90, p. 025008, 2018.
  * [2] M. Bohman, A. Mooser, G. Schneider, N. Schön, M. Wiesinger, J. Harrington, T. Higuchi, H. Nagahama, C. Smorra, S. Sellner, K. Blaum, Y. Matsuda, W. Quint, J. Walz, and S. Ulmer, “Sympathetic cooling of protons and antiprotons with a common endcap $\mathrm{Penning}$ trap,” Journal of Modern Optics, vol. 65, no. 5-6, pp. 568–576, 2018.
  * [3] B. Tu, F. Hahne, I. Arapoglou, A. Egl, F. Heiße, M. Höcker, C. König, J. Morgner, T. Sailer, A. Weigel, R. Wolf, and S. Sturm, “Tank-circuit assisted coupling method for sympathetic laser cooling,” Advanced Quantum Technologies, vol. 4, no. 7, p. 2100029, 2021.
  * [4] C. J. Baker, W. Bertsche, A. Capra, C. L. Cesar, M. Charlton, A. C. Mathad, S. Eriksson, A. Evans, N. Evetts, S. Fabbri, J. Fajans, T. Friesen, M. C. Fujiwara, P. Grandemange, P. Granum, J. S. Hangst, M. E. Hayden, D. Hodgkinson, C. A. Isaac, M. A. Johnson, J. M. Jones, S. A. Jones, S. Jonsell, L. Kurchaninov, N. Madsen, D. Maxwell, J. T. K. McKenna, S. Menary, T. Momose, P. Mullan, K. Olchanski, A. Olin, J. Peszka, A. Powell, P. Pusa, C. Ø. Rasmussen, F. Robicheaux, R. L. Sacramento, M. Sameed, E. Sarid, D. M. Silveira, G. Stutter, C. So, T. D. Tharp, R. I. Thompson, D. P. van der Werf, and J. S. Wurtele, “Sympathetic cooling of positrons to cryogenic temperatures for antihydrogen production,” Nature Communications, vol. 12, no. 1, p. 6139, 2021.
  * [5] N. Madsen, F. Robicheaux, and S. Jonsell, “Antihydrogen trapping assisted by sympathetically cooled positrons,” New Journal of Physics, vol. 16, no. 6, p. 063046, 2014.
  * [6] R. X. Schüssler, H. Bekker, M. Braß, H. Cakir, J. R. Crespo López-Urrutia, M. Door, P. Filianin, Z. Harman, M. W. Haverkort, W. J. Huang, P. Indelicato, C. H. Keitel, C. M. König, K. Kromer, M. Müller, Y. N. Novikov, A. Rischka, C. Schweiger, S. Sturm, S. Ulmer, S. Eliseev, and K. Blaum, “Detection of metastable electronic states by Penning trap mass spectrometry,” Nature, vol. 581, no. 7806, pp. 42–46, 2020.
  * [7] S. M. Brewer, J.-S. Chen, A. M. Hankin, E. R. Clements, C. W. Chou, D. J. Wineland, D. B. Hume, and D. R. Leibrandt, “${}^{27}{\mathrm{Al}}^{+}$ Quantum-Logic Clock with a Systematic Uncertainty below ${10}^{-18}$,” Phys. Rev. Lett., vol. 123, p. 033201, 2019\.
  * [8] P. Micke, T. Leopold, S. A. King, E. Benkler, L. J. Spieß, L. Schmöger, M. Schwarz, J. R. Crespo López-Urrutia, and P. O. Schmidt, “Coherent laser spectroscopy of highly charged ions using quantum logic,” Nature, vol. 578, no. 7793, pp. 60–65, 2020.
  * [9] M. Harlander, R. Lechner, M. Brownnutt, R. Blatt, and W. Hänsel, “Trapped-ion antennae for the transmission of quantum information,” Nature, vol. 471, no. 7337, pp. 200–203, 2011.
  * [10] K. R. Brown, C. Ospelkaus, Y. Colombe, A. C. Wilson, D. Leibfried, and D. J. Wineland, “Coupled quantized mechanical oscillators,” Nature, vol. 471, no. 7337, pp. 196–199, 2011.
  * [11] M. Schuh, F. Heiße, T. Eronen, J. Ketter, F. Köhler-Langes, S. Rau, T. Segal, W. Quint, S. Sturm, and K. Blaum, “Image charge shift in high-precision Penning traps,” Phys. Rev. A, vol. 100, p. 023411, 2019\.
  * [12] M. Bohman, V. Grunhofer, C. Smorra, M. Wiesinger, C. Will, M. J. Borchert, J. A. Devlin, S. Erlewein, M. Fleck, S. Gavranovic, J. Harrington, B. Latacz, A. Mooser, D. Popper, E. Wursten, K. Blaum, Y. Matsuda, C. Ospelkaus, W. Quint, J. Walz, S. Ulmer, and B. A. S. E. Collaboration, “Sympathetic cooling of a trapped proton mediated by an lc circuit,” Nature, vol. 596, no. 7873, pp. 514–518, 2021.
  * [13] H. Nagahama, G. Schneider, A. Mooser, C. Smorra, S. Sellner, J. Harrington, T. Higuchi, M. Borchert, T. Tanaka, M. Besirli, K. Blaum, Y. Matsuda, C. Ospelkaus, W. Quint, J. Walz, Y. Yamazaki, and S. Ulmer, “Highly sensitive superconducting circuits at $\sim 700$ kHz with tunable quality factors for image-current detection of single trapped antiprotons,” Review of Scientific Instruments, vol. 87, no. 11, p. 113305, 2016.
  * [14] D. An, A. M. Alonso, C. Matthiesen, and H. Häffner, “Coupling two laser-cooled ions via a room-temperature conductor,” 2021.  arXiv:2107.00851.
  * [15] C. Smorra, K. Blaum, L. Bojtar, M. Borchert, K. Franke, T. Higuchi, N. Leefer, H. Nagahama, Y. Matsuda, A. Mooser, M. Niemann, C. Ospelkaus, W. Quint, G. Schneider, S. Sellner, T. Tanaka, S. Van Gorp, J. Walz, Y. Yamazaki, and S. Ulmer, “$\mathrm{BASE}$–the baryon antibaryon symmetry experiment,” The European Physical Journal Special Topics, vol. 224, no. 16, pp. 3055–3108, 2015.
  * [16] S. Ulmer, C. Smorra, A. Mooser, K. Franke, H. Nagahama, G. Schneider, T. Higuchi, S. Van Gorp, K. Blaum, Y. Matsuda, W. Quint, J. Walz, and Y. Yamazaki, “High-precision comparison of the antiproton-to-proton charge-to-mass ratio,” Nature, vol. 524, no. 7564, pp. 196–199, 2015.
  * [17] G. Schneider, A. Mooser, M. Bohman, N. Schön, J. Harrington, T. Higuchi, H. Nagahama, S. Sellner, C. Smorra, K. Blaum, Y. Matsuda, W. Quint, J. Walz, and S. Ulmer, “Double-trap measurement of the proton magnetic moment at 0.3 parts per billion precision,” Science, vol. 358, no. 6366, pp. 1081–1084, 2017.
  * [18] C. Smorra, S. Sellner, M. J. Borchert, J. A. Harrington, T. Higuchi, H. Nagahama, T. Tanaka, A. Mooser, G. Schneider, M. Bohman, K. Blaum, Y. Matsuda, C. Ospelkaus, W. Quint, J. Walz, Y. Yamazaki, and S. Ulmer, “A parts-per-billion measurement of the antiproton magnetic moment,” Nature, vol. 550, no. 7676, pp. 371–374, 2017.
  * [19] S. Sturm, I. Arapoglou, A. Egl, M. Höcker, S. Kraemer, T. Sailer, B. Tu, A. Weigel, R. Wolf, J. López-Urrutia, and K. Blaum, “The $\mathrm{ALPHATRAP}$ experiment,” The European Physical Journal Special Topics, vol. 227, no. 13, pp. 1425–1491, 2019.
  * [20] P. Pérez, D. Banerjee, F. Biraben, D. Brook-Roberge, M. Charlton, P. Cladé, P. Comini, P. Crivelli, O. Dalkarov, P. Debu, A. Douillet, G. Dufour, P. Dupré, S. Eriksson, P. Froelich, P. Grandemange, S. Guellati, R. Guérout, J. M. Heinrich, P. A. Hervieux, L. Hilico, A. Husson, P. Indelicato, S. Jonsell, J. P. Karr, K. Khabarova, N. Kolachevsky, N. Kuroda, A. Lambrecht, A. M. M. Leite, L. Liszkay, D. Lunney, N. Madsen, G. Manfredi, B. Mansoulié, Y. Matsuda, A. Mohri, T. Mortensen, Y. Nagashima, V. Nesvizhevsky, F. Nez, C. Regenfus, J. M. Rey, J. M. Reymond, S. Reynaud, A. Rubbia, Y. Sacquin, F. Schmidt-Kaler, N. Sillitoe, M. Staszczak, C. I. Szabo-Foster, H. Torii, B. Vallage, M. Valdes, D. P. Van der Werf, A. Voronin, J. Walz, S. Wolf, S. Wronka, and Y. Yamazaki, “The GBAR antimatter gravity experiment,” Hyperfine Interactions, vol. 233, no. 1, pp. 21–27, 2015.
  * [21] C. Amsler, M. Antonello, A. Belov, G. Bonomi, R. S. Brusa, M. Caccia, A. Camper, R. Caravita, F. Castelli, P. Cheinet, D. Comparat, G. Consolati, A. Demetrio, L. Di Noto, M. Doser, M. Fanì, R. Ferragut, J. Fesel, S. Gerber, M. Giammarchi, A. Gligorova, L. T. Glöggler, F. Guatieri, S. Haider, A. Hinterberger, A. Kellerbauer, O. Khalidova, D. Krasnický, V. Lagomarsino, C. Malbrunot, S. Mariazzi, V. Matveev, S. Müller, G. Nebbia, P. Nedelec, L. Nowak, M. Oberthaler, E. Oswald, D. Pagano, L. Penasa, V. Petracek, L. Povolo, F. Prelz, M. Prevedelli, B. Rienäcker, O. Røhne, A. Rotondi, H. Sandaker, R. Santoro, G. Testera, I. Tietje, V. Toso, T. Wolz, P. Yzombard, C. Zimmer, and N. Zurlo, “Pulsed production of antihydrogen,” Communications Physics, vol. 4, no. 1, p. 19, 2021.
  * [22] D. Rodríguez, “A quantum sensor for high-performance mass spectrometry,” Applied Physics B, vol. 107, no. 4, pp. 1031–1042, 2012.
  * [23] C. W. Chou, D. B. Hume, T. Rosenband, and D. J. Wineland, “Optical Clocks and Relativity,” Science, vol. 329, no. 5999, pp. 1630–1633, 2010.
  * [24] N. Huntemann, C. Sanner, B. Lipphardt, C. Tamm, and E. Peik, “Single-ion atomic clock with $3\times{}{10}^{-18}$ systematic uncertainty,” Phys. Rev. Lett., vol. 116, p. 063001, 2016.
  * [25] G. Schneider, 300 ppt Measurement of the Proton g-Factor.  PhD thesis, Johannes Gutenberg University Mainz, 2017.
  * [26] G. Gabrielse, L. Haarsma, and S. L. Rolston, “Open-endcap $\mathrm{Penning}$ traps for high precision experiments,” International Journal of Mass Spectrometry and Ion Processes, vol. 88, no. 2, pp. 319 – 332, 1989.
  * [27] L. S. Brown and G. Gabrielse, “Geonium theory: Physics of a single electron or ion in a Penning trap,” Rev. Mod. Phys., vol. 58, pp. 233–311, Jan 1986\.
  * [28] K. Blaum, Y. N. Novikov, and G. Werth, “Penning traps as a versatile tool for precise experiments in fundamental physics,” Contemporary Physics, vol. 51, no. 2, pp. 149–175, 2010.
  * [29] D. J. Wineland and H. G. Dehmelt, “Principles of the stored ion calorimeter,” Journal of Applied Physics, vol. 46, no. 2, pp. 919–930, 1975.
  * [30] W. Shockley, “Currents to Conductors Induced by a Moving Point Charge,” Journal of Applied Physics, vol. 9, no. 10, pp. 635–636, 1938.
  * [31] S. Ulmer, H. Kracke, K. Blaum, S. Kreim, A. Mooser, W. Quint, C. C. Rodegheri, and J. Walz, “The quality factor of a superconducting rf resonator in a magnetic field,” Review of Scientific Instruments, vol. 80, no. 12, p. 123302, 2009.
  * [32] E. A. Cornell, R. M. Weisskoff, K. R. Boyce, and D. E. Pritchard, “Mode coupling in a $\mathrm{Penning}$ trap: $\pi$ pulses and a classical avoided crossing,” Phys. Rev. A, vol. 41, pp. 312–315, 1990\.
  * [33] S. Sturm, A. Wagner, B. Schabinger, and K. Blaum, “Phase-sensitive cyclotron frequency measurements at ultralow energies,” Phys. Rev. Lett., vol. 107, p. 143003, 2011.
  * [34] L. S. Brown, “Line Shape for a Precise Measurement of the Electron’s Magnetic Moment,” Phys. Rev. Lett., vol. 52, pp. 2013–2015, 1984.
  * [35] H. Häffner, T. Beier, S. Djekić, N. Hermanspahn, H.-J. Kluge, W. Quint, S. Stahl, J. Verdú, T. Valenzuela, and G. Werth, “Double Penning trap technique for precise g factor determinations in highly charged ions,” The European Physical Journal D - Atomic, Molecular, Optical and Plasma Physics, vol. 22, no. 2, pp. 163–182, 2003.
  * [36] A. Mooser, S. Bräuninger, K. Franke, H. Kracke, C. Leiteritz, C. Rodegheri, H. Nagahama, G. Schneider, C. Smorra, K. Blaum, Y. Matsuda, W. Quint, J. Walz, Y. Yamazaki, and S. Ulmer, “Demonstration of the double Penning Trap technique with a single proton,” Physics Letters B, vol. 723, no. 1, pp. 78–81, 2013.
  * [37] A. Mooser, S. Ulmer, K. Blaum, K. Franke, H. Kracke, C. Leiteritz, W. Quint, C. C. Rodegheri, C. Smorra, and J. Walz, “Direct high-precision measurement of the magnetic moment of the proton,” Nature, vol. 509, no. 7502, pp. 596–599, 2014.
  * [38] C. Smorra, A. Mooser, M. Besirli, M. Bohman, M. Borchert, J. Harrington, T. Higuchi, H. Nagahama, G. Schneider, S. Sellner, T. Tanaka, K. Blaum, Y. Matsuda, C. Ospelkaus, W. Quint, J. Walz, Y. Yamazaki, and S. Ulmer, “Observation of individual spin quantum transitions of a single antiproton,” Physics Letters B, vol. 769, pp. 1–6, 2017.
  * [39] B. D’Urso, R. Van Handel, B. Odom, D. Hanneke, and G. Gabrielse, “Single-particle self-excited oscillator,” Phys. Rev. Lett., vol. 94, p. 113002, 2005.
  * [40] S. Ulmer, K. Blaum, H. Kracke, A. Mooser, W. Quint, C. Rodegheri, and J. Walz, “A cryogenic detection system at 28.9 MHz for the non-destructive observation of a single proton at low particle energy,” Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, vol. 705, pp. 55–60, 2013\.
  * [41] M. Bohman, Sympathetic Cooling of a Proton with Resonant Image Current Coupling.  PhD thesis, Ruperto-Carola University of Heidelberg, 2020.
  * [42] M. Wiesinger.  PhD thesis, Ruperto-Carola University of Heidelberg.  in preparation.
  * [43] D. J. Heinzen and D. J. Wineland, “Quantum-limited cooling and detection of radio-frequency oscillations by laser-cooled ions,” Physical Review A, vol. 42, no. 5, p. 2977, 1990.
  * [44] H. Yoshida, “Construction of higher order symplectic integrators,” Physics Letters A, vol. 150, no. 5, pp. 262 – 268, 1990.
  * [45] H. J. Metcalf and P. Van der Straten, “Laser cooling and trapping of neutral atoms,” The Optics Encyclopedia: Basic Foundations and Practical Applications, 2007.
  * [46] W. M. Itano and D. J. Wineland, “Laser cooling of ions stored in harmonic and $\mathrm{Penning}$ traps,” Phys. Rev. A, vol. 25, pp. 35–54, 1982.
  * [47] R Core Team, R: A Language and Environment for Statistical Computing.  R Foundation for Statistical Computing, Vienna, Austria, 2020.
  * [48] D. Eddelbuettel and R. François, “Rcpp: Seamless R and C++ integration,” Journal of Statistical Software, vol. 40, no. 8, pp. 1–18, 2011.
  * [49] Microsoft and S. Weston, foreach: Provides Foreach Looping Construct, 2020\.  R package version 1.5.1.
  * [50] Microsoft Corporation and S. Weston, doParallel: Foreach Parallel Adaptor for the ’parallel’ Package, 2020.  R package version 1.0.16.
  * [51] M. J. Borchert, P. E. Blessing, J. A. Devlin, J. A. Harrington, T. Higuchi, J. Morgner, C. Smorra, E. Wursten, M. Bohman, M. Wiesinger, A. Mooser, K. Blaum, Y. Matsuda, C. Ospelkaus, W. Quint, J. Walz, Y. Yamazaki, and S. Ulmer, “Measurement of ultralow heating rates of a single antiproton in a cryogenic $\mathrm{Penning}$ trap,” Phys. Rev. Lett., vol. 122, p. 043201, 2019.
  * [52] A. Mooser, H. Kracke, K. Blaum, S. A. Bräuninger, K. Franke, C. Leiteritz, W. Quint, C. C. Rodegheri, S. Ulmer, and J. Walz, “Resolution of single spin flips of a single proton,” Phys. Rev. Lett., vol. 110, p. 140405, 2013\.
  * [53] A. Mooser, A. Rischka, A. Schneider, K. Blaum, S. Ulmer, and J. Walz, “A new experiment for the measurement of the g-factors of 3He+ and 3He2+,” Journal of Physics: Conference Series, vol. 1138, p. 012004, 2018.
  * [54] E. G. Myers, “The most precise atomic mass measurements in Penning traps,” International Journal of Mass Spectrometry, vol. 349-350, pp. 107–122, 2013\.
  * [55] K. Blaum, “High-accuracy mass spectrometry with stored ions,” Physics Reports, vol. 425, no. 1, pp. 1–78, 2006.
  * [56] T. Eronen and A. Jokinen, “High-precision atomic mass measurements for a CKM unitarity test,” International Journal of Mass Spectrometry, vol. 349-350, pp. 69–73, 2013.  100 years of Mass Spectrometry.
  * [57] W. B. Cairncross, D. N. Gresh, M. Grau, K. C. Cossel, T. S. Roussy, Y. Ni, Y. Zhou, J. Ye, and E. A. Cornell, “Precision Measurement of the Electron’s Electric Dipole Moment Using Trapped Molecular Ions,” Phys. Rev. Lett., vol. 119, p. 153001, 2017.
  * [58] C. D. Bruzewicz, J. Chiaverini, R. McConnell, and J. M. Sage, “Trapped-ion quantum computing: Progress and challenges,” Applied Physics Reviews, vol. 6, no. 2, p. 021314, 2019.
  * [59] J. F. Goodwin, G. Stutter, R. C. Thompson, and D. M. Segal, “Resolved-Sideband Laser Cooling in a Penning Trap,” Phys. Rev. Lett., vol. 116, p. 143002, 2016.
  * [60] P. Hrmo, M. K. Joshi, V. Jarlaud, O. Corfield, and R. C. Thompson, “Sideband cooling of the radial modes of motion of a single ion in a Penning trap,” Phys. Rev. A, vol. 100, p. 043414, 2019.