# Buffer gas cooling of ions in time-dependent traps using ultracold atoms

E. Trimby  Van der Waals-Zeeman Institute, Institute of Physics, University of
Amsterdam, 1098 XH Amsterdam, Netherlands  H. Hirzler  Van der Waals-Zeeman
Institute, Institute of Physics, University of Amsterdam, 1098 XH Amsterdam,
Netherlands  H. Fürst  Physikalisch-Technische Bundesanstalt, Bundesallee 100,
38116 Braunschweig, Germany  Institut für Quantenoptik, Leibniz Universität
Hannover, Welfengarten 1, 30167 Hannover, Germany  A. Safavi-Naini  Van der
Waals-Zeeman Institute, Institute of Physics, University of Amsterdam, 1098 XH
Amsterdam, Netherlands  QuSoft, Science Park 123, 1098 XG Amsterdam, the
Netherlands  R. Gerritsma  Van der Waals-Zeeman Institute, Institute of
Physics, University of Amsterdam, 1098 XH Amsterdam, Netherlands  QuSoft,
Science Park 123, 1098 XG Amsterdam, the Netherlands  R. S. Lous  Van der
Waals-Zeeman Institute, Institute of Physics, University of Amsterdam, 1098 XH
Amsterdam, Netherlands

###### Abstract

For exploration of quantum effects with hybrid atom-ion systems, reaching
ultracold temperatures is the major limiting factor. In this work, we present
results on numerical simulations of trapped ion buffer gas cooling using an
ultracold atomic gas in a large number of experimentally realistic scenarios.
We explore the suppression of micromotion-induced heating effects through
optimization of trap parameters for various radio-frequency (rf) traps and rf
driving schemes including linear and octupole traps, digital Paul traps,
rotating traps and hybrid optical/rf traps. We find that very similar ion
energies can be reached in all of them even when considering experimental
imperfections that cause so-called excess micromotion. Moreover we look into a
quantum description of the system and show that quantum mechanics cannot save
the ion from micromotion-induced heating in an atom-ion collision. The results
suggest that buffer gas cooling can be used to reach close to the ion’s
groundstate of motion and is even competitive when compared to some sub-
Doppler cooling techniques such as Sisyphus cooling. Thus, buffer gas cooling
is a viable alternative for ions that are not amenable to laser cooling, a
result that may be of interest for studies into quantum chemistry and
precision spectroscopy.

##  I Introduction

By immersing a trapped ion in a cold atomic bath, one creates a system that
benefits both from the spatial localization and addressability of the ion, as
well as from the scalability and ultracold temperatures of the atom cloud
Tomzareview ; Harter2014CAIrev ; Cote2016UHAI . The result is a unique
combination of two well-controllable platforms linked together by the
intermediate range ion-atom interaction, with applications in impurity physics
Ratschbacher2013DSI ; Schmid2010DCT ; Furst2018DSI ; Kollath2007STM ;
Astrak2021IPBEC ; Dieterle2021TSI ; Hirzler2020CNC ; Zipkes2010TSI ;
Oghittu2021doa , cold chemistry Ratschbacher2012CCR ; Haze2015CEC ;
Sikorsky2018SCA ; Mohammadi2021LDC  and quantum technology Bissbort2013ESS ;
Doerk2010AIQ . Buffer gas cooling of trapped ions has long been used to obtain
cold samples Sauder1968TEC ; Chen2014NGS ; Zipkes2011KST ; Meir2016DGS .
However, in efforts to reach quantized ion motion it has been outperformed by
sub-Doppler laser cooling techniques, which have allowed small ion crystals to
be cooled to their ground state of motion Ioncooling ; Morigi2000GSLC .
Meanwhile, developments in laser and evaporative cooling techniques have
opened up the possibility to prepare long-lived, ultracold atomic gases below
about 1 $\mu$K, making them excellent buffer gases for cooling single ions.
This progress has allowed buffer gas cooling to become a promising route
towards ultracold atom-ion mixtures and exploring the realm of quantum effects
within them.

Reaching the lowest possible temperatures in a hybrid atom-ion system is
crucial to taking full advantage of the research opportunities it has to
offer. In particular, an ion cooled to the regime of quantized motion would
make an excellent probe to monitor interactions within the atomic bath and
study impurity physics. Furthermore, cooling of both the atomic bath and the
ion to atom-ion collision energies below the quantum limit would allow tuning
of their interaction via $s$-wave Feshbach resonances Idziaszek2011MCQ ;
Tomza2015CIY , similar to those widely used in neutral atom systems Chin2010FR
and recently observed in atom-ion systems SchaetzFRs . In previous work, we
demonstrated that buffer gas cooling can be used to cool an atom-ion system to
the edge of the $s$-wave collision energy, where the ion reached an average of
3.7 motional quanta Nature . However, efforts to reach deeper into the quantum
regime or to buffer gas cool an ion to its motional ground state have remained
elusive.

A major obstacle limiting buffer gas cooling of an ion in a radiofrequency
(rf) trap is the heating effect of the time-dependent potential. In such a
trap, energy stored in the ion’s micromotion can be released during an atom-
ion collision, heating the system Cetina . This competes with the desired
cooling effects of the buffer gas collisions. As a result, the temperature of
the ion after thermalization is always higher than that of the buffer gas in
which it is immersed. Reduction of atom-ion collision energies to the quantum
regime are therefore predominantly limited by the kinetic energy of the ion,
when trapping in a rf ion trap. For this reason, minimizing the ion’s excess
micromotion (EMM) through compensation of experimental imperfections is
critical to buffer gas cooling experiments HAF . Furthermore, choosing an
atom-ion species combination with a large mass ratio greatly suppresses the
effects of micromotion-induced heating Cetina ; Haze2018CDS . Therefore, the
coldest atom-ion collision energies and subsequent observation of quantum
effects have been reported for Li/Yb+ Nature  and Li/Ba+ SchaetzFRs  systems.
Attempts to avoid these micromotion heating effects altogether by trapping
ions in other types of trap, for example optical traps, are ongoing, yet offer
different challenges arxivKarpa . For an ion in a Paul trap, the question
remains: what are the lowest temperatures achievable in a rf-trapped ion-atom
system?

In this paper, we numerically simulate buffer gas cooling in time-dependent
traps, showing that the optimization of trap parameters is crucial to reaching
the lowest possible energies. We focus on the Li/Yb+ species combination, and
find that parameter optimization can allow cooling to collision energies
approximately two times smaller than what has previously been achieved
experimentally Nature . For these settings, we expect an occupancy of about
one motional quanta on average, putting buffer gas cooling on par with sub-
Doppler cooling techniques such as Sisyphus cooling Ejtemaee20173DS . We study
the collision energies that can be achieved by buffer gas cooling in
alternative rf traps including hybrid rf-optical traps, digital Paul traps,
rotating rf-traps and hexapole and octupole traps. We find that all
investigated traps can be used to achieve the same low collision energies as
in a linear Paul trap. The only investigated traps that offer a clear
advantage over the linear Paul trap are the rf-optical hybrid trap and
multipole traps, in which even lower temperatures can be reached. However
these depend on the use of ambitious experimental parameters and, in the case
of the hybrid trap, suppression of photon scattering effects which could
otherwise lead to heating of the atom cloud. All simulations show that within
these time-dependent traps, we can reach the energy regime where a quantum
mechanical treatment of the atom-ion interaction becomes necessary. Therefore,
we also discuss how to describe the atom-ion collision within a quantum
description and show that the micromotion-induced heating survives.

The paper is organised as follows. In section II we outline the method with
which we classically simulate the buffer gas cooling process, indicating how
we extract and compare the ion kinetic energies and atom-ion collision
energies reached in these simulations. In section III we detail the time-
dependent potential used to confine an ion in a linear Paul trap, and show
simulation results for the ion energies that can be reached in such a trap
with variable parameters. In section IV, we consider other types of time-
dependent ion traps including digital, rotating and multipole traps, comparing
buffer gas cooling simulations in each of these to the results of the linear
Paul trap. We make a similar comparison in section V, now focusing on a hybrid
optical-rf trap. Finally, in section VI, we present a quantum description of
the atom-ion interaction to support our results also in the regime of low
kinetic energies of the ion.

##  II Method

For our investigation into buffer gas cooling, we simulate repeated collisions
between a trapped ion and atoms from an ultracold cloud. During these
simulations, there are two effects that dictate the motion of the colliding
particles as they are propagated towards each other in time. These are the
atom-ion interaction $V_{\text{ai}}(r)$, and the time-dependent potential used
to trap the ion $V_{\text{trap}}(t,r)$. The latter is adjusted to describe the
rf trap we wish to investigate. Our simulations follow the method outlined in
ref. HAF , which has been shown to be in good agreement with experimental
results Nature . We adapt this process to allow for a broader choice of
$V_{\text{trap}}(t,r)$, corresponding to a range of rf ion traps. We study the
role played by $V_{\text{trap}}(t,r)$, and its defining parameters, in
dictating the properties of the simulated ion after buffer gas cooling.

The atom-ion interaction consists of a long-range induced-dipole interaction,
in which the charge of the ion polarizes the atom, giving rise to an
attraction between them, as well as a short-range repulsion. The interaction
is described in our simulation as

$V_{\text{ai}}(r)=-\frac{C_{\text{4}}}{r^{4}}+\frac{C_{\text{6}}}{r^{6}}\,,$
(1)

with $r$ the separation of the colliding particles. Here, $C_{4}$ and $C_{6}$
are the long-range attraction and short-range repulsion interaction
coefficients, respectively. The choice of $C_{6}$ dictates the point at which
the atom-ion interaction becomes repulsive. It was shown in HAF  that its
exact value is unimportant as long as the $r^{-6}$ term starts to dominate at
a short enough length scale of $\sim$1 nm. Meanwhile, the value of $C_{4}$
depends on the polarizability of the species of buffer gas atoms. To describe
buffer gas cooling of an ion to the mK range, the short-range repulsion is
enough Chen2014NGS . However for cooling beyond this, the induced-dipole
interaction has been shown to play a significant role Cetina , as it is the
mechanism by which the ion can be pulled from the trap centre into a region
where micromotion heating limits its final temperature. Both terms are
therefore crucial to our simulation of atom-ion collisions at the $\mu$K
energy level.

During each simulated atom-ion collision, we use a fourth-order step-size-
adaptive Runge-Kutta algorithm to propagate one atom towards the confined ion,
under the influence of $V_{\text{trap}}(t,r)$ and $V_{\text{ai}}(r)$. We
assume the atom cloud is dilute enough to ignore non-binary collisions between
the ion and the atoms. The ion’s initial kinetic energy is set to 0 K. The
atom position is chosen at random on a sphere centred around the ion at the
start of each collision. The sphere size is set to be large enough to
accommodate large ion motion if and when the ion is heated. The atom’s
velocity in the radial (angular) direction is also selected at random from a
Weibull (Gaussian) distribution corresponding to a chosen buffer gas
temperature in the $\mu$K range. The two particles are then propagated until
the atom leaves the sphere. After each simulated collision we record the ion’s
average kinetic energy ($E_{\text{kin}}$) and motional properties. These are
not reset after each collision. We investigate the final values achieved after
the ion has thermalized, usually after some hundreds of collisions. To stay
within a reasonable simulation time, the temperature of the buffer gas is set
to 2 $\mathrm{\mu K}$, which corresponds to the experimental values of ref.
Nature .

An example of the typical ion thermalization data obtained in our simulation
is shown in Figure 1. For each simulation, we run at least 200 repeats of the
thermalization process, averaging over them to find the mean ion energy
reached after the $n$th collision (green line). The blue error bars correspond
to the standard deviation in these energies. In general, higher ion energies
are accompanied by larger error bars. This is because, dependent on the mass
ratio of the atom-ion species combination, the ion energy statistics after
multiple collisions follow either a Tsallis or Maxwell-Boltzmann distribution
DeVoe2009PLD ; Rouse2017SSE ; Nature . In both cases, the higher the mean ion
energy, the larger the statistical spread of ion energies about this mean, as
reflected in the error bars. To extract the final steady-state ion energy, we
perform a weighted least-squares fit to our data. The weighting of the fit is
determined by the uncertainty at each data point. We fit according to the
equation

$E_{\text{kin,i}}(n)=(E_{\text{kin,f}}-E_{\text{kin,0}})(1-e^{-n\Gamma})+E_{\text{kin,0}}\,.$
(2)

Here, $E_{\text{kin,0}}$ is the initial kinetic energy of the ion, $\Gamma$
indicates the rate of thermalization and $E_{\text{kin,f}}$ is the kinetic
energy reached by the ion at steady state. The latter two are fitting
parameters. In figure 1, the fit is shown by the black curve, and the steady-
state kinetic energy is indicated by the grey, dashed line. Note that $n$ in
this plot includes both Langevin and glancing collisions, meaning that the
value of $\Gamma$ extracted from the fit depends strongly on the initial atom-
ion separation. However, within the range used in our simulation, this
separation does not influence $E_{\text{kin,f}}$ as shown in ref. HAF .

Figure 1: Typical thermalization curve when simulating consecutive atom-ion
collisions. The green line shows the average temperature reached after each
collision, averaged over 221 runs of the thermalization process. The blue
shaded area indicates the standard deviation about this average value and
represents the width of the distribution of temperatures after the $n$th
collision. The black line shows the fit of equation (2) to the averaged data,
with the grey dashed line indicating the $E_{\text{kin,f}}$ found from this
fit.

We focus our simulations on the Li/Yb+ species combination, due to its proved
effectiveness in reaching $s$-wave collision energies in buffer gas cooling
experiments Nature . The species choice has implications for the atom-ion
interaction strength. Choosing 6Li/171Yb+ corresponds to setting
$C_{4}=1.121\times 10^{-56}\,\mathrm{Jm}^{4}$. For the short-range repulsion
we use $C_{6}=2.804\times 10^{-75}\,\mathrm{Jm}^{6}$. The induced-dipole
interaction has a characteristic range $R^{*}=\sqrt{\frac{\mu
C_{4}}{\hbar^{2}}}$, with a corresponding energy scale given by
$E^{*}=\frac{\hbar^{2}}{2\mu(R^{*})^{2}}$, which also represents the s-wave
energy limit for the system. In a 6Li/171Yb+ system, these take the values
69.75 nm and 8.58 $\mu$K, respectively. To reach deep into the quantum regime
with buffer gas cooling, the kinetic energy of the ion must be reduced such
that the collision energy of the atom-ion system
$E_{\text{col}}=\frac{\mu}{m_{\text{i}}}E_{\text{kin,i}}+\frac{\mu}{m_{\text{a}}}E_{\text{kin,a}}$
is significantly lower than $E^{*}$. Here, $\mu$ is the reduced mass.

##  III The linear Paul trap

In an ideal linear Paul trap, the time-dependent potential can be written as

$V_{\text{trap}}(t,r)=\frac{-v_{\text{dc}}}{R_{\text{dc}}^{2}}\sum_{j=x,y,z}\alpha_{j}r_{j}^{2}+\frac{v_{\text{rf}}}{2R_{\text{rf}}^{2}}\cos{(\Omega_{\text{rf}}t)}\sum_{j=x,y,z}\alpha_{j}^{\prime}r_{j}^{2}\,,$
(3)

with $\alpha_{x}=\alpha_{y}\approx-\alpha_{z}/2=-0.5$ and
$\alpha_{x}^{\prime}=-\alpha_{y}^{\prime}=1$, $\alpha_{z}^{\prime}=0$
Leibfried ; HAF . Here, $v_{\text{dc}}$ and $v_{\text{rf}}$ correspond to the
direct-current (dc) and radio-frequency (rf) voltages applied to the trap
electrodes. Meanwhile, $R_{\text{dc}}$ and $R_{\text{rf}}$ describe the fixed
geometry of the trap and are approximately half the separation between
opposing dc and rf electrodes, respectively. The frequency at which the rf
voltage oscillates is given by $\Omega_{\text{rf}}$. The two terms of equation
(3) are commonly referred to as $V_{\text{trap}}^{\text{dc}}(r)$ and
$V_{\text{trap}}^{\text{rf}}(t,r)$, respectively. The resulting ion motion is
described by the Mathieu equations,
$\ddot{r}_{j}=-(a_{j}+2q_{j}\cos{(\Omega_{\text{rf}}t)})\frac{\Omega_{\text{rf}}^{2}}{4}r_{j}$,
with $a_{j}$, $q_{j}$ the stability parameters of the ion trap for all three
directions ($j=x,y,z$). The stability parameters can be defined in terms of
the trap geometry and voltage signals as $a_{x,y}\approx-
a_{z}/2=4ev_{\text{dc}}/(m_{\text{i}}R_{\text{dc}}^{2}\Omega_{\text{rf}}^{2})$,
and $q\equiv
q_{x}=-q_{y}=2ev_{\text{rf}}/(m_{\text{i}}R_{\text{rf}}^{2}\Omega_{\text{rf}}^{2})$,
$q_{z}$=0, with $e$ the charge of the ion. The stability parameters combined
with $\Omega_{\text{rf}}$ determine the frequency
$\omega_{\text{sec},j}\approx\frac{\Omega_{\text{rf}}}{2}\sqrt{a_{j}+\frac{q_{j}^{2}}{2}}$
of the trapped ion’s secular motion.

In an ideal trap, the ion’s energy is mainly given by the kinetic energy,
which is averaged over one period of the ion’s secular motion. It is defined
as

$\bar{E}_{\text{kin,j}}\approx\frac{1}{4}m_{\text{i}}A_{j}^{2}\left(\omega_{\text{sec,}j}^{2}+\frac{1}{8}q_{j}^{2}\Omega_{\text{rf}}\right)\,,$
(4)

and depends on the secular frequency and amplitude $A_{j}$ of the ion motion,
and the trap parameters $q$ and $\Omega_{\text{rf}}$. The latter two can be
easily changed in experiments by changing the frequency and amplitude of the
rf signal applied to the trap electrodes, and will be studied in our work.
Typical numbers used in our Li/Yb+ experiment are $q=0.5$,
$\Omega_{\text{rf}}=2\pi\times 2$ MHz and $v_{\text{dc}}$ up to 6.3 V,
corresponding to $\omega_{\text{sec,}x}$ and $\omega_{\text{sec,}y}$ up to
$2\pi\times$350 kHz, and $\omega_{\text{sec,}z}$ up to $2\pi\times$120 kHz.
The size of our trap is $R_{\text{dc}}=5$ mm and $R_{\text{rf}}=1.5$ mm, as
outlined in ref. setup .

The ion’s kinetic energy is connected to the number of excited motional quanta
$n_{j}$ of the ion in each direction $j$ by the following equations

$T_{\text{sec,}j}=\hbar\omega_{\text{sec,}j}\left(n_{j}+\frac{1}{2}\right)/k_{\text{B}}\,,$
(5)

and

$\sum_{j=x,y,z}\bar{E}_{\text{kin},j}\approx\frac{k_{\text{B}}}{2}\left(\sum_{j=x,y,z}T_{\text{sec,}j}+T_{\text{sec,}x}+T_{\text{sec,}y}\right)\,.$
(6)

Here $T_{\text{sec,}j}$, represents the secular temperature in direction $j$
and $k_{\text{B}}$ is the Boltzmann constant. The kinetic energy contains both
secular motion (first term) and intrinsic micromotion (second and third terms)
contributions Berkeland . Intrinsic micromotion (IMM) is driven by the rf
field, and so depends on the radial temperature of the ion only. In our
simulations, we focus on the average number of excited motional quanta in the
radial direction ($\bar{n}=(n_{x}+n_{y})/2$), as the axial motion of the ion
is largely unaffected by variations in our parameters of interest, $q$ and
$\Omega_{\text{rf}}$.

In addition to $q$ and $\Omega_{\text{rf}}$, the ion’s kinetic energy is also
influenced by trap imperfections, which can lead to excess micromotion (EMM).
Although trap-dependent, in a realistic simulation of the linear Paul trap
these effects should be included HAF ; Nature . There are three forms of EMM
which all scale differently with $q$ and $\Omega_{\text{rf}}$. Axial EMM
arises from the pickup of the rf signal on the dc electrodes, and contributes
a kinetic energy ($E_{\text{ax}}$) that scales with $q^{2}$ and
$\Omega_{\text{rf}}^{-2}$. Phase EMM, $E_{\text{ph}}$, results from a phase
difference between the rf voltage signals on opposing electrodes, and scales
with $q^{2}$ and $\Omega_{\text{rf}}^{4}$. Finally, $E_{\text{rad}}$ scales
with $q^{-2}$ and $\Omega_{\text{rf}}^{-2}$, and arises when stray electric
fields in the radial direction of the trap displace the ion from the trap
centre. The ion’s total kinetic energy is therefore

$E_{\text{kin}}=\sum_{j=x,y,z}\bar{E}_{\text{kin},j}+bE_{\text{rad}}+cE_{\text{ax}}+dE_{\text{ph}}+g\,.$
(7)

Here, $b,c,d$ are fitting parameters indicating the contributions of the
experiment-specific energies $E_{\text{rad}}$, $E_{\text{ax}}$ and
$E_{\text{ph}}$. The fitting parameter $g$ is added when an overall offset of
$E_{\text{kin}}$ needs to be taken into account. In experiments, we can use
ion position tracking or Ramsey microwave spectroscopy to measure stray fields
to an accuracy of $\sim$0.01 Vm-1 Nature . It is feasible that radial stray
fields can be compensated down to 0.05 Vm-1. Note that the finite size of the
trap can lead to $q_{z}\neq 0$, which can give rise to additional EMM.
However, since $q_{z}$ is much smaller than $q$ and its EMM effects can be
minimised by selection of an appropriate ion position along the trap axis, we
assume it is small enough to be ignored in our simulations.

We investigate the values that $E_{\text{kin}}$ and $\bar{n}$ can reach after
buffer gas cooling by simulating collisions between 6Li atoms and an 171Yb+
ion confined in a linear Paul trap with varying levels of EMM. The temperature
of the Li cloud is set to 2 $\mathrm{\mu K}$. We vary $q$ and
$\Omega_{\text{rf}}$ independently, to explore their impact, setting
$\Omega_{\text{rf}}=2\pi\times 2$ MHz while $q$ is varied and $q=0.219$ while
$\Omega_{\text{rf}}$ is varied. The simulated trap shares its geometry with
that of ref. setup  and has $v_{\text{dc}}=0.787$ V such that
$\omega_{\text{sec,}z}=2\pi\times 4.24$ kHz. We include experimentally
feasible levels of axial and phase EMM corresponding to an axial rf pickup of
2.02 Vm-1 ($E_{\text{ax}}=42.4$ $\mu$K) and a phase of 0.041 millirad
($E_{\text{ph}}=9.18$ $\mu$K) when $q=0.219$ and
$\Omega_{\text{rf}}=2\pi\times 2$ MHz HAF ; Harter ; setup . We also vary the
strength of the radial stray field around experimentally realistic values.

Figure 2: Calculated ion energy and motion after buffer gas cooling in a
linear Paul trap. The ion’s steady-state kinetic energy $E_{\text{kin}}$ and
number of excited motional quanta $\bar{n}$ are shown for cooling inside a
trap of variable stability parameter $q$ and radio frequency
$\Omega_{\text{rf}}$. Blue, orange and green data corresponds to a trap with a
radial stray field of strength 0 Vm-1, 0.05 Vm-1 and 0.1 Vm-1, respectively.
Red data points in a and b show experimental results from ref. Nature , while
the red dashed lines in c and d indicate the value of $\Omega_{\text{rf}}$
used in the same experiment. The lines in a and b correspond to equation (7),
fit to the data using fitting parameters $b,c,d,g$ and $A_{x,y,z}$. The fit
values of these parameters can be found in Appendix A. Errors are standard
errors and are sometimes smaller than the point size.

Our simulations show that the optimal values of $q$ and $\Omega_{\text{rf}}$
for buffer gas cooling depend on the radial stray fields present in our
system. This is shown in figure 2. Blue, orange and green data corresponds to
a trap with a radial stray field of strength 0 Vm-1, 0.05 Vm-1 and 0.1 Vm-1,
respectively. Figures 2a and b show how the ion energy $E_{\text{kin}}$ varies
with $q$ and $\Omega_{\text{rf}}$. In general, the optimal trap parameters for
minimising the ion energy take non-zero values, and depend on the strength of
the radial stray field. During atom-ion collisions, the trap’s rf field does
work to heat the ion mediated by the long-range attractive atom-ion
interaction Cetina . This heating competes with buffer gas cooling and leads
to a trade-off between two effects. On the one hand, high values of $q$ and
$\Omega_{\text{rf}}$ provide tight ion confinement, meaning that the pull from
the atoms does not significantly alter its motion. On the other hand, low
values of $q$ and $\Omega_{\text{rf}}$ will decrease the energy of the rf
electric field, lowering the energy that it can donate to the ion during an
atom-ion collision. The optimal trap parameters therefore provide a trade-off
between these effects. Moreover, EMM provides additional trap-parameter-
dependent heating. The higher the field-strength of stray radial electric
fields that additionally pull the ion from the Paul trap centre, the higher
the favoured optimal values of $q$ and $\Omega_{\text{rf}}$. Appropriate
selection of $q$ is particularly important for optimal buffer gas cooling.
This can be seen by the steeper slope of the fit lines in Fig 2a, compared to
b.

The lowest achievable kinetic energy depends significantly on the possibility
to reduce electric stray fields and optimal choice of $q$ and
$\Omega_{\text{rf}}$. For $E_{\text{kin}}$ we find optimal values by fitting
the data points using equation (7). The fit values can be found in Appendix A.
For the chosen parameters, the lowest ion kinetic energy achievable by buffer
gas cooling in our Paul trap is $64(1)\,\mu$K, which can be achieved using
trap parameters $q=$0.147 and $\Omega_{\text{rf}}=2.28$ MHz. In a 2 $\mu$K
atom cloud, this corresponds to $E_{\text{col}}=0.59\times E^{*}$,
significantly below the $s$-wave limit. It is also two times lower than the
lowest collision energy achieved in previous experiments Nature , which is
indicated by the red data point in a. This experiment yielded collision
energies $1.15\times E^{*}$ by buffer gas cooling an ion to kinetic energies
of 193 $\mu$K. The lowest achievable ion kinetic energies indicated by our
simulation are therefore roughly 3 times colder than what has been achieved in
experiments. Note that the discrepancy between the experimental data point and
the simulation predictions can be explained by an additional cooling step of
adiabatic decompression that was used in the plotted experiment, but is not
included in our simulation. Our results suggest that with careful parameter
optimization, this final step is not necessary for reaching low temperatures.
The value 64 $\mu$K is a lower limit on what can be achieved for the ion in
this setup with an atomic bath of 2 $\mu$K. Experiments will be limited to
higher energies due to background heating from technical noise setup , which
varies across experiments and is not included in our simulations. Quantum
effects may also play a role in reducing the rate of buffer gas cooling at
very low temperatures, as the rate of atom-ion collisions is expected to be
lower once $E_{\text{col}}$ reaches below the $s$-wave energy Nature . More
details on a quantum treatment of the atom-ion interaction can be found in
section VI.

Depending on the stray fields, $\bar{n}$ is minimized for different values of
the trap parameters to $E_{\text{kin}}$. This can be seen when comparing
figure 2a and b to c and d. Each of these variables is affected differently by
sources of EMM. Therefore, to optimize buffer gas cooling experiments, one
should consider whether to prioritize reaching quantized ion motion or
$s$-wave collision energies to inform the choice of trap parameters. The
lowest average number of excited motional quanta to which the ion can be
cooled by an atom cloud at 2 $\mu$K in our setup is 0.92(7), at a stability
parameter $q=0.26$. This is 4 times smaller than what has been achieved in
buffer gas cooling experiments, as indicated by the red data point in c. It
allows us to reach quantized ion motion, making buffer gas cooling competitive
with some sub-Doppler laser cooling techniques, such as Sisyphus cooling
Ejtemaee20173DS . However, it is not capable of reaching $\bar{n}\ll 1$, as is
possible using, for example, resolved sideband cooling Ioncooling .

##  IV Other types of rf ion traps

Since the ion’s final kinetic energy is limited by the trap parameters and
excess micromotion in a linear Paul trap, this raises the question of whether
other trap configurations are better suited for buffer gas cooling. Therefore,
in this section, we simulate a digital ion trap (DIT), multipole ion trap
(MIT) and rotating ion trap (RIT). We investigate how the dependence of
$E_{\text{kin}}$ on the trap parameters may differ for different traps and,
where relevant, consider the roles played by new parameters that are unique to
certain traps.

###  IV.1 Digital ion trap

The DIT is a modification of the previously-discussed linear Paul trap,
whereby the rf voltage is driven digitally (stepwise constant) instead of
analogue (sinusoidally). The switching of the voltage leads to times where the
ion dynamics are that of a free particle as the rf voltage is zero and this
might lead to more efficient buffer gas cooling in these periods. However,
collisions occurring during a voltage switch will experience a violent change
in potential, which may result in larger ion heating than in the traditional
analogue Paul trap.

Similar to equation (3), the time-dependent potential of a DIT can be
described by

$V_{\text{trap}}(t,r)=V_{\text{trap}}^{\text{dc}}(r)+\frac{v_{\text{rf}}}{2R_{\text{rf}}^{2}}P_{\text{DIT}}(t)\sum_{j=x,y,z}\alpha_{j}^{\prime}r_{j}^{2}\,.$
(8)

Here, the variables are equivalent to those in equation (3) apart from the
shape of the rf signal $P_{\text{DIT}}(t)$, which is defined as

$P_{\text{DIT}}(t)=\begin{cases}1&\text{when }t_{r}\leq\tau\,T/2\,,\\\
0&\text{when }\tau\,T/2<t_{r}\leq(1-\tau)\,T/2\,,\\\ -1&\text{when
}(1-\tau)\,T/2<t_{r}\leq(1+\tau)\,T/2\,,\\\ 0&\text{when
}(1+\tau)\,T/2<t_{r}\leq(2-\tau)\,T/2\,,\\\ 1&\text{when
}(2-\tau)\,T/2<t_{r}\leq T\,.\end{cases}$ (9)

Here, $T=2\pi/\Omega_{\text{rf}}$ is the period of the signal, $t_{r}$ is the
remainder $t=\kappa\times T+t_{r}$ Deb , with $\kappa$ an integer, and $\tau$
is the fractional pulse width. The digital voltage driving is shown in Figure
3a. For a DIT, $\tau$ adds an additional trap parameter that affects the
stability of the trap KjaergaardDrewsen . Approximating the digital signal
with the first term of the corresponding Fourier series Bandelow , the $q$
parameter for a DIT becomes
$q_{x,y,\text{DIT}}=16ev_{\text{rf}}\sin{(\tau\pi)}/(\pi
m_{\text{i}}R^{2}_{\text{rf}}\Omega_{\text{rf}}^{2})$.

We investigate how $E_{\text{kin}}$ varies when buffer gas cooling an ion in a
DIT, dependent on the temperature of the atomic bath $T_{\text{a}}$, and the
trap parameters $q$ and $\tau$. The results can be seen in figure 3b and c. In
these simulations, we use the same geometry, trap parameters and atomic bath
temperature as in the linear Paul trap simulations, except where otherwise
stated. In figure 3b, we choose $\tau=0.285$ and $v_{\text{rf}}=34.467$ V such
that we have $q=0.219$, to match the previous simulations. In figure 3c, we
vary $q$ and $\tau$ while keeping the trap geometry constant. For a fixed
$\tau$, the variation in $q$ corresponds only to a change in the voltage of
the digital pulses $v_{\text{rf}}$. The black lines show the trends as given
by an analogue Paul trap with the same $q$, $\Omega_{\text{rf}}$ and EMM
parameters. Radial stray field strengths are set to 0 or 0.1 Vm-1 in these
simulations, while sources of axial and phase MM are not included.

Our simulation shows that buffer gas cooling using a digital rf driving
results in equal ion temperatures to those in an analogue linear Paul trap. In
figure 3b, DIT results are in excellent agreement with the analogue Paul trap
trend line. Therefore, for the chosen $\tau$ and $q$ of this plot, the
energies reached in the DIT are identical to those of an analogue trap,
irrespective of the atomic bath temperature. In figure 3c, we see that in a
DIT of variable $q$, the data points corresponding to the lowest
$E_{\text{kin}}$ also lie on the trend line of the analogue trap. The range of
$q$ for which the DIT and analogue trends match is affected by the fractional
pulse width $\tau$. While $\tau$ provides an extra degree of freedom for
tuning the trap stability, its tunability does not lead to lower ion kinetic
energies than are achievable with an analogue Paul trap. The fact that the DIT
results are so similar to those of the analogue Paul trap suggests that the
time scale of an atom-ion collision is longer than the period of the rf trap,
such that the shape of the rf voltage signal during the collision period is
unimportant Cetina .

Figure 3: Ion energy after buffer gas cooling in a DIT. a Analogue (black,
dashed) versus digital (purple and red) driving of the rf voltage. The
fractional pulse width $\tau$ defines the driving. Digital signals are shown
for $\tau=0.285$ (purple) and $0.5$ (red). b The kinetic energy reached for
variable buffer gas temperatures, for $q=0.219$ and $\tau=0.285$. c The
kinetic energy reached for variable trap parameter $q$ for $\tau=0.285$
(purple), $0.4$ (green) and $0.5$ (red). The black dotted lines represent the
trend in an analogue linear Paul trap. Errors are standard errors and are
sometimes smaller than the point size.

###  IV.2 Rotating ion trap

In a RIT, ion confinement is provided by a rotating potential, instead of the
oscillating potential typical for linear Paul traps.The ion motion is
described by four compoments, instead of the infinite sum of Fourier
components needed for a linear Paul trap. Since fast MM plays an important
role in atom-ion collisions, we may therefore expect a different outcome for
buffer gas cooling when comparing the two traps.

A RIT time-dependent trapping potential can be written as HasegawaBollinger ;
rotsaddle

$V_{\text{trap}}(t,r)=V_{\text{trap}}^{\text{dc}}(r)+\frac{v_{\text{rf}}}{2R_{\text{rf}}^{2}}[(r_{x}\cos{\Omega_{\text{rf}}t}-r_{y}\sin{\Omega_{\text{rf}}t})^{2}-(r_{x}\sin{\Omega_{\text{rf}}t}+r_{y}\cos{\Omega_{\text{rf}}t})^{2}]\,.$
(10)

Here, variables are equivalent to those in equation (3), yet the rf driving
part has changed. The resulting ion motion is then
$\ddot{r}_{x}=-(a_{x}r_{x}+2q_{x}(r_{x}\cos{\Omega_{\text{rf}}t}-r_{y}\sin{\Omega_{\text{rf}}t}))\frac{\Omega_{\text{rf}}^{2}}{4}$,
$\ddot{r}_{y}=-(a_{y}r_{y}+2q_{y}(r_{y}\cos{\Omega_{\text{rf}}t}+r_{x}\sin{\Omega_{\text{rf}}t}))\frac{\Omega_{\text{rf}}^{2}}{4}$,
$\ddot{r}_{z}=-a_{z}r_{z}\frac{\Omega_{\text{rf}}^{2}}{4}$. An important
consequence of this is that the ion’s motion is now circular, instead of
oscillating, and consists of four Fourier components: two secular motions and
two IMM components. The frequency of the secular motion in the RIT can be
written to first order in $q$ as

$\displaystyle\omega_{\text{sec,RIT}}\approx\pm\frac{\Omega_{\text{rf}}}{2}\sqrt{a+q^{2}}\,.$
(11)

Here $a=a_{x}=a_{y}$, and
$q=q_{x}=-q_{y}=2ev_{\text{rf}}/(m_{\text{i}}R^{2}_{\text{rf}}\Omega_{\text{rf}}^{2})$.
Note that if $a$ is very small, the secular frequency differs from that of a
linear ion trap by a factor $\sqrt{2}$. Therefore, the ion confinement in a
RIT is twice as tight as in a linear trap with the same trap parameters.

We simulate buffer gas cooling of an ion in a RIT of variable $q$ and
$\Omega_{\text{rf}}$, and explore how the dependence of $E_{\text{kin}}$ on
these parameters compares to the case of the linear Paul trap. Results can be
seen in figure 4. We set the same ion trap and atom cloud parameters as in our
linear trap simulations. We simulate radial stray field strengths of 0 and 0.1
Vm-1 and do not include sources of axial and phase EMM. We set the stability
parameter $q_{\text{set,RIT}}$ in terms of the trap geometry and driving
signal as
$q_{\text{set,RIT}}=q_{\text{lin}}=2ev_{\text{rf}}/(m_{\text{i}}R^{2}_{\text{rf}}\Omega_{\text{rf}}^{2})$,
which is the same definition as used in the linear Paul trap. However, due to
the rotation of the potential, its effect on the ion energy differs from that
of $q_{\text{lin}}$. We simulate buffer gas cooling in a RIT with and without
a stray electric field of 0.1 Vm-1, as shown by the blue circles and triangles
in figure 4. The data is compared to the equivalent linear trap trends,
represented by the black dotted and dashed lines.

Figure 4: Ion energy after buffer gas cooling in a RIT. Results are shown for
traps of various $q$ (a) and $\Omega_{\text{rf}}$ (b), with (circles) and
without (triangles) a radial stray field of 0.1V/m. The black dotted and
dashed lines represent the trend in a linear Paul trap with similar trap
parameters. Note that the black lines have been adjusted according to the
relation $q_{\text{eff,RIT}}=\sqrt{2}q_{\text{lin}}$ (see text). Errors are
standard errors and are sometimes smaller than the point size.

We find good agreement between the RIT data and the linear Paul trap trend at
low $q$ when using $q_{\text{eff,RIT}}=\sqrt{2}q_{\text{lin}}$ to rescale the
latter. Here, $q_{\text{eff,RIT}}$ refers to the effective stability parameter
of the RIT and $q_{\text{lin}}$ is that of the linear trap. In figure 4a the
x-axis of the black trend line is rescaled by $\sqrt{2}$ to obtain the match
with the RIT data at low $q$, to within 3 $\mu$K. The black line in b is also
in good agreement with the RIT data when we use the same scaling of
$q_{\text{lin}}$. The $E_{\text{kin}}$ dependence on $\Omega_{\text{rf}}$ and
$q$ after this adjustment is thus similar for both traps, and significantly
lower energies cannot be reached in the RIT. In figure 4a, the minimum
$E_{\text{kin}}$ is achieved in the low-$q$ region. Only at high $q$ do the
linear and RIT data differ significantly, and in this limit $E_{\text{kin}}$
is higher in the RIT. Note that the energies reached in this graph cannot be
compared to those in figure 2, as the latter includes sources of axial and
phase MM, making the system inherently hotter. We conclude that although the
$q_{\text{set,RIT}}$ we set in the simulation is defined identically in
$v_{\text{rf}}$, $\Omega_{\text{rf}}$ and $R_{\text{rf}}$ to $q_{\text{lin}}$,
the effect of the rotating potential is that of a linear trap that is a factor
$\sqrt{2}$ higher in $q$. In a RIT, we can therefore use a lower-amplitude
voltage signal $v_{\text{rf}}$ to achieve similar ion temperatures as in a
linear Paul trap.

###  IV.3 Multipole ion trap

The benefits of using a MIT for buffer gas cooling of molecular and atomic
ions on the $\sim$ K-mK level has been extensively explored Wester2009RMT ;
Asvany2009NSK ; Notzold2020TMI . It has been shown that an increased number of
trap poles minimises rf heating effects, allowing more efficient buffer gas
cooling than would otherwise be achievable, as well as stable cooling of
systems with a smaller atom-ion mass ratio than is possible in quadrupole
traps HoltkemeierPRL . Here we investigate whether the same benefits extend to
cooling of the high mass-ratio Li/Yb+ system, beyond the lowest possible
energies that can be reached in a quadrupole trap.

In a MIT, the four electrodes of the Paul trap are replaced by a higher number
of electrodes, which are used to create an rf potential with $N$ poles. This
requires $N$ ground and $N$ rf electrodes. For a Paul trap $N=2$, while a
hexapole trap uses $N=3$ and an octupole trap has $N=4$. The resulting
potential can be written as MultiTextbook

$V_{\text{trap}}(t,r)=V_{\text{trap}}^{\text{dc}}(r)+\frac{v_{\text{rf}}}{2R_{\text{rf}}^{2}}r_{\perp}^{N}\cos{N\phi}\,.$
(12)

Here we use cylindrical coordinates $r_{\perp}=\sqrt{r_{x}^{2}+r_{y}^{2}}$ and
$\phi=\tan^{-1}(r_{y}/r_{x})$. The resulting motion in each direction
$j=x,y,z$ will depend on $N$ and can be expressed as
$\ddot{r_{x}}=-(a_{x}+\frac{N}{(N-1)}\eta_{x}\cos{\Omega_{\text{rf}}t})\frac{\Omega_{\text{rf}}^{2}}{4}r_{\perp}\cos{[(N-1)\phi]}$,
$\ddot{r_{y}}=-(a_{y}+\frac{N}{(N-1)}\eta_{y}\cos{\Omega_{\text{rf}}t})\frac{\Omega_{\text{rf}}^{2}}{4}r_{\perp}\sin{[(N-1)\phi]}$
and $\ddot{r}_{z}=-a_{z}r_{z}\frac{\Omega_{\text{rf}}^{2}}{4}$. Here, all
parameters are defined as in equation (3), except now the rf-dependent motion
is written in terms of a generalized stability parameter $\eta_{j}$. We define
this in terms of $q$ as

$\eta_{j}(r_{\perp})=q_{j}(N-1)\left(\frac{r_{\perp}}{R_{\text{rf}}}\right)^{(N-2)}\,,$
(13)

such that $\eta_{x}=-\eta_{y}=q$ when $N=2$. Here,
$q=q_{x}=-q_{y}=2ev_{\text{rf}}/(m_{\text{i}}R^{2}_{\text{rf}}\Omega_{\text{rf}}^{2})$
and $q_{z}=0$. For $N>2$, $\eta_{j}$ has a $r_{\perp}$-dependence that is not
present in a quadrupole trap. Depending on its energy, the ion oscillates
around the trap axis with an amplitude $A_{\perp}$. Therefore, in a MIT used
for buffer gas cooling, the stability parameter $\eta_{j}$ changes throughout
the cooling process as the ion’s energy decreases. In experiments, to ensure
that $\eta(A_{\perp})$ is kept roughly constant, changes to $A_{\perp}$ can be
compensated by ramping the other trap parameters, for instance $v_{\text{rf}}$
can be easily ramped.

We simulate buffer gas cooling of an ion in a hexapole ($N=3$) and octupole
($N=4$) trap of variable stability. We use the same trap parameters as in the
Paul trap from section III, except for $v_{\text{rf}}$, which we use to tune
the trap stability $\eta_{j}$. We set $v_{\text{rf}}$ to define
$\eta_{j}(r_{\perp})=\eta$ at a chosen ion position $r_{\perp}=100$ nm. We
repeat the simulations with and without a radial stray electric field of 0.05
Vm-1. Attempts were made to simulate the process at stronger fields, however
for $N>2$ this frequently resulted in a very unstable trap, as the ion was
pulled to high $r_{\perp}$. Propagating the ion under these conditions is
beyond the capability of our simulation. Similarly, for simulations with $N=4$
and $\eta>0.2$, frequent stability problems were encountered, making it
difficult to repeat the simulations. Note that these simulations do not
include sources of axial and phase EMM.

The greater the number of poles of the MIT, the more efficient the buffer gas
cooling process. Figure 5 shows the variation of $E_{\text{kin}}$ with $\eta$
for each trap, with (b) and without (a) stray electric fields. We vary $\eta$
between 0.1 and 0.5. Black lines show results for a quadrupole trap ($N=2$),
while red and olive data points show results for the hexapole and octupole
traps, respectively. In a, the trends of the hexapole and octupole data follow
similar shapes to the lines of the quadrupole trap, but with a shallower
gradient and an overall lower $E_{\text{kin}}$. Meanwhile in b we see
indications that the optimal value of $\eta$ in an octupole trap is higher
than in a quadrupole trap, and gives access to lower energies. The gradients
of the MIT trends in b are also shallower than in the quadrupole trap. The
lowest temperatures are therefore reached in traps of higher $N$, and the
consequences of a sub-optimal choice of $\eta$ are most severe for traps with
fewer poles.

To achieve the lowest possible ion energies in a MIT of high $N$ requires
well-compensated stray electric fields and a high-amplitude voltage signal.
This makes the use of high-order traps for buffer gas cooling practically
challenging. When cooled to the $\mu$K range, an ion in the simulated trap
will oscillate with an amplitude on the scale of $10^{-4}\times
R_{\text{rf}}$. Since $\eta_{j}$ scales with
$(r_{\perp}/R_{\text{rf}}^{2})^{(N-2)}$, traps of increasing $N$ require
increasing $v_{\text{rf}}$ to maintain a reasonable stability. For example, an
octupole trap with $\eta=0.2$ requires $v_{\text{rf}}=4.72$ GV, while the same
stability parameter can be achieved in a quadrupole trap with
$v_{\text{rf}}=62.95$ V. This also has severe implications for axial EMM in
the trap, as these voltages are likely to lead to significant rf pickup on the
dc electrodes. Even with a suitable choice of $v_{\text{rf}}$, the
$N$-dependent scaling of $\eta$ with $r_{\perp}$ means that traps with high
$N$ are more likely to become unstable when stray electric fields are present.
These practical limitations should be considered when selecting how many poles
would make a MIT most suitable for buffer gas cooling.

Figure 5: Ion energy after buffer gas cooling in a multipole ion trap. Results
are shown for an ion in a hexapole (red) and octupole (olive) trap, without
(a) and with (b) a radial stray field of 0.05 V/m. The dotted and dash-dotted
lines are the fit to the data from the corresponding quadrupole trap without
and with the radial stray field, respectively. Errors are standard errors and
are sometimes smaller than the point size.

##  V Hybrid optical-Paul trap

Another possible extension of the Paul trap is to combine it with an optical
tweezer for the ion. This hybrid optical-Paul trap would give an additional
confinement for the ion which could benefit the attainable ion kinetic
energies and suppress the impact of stray fields. Several approaches of using
optical ion trapping in atom-ion setups have been experimentally explored
Schaetz ; Karpa ; Sias , as an optical trap avoids the ion-heating effects
present in time-dependent trapping potentials. However, the optical trap for
an ion is typically less deep than an rf trap, meaning ions that gain energy
due to inelastic collisions are more easily lost. Here, we therefore focus on
a hybrid approach, designed to combine the benefits of rf and optical traps.
Unlike the traps discussed so far, a deep optical-rf hybrid trap will also
affect the density and temperature of the atom cloud. This leads to a tightly-
compressed buffer gas and off-resonant heating of the atoms. However, here we
focus on the effect on the ion alone, and assume in our simulations that the
atoms are unaffected by the optical potential.

The modeled hybrid optical-Paul trap consists of a quadrupole linear Paul trap
with an optical tweezer directed along its axis for additional ion
confinement. The setup is shown in the inset of Figure 6. This corresponds to
using the rf potential of (3) and adding to it a Gaussian potential

$V_{\text{opt}}(r_{\perp},r_{z})=-U\left(\frac{1}{1+\left(\frac{r_{z}}{z_{\text{R}}}\right)^{2}}\right)\exp\left({-\frac{2r_{\perp}^{2}}{w_{0}^{2}\left(1+\left(\frac{r_{z}}{z_{\text{R}}}\right)^{2}\right)}}\right)\,.$
(14)

Here $r_{\perp}=\sqrt{r_{x}^{2}+r_{y}^{2}}$, $w_{0}$ is the beam waist of our
optical tweezer, $z_{\text{R}}=\frac{\pi w_{0}^{2}}{\lambda}$ is the Rayleigh
length, and $\lambda$ is the frequency of the tweezer beam. The effective trap
depth $U$ is determined by the beam power $P$ and the polarizability $\alpha$
of the species in the trap. It is defined as $U=\frac{\alpha
P}{\pi\epsilon_{0}cw_{0}^{2}}$, with $\epsilon_{0}$ the permittivity of a
vacuum and $c$ the speed of light. The effect of the optical potential on the
ion motion is to give an additional position-dependent force towards the trap
centre

$F_{\text{opt},j}=-\frac{\text{d}V_{\text{opt}}}{\text{d}r_{j}}=-m_{\text{i}}\omega_{\text{opt,}j}^{2}r_{j}\,.$
(15)

Here $\omega_{\text{opt,}j}$ is the trapping frequency of the optical trap in
each direction
${\omega_{\text{opt,}x,y}=\sqrt{\frac{4U}{m_{\text{i}}w_{0}^{2}}}}$,
$\omega_{\text{opt,}z}=\sqrt{\frac{2U\lambda^{2}}{m_{\text{i}}\pi^{2}w_{0}^{4}}}$
Grimmreview . The confinement of the tweezer in the axial direction is
sufficiently weak for us to assume that $\omega_{\text{opt,}z}$ plays no
significant role here.

We simulate buffer gas cooling of an ion in this hybrid optical-Paul trap, as
a function of $T_{\text{a}}$ and $q$ for varying optical confinements. The
contribution from the linear Paul trap is taken with identical trap parameters
and geometry to those used in section III. We choose a stray radial electric
field of either 0 or 0.1 Vm-1. For the former, no other EMM effects are
included. The optical tweezer has a wavelength of 1070 nm, a variable power
set to 0.423, 4.23 and 2.12 W and a variable beam waist set to 2, 2 and 1
$\mu$m, respectively. The Stark shift of a 171Yb+ ion in such a tweezer gives
it a dynamic polarizability $\alpha$ of approximately 66 au (1.09$\times
10^{-39}$ Jm2V-2) Yb+polarizability . The chosen tweezer parameters therefore
correspond to respective trapping frequencies
$\omega_{\text{opt,}x,y}/\omega_{\text{sec,}x,y}=0.230,0.728$ and $2.05$ for
the 171Yb+ ion. Here, trapping frequencies are given as a ratio of the linear
Paul trap’s secular frequency, as defined in section III.

Our simulations show that adding an optical tweezer aids in decreasing the
effects of radial stray fields on the ion cooling. Therefore at low $q$, the
ion can be cooled to temperatures significantly lower than those accessible in
a Paul trap alone. Figure 6a shows the simulation data of $E_{\text{kin}}$ for
variable $q$ in a hybrid trap with stray electric fields of 0.1Vm-1. The data
is compared to the trend lines from a pure Paul trap with varying levels of
stray fields. All lines and data points in this plot also include
$E_{\text{ax}}=42.4$ $\mu$K and $E_{\text{ph}}=9.18$ $\mu$K at $q=0.219$ and
$\Omega_{\text{rf}}=2\pi\times 2$ MHz. For a hybrid trap with increasing
$\omega_{\text{opt}}$, the data points tend increasingly towards the trend
line of an rf trap with no stray electric fields. In other words, the tighter
the optical tweezer, the smaller the heating effects of radial EMM, allowing
the ion to reach much lower temperatures than in an equivalent regular Paul
trap. The same suppression of EMM effects is seen when analysing simulations
of a hybrid trap with variable $\Omega_{\text{rf}}$.

Figure 6: Ion energy after buffer gas cooling in an optical-rf hybrid trap.
The tweezer has a radial frequency $\omega_{\text{opt}}/\omega_{\text{sec}}$
of 0.23 (brown), 0.728 (orange) and 2.05 (pink). a Variation with trap
parameter q. The black trend lines are simulations for a Paul trap alone with
varying radial stray fields of 0.1 (black, dotted), 0.05 (black, dash-dotted)
and 0 Vm-1 (black, dashed). In this plot, the stray electric field for the
data points is 0.1 Vm-1. Inset: illustration of a hybrid optical-rf trap. b
Varying buffer gas temperature. The black, dashed line represents a Paul trap
without radial stray fields, while the grey dash-dotted line represents a
unity gradient. Data points are taken for a setup with no radial stray field.
Errors are standard errors and are sometimes smaller than the point size.

A tighter confinement of the optical tweezers leads to a more efficient buffer
gas cooling process. Figure 6b shows the linear relationship between the ion
energies reached in a hybrid trap of variable optical depth, and the
temperature $T_{\text{a}}$ of the buffer gas in which it is submerged. We also
plot the same relationship for a Paul trap without an optical tweezer as well
as a line with unity gradient, for comparison. The latter corresponds to a
static trap. We find that the hybrid trap results follow lines of increasingly
shallow gradient for an increasingly tight optical tweezer. This reflects the
time-independent tweezer potential becoming increasingly dominant over rf
contributions in determining the energy reached by the ion. The value of
$E_{\text{kin,i}}$ at $T_{\text{a}}=0$ K also decreases for an increasingly
tight tweezer.

To benefit from the optical tweezer, it must be perfectly aligned with the
Paul trap axis. Figure 7 shows the simulation of buffer gas cooling in a
hybrid trap in which the optical tweezer runs parallel to the Paul trap axis,
with a varying radial offset $\Delta$ between them. For comparison, we also
indicate the ion energy reached in an equivalent Paul trap with no optical
tweezer. Offsets of 50 nm or greater lead to ion energies above this value,
and the greater the offset, the higher the ion energy. For misalignment, the
effect of the tweezer is to pull the ion from the Paul trap axis, providing an
additional source of excess micromotion, akin to a radial stray field.

Figure 7: Ion energy after buffer gas cooling in a displaced rf-optical hybrid
trap. Orange data points mark the temperatures reached for a variable radial
offset ($\Delta$) between the Paul trap axis and the optical beam. The black
dashed line shows the energy achieved when simulating the same process with
the dipole beam switched off. Errors are standard errors and are sometimes
smaller than the point size.

Another important limitation for practical implementation of the optical-rf
hybrid trap is the effect of the optical potential on the atoms. At the high
trapping frequencies required to observe a significant benefit of the tweezer
on the ion, heating and confinement of the atoms become significant. If we
wish to trap an $\mathrm{Yb}^{+}$ ion in a tweezer with $\lambda=1070$ nm and
$\omega_{\text{opt,}x,y}=2.05\times\omega_{\text{sec,}x,y}$, a cloud of Li
atoms in the path of the beam will experience a trap depth of 82 mK
Grimmreview . This will lead to a tightly-compressed atom cloud. Combined with
the high photon-intensity of the tweezer, these atom densities give a
possibility for unwanted chemistry TomzaPhotoa ; Mohammadi2021LDC . The same
tweezer would also lead to heating of the Li atoms due to photon scattering at
a rate of 128 $\mu$Ks-1, significantly reducing the efficiency of buffer gas
cooling. To address the problem of trap depth, it may be possible that for
some atom/ion species combinations one could confine the ion in an optical
tweezer with a wavelength equal to a tune-out wavelength of the atoms, as has
been proposed for multi-species atom experiments LeBlanc2007SSO . However for
our Li/Yb+ species, this is not possible, as the only Li tune-out wavelength
that is red-detuned for Yb+ is too close to the D1 and D2 lines of Li
Lituneout . Suitable combinations of bichromatic optical dipole traps can also
be used for all-optical trapping of both atoms and ions arxivKarpa . We
consider this as an alternative method of reducing the effect of the tweezer
on the atom cloud in an optical-rf hybrid trap. However, the cooling in a
bichromatic trap is highly sensitive to relative position fluctuations between
the two trapping beams. Furthermore, for our species and desired trap depth,
the wavelength and intensity requirements cannot be met simultaneously with
the requirement for a large detuning of each beam from the atom and ion
transition lines, to avoid off-resonant heating. In fact, the heating problem
would be intensified by the higher-frequency beam.

Figure 8: Energy of one ion in a two-ion crystal after buffer gas cooling,
with the second ion confined by an optical tweezer outside of the atom bath.
Data points show results for a tweezer of varying depth. For comparison, the
black lines show the temperature reached by an ion in a Paul trap without a
tweezer for the single ion case (black, dashed) and two-ion case (black,
solid). Errors are standard errors. The grey shaded regions indicate standard
errors on the lines.

A way to avoid the negative effects of the tweezer on the atoms may be to use
a two-ion crystal, with one ion submerged in the atomic bath and the other in
the tweezer. Using the ion chain’s collective modes of oscillation, the pinned
ion could then be used to sympathetically enhance the cooling of the ion in
the buffer gas. We simulate this option and show the results in figure 8. It
shows the energy reached by the untweezed ion in the crystal, dependent on the
trapping frequency of the optical tweezer. We also show for comparison
$E_{\text{kin}}$ for a single ion (dashed line) and a two-ion crystal (solid
line) after buffer gas cooling in a Paul trap without an optical tweezer. The
results show that the higher the trapping frequency of the optical tweezer,
the lower the achievable ion energies. However, these energies tend towards a
minimum equal to the energy of a single ion in a Paul trap, for large tweezer
depths. The effect of the tweezer on the two-ion crystal is to decouple the
motion of the two ions, thereby giving no additional cooling benefits.

##  VI The quantum dynamics of micromotion-induced heating

Up until now, we have used classical physics to describe the dynamics of the
trapped ion in the buffer gas. However, at the simulated temperatures, quantum
effects may play a role. In particular, the angular momentum in the atom-ion
collision and the energy of the ion in its trap will both be quantized.
Furthermore, the dispersion of the atomic wavepacket may become much larger
than the micromotion amplitude of the ion such that it cannot be resolved at
all in a collision.

Specifically, when looking into micromotion-induced heating due to the time-
dependence of the trap, the classical and quantum mechanical picture might
differ. In the classical simulations of colliding atoms and trapped ions, the
heating stems from a temporary binding between the ion and atom during which
multiple collisions can occur Cetina . At some point, and depending on the
phase of the rf trap, the ion and atom can violently separate again. During
the whole process, energy can be absorbed from the rf field, leading to
heating of the system. One may wonder whether this behaviour survives in a
quantum treatment, in which binding can only occur in discrete molecular
states with energy level spacings that may be larger than the collision
energy. In this section we answer the question: Can quantum mechanics save us
from micromotion-induced heating in an atom-ion collision?

One approach is to consider the same case that we studied in the classical
regime, but this time launching an atomic wavepacket at an ionic wavepacket
that is held in a (quantized) rf Paul trap. Floquet theory allows us to solve
the Schrödinger equation, similar to previous theoretical investigations into
mechanically-controlled interactions between trapped atoms and ions
Idziaszek2007CCS ; Idziaszek2011MQD ; Nguyen2012MMT ; Joger2014QDA ;
Gerritsma2012BJJ  and individually-trapped atoms Krych2009CCT . However, even
if we restrict ourselves to one dimension to reduce the complexity, the
required Floquet Hilbert space is very large and convergence is very slow.
Instead, we follow a qualitative analysis of the problem, showing that quantum
mechanics cannot eliminate micromotion-induced heating.

###  VI.1 Quantum description of an ion in a Paul trap

We start by studying the ion motion quantum mechanically, when the ion is
confined by an rf-potential $\hat{V}_{\rm
rf}(t)=\frac{1}{8}m_{\text{i}}\Omega_{\rm rf}\left(a+2q\cos\left(\Omega_{\rm
rf}t\right)\right)\hat{x}_{\text{i}}^{2}$. Its motion can be described by the
Hamiltonian $\hat{H}_{\rm
i}(t)=\hat{p}_{\text{i}}^{2}/2m_{\text{i}}+\hat{V}_{\rm rf}(t)$, with
wavefunction $|\Psi_{\text{i}}(t)\rangle$ and momentum $\hat{p}_{\text{i}}$.
Using the Cook and Shankland transformation the wavefunction of the ion can be
written as $|\Psi_{\rm
CS}(t)\rangle=\exp\left[-i/(4\hbar)m_{\text{i}}q\Omega_{\rm
rf}\hat{x}^{2}\sin\left(\Omega_{\rm
rf}t\right)\right]|\Psi_{\text{i}}(t)\rangle$ Cook1985QTP . The fast dynamics
is now contained in the exponential factor and the slow-varying part of the
ion wavefunction evolves under the Hamiltonian $\hat{H}_{\rm i}(t)$. The rf-
potential $\hat{V}_{\rm rf}(t)$ can be separated into a secular and intrinsic
part by

$\hat{V}_{\rm
rf}(t)=\frac{1}{2}m_{\text{i}}\omega_{\text{sec,i}}^{2}\hat{x}_{\text{i}}^{2}+\hat{H}_{\text{mm}}(t)\,,$
(16)

where $\omega_{\text{sec,i}}$ takes the same value as in our classical
equations (see section III). Here, the ion intrinsic micromotion Hamiltonian
$\hat{H}_{\text{mm}}(t)$ is given by

$\hat{H}_{\text{mm}}(t)=-m_{\text{i}}g^{2}\omega_{\text{sec,i}}^{2}\hat{x}_{\text{i}}^{2}\cos(2\Omega
t)-g\omega_{\text{sec,i}}\\{\hat{x}_{\text{i}},\hat{p}_{\text{i}}\\}\sin(\Omega_{\rm
rf}t)\,.$ (17)

Furthermore, $g=[2(1+2a/q^{2})]^{-1/2}$ and $\\{.,.\\}$ denotes the anti-
commutator. Note that excess micromotion is not included in our quantum study.

Typically in analysis of the trapped ion, one would proceed with applying the
secular approximation. The basis for the secular approximation is to neglect
$\hat{H}_{\text{mm}}(t)$ such that the dynamics of the ion is given by a
static harmonic oscillator alone. Note that the relative size of the terms in
$\hat{H}_{\text{mm}}(t)$ are of a similar size to those of the static harmonic
oscillator and in fact, the mathematical basis for the secular approximation
is quite delicate. The reason it works at all is the structure of the
eigenstates of the harmonic potential. The time-dependent part
$\hat{H}_{\text{mm}}(t)$ contains terms $\propto\hat{x}_{\text{i}}^{2}$.
Taking the Fock states of the static harmonic oscillator as a convenient basis
in which to study the Hamiltonian $\hat{H}_{\rm i}(t)$, we see that strict
selection rules apply, i.e. $\langle n|\hat{x}_{\text{i}}^{2}|m\rangle=0$
except for $n=m\pm 2$ and $n=m$, with $|n\rangle$ denoting the $n$-th Fock
state. Since $\Omega_{\text{rf}}\gg\omega_{\text{sec,i}}$ such transitions
cannot be resonant. Therefore, the only effect of $\hat{H}_{\text{mm}}(t)$
here is to cause a small increase in the effective secular trapfrequency that
becomes more prominent for large $q$.

However, any anharmonicity in the trapping potential can cause a serious
breakdown of the secular approximation. This is because transitions
$|n\rangle\rightarrow|m\rangle$ close to the resonance condition
$|\omega_{\text{sec,i}}(n-m)|=\Omega_{\text{rf}}$ are allowed in anharmonic
traps. The observation of such parametric resonances in rf traps, in which
ions show rapid heating, dates back a long time Alheit1995OIP ;
Pedregosa2010ACR ; DengDBS2015 . Thus, for anharmonic potentials, the
contribution of $\hat{H}_{\text{mm}}(t)$ can not be ignored. A similar effect
is expected when describing an ion in the presence of a buffer gas, which adds
an additional atom-ion interaction potential.

###  VI.2 Quantum description of atom-ion interactions

Now we add the atom to our quantum description. Then, the Hamiltonian is given
by

$\hat{H}(t)=\hat{H}_{\text{a}}+\hat{H}_{\text{i}}(t)+\hat{V}_{\text{ai}}\\\
=\frac{\hat{p}_{\text{a}}^{2}}{2m_{\text{a}}}+\frac{\hat{p}_{\text{i}}^{2}}{2m_{\text{i}}}+\hat{V}_{\text{ai}}+\hat{V}_{\rm
rf}(t)\,,$ (18)

where $\hat{V}_{\rm ai}=-C_{4}/\hat{r}^{4}$ is the atom-ion interaction
potential, $r$ is the separation between the atom and the ion and $C_{4}$
takes the same value as that in our classical simulations (see section II). We
transform to the relative ($r$) and centre of mass ($R$) coordinates, to
obtain the atom-ion dynamics Joger2014QDA . In these coordinates, eq. (18) can
be written as

$\displaystyle\hat{H}(t)$
$\displaystyle=\hat{H}_{\text{R}}(t)+\hat{H}_{\text{r}}(t)+\hat{H}_{Rr}(t)\,,$
$\displaystyle\hat{H}_{R}(t)$
$\displaystyle=-\frac{\hbar^{2}}{2M}\frac{\partial^{2}}{\partial
R^{2}}+\frac{1}{2}M\omega_{R}^{2}\hat{R}^{2}+\hat{H}^{RR}_{\text{mm}}(t)\,,$
(19) $\displaystyle\hat{H}_{r}(t)$
$\displaystyle=-\frac{\hbar^{2}}{2\mu}\frac{\partial^{2}}{\partial
r^{2}}+\frac{1}{2}\mu\omega_{r}^{2}\hat{r}^{2}-\frac{C_{4}}{\hat{r}^{4}}+\hat{H}^{rr}_{\text{mm}}(t)\,,$
(20) $\displaystyle\hat{H}_{Rr}(t)$
$\displaystyle=\mu\omega_{\text{sec,i}}^{2}\hat{R}\hat{r}+\hat{H}^{Rr}_{\text{mm}}(t)\,,$
(21)

with $\omega_{R}=\omega_{\text{sec,i}}\sqrt{m_{\text{i}}/M}$ and
$\omega_{r}=\omega_{\text{sec,i}}\sqrt{m_{\text{a}}/M}$ denoting the centre of
mass and relative frequencies of the system, respectively. The micromotion
Hamiltonians are

$\displaystyle\hat{H}^{RR}_{\text{mm}}(t)$
$\displaystyle=-Mg^{2}\omega_{R}^{2}\hat{R}^{2}\cos(2\Omega
t)-g\omega_{R}\\{\hat{R},\hat{P}\\}\sin(\Omega_{\text{rf}}t)\,,$ (22)
$\displaystyle\hat{H}^{rr}_{\text{mm}}(t)$ $\displaystyle=-\mu
g^{2}\omega_{r}^{2}\hat{r}^{2}\cos(2\Omega
t)-g\omega_{r}\\{\hat{r},\hat{p}\\}\sin(\Omega_{\text{rf}}t)\,,$ (23)
$\displaystyle\hat{H}^{Rr}_{\text{mm}}(t)$ $\displaystyle=-\mu
g^{2}\omega_{\text{sec,i}}^{2}\hat{r}\hat{R}\cos(2\Omega
t)-g\omega_{\text{sec,i}}\left(\frac{\mu}{M}\\{\hat{r},\hat{P}\\}+\\{\hat{R},\hat{p}\\}\right)\sin(\Omega_{\text{rf}}t)\,.$
(24)

Here $\hat{P}$ and $\hat{p}$ are the centre of mass and relative momenta,
respectively.

To understand where heating can arise in this system, we must look at each of
the above equations individually. Let us begin with the centre of mass
Hamiltonian eq. (19). Excluding micromotion effects (i.e.
$\hat{H}^{RR}_{\text{mm}}(t)=0$), the centre of mass motion can be treated as
that of a harmonic oscillator, with solutions that are Fock states
$|n_{R}\rangle$ and energies $n_{R}\hbar\omega_{R}$. If we include
micromotion, $\hat{H}_{R}(t)$ resembles the Hamiltonian
$\hat{H}_{\text{i}}(t)$ of a single ion in a Paul trap, as described in
section VI.1, now with $m_{\text{i}}\rightarrow M$ and
$\omega_{\text{sec,i}}\rightarrow\omega_{R}$. Since this Hamiltonian is
without anharmonicity, it describes a stably trapped particle for which we may
employ the secular approximation. As a result, looking at the centre of mass
motion alone, it would seem that no significant heating is expected.

The relative Hamiltonian, eq. (20), takes a similar form to that of the centre
of mass, except it includes the additional static potential
$\hat{V}_{\text{ai}}$. Provided that there is no micromotion (i.e.
$\hat{H}^{rr}_{\text{mm}}(t)=0$), this static term does not give rise to
heating effects. However, it does mean that, unlike the centre of mass
Hamiltonian, the solutions to the unperturbed relative Hamiltonian are not
Fock states. Instead the dynamics must be obtained by solving the Schrödinger
equation by means of quantum defect theory Joger2014QDA . In the limit where
$r\rightarrow 0$, the energy is dominated by the atom-ion interaction, and we
obtain even and odd solutions given by Idziaszek2007CCS ; Idziaszek2011MQD

$\displaystyle\tilde{\psi}_{e}(r)$ $\displaystyle\propto$
$\displaystyle|r|\sin\left(\frac{R^{*}}{|r|}+\phi_{e}\right)\,,$ (25)
$\displaystyle\tilde{\psi}_{o}(r)$ $\displaystyle\propto$ $\displaystyle
r\sin\left(\frac{R^{*}}{|r|}+\phi_{o}\right)\,,$ (26)

where we set the even and odd short range phases as $\phi_{e}=\pi/4$ and
$\phi_{o}=-\pi/4$. We can use Eqs. (25),(26) as boundary conditions in a
Numerov method to find the relative eigenstates $|r\rangle$ and energies
$\epsilon_{r}$ of (20). The eigenstates can be subdivided in two types. On the
one hand we find many states that are mostly confined by the harmonic part of
(20) and we find a number of states that are deeply bound by the
$\hat{V}_{\text{ai}}$ term, which we identify as molecular states. Note that
the basis states of the total unperturbed Hamiltonian (18)
($\hat{H}_{\text{mm}}(t)=0$) are therefore given by
$|\Phi_{rn}^{\text{(b)}}\rangle=|r\rangle\otimes|n_{R}\rangle$.

When including micromotion in eq. (20), we again see similarities to eq. (19),
except for the addition of $\hat{V}_{\text{ai}}$. The Hamiltonian for the
relative coordinate resembles that of a single ion in a Paul trap with
$m_{\text{i}}\rightarrow\mu$ and $\omega_{\text{sec,i}}\rightarrow\omega_{r}$,
but with $\hat{V}_{\text{ai}}$ acting as an additional anharmonic term. This
anharmonicity becomes a problem now that we include micromotion, as the time-
dependent terms will cause parametric heating in analogy to heating observed
in anharmonic rf traps Alheit1995OIP ; Pedregosa2010ACR ; DengDBS2015 . Note
that due to the dense spectrum of nearly-free scattering states, there will
always be (near) resonant transitions available for the parametric heating.

Finally, we consider the role of eq. (21), which includes contributions from
both the centre of mass and relative motions. Its contribution to the total
Hamiltonian (18) can be interpreted as a further anharmonicity, in addition to
$\hat{V}_{\text{ai}}$, with implications for both the relative and centre of
mass dynamics. This anharmonicity is not a problem until we include non-zero
micromotion terms, as the combination of micromotion and an anharmonic
potential gives rise to allowed transitions close to a resonance. In the case
of the total Hamiltonian (18), the crossterm $\hat{H}^{Rr}_{\text{mm}}(t)$
drives transitions
$|n_{R}\rangle|r\rangle\rightarrow|n_{R}+1\rangle|r^{\prime}\rangle$ that
follow the resonance condition
$|\epsilon_{r}-\epsilon_{r}^{\prime}|=\hbar\omega_{\text{i}}+k\hbar\Omega$
with $k=0,\pm 1,\pm 2$. This leads to heating in both the relative and centre
of mass coordinates. Physically, it can be interpreted as the system absorbing
and emitting quanta of energy $\hbar\Omega$ from the trap driving. By
investigating the roles of $\hat{H}_{Rr}(t)$ and $\hat{V}_{\text{ai}}$ in this
way, it becomes clear that quantum mechanics is unlikely to save us from
micromotion-induced heating.

Having established the sources of anharmonicity in the system, we now further
investigate the micromotion terms of (18), which perturb the dynamics of the
system and combine with the anharmonic potential to give rise to parametric
heating. In particular, we will investigate the expected scale of this
micromotion perturbation, by looking at the full hamiltonian and calculating
the matrix elements $\langle n_{R}|\langle
r|H_{\text{mm,1}}^{\text{max}}|r\rangle|n_{R}\rangle$. Here $H_{\text{mm,1}}$
denotes the prefactor to the terms $\propto\sin\Omega_{\text{rf}}t$ that
dominate the micromotion Hamiltonians $\hat{H}^{RR}_{\text{mm}}(t)$,
$\hat{H}^{rr}_{\text{mm}}(t)$ and $\hat{H}^{Rr}_{\text{mm}}(t)$. Joger2014QDA
. We consider the realistic scenario where $\Omega_{\text{rf}}=2\pi\times 2$
MHz, $a=0$ and $q=0.5$, similar to the experimental parameters of ref. Nature
. The results are shown in Fig. 9.

Figure 9: Matrixelements $D_{nr,n^{\prime}r^{\prime}}=|\langle n_{R}|\langle
r|H_{\text{mm,1}}|r^{\prime}\rangle|n^{\prime}_{R}\rangle|$ versus the energy
difference $\delta E=|E_{nr}-E_{n^{\prime}r^{\prime}}|$. Data is shown for a
system of 6Li and 171Yb+, confined in a Paul trap with parameters
$\Omega_{\text{rf}}=2\pi\times 2$ MHz, $a=0$ and $q=0.5$.

We see that the couplings due to micromotion can be up to $\sim
0.1\hbar\omega_{\text{sec,i}}$ for an energy difference $\delta
E\sim\hbar\Omega_{\text{rf}}$. This energy scale is comparable to the
fractional change of kinetic energy $K$ for a classical hardcore collision.
Assuming the atom is at rest initially and the initial ion kinetic energy is
$K_{1}$, we get for the final ion kinetic energy
$K_{2}=\left(1-\frac{2\beta}{(1+\beta)^{2}}\right)K_{1}=0.93K_{1}$, with
$\beta=m_{\text{a}}/m_{\text{i}}$. For a cold ion
$K_{1}\sim\hbar\omega_{\text{sec,i}}$, giving $K_{1}-K_{2}\sim
0.07\hbar\omega_{\text{sec,i}}$. We therefore conclude that buffer gas cooling
to the ion ground state is not possible, even in the quantum picture, as
micromotion-induced heating will impose a limit on the ion temperature,
similar in scale to that seen in classical analysis.

##  VII Conclusion and Outlook

In this work, we demonstrate that careful optimization of ion trap parameters
is crucial for reaching cold temperatures when buffer gas cooling ions with
ultracold atoms in time-dependent traps. We show that, under appropriate
conditions, buffer gas cooling should allow one to reach the regime of
quantized ion motion. For the Li/Yb+ system, parameter optimization can allow
buffer gas cooling to ion kinetic energies three times colder than the
previous best experimental values, corresponding to collision energies
approximately 0.6 times the $s$-wave limit. This makes buffer gas cooling
capable of outperforming laser Doppler cooling in some situations, and it is
even competitive when compared to some sub-Doppler cooling techniques such as
Sishyphus cooling. These low energies are achievable by suppression of
micromotion-induced heating effects, through optimization of $q$ and
$\Omega_{\text{rf}}$, and by compensation of trap imperfections to low, but
experimentally realistic, levels. Additionally, we benefit from a large ion-
atom mass ratio. For equal mass systems, similar trap parameter dependencies
were found for a linear Paul trap Pinkas2020EIT , however the achievable
temperatures for such systems remain far from the $s$-wave regime.

We further demonstrate that the lowest possible energies achievable in a
linear Paul trap can also be reached through similar parameter optimization in
a range of rf traps. In the case of the multipole and hybrid optical-rf traps,
even lower temperatures can be reached, albeit with the requirement for
challenging experimental parameters and, for the hybrid trap, a method of
avoiding photon-scattering heating effects. In addition, we provide a quantum
description of the atom-ion interaction, and argue that micromotion-induced
heating continues to play a significant role in this regime.

Our study shows a route towards reaching deeper into the quantum regime with
time-dependent traps, and presents buffer gas cooling as a feasible
alternative to laser cooling techniques. Especially for studies of quantum
chemistry and precision spectroscopy, cold controlled collisions in hybrid
atom-ion systems are important. Moreover, in the quantum regime one can
benefit from the tunability of the atom-ion interaction through Feshbach
resonances Idziaszek2011MCQ ; Tomza2015CIY ; SchaetzFRs  and use the ground
state cooled-ion for high-precision studies of charged impurity physics
Astrak2021IPBEC ; Oghittu2021doa .

##  VIII Acknowledgements

This work was supported by the Netherlands Organization for Scientific
Research (Vidi Grant 680-47-538 and Start-up grant 740.018.008, and Vrije
Programma 680.92.18.05 (R.G.)). R.S.L. acknowledges funding from the European
Unions Horizon 2020 research and innovation programme under the Marie
Sklodowska-Curie grant agreement No 895473. A.S.N is supported by the Dutch
Research Council (NWO/OCW), as part of the Quantum Software Consortium
programme (project number 024.003.037).

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##  X Appendix A

Fit parameter | Trend in $q$ | Trend in $\Omega_{\text{rf}}$
---|---|---
$A_{x}$ | 2.483$\times 10^{-8}$ | 1.335$\times 10^{-8}$
$A_{y}$ | 2.488$\times 10^{-8}$ | 1.335$\times 10^{-8}$
$A_{z}$ | 1.859$\times 10^{-7}$ | 1.335$\times 10^{-7}$
$b$ | 1.782 | 1.690
$c$ | 1.689 | 1.704
$d$ | 1.688 | 1.771
$g$ | 0 | 9.871
Table 1: Fit parameters for the ion energy and motion after buffer gas cooling
in a linear Paul trap. These parameters together with equation (7) are used to
obtain the lines in Fig 2a and b for varying radial micromotion contributions.
Note that these results were obtained not only by fitting to the data shown in
figure 2, but also to data from similar simulations that excluded one or more
forms of EMM.