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Indirect Cooling of Weakly Coupled Trapped-Ion Mechanical Oscillators
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Indirect Cooling of Weakly Coupled Trapped-Ion Mechanical Oscillators
Pan-Yu Hou ,1,2,Jenny J. Wu,1,2Stephen D. Erickson ,1,2,?Giorgio Zarantonello ,1,2,?Adam D. Brandt ,1
Daniel C. Cole,1,§Andrew C. Wilson,1Daniel H. Slichter,1and Dietrich Leibfried1,∥
1National Institute of Standards and Technology, Boulder, Colorado 80305, USA
2Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
(Received 10 August 2023; accepted 4 March 2024; published 2 April 2024)
Cooling the motion of trapped ions to near the quantum ground state is crucial for many applications in
quantum information processing and quantum metrology. However, certain motional modes of trapped-ion
crystals can be difficult to cool due to weak or zero interaction between the modes and the coolingradiation, typically laser beams. We overcome this challenge by coupling a mode that interacts weakly with
cooling radiation to one that interacts strongly with cooling radiation using parametric modulation of the
trapping potential, thereby enabling indirect cooling of the weakly interacting mode. In this way, wedemonstrate near-ground-state cooling of motional modes with weak or zero cooling radiation interaction
in multi-ion crystals of the same and mixed ion species, specifically
9Be?-9Be?,9Be?-25Mg?, and
9Be?-25Mg?-9Be?crystals. This approach can be generally applied to any Coulomb crystal where certain
motional modes cannot be directly cooled efficiently, including crystals containing molecular ions, highly
charged ions, charged fundamental particles, or charged macroscopic objects.
DOI: 10.1103/PhysRevX.14.021003 Subject Areas: Atomic and Molecular Physics,
Quantum Physics, Quantum Information
I. INTRODUCTION
Trapped-ion systems are a leading platform for quantum
information processing and quantum metrology because oftheir long coherence times [1], the ability to perform high-
fidelity quantum state preparation and measurement [2–8],
and high-fidelity coherent quantum operations [3,9–13].
Coupling different ion species using quantum-logic-basedtechniques [14]can enable experiments on ion species that
lack convenient optical transitions for cooling, state prepa-
ration, and measurement. This method has been usedfor precision spectroscopy and quantum metrology ofion-based optical clocks [15,16] , molecular ions [17,18] ,
and highly charged ions [19,20] , to enable novel frequency
standards and tests of fundamental physics.
Many applications require cooling the ion motion to near
the ground state, which is typically accomplished usinginteractions between internal “spin”states and the motion
combined with a dissipative process. Spin-motion inter-actions can be realized with laser beams [21,22] or
magnetic field gradients [23–25], with the dissipation
necessary for cooling introduced by the spontaneous
emission during laser-driven repumping. In some instances,
ions cannot be directly laser cooled, because they lacksuitable transitions or because their internal states containquantum information that must be preserved. In these cases,cotrapped ion(s) of the same or different species, referred toas coolant ion(s), can be used for sympathetic cooling ofcollective motional modes (normal modes) of the whole ioncrystal [26]. The cooling of a given motional mode due to
interaction with radiation competes with motional heating
due to the environment, resulting in a steady state with anonzero average mode occupation. The cooling and heatingrates determine this steady-state occupation and the timerequired to achieve a steady state. However, some motionalmodes have small cooling rates due to weak interactionswith the available cooling radiation because of either
geometrical constraints on the cooling radiation and/or
small or no participation of coolant ions in the modes.Inefficient cooling can lead to high steady-state motionaloccupation that can limit gate fidelities or experimentalprecision.
panyu.hou@colorado.edu
Present address: Center for Quantum Information, Institute forInterdisciplinary Information Sciences, Tsinghua University,Beijing 100084, People ’s Republic of China.
?Present address: Quantinuum, Broomfield, Colorado 80021,
USA.
?Present address: QUDORA Technologies GmbH, Braun-
schweig, Germany and Institute of Quantum Optics, LeibnizUniversity Hannover, Hannover, Germany.
§Present address: ColdQuanta, Inc., Boulder, Colorado 80301,
USA.
∥dietrich.leibfried@nist.gov
Published by the American Physical Society under the terms of
theCreative Commons Attribution 4.0 International license.
Further distribution of this work must maintain attribution to
the author(s) and the published article ’s title, journal citation,
and DOI.PHYSICAL REVIEW X 14,021003 (2024)
2160-3308 =24=14(2) =021003(17) 021003-1 Published by the American Physical SocietyHere, we perform indirect cooling of motional modes
whose direct cooling rate from radiation is small or zero.
We coherently couple these weakly cooled modes (WCMs)to strongly cooled modes (SCMs) with a large direct
cooling rate, using parametric modulation of the trapping
potential recently demonstrated in Ref. [27]. Indirect
cooling is accomplished either by repeatedly exchanging
the WCM and SCM states and recooling the SCM or by
cooling the SCM while simultaneously coupling it to theWCM. Our work builds on previous demonstrations in ionsand other platforms. For instance, the “axialization ”tech-
nique is widely used in Penning traps to improve cooling of
magnetron motion [28,29] , and ground-state cooling of
mechanical resonators has been accomplished by coupling
to a cavity [30–32], among other works [33–37]. We extend
this method to the cooling of normal modes of multi-ioncrystals composed of same and mixed species ions in rf
Paul traps.
The manuscript is organized as follows: Section II
describes why some modes are weakly cooled in common
experimental situations. We then introduce and discuss two
different schemes for indirect cooling of WCMs in Sec. III
and describe the apparatus in Sec. IV. Experimental results
on indirect near-ground-state cooling of WCMs in
9Be?-9Be?,9Be?-25Mg?, and9Be?-25Mg?-9Be?crystals
are reported in Secs. V,VI, and VII, respectively.
II. ORIGINS OF WEAKLY COOLED MODES
WCMs often arise due to geometric relationships
between the ion crystal and the cooling radiation, for
example, if the effective wave vector of the cooling
radiation has small or zero projection onto a direction ofion motion in a mode of interest. This encompasses
multiple cooling modalities, referring to the wave vector
of a single laser beam for Doppler cooling [21,38] or
resolved sideband cooling [39], the difference wave vector
of two laser beams in Raman sideband cooling [21,40] or
electromagnetically induced transparency cooling [41,42] ,
or the direction of a magnetic field gradient used in rf ormicrowave sideband cooling [21,25,43,44] . Often, laser
beams or magnetic field gradients are deliberately set up
with an effective wave vector projection that predominatelyaligns with the axes of specific motional modes, which are
used for coherent operations to produce ion-motion or
ion-ion entanglement, and small projection onto other“spectator ”motional modes, to reduce the contributions
of spectator modes to the Debye-Waller effect [21].
However, the anharmonicity of the Coulomb interactioncan lead to cross-Kerr-type coupling with spectator modes,
resulting in motional decoherence through mutual
motional-state-dependent frequency shifts if the spectatormodes are not cold [45,46] . This has been identified as one
of the leading error sources in some two-qubit entangling
gates [10,12] and quantum logic spectroscopy experiments
[47]. Cooling these spectator modes directly to improveperformance often requires additional laser beams or
magnetic field gradients.
Inefficient cooling can also occur for modes in which the
coolant ions do not participate strongly. The mode partici-
pation depends on the ion species and the crystal configu-ration and is denoted as ξ
?i?
j;mfor the component of the jth ion
in the normalized mass-weighted eigenvector of the mth
mode along the ith spatial axis [41]. The motion of different
species in the same potential well becomes more decoupledas their charge-to-mass ratios become more mismatched.
This decoupling means that some motional modes have large
participations from ions of only one species but not the otherspecies. This poses a challenge for sympathetic cooling, as
modes without large participation from the coolant species
cannot be cooled efficiently, and has been investigated forquantum information processing [48–52]and precision
spectroscopy [53]. It is particularly problematic for modes
in the directions perpendicular to the axis of a linear mixed-species ion crystal (radial modes), where participations can
be highly imbalanced between species [52–54]. A different
approach, based on algorithmic cooling, has been demon-strated for ground-state cooling of WCMs in a crystalcontaining highly charged ions [54]. This method requires
driving sideband transitions on the spectroscopy ions, limit-
ing the scope of applicability of the technique.
The symmetry of the ion crystal can give rise to small or
vanishing mode participations for specific ions. For exam-ple, in a crystal that is mirror symmetric around its center
and consists of an odd number of ions, the center ion is
completely decoupled from all normal modes that haveeven parity under reflection across the plane of symmetry
and cannot be used to directly cool these modes. In the
simplest case, a symmetric three-ion crystal, the twooutside ions oscillate out of phase while the middle ion
does not participate in the axial and two radial out-of-phase
modes. If the middle ion is the only coolant ion, thesemodes cannot be cooled directly.
The ability to sympathetically cool the out-of-phase
modes of symmetric three-ion crystals can be importantto the quantum charge-coupled device architecture for
trapped ion quantum computing [21,55,56] . A three-ion
“data-helper-data ”crystal could be used to implement
entangling gates between the two data qubit ions on the
sides while still containing one helper ion in the middle for
sympathetic cooling and other auxiliary operations. Theirout-of-phase modes are particularly appealing candidates
for mediating high-fidelity two-qubit gates [57], as they
often have much lower heating rates than modes where thecenter of mass of the crystal moves as well as large
participations for data ions. However, since this helper
ion does not participate in the out-of-phase modes, indirectcooling techniques are required. Out-of-phase modes oftwo-ion crystals have been used to demonstrate the highest-
fidelity entangling gates, with Bell-state fidelities of
approximately 99.9% [10,12,13] .PAN-YU HOU et al. PHYS. REV . X 14,021003 (2024)
021003-2III. INDIRECT COOLING SCHEMES
We begin with a brief description of the principle of
coherent coupling between motional modes. More detailsabout this technique can be found in Ref. [27]. Consider a
crystal composed of Nions possessing 3Nharmonic
oscillator normal modes [26], including a WCM at fre-
quency ω
walong direction iwand a SCM at ωsalong is.W e
assume that the WCM and SCM can be coherently coupledby an exchange interaction of the form
H
c??gw;s??w??s??w?s??; ?1?
where ?is the reduced Planck constant, gw;sis the coupling
rate between the modes, and the creation and annihilation
operators are, respectively, ?w?and ?wfor the WCM and ?s?
and ?sfor the SCM. An exchange coupling of the desired
form can be realized by using the trap electrodes to apply atime-dependent potential modulation:
U?r;t??U?r?cos?δ
w;st?? 2?
to the ions, which oscillates at the mode frequency differ-
ence δw;s?jωw?ωsjand has a suitable spatial variation to
achieve a nonzero coupling strength. The coupling strength
gw;sis a sum over the contributions gjfrom all Nions:
gw;s?XN
j?1gj?XN
j?1Qjαj
4Mj???????????ωwωsp ξ?iw?
j;wξ?is?
j;s; ?3?
where QjandMjare the charge and mass, respectively, of
thejth ion and
αj??2U
?iw?is/C12/C12/C12/C12
r?rj;0?4?
is the potential curvature at the equilibrium position rj;0of
thejth ion. Finite gw;srequires that the product of mass-
weighted mode participations ξ?iw?
j;wξ?is?
j;sis nonzero for at
least one of the ions, and, ideally, the curvatures at the ion
equilibrium positions should be chosen such that all gjadd
constructively [27].
Transforming Eq. (1)to the interaction frame with
respect to the Hamiltonian for the uncoupled modes, the
creation and annihilation operators acquire a periodic timedependence ?w
??t?and ?s??t?[27]. We can write an arbitrary
pure state of the motional modes at time tasjΨ?t?isjΦ?t?iw
in terms of these operators as
jΨ?t?isjΦ?t?iw?X∞
m;n?0cmn??????????
m!n!p ??s??t?/C138m??w??t?/C138nj0isj0iw;?5?
where the constants cmnare determined by the initial state
att?0and the time dependence is fully captured by thecreation and annihilation operators. For particular times
τk?kπ=?2gw;s?, with kan odd positive integer, the
operators can be written as
?w??τk??ieikπ=2?s??0?;
?s??τk??ieikπ=2?w??0?: ?6?
At these times τk, the populations of the motional modes
are swapped: The populations jhΨ?τk?jnisj2of the SCM at
τkin its number basis jnisare equal to the populations
jhΦ?0?jniwj2of the WCM at t?0in its number basis jniw,
and vice versa. When kis an even positive integer, the
operators are the same as they were at t?0up to a phase
factor:
?w??τk??eikπ=2?w??0?;
?s??τk??eikπ=2?s??0?; ?7?
and the motional mode populations have been swapped
back to their original mode initial states jhΨ?τk?jnisj2?
jhΨ?0?jnisj2andjhΦ?τk?jniwj2?j hΦ?0?jniwj2.
Thus, the effect of the mode coupling interaction in
Eq.(1)is to swap the mode populations back and forth
between the SCM and WCM, with the duration of a single
swap given by τw;s?π=?2gw;s?.I f ωs≠ωw, the total
energy in the two modes is different between swappedand unswapped configurations provided they are not both
in their ground states. The necessary energy difference is
supplied or absorbed by the external drive that creates theparametric modulation of the potential.
We point out that the swapping operations introduce a
number-state-dependent phase factor on the complex
amplitudes hΨ?t?jni
sandhΦ?t?jniwthat may affect coher-
ent operations [27] but can be ignored for thermal state
distributions during cooling.
Figure 1illustrates two schemes for indirect cooling of
WCMs. Figures 1(a)–1(d) depict a pulsed scheme, where
the SCM and WCM start out with initial average motional
occupations ?ni;wand ?ni;s, illustrated in Fig. 1(a). The SCM
is then cooled in Fig. 1(b) to low motional occupation
?ns≈0. A coupling pulse of duration τw;sin Fig. 1(c)then
swaps the mode occupations such that ?nw≈0and
?ns??ni;w. A second round of cooling on the SCM in
Fig.1(d) brings its average occupation to ?ns≈0again. In
this way, the occupations of both modes can be reduced
compared to their initial values and approach their cooling
limit after one or several repetitions of this sequence.Multiple repetitions can suppress the effects of photon
recoil heating and incomplete motional exchanges.
Practically, τ
w;sshould be made as short as possible to
minimize anomalous heating during swaps while making
sure that the swap pulses themselves do not cause sub-
stantial excitation in any mode.INDIRECT COOLING OF WEAKLY COUPLED TRAPPED-ION … PHYS. REV . X 14,021003 (2024)
021003-3Figure 1(e)shows a second, continuous scheme, where
the cooling of the SCM and a continuous coupling betweenSCM and WCM are applied simultaneously. The param-
eters of the cooling and the coupling need to be jointly
optimized, because the mode coupling results in two new
eigenmodes (dressed modes) with ladder operators
?1=???
2p
???w/C6?s?, split in frequency by 2g
w;s. The optimal
frequency for monochromatic cooling radiation is centered
between these two cooling resonances so that the two
dressed modes can be cooled at equal rates. Cooling is most
efficient if the coupling rate is comparable to the cooling
rate. Weaker coupling slows the cooling of the WCM,while stronger coupling increases the frequency splitting of
the dressed modes and, thus, the detuning from the cooling
radiation, which slows down the cooling of both modes.
Because the minimum occupation is set by the competition
between heating and cooling rates for the two modes, the
detuning of the cooling radiation in the continuous scheme
may result in a higher occupation compared to the pulsed
scheme. The effects of mode splitting can be alleviated by
taking advantage of methods with large cooling rate andwide bandwidth, such as electromagnetically-induced-
transparency cooling [41].
Imperfections of the coupling potential can also affect
cooling. Residual oscillating potential gradients lead to
driven motion at δ
w;s[34], which is analogous to micro-
motion due to the rf trapping field [58]and can reduce the
coupling of the cooling radiation to the ions. Undesiredcurvatures, which are unavoidable due to the Laplace
equation, can cause small shifts to the mode frequencies,
but this can be accounted for by adjusting the frequency ofcooling radiation.
Both the simultaneous and pulsed schemes should be
compatible with a wide variety of laser cooling techniques
[41,59 –61], and the initial mode occupations can be
arbitrarily high as long as the ions remain in a crystalwith a well-defined mode structure. Even if the initial
temperature of the WCM is very high, it can be cooled
using at least one of these schemes at a rate that approachesthe rate achievable for cooling the SCM alone. Thesimultaneous scheme is simpler to implement and reachesthe same occupation more quickly compared to the pulsed
scheme, if the coupling potential is nearly perfect and the
coupling rate can be made comparable to the cooling rate ofthe SCM. The pulsed scheme can achieve lower occupa-tions and is more robust against coupling potential imper-
fections. If a certain WCM is the mode of interest, cooling
using the simultaneous scheme can be followed by acooling pulse on the SCM with the coupling turned offand a final swap to achieve the lowest possible occupationon this particular WCM.
It is worth noting that motional modes of single ions
confined in adjacent potential wells are coupled if theirmotional frequencies are resonant [62,63] . A similar idea of
indirect cooling of single ions based on this technique has
been proposed [62,64] and recently demonstrated using
two identical ions [65].
IV. EXPERIMENTAL SETUP
We trap
9Be?and25Mg?ions in a segmented linear Paul
trap[66] in linear Coulomb crystals. The direction of the
trap axis is denoted as zin Fig. 2(a), and the confinement
along this direction is chosen to be sufficiently weaker than
that in the perpendicular directions such that multi-ion
crystals align with the trap axis. The time-independentstatic potentials and rf pseudopotentials are approximatelyharmonic in all three dimensions, and the other twoprincipal axes are denoted with xandy. The coordinate
origin is chosen to coincide with the total potential
minimum.
Mixed-species ion crystals are Doppler cooled on both
species simultaneously and then just on the Be
?ions to
achieve lower mode occupations due to the narrower
linewidth of the Be?excited states. After Doppler cooling,
Be?is optically pumped into the qubit state j↓iB≡
2S1=2jF?2;mF?2iBand Mg?into the qubit state(a)
Cooling
SwapWCM
ωwSCM
ωs
CouplingSimultaneous coolingPulsed cooling
Cooling
Cooling(b)
(c)
(d)
(e)
FIG. 1. Two cooling schemes. (a) –(d) Pulsed cooling sequence.
(a) Initial number state distributions of both motional modes.(b) The SCM is cooled to near its ground state. (c) A couplingpulse swaps the mode populations. (d) The SCM is cooled again.This sequence can be repeated and interleaved with direct orindirect cooling of other modes. (e) Simultaneous coolingsequence. The cooling of the SCM and the coupling betweenmodes occur simultaneously.PAN-YU HOU et al. PHYS. REV . X 14,021003 (2024)
021003-4j↓iM≡2S1=2jF?3;mF?3iM. We can drive transitions
among all hyperfine states within the2S1=2ground level of
the9Be?and25Mg?ions using microwave fields near 1.2
and 1.7 GHz, respectively, applied to antennas outside thevacuum apparatus that address the ions globally. An
external magnetic field of 11.9 mT, chosen so that the
frequency of the
2S1=2jF?2;mF?0iB?2S1=2jF?1;
mF?1iB“clock ”transition is first-order insensitive to
magnetic field fluctuations, lifts the degeneracy between
the states in each hyperfine manifold. For most of our
experiments, we use j↑iB≡2S1=2j1;1iBand j↑iM≡
2S1=2j2;2iMas the other qubit state. The qubit state is
read out using state-dependent fluorescence, after first
“shelving ”populations in the j↑iB=Mstates to the “dark”
states2S1=2j1;?1iBand2S1=2j2;?2iMby a series of micro-
wave pulses [67]. Fluorescence photons from both species
are collected using an achromatic objective and counted
with a photomultiplier tube (PMT), with each species readout sequentially. Qubit state determination is performed by
thresholding the number of PMT counts (for single ions ofone species) or by maximum-likelihood estimation based
on count histograms (for two ions of one species).
As illustrated in Fig. 2(a), three laser beams (solid red
arrows) globally address the Be
?ions for Raman transitions.
The wave vector difference ΔkBe;12?kBe;1?kBe;2between
beams 1 and 2, aligned with the trap ( z) axis (horizontal
dashed red arrow), is selected to manipulate normal modes in
the axial direction by coupling them to the hyperfine ground
states of the Be?ions. Radial modes are coupled to the qubit
states using beams 2 and 3 with ΔkBe;23?kBe;2?kBe;3
(vertical dashed red arrow), which is at an angle of approx-
imately 45° to both xandyin the xyplane. The wave vector
difference of the two Mg?Raman beams (solid green arrows)
couples hyperfine states of Mg?ions to the axial modes only.
We perform continuous sideband cooling (CSBC) [21] on
relevant normal modes where the relevant states and laser
couplings are shown in Fig. 2(b) for Be?and Fig. 2(c)for
Mg?. Raman beams resonant with a first- or second-order
red-sideband (RSB) transition j↓iB=Mjn?li?j↑iB=Mjni
(l?1, 2) are simultaneously applied with two resonant light
fields repumping the ions through the excited2P1=2state (for
both Be?and Mg?)o r2P3=2state (for Mg?)b a c kt o
j↓iB=Mjni, predominately without changing the motional
state [21]. The second-order RSB ( l?2) is used for cooling
high number states which have very low Rabi frequencies on
the first-order RSB transition [21].
Coupling potentials are produced by 12 control electrodes
closest to the ions (also see Appendix B) that are driven by
voltages oscillating near δw;swith individual amplitudes
Vi?i?1;…;12?, applied to the electrodes via a two-stage
low-pass filter with a 3 dB corner frequency of about 50 kHz.
Amplitudes Viare calculated from simulations of the electric
potential produced by the electrodes at the position of the
ions to generate the desired local potential curvatures while
minimizing potential gradients. The coupling potential
U1?r?has nonzero curvature elements ?2U1=??x?z?and
?2U1=??y?z?at the equilibrium positions of a Be?-Be?or
Be?-Mg?two-ion crystal which couple axial modes with
radial modes, while coupling potential U2?r?is approxi-
mately proportional to z3and serves to couple the axial
modes in a Be?-Mg?-Be?crystal. To reduce off-resonant
driving of motional modes, the coupling pulse amplitudes
ramp up with an approximate sine-squared envelope in
τr?20μs?2π=δw;s, stay constant for τc, and ramp back
to zero using the time reversal of the ramp up. The shaped
pulse area is equal to that of a square pulse with a duration
τ?τr?τc(see Appendix Bfor more details).
V. COOLING WCMs IN A SAME-SPECIES
CRYSTAL
We begin by demonstrating indirect near-ground-state
cooling of two radial modes of a Be?-Be?crystal [see
(a)
kMg,1 kMg,2kBe,1 kBe,2kBe,3
ΔkBe,23
ΔkMg,12
ΔkBe,12
|F=2,mF=1(b)
Repump9Be+2P1/2
2S1/2(c)
Repump
Raman25Mg+
2S1/22P3/2
2P1/2
|F=3,mF=2Raman9Be+ 25Mg+zx+y
x-y
FIG. 2. Laser beam setup and level schemes. (a) Wave vectors
of9Be?(solid red arrows) and25Mg?(solid green arrows)
Raman beams relative to a9Be?-25Mg?ion crystal oriented
along the axial ( z) direction. The two radial directions ( x,y) are
oriented at /C645° to the plane spanned by the wave vectors. Wave
vector differences are also indicated (dashed red and greenarrows). The beams are global, illuminating all ions in thecrystal. (b) Relevant states and laser couplings for resolved
sideband cooling on a certain normal mode of
9Be?. Raman
beams (red arrows) drive red-sideband transitions, while resonantrepumping light (dark gray arrows) and decay of the P-level
continuously return the internal state to j↓i(gray wavy lines) and
the other two states in the
2S1=2level (not shown). (c) Corre-
sponding states and laser couplings for25Mg?.INDIRECT COOLING OF WEAKLY COUPLED TRAPPED-ION … PHYS. REV . X 14,021003 (2024)
021003-5Fig. 3(a)] using Raman beams whose wave vector differ-
ence has no component along either radial mode. The wave
vector difference ΔkBe;12of the Raman beams for sideband
cooling is parallel to z. The zstretch (out-of-phase) mode zs
atωzs?2π×6.304?1?MHz is an SCM, while the xandy
rocking modes xrandyr(also out-of-phase modes), with
ωxr?2π×7.483?1?MHz and ωyr?2π×6.437?1?MHz,
are WCMs, since the cooling radiation wave vector is
orthogonal to the mode directions. We use U1?r;t??
U1?r?cos?δt?to realize the required couplings between
modes. The xzandyzcurvatures of U1?r;t?enable the
respective couplings between the axial stretch and radial
rocking modes. As the products of mode participations of
the two ions have the same sign, U1?r;t?is designed to
have similar values of the xzandyzcurvatures at the two
ion positions so that their contributions add constructively
in Eq. (3).
To calibrate the coupling between the zsandyrmodes,
all six normal modes are sideband cooled close to their
ground states using both pairs of Be?Raman beams. Aftera microwave πpulse on the j↓iB?j↑iBtransition,
approximately 1.4 phonons on average are injected into
theyrmode by a pulse on j↑iBjni?j↓iBjn?1iwith
Raman beams 2 and 3. A repump pulse resets the hyperfine
state of the two Be?ions to j↓↓iB. A coupling pulse of
U1?r;t?with variable modulation frequency δand duration
τcoherently swaps the phonons between the yrmode and
thezsmode. After the coupling pulse, we apply an RSB
pulse resonant with either the yror the zsmode followed
by state-dependent fluorescence detection. The averagenumber of phonons in either mode can be approximated by
the average number of dark ions Dthat is computed from
the average fluorescence counts Cas
D?2C2?C
C2?C0; ?8?
where C2andC0are the average count rates for two or zero
ions in the bright state, respectively, based on an indepen-dent calibration.
Figures 3(b)and3(c)show scans of δandτ, respectively,
for optimum values of the other parameter. We see near-complete exchange of motional occupations between themodes for a particular value of δin Fig. 3(b), while a scan
ofτat this value of δswaps the populations multiple times
in Fig. 3(c), with the first complete swap at 63μs. The
experimental data are fit to theoretical expressions (solidlines; see Appendixes AandC), yielding δ
yr;zs?2π×
0.139?1?MHz≈ωyr?ωzsand a single swap duration of
τyr;zs?63?1?μs.
The coupling between the zsand the xrmodes is
calibrated in a similar manner [Figs. 3(e) and 3(f)].
However, coupling of two modes with a frequency differ-ence above 1 MHz is hampered in our experimental
apparatus due to the low-pass filters on the control
electrode inputs. To obtain stronger coupling between thesemodes, we bring their frequencies closer together forthe exchange, adiabatically ramping down the trap rfamplitude over 100μs before the coupling pulse and then
ramping up again over 100μs after the coupling pulse. At
the lower trap rf amplitude, the xrmode frequency
becomes 2π×6.150?1?MHz, and the zsfrequency is
slightly shifted to 2π×6.294?1?MHz, bringing the mode
frequency difference into the desired range. The yrmode
frequency is approximately 2π×4.83MHz. We extract
values of modulation frequency δ
xr;zs?2π×0.139?1?MHz
and swap duration τxr;zs?106?1?μs and observe no
appreciable motional excitation due to the ramping of
the trap rf amplitude.
After initial Doppler cooling, we implement the pulsed
cooling scheme using Nrepetitions of the sequence shown
in Fig. 4(a). The sequence begins with a 270μs CSBC
pulse on the zsmode and the center of mass modes. To cool
thexrmode, we lower the trap rf amplitude as previously
described, swap the zsandxrmode populations, and return
(a)
(d)(c)
(e)zsBe+Be+
zsxr
/(2) (MHz) (s)400 200 0 0.12(b)xr/yr
0.14 0.16300 0010 02Average dark ions D
0.51.0
0.12 0.14 0.16
/(2) (MHz) (s)00.01.5Average dark ions D
0.51.0
0.01.5zsyr
xz-swapyz-swap
FIG. 3. Mode coupling characterization in a Be?-Be?crystal
(see the text for details). (a) Mode participations of the zs,xr, and
yrmodes. The length and direction of the black arrows represent
the normal mode vector component of each ion, which is eitheralong or transverse to the trap axis. Calibration data for thecoupling between the zsandyr(b),(c) and between zsand
xr(d),(e). Blue and orange points show the average number of
dark ions [Eq. (8)] after a mode coupling pulse and an RSB pulse
on the xr(oryr) and zsmodes, respectively. A larger number of
dark ions corresponds to a larger average phonon number beforethe RSB pulse. Solid lines are fits to theory (see Appendix C).
Dashed vertical lines indicate the duration of single swapoperations. Each data point is obtained from 300 experimentswith a 68% confidence error bar, which is smaller than the plotsymbols in some cases.PAN-YU HOU et al. PHYS. REV . X 14,021003 (2024)
021003-6the trap rf amplitude to the initial value. We then apply
another CSBC pulse of 270μs to recool the zsmode, swap
thezsandyrpopulations, and then perform a final CSBC
pulse of the zsmode. Each sequence has a total duration of
approximately 1.34 ms. After cooling, we use RSB and
blue-sideband (BSB) analysis pulses to determine theaverage motional occupation ?nin a given mode, assuming
a thermal state of motion [68]; we select one of the three
modes involved to probe in each experimental trial.Figure 4(b)shows ?nfor all three modes versus the number
of cooling cycles N(bottom axis) or total cooling duration
(top axis).
The radial rocking modes xrandyrare naturally coupled
to the axial stretch mode zsdue to the nonlinearity of the
Coulomb interaction, which can be expressed as a cross-
Kerr coupling [45,46] . The Hamiltonian can be written as
H
K???χzs;xr ?nzs?nxr?χzs;yr ?nzs?nyr?; ?9?
where the f?ngare number operators for the motional modes
and the fχgare the Kerr coupling strengths. The presence
of nonzero motional population in either xroryrcauses
frequency shifts on the zsmode, and vice versa. If the radial
rocking modes are in thermal states of motion, this
coupling causes dephasing of the zsmode, which could
impact the fidelity of an entangling operation mediated by
thezsmode, or precision spectroscopy using the zsmode.
Using Eq. (16) in Ref. [46], the Kerr coupling rates in our
experiments are calculated to be χzs;xr?2π×75.86?5?Hz
andχzs;yr?2π×95.4?7?Hz.
We characterize this cross-Kerr dephasing effect exper-
imentally by performing sideband spectroscopy on the zsmode. We prepare the two ions in j↑↑iBand the zsmode
close to the ground state. We then drive a zsRSB πpulse of
duration 1.8 ms on the clock transition j↑iB?j2;0iB
using weak Raman beams, followed by shelving and state-
dependent fluorescence detection. We choose the clock
transition in this instance so that qubit dephasing does not
contribute appreciably to the measured transition linewidth.
With the rocking modes indirectly cooled to near their
ground states, we observe an approximately Fourier-limited
resonance when scanning the frequency of the sideband
pulse, as seen in blue in Fig. 4(c). With only Doppler
cooling of the rocking modes (orange squares), thesideband resonance is shifted to higher frequencies by
approximately 250 Hz, broadened, and reduced in contrast
due to averaging of the cross-Kerr coupling over the
thermal occupations of the rocking modes. We fit the data
to a model including the cross-Kerr couplings (seeAppendix D). Our demonstration shows that the detrimen-
tal effect from the cross-Kerr couplings can be suppressed
without the need for extra laser beams or magnetic field
gradients.
VI. COOLING ION CRYSTALS WITH DIFFERENT
CHARGE-TO-MASS RATIOS
The approximate 9∶25mass ratio in a
9Be?-25Mg?
crystal leads to very unequal participation of the two
species in all normal modes, as shown in Fig. 5(a).I n
the axial zout-of-phase ( zo) mode the Be?participation
dominates, with jξ?z?
Be;zoj≈0.930andjξ?z?
Mg;zoj≈0.368. The
roles are reversed and more extreme in the radial xandy
out-of-phase modes ( xoandyo) with jξ?x?=?y?
Be;xo=yoj≈0.022
andjξ?x?=?y?
Mg;xo=yoj≈0.999; this large asymmetry means that
thexoandyomodes are WCMs if cooling is performed
using Be?.
We perform indirect cooling of the xomode [ ωxo?
2π×4.48?2?MHz] and yo mode [ ωyo?2π×
4.04?3?MHz] by coupling them to the zomode
[ωzo?2π×4.722?1?MHz] with U1?r;t?. The couplings
are calibrated with a similar experimental sequence as for
the Be?-Be?crystal, illustrated in Fig. 5(b). We prepare the
zomode in j1izoand one romode in j0iro,ro∈fxo; yo g.
This is accomplished by iterating Be?sideband cooling
and a coupling pulse multiple times until both modes are
close to the ground states and then injecting a phonon into
zo. Next, we apply a coupling pulse with variable fre-
quency δ, shown in Figs. 5(c) and 5(e), or variable
duration τwith the coupling on resonance, plotted in
Figs. 5(d) and5(f). Since the Be?ion hardly participates
in the romodes, we measure only the final state of zo, with
an RSB πpulse j↓iBj1izo→j↑iBj0izofollowed by Be?
fluorescence detection. We measure the probability
P?j↑iB?of being in j↑iB, which approximately equals
the probability of finding the single phonon in the zomode.14(a)
(b)
80 1 0 2 4 6801 02461 2t (ms)
Cooling cycle N(c)
0.40.8Average dark ions D
01.21.62.0
f-f0 (kHz)5.00 .1 -1.5 -1.0 -0.5 0 1.5Dop
cool×N
zs/xr/yr
RSB/BSBReduce
trap rf
ampCSBC
zs
&COM sBe
DetRaise
trap rf
ampCSBC
zsyz-
swapCSBC
zs
zs
xr
yr
10-1100
10-2xz-
swap
FIG. 4. Be?-Be?cooling results. (a) Experimental sequence for
ground-state cooling of the xrandyrmodes via coupling to the
zsmode. (b) Mean occupation ?nof the three modes versus
number of cooling cycles N(bottom) and cooling duration (top).
Data points of xrandyrare laterally offset from nominal N
values for legibility. (c) Sideband spectra of the zsmode with
(blue) and without (orange) indirect ground-state cooling of thexrandyrmodes. Lines are the fits to a theory model accounting
for cross-Kerr coupling between modes (see the main text fordetails). Each data point is obtained from 300 experiments with a68% confidence error bar, which is smaller than the plot symbolsfor some points in (b) and (c).INDIRECT COOLING OF WEAKLY COUPLED TRAPPED-ION … PHYS. REV . X 14,021003 (2024)
021003-7Data from a frequency scan of the yo-zocoupling is
shown in Fig. 5(c) (blue circles). The fit (solid blue line)
yields an exchange resonance frequency of δyo;zo ?
2π×0.7116 ?1?MHz≈ωzo?ωyo. Figure 5(d) shows the
yz-coupling dynamics when driven on resonance with an
amplitude approximately twice as large as what was used in
the frequency scan for faster indirect cooling. The resulting
exchange dynamics are fit to a decaying sinusoid to yield
the single swap time τyo;zo ?49.5?5?μs. The reduced
contrast of approximately 0.8 is limited by the initial
occupation of the yomode, which is predominately caused
by its high heating rate (determined independently). The
corresponding data and fits for the xo-zocoupling
are shown in Figs. 5(e) and5(f). The fits yields δzo;xo?
2π×0.2485 ?1?MHz≈ωzo?ωxoand τxo;zo ?47.6?9?μs.
Before scanning the coupling duration, the xomode is
cooled much closer to the ground state than the yomode
was in the previously described experiments. This results inthe relatively higher contrast of 0.88(4) for the data in
Fig.5(f).After Doppler cooling, all three out-of-phase modes are
cooled to near the ground state with repetitions of the
sequence shown in parentheses in Fig. 6(a). We use a
duration of 50μs for swap pulses and CSBC is performed
on Be
?with a 75μs pulse on zo, followed by a 120μs
pulse on the in-phase axial mode to suppress the Debye-
Waller effect from occupation in this mode. The cooling
sequence takes 455μs and is repeated Ntimes before
determining the mode occupations. We measure ?nof the xo
andyomodes by swapping the state to zobefore perform-
ing sideband analysis with Be?. Figure 6(b) shows the
occupation ?nof the three out-of-phase modes for different
numbers of cooling cycles N(bottom axis) or cooling time
(top axis). All three modes reach steady state for N> 10
cycles, with ?n?f0.03?1?;0.23?2?;0.11?1?gfor the
fxo; yo; zo gmodes, respectively, at N?10. The mode
heating rates are characterized by cooling ( N?10) and
adding a variable delay time before sideband analysis. In
Fig.6(c), the ?nof the three modes is shown as a function of
the delay. The heating rates are the slopes of linear fits to ?n
versus delay time for each mode, shown as solid lines and
yielding heating rates of f5?5?;330?30?;20?7?gquanta per
second for xo,yo, and zo, respectively. The steady-state
mode occupation is substantially higher for yothanxoand
zobecause of its much higher heating rate. The zomode
has a higher final ?nthan the indirectly cooled xomode,
because the last cooling cycle swaps a thermal state of
?n≈0.2from the yomode into the zomode and the last
CSBC pulse is not long enough for zoto reach its steady
state of ?n?0.02?1?. Increasing the duration of the last
, RSB
-pulse
|0|1|B Be+
zo
xo/yozo
xo/yoBe+Mg+
zo1.0
0.8
0.6
0.4
0.2
0.037.0 17.02 7.0 07.096.0
/(2) (MHz) (s)yz-swap
002 0010 51 05 0
1.0
0.8
0.6
0.40.2
0.072.0 52.06 2.0 42.032.0
/(2) (MHz) (s)xz-swap
002 0010 51 05 0zo(a) (b)
(c)
(e)(d)
(f)
zozo
FIG. 5. Mode coupling characterization in a Be?-Mg?crystal.
(a) Ion normal mode directions and participations in the zoand
xo=yo modes. The length and direction of the black arrows
represent the normal mode vector component of each ion. The
arrow for Be?inxo=yo is barely visible. (b) Circuit for coupling
calibration. The Be?RSB πpulse couples only to the zomode
efficiently due to the weak participation of Be?in the xoandyo
modes. Calibration results for the yo-zo(c),(d) and xo-zo(e),(f)
coupling. Vertical dashed lines denote the single-swap duration.Lines in (c) –(f) are from theory expressions fitted to the data (see
Appendixes AandC). Each data point is obtained from 300
experiments with a 68% confidence error bar, which is smallerthan the plot symbols for most points.
yoxozoyz-
swapBe
CSBCDop
cool×N
Delayyz/xz -
swapzo
RSB/BSB
pulse Be
CSBCxz-
swapBe
CSBCBe
Det(a)
(b) (c)
N Delay (ms)0246810 15 2001.8 4.6 6.8 9.1
0 1 2 3 4 510-1100
10-1100
yoxozot (ms)
FIG. 6. Be?-Mg?cooling results. (a) Experimental sequence for
cooling and heating measurements of three out-of-phase modes.Thexoandyomodes are indirectly cooled and measured by
swapping their states to the zomode. (b) Mean occupation number
?nversus number of cooling cycles N(bottom) and cooling
duration (top) for the three out-of-phase modes. (c) Occupations
?nafter ten cooling cycles followed by a variable delay for the three
out-of-phase modes. Solid lines are fits to a linear increase inaverage occupation over time, corresponding to constant heatingrates. Each data point is obtained from 300 experiments with a68% confidence error bar, which is smaller than the plot symbolsfor some points.PAN-YU HOU et al. PHYS. REV . X 14,021003 (2024)
021003-8CSBC pulse can reduce the final occupation of the zomode
at the expense of increasing the occupation of the other
modes that heat up during this pulse.
VII. MODES WITH NO PARTICIPATION
OF THE COOLANT ION
The participation of a specific ion in a normal mode can
be exactly zero, for example, due to symmetry. A crystal
that has reflection symmetry around its center has axialmodes that are either odd or even under reflection. If thecrystal consists of an odd number Nof ions, the center ion
has zero participation for all ?N?1?=2normal modes that
have even parity under reflection through the center of thecrystal and is, thus, completely decoupled from these
modes. Here, we investigate sympathetic cooling of allaxial modes of a Be
?-Mg?-Be?crystal with cooling light
that interacts with only the middle Mg?ion. The three
axial modes are the in-phase ( ip), stretch ( st), and alter-
nating ( al) modes, with frequencies fωip;ωst;ωalg? 2π×
f1.501?1?;3.374?1?;3.655?1?gMHz and eigenmode par-
ticipations as shown in Fig. 7(a). The magnitudes of the
Mg?participations for fip; st; al gare approximately
f0.83;0;0.56g, respectively, vanishing exactly for the st
mode. We can use U2?r;t?to couple the stmode with the
almode, which has significant Mg?participation. The
coupling potential U2?r;t?contains a cubic term propor-
tional to z3resulting in opposite ?2U2=?z2curvatures for
the two Be?ions, whose contributions in Eq. (3), thus, add
constructively and provide a nonzero coupling rate. We
calibrate the coupling similarly to the experiments dis-
cussed above and find that the coupling drive is resonantforδ
st;al?2π×0.2834 ?1?MHz≈ωal?ωst.Cooling of all axial modes to near their ground states
using Mg?is accomplished by alternating CSBC and mode
coupling using the sequence shown in the upper row inFig.7(b), with the white box representing either the pulsed
or continuous scheme, as detailed in the lower row. In the
pulsed cooling demonstration, we use a 100μs swap pulse
that exchanges occupation between the standalmodes
[27]. After Doppler cooling, a 300μsM g
?CSBC pulse on
the second-order RSB of the ipmode cools population in
high number states and is followed by first-order RSB
pulses of 80 and 150μs duration to perform CSBC of the
ipmode and almodes, respectively. These latter pulses are
repeated eight times. Then, a swap pulse exchanges the
occupations of the stmode and the almode. The ipandal
modes are cooled by iterating 20μs CSBC pulses and
150μs CSBC pulses on the two modes, respectively, eight
times. Because Mg?ideally does not participate in the st
mode, photon recoil during cooling of the ipandalmodes
does not heat the stmode significantly [27]. We repeat this
cooling sequence Ntimes and perform sideband analysis to
measure ?n. The occupation of the stmode is characterized
by swapping its occupation to the almode and then
determining ?nof the almode. The ?nof all axial modes
are shown in Fig. 7(c) as a function of the number of
cooling cycles. All three modes are cooled close to
their ground states, ?n?f0.17?2?;0.03?1?;0.01?1?gfor
fip; st; al g, respectively, with just a single cooling cycle
with a duration of about 3.5 ms. Slightly lower occupation
can be achieved with more cycles, for example, ?n?
f0.10?2?;0.018?6?;0.009?4?gwith N?3. The ?nof the
stmode is slightly higher than that of the almode due to
heating after the final swap and during the extra swap pulse
needed for indirect characterization.
(b) (a)
ip
alstBe+Mg+Be+Dop
coolMg RSB/BSB
pulse Mg
Detst-alswapMg SBC
×N
Mg CSBC st-al
swapMg CSBC
ip and al ip and al Mg SBC
(pulsed)×M
Mg
CSBC
ipst-al coupling
Mg CSBC alMg
CSBC
ip and al Mg
CSBC
ipMg SBC
(simulta-
neous)
(c)
10-210-1100101
0 1 2 30 3.5 7 10.5
Nip
alstt (ms)
0 510 15
M20 2510-1100101
10-2(d)
02.7 4.4 6.17.8 9.5t (ms)
10-1100
r0 (kHz)4 123st(e)
ip
alst
FIG. 7. Sympathetic cooling of a Be?-Mg?-Be?crystal. (a) Mode participation of the three axial modes. (b) Experimental sequences
for ground-state cooling the axial modes on Mg?. The stmode is indirectly cooled to near ground state using either the pulsed or
simultaneous scheme and indirectly measured by swapping its state to the almode. (c),(d) Plots of mean occupation ?nof the axial modes
versus cooling cycle (bottom axis) and duration (top axis) when using (c) the pulsed scheme or (d) the simultaneous scheme with acoupling rate r
0?1.29kHz. (e) Mean occupation of the stmode after M?25versus coupling rate. Each data point is obtained from
300 experiments with a 68% confidence error bar, which is smaller than the plot symbols in some cases.INDIRECT COOLING OF WEAKLY COUPLED TRAPPED-ION … PHYS. REV . X 14,021003 (2024)
021003-9Thestandalmodes can also be cooled by simulta-
neously applying CSBC on the almode and driving
U2?r;t?to exchange mode occupations. Electric fields at
the Mg?position from imperfections in generating U2?r;t?
can drive Mg?to oscillate at the parametric drive fre-
quency, thus reducing the Rabi frequency of the coolinglight. We minimize these extraneous electric fields and alsooptimize the RSB frequency to give minimum ?nof the st
mode for a given cooling duration (see Appendix Efor
more details). The cooling sequence is shown in the lowerrow in Fig. 7(b), consisting of an initial 40μs CSBC pulse
on the first-order RSB of the ipmode followed by a 225μs
second-order RSB pulse. The standalmodes are then
cooled simultaneously by driving both the mode coupling
and the first-order RSB of the almode simultaneously for
300μs, followed by a 40μs CSBC pulse on the ipmode.
This latter sequence is repeated Mtimes. Since the ipmode
occupation increases during the cooling of other modes dueto anomalous heating and additional excitation from
imperfect coupling pulses, we perform final cooling pulses
of750μs total duration on the ipandalmodes before
analysis. For M?0, no sideband cooling is implemented
and the ?nafter Doppler cooling is shown.
Figure 7(d) shows the occupations ?nversus Mfor the
three axial modes when the alandstmodes are coupled at
a rate of r
0?2gst;al?1.29?2?kHz. Because of the
precooling of the ipmode and the final cooling sequence
foripandalmodes, these modes reach their steady state of
?nip?0.16?2?and ?nal?0.013?5?for all M> 0. The st
mode occupation decreases as Mincreases with ?nst?
0.05?1?atM?25. The final occupation of the stmode is
affected by the coupling rate. Figure 7(e)shows ?nstatM?
25as a function of r0. Faster motional exchange reduces ?nst
until the imperfections in the parametric drive lead to
stronger driven motion of Mg?which reduces the RSB
Rabi rate during CSBC cooling. In our demonstration, thelowest stmode ?nis achieved for r
0≈1.29kHz.
VIII. CONCLUSIONS
We have demonstrated indirect cooling of weakly cooled
motional modes in a multi-ion crystal by parametrically
coupling them to modes that interact with the coolingradiation more strongly. Indirect cooling has immediateapplications in quantum information processing and pre-cision measurements based on quantum logic. It can beextended to more general Coulomb crystals and potentially
enable efficient cooling of a broad range of atomic and
molecular species and charge-to-mass ratios, includingcharged fundamental particles, light molecular ions
[69–71], heavy molecular ions [72–75], highly charged
atomic ions [76], and charged mesoscopic objects [77].
The required coupling potentials need to be finely tuned
to accommodate the more complicated mode structures of
large ion crystals, and this will become easier with smallertraps and more control electrodes. In mixed-species crystalswith large charge-to-mass ratio mismatches, it may befavorable to apply multiple couplings sequentially to swappopulation between a WCM and a SCM via intermediatemodes if the WCM cannot be directly coupled to any SCM
with sufficient rate. The simultaneous cooling scheme is
currently limited by residual driven motion and anomalousheating. Driven motion can be caused by several technicalsources, and the dominant cause needs further investiga-tion. We find that the pulsed cooling scheme outperformssimultaneous cooling in the presence of such imperfections.
Although our demonstrations are conducted after Doppler
cooling of all modes, the demonstrated methods can workat considerably higher motional occupations, as long as theions remain in a crystal with well-defined mode structures.Mode coupling during Doppler cooling may be used to coolWCMs to an occupation set by the Doppler limit of the
SCM to which they are coupled, even if the initial
occupation of the WCM is very high due to geometry orsymmetry.
ACKNOWLEDGMENTS
We thank Justin Niedermeyer, Nathan Lysne, and Yu Liu
for their helpful comments on the manuscript. P.-Y . H.,J. J. W., S. D. E., and G. Z. thank the Professional ResearchExperience Program (PREP) operated jointly by NIST andthe University of Colorado. S. D. E. acknowledges support
from the National Science Foundation Graduate Research
Fellowship under Grant No. DGE 1650115. D. C. C. andA. D. B. acknowledge support from National ResearchCouncil Postdoctoral Fellowships. This work was sup-ported by IARPA and the NIST Quantum InformationProgram.
APPENDIX A: NORMAL MODES AND MODE
COUPLINGS OF ION CRYSTALS
Table Ilists the crystals used in this work along with
frequencies and normalized eigenvectors (participations)for the modes used in the experimental demonstrations.Tables IIandIIIshow the fitted parameters from the mode
coupling characterization; see Appendix C.PAN-YU HOU et al. PHYS. REV . X 14,021003 (2024)
021003-10APPENDIX B: MODE COUPLING POTENTIAL
GENERATION AND CONTROL
Experiments are performed in a segmented linear Paul
trap consisting of two stacked wafers with rf and controlelectrodes (see details in Ref. [66]). The trap electrode
layout near the ion crystals is shown in Figs. 8(a)and8(b).
The rf electrodes are not shown in Fig. 8(b). Time-varying
coupling potentials are produced by independent arbitrarywaveform generators (AWGs) [78] and applied to the 12
control electrodes through two-stage low-pass filters tosuppress noise at the motional frequencies. The oscillating
signals are added to the static voltages that produce the
axial confinement. The AWGs operate at a 50 MHz clockrate with synchronized phases. The mode coupling isineffective for frequency differences larger than 1 MHzdue to attenuation from the low-pass filters and the 1 MHzbandwidth of the AWG output amplifiers. The amplitude
envelope of coupling pulses varies smoothly to suppressoff-resonant excitation of nearby normal modes. The pulse
amplitude ramps up as approximately sin ?2πft?
2(with f?
12.5kHz and 0≤t≤20μs) at the beginning of the pulse
and ramps back to zero using the time reversal of the
ramp up.
We calculate the desired AWG amplitudes by simulating
the trap potential [66]to generate a potential for which the
desired curvatures at the positions of the ions are maxi-
mized while unwanted fields and curvatures are minimized.To couple an axial mode with a radial mode, we maximizethe curvatures ?
2U=?x?zand ?2U=?y?zfor the potential
U1?r?. To couple the axial modes in the Be?-Mg?-Be?
crystal, U2?r?has maximized cubic derivatives ?3U=?z3at
the position of the center Mg?ion. Unwanted terms include
electric fields ??U=?i, which displace and potentially heatTABLE I. Characteristics of the relevant normal modes of the ion crystals used in this work. When the trap rf amplitude ramps down,
thexrandyrmodes of a Be?-Be?crystal are at 6.150(1) MHz and approximately 4.83 MHz, respectively.
Crystal Mode Freq (MHz) Participation
Be?-Be?zs 6.304(1) 0.707, ?0.707
Be?-Be?xr 7.483(1) 0.707, ?0.707
Be?-Be?yr 6.437(1) 0.707, ?0.707
Be?-Mg?zo 4.722(1) 0.930, ?0.368
Be?-Mg?xo 4.04(3) 0.022, ?0.999
Be?-Mg?yo 4.48(2) 0.022, ?0.999
Be?-Mg?-Be?ip 1.501(1) 0.396, 0.828, 0.396
Be?-Mg?-Be?st 3.374(1) ?0.707, 0, 0.707
Be?-Mg?-Be?al 3.655(1) 0.586, ?0.560, 0.586
TABLE II. Fit parameters for frequency-scan data of various mode couplings. The results shown in Figs. 3(b)and3(d)(Be?-Be?) and
Figs. 5(c) and5(e) (Be?-Mg?) are fitted to Eq. (C1). The fitted values and uncertainties for all parameters are listed.
Crystal Coupled modes Measured mode Ar 0=?2π?(kHz) τ(μs) δw;s=2π(MHz) P0
Be?-Be?zs-yr zs 1.2(1) 7(2) 67(5) 0.1394(3) 0.07(3)
Be?-Be?zs-yr yr ?1.29?7? 8(1) 65(3) 0.1393(2) 1.36(2)
Be?-Be?zs-xr zs 1.1(2) 4(1) 96(5) 0.1386(1) 0.21(1)
Be?-Be?zs-xr xr ?1.1?1? 3.6(6) 84(3) 0.1386(2) 1.34(2)
Be?-Mg?zo-yo zo ?0.79?3? 5.2(4) 101(3) 0.7116(1) 0.944(7)
Be?-Mg?zo-xo zo ?0.97?2? 5.4(3) 98(2) 0.2485(1) 0.976(5)
TABLE III. Fit parameters for time-scan data of various mode couplings. The results shown in Figs. 3(c)and3(e)(Be?-Be?) and
Figs. 5(d)and5(f)(Be?-Mg?) are fitted to c?τ??Asin?r0τ?exp??γτ?=2?y0. The values of r0for the Be?-Mg?crystal are larger than
those in Table IIbecause stronger coupling drives are used.
Crystal Coupled modes Measured mode Ar 0=?2π?(kHz) ?γ (ms) y0
Be?-Be?zs-yr zs 1.20(6) 7.84(6) ?1.38?6? 1.5(8) 0.67(1)
Be?-Be?zs-yr yr 1.34(4) 7.91(4) 1.66(4) 2.6(1.6) 0.69(1)
Be?-Be?zs-xr zs 1.12(8) 4.70(5) ?1.42?8? 1.3(5) 0.68(1)
Be?-Be?zs-xr xr 1.10(6) 4.77(6) 1.64(6) 3.7(3.0) 0.78(1)
Be?-Mg?zo-yo zo 0.78(6) 10.1(1) 1.58(1) 1.4(1.1) 0.514(9)
Be?-Mg?zo-xo zo 0.88(4) 10.5(1) 1.42(6) 16(83) 0.502(6)INDIRECT COOLING OF WEAKLY COUPLED TRAPPED-ION … PHYS. REV . X 14,021003 (2024)
021003-11the ions, and curvatures ?2U=?i2, which modulate the
motional frequencies, i∈fx; y; z g. We neglect higher-order
derivatives of the potential, since their effects are sup-pressed if the extent of the Coulomb crystal is smallcompared to the distance to the nearest electrodes.
APPENDIX C: LINE SHAPE
OF MODE COUPLINGS
We determine the mode coupling resonant frequency δ
w;s
by fitting the frequency-scan data to the equation
c?δ??Asin2?r?δ?τ=2/C138=?r?δ?=r0/C1382?P0; ?C1?
withr?δ???????????????????????????????? ?
r2
0??δ?δw;s?2q
. We derive this expression
from the following mode coupling Hamiltonian:Hc??gw;s?ei?δt????w??s?e?i?δt????s?w??; ?C2?
where δis the detuning of the coupling drive from the
frequency difference of two modes δw;s?ωw?ωs. In the
calibration experiments, we fix the coupling pulse duration
τto approximately a single exchange time π=?2gw;s?and
sweep δacross the resonance. We then read out the
motional information through the internal state of eithera single ion or two ions. We discuss these cases sepa-rately below.
1. Readout with a single ion
When reading out motional information using a single
ion, the two motional modes are prepared in j1i
wj0is. Since
Hcconserves the total phonon number, the initial state can
couple only to j0iwj1is. Following a coupling pulse, the
motional state can be expressed as
jψi?c01j0iwj1is?c10j1iwj0is: ?C3?
By solving the differential equations
i˙c10?gw;seiδtc01;
i˙c01?gw;se?iδtc10; ?C4?
with initial conditions c10?t?0??1andc01?t?0??0,
we obtain
c01?ie??i=2?δt?2gw;s?sin?rt=2?
r;
c10?eiδt=2/C20
cos?rt=2??iδsin?rt=2?
r/C21
; ?C5?
where r?????????????????????
4g2w;s?δ2q
. The state populations are
p01?jc01j2?4g2w;ssin2?rt=2?
r2;
p10?jc10j2?1?4g2w;ssin2?rt=2?
r2: ?C6?
AB S B πpulse on mode wdrives j↑i?c10j1iwj0is?
c01j0iwj1is?toc10j↓ij0iwj0is?c01j↑ij0iwj1is, which
results in the probability of finding the ion in j↓i,
p?j↓i? ?p10. A BSB πpulse on mode syields p?j↓i? ?
p01by similar logic. Both populations in Eq. (C6) can be
reexpressed in the form of Eq. (C1) with suitable parameter
substitutions.
2. Readout with two ions
In the experiments where motional information is read
out through two globally addressed ions, the crystal isinitially prepared in j↑↑ij0i
wj0is. A RSB pulse on mode w
with a duration of π=????
6p
Ω0?is applied resulting in the
(c)(a)
(d)
(e) (f)1.0
0.8
0.6
0.4
0.2
0.0Relative Rabi rate , |J0|Modulation index
01234 5
r0 (kHz) 0 0.31 0.62 0.93 1.24 1.551.00 kHz0.50 kHz0.25 kHzr0=0 kHz
0.75 kHz1.0
0.8
0.6
0.40.2
0.0P(|M)
1.0
0.8
0.6
0.4
0.2
0.0P(|M)t (s)0 20 40 60 80 100z1 357 911
2 468 1012Top view
xySide view
Trap rfTrap rf(b)
t (s)0 20 40 60 80 1001.32 kHz0.66 kHzr0=0 kHz
1.00
0.98
0.96
0.94
0.92Relative Rabi rate , |J0|Modulation index
0 0.1 0.2 0.3 0.4 0.5 0.6
r0 (kHz) 0 0.50 1.0 1.5
FIG. 8. (a) Side and (b) top views of trap electrode structure
near the ions for the two-layer rf Paul trap used in this work. Therf electrodes are not shown in the top view to enable all thesegmented dc electrodes to be seen. (c) –(f) Characterization of
Mg
?driven motion from U2?r;t?(c),(d) before and (e),(f) after
minimizing the electric fields. (c),(e) Rabi oscillations of an Mg?
Raman sideband transition while the parametric drive is on at
different drive amplitudes. The ratio of the Rabi frequency withparametric drive on and off is plotted versus the correspondingon-resonance mode exchange rates r
0(blue dots) in (d) and (f),
respectively, and compared to the absolute value of the Besselfunction J
0versus modulation index (red line). Error bars
represent 68% confidence intervals.PAN-YU HOU et al. PHYS. REV . X 14,021003 (2024)
021003-12state ?2???
2p
=3?j↓↓ij2iwj0is?1
3j↑↑ij0iwj0is, where Ω0is
the ground-state sideband Rabi rate for a single ion. A
subsequent repump pulse resets the two ions to j↓↓iand
also destroys the coherence of the two components,
resulting in a mixture of two states:
ρ?1
9jψihψj?8
9j?ih?j; ?C7?
where jψi?j↓↓ij0iwj0isand j?i?j↓↓ij2iwj0is. The
first component jψihψjdoes not evolve under the mode
coupling or the second RSB pulse which is applied laterfor readout. For the second component, j?ievolves to
j?
ex?t? i?j↓↓i?c20j0iwj0is?c11j1iwj1is?c02j0iwj2is?
during mode coupling. The evolution of the coefficients inj?
ex?t?iis governed by
i˙c11????
2p
gw;s?eiδtc20?e?iδtc02?;
i˙c20????
2p
gw;se?iδtc11;
i˙c02????
2p
gw;seiδtc11: ?C8?
By solving these equations with the initial conditions
c02?t?0??0,c11?t?0??0, and c20?t?0??1,w e
obtain
c02?2eiδtg2w;s??1?cos?rt?/C138
r2;
c11????
2p
gw;sfδ?1?cos?rt?/C138?irsin?rt?g
r2;
c20?e?iδt?2g2w;s??2g2w;s?δ2?cos?rt??iδrsin?rt?/C138
r2:?C9?
After the exchange, the system further evolves under a RSB
pulse on either mode wors, governed by
i˙cj↓↓ij2i????
2p
Ω0?cj↓↑ij1i?cj↑↓ij1i?;
i˙cj↓↓ij1i?Ω0?cj↓↑ij0i?cj↑↓ij0i?;
i˙cj↓↑ij1i?Ω0????
2p
cj↓↓ij2i?cj↑↑ij0i?;
i˙cj↑↓ij1i?Ω0????
2p
cj↓↓ij2i?cj↑↑ij0i?;
i˙cj↓↑ij0i?Ω0cj↓↓ij1i;
i˙cj↑↓ij0i?Ω0cj↓↓ij1i;
i˙cj↑↑ij0i?Ω0?cj↓↑ij1i?cj↑↓ij1i?: ?C10?
In this notation, we omit the mode not involved in the RSB
pulse, as it has no impact on the final population of theinternal state. We denote the coefficient of the reduced state
ji; jijni, where i; j∈f↑;↓gandn∈f0;1;2g,a sc
ji;jijni.
When applying a RSB analysis pulse with duration of
π=????
6p
Ω0?on mode w, the initial conditions are set as
cj↓↑ij2iw?0??c20and cj↓↓ij1iw?0??c11, and all othercoefficients are set to zero. The component j↓↓ij0iwj2is
inj?exidoes not evolve under the RSB pulse on mode w.
We solve Eqs. (C10) and calculate the probabilities
pji;jijniw?t??jcji;jijniw?t?j2for the relevant states:
pj↓↓ij2iw??2g2w;s?δ2?2g2w;scos?rt?/C1382
9r4;
pj↓↓ij1iw?8g2w;s
r4cos2/C18π???
3p/C19
?2g2w;s?δ2?2g2w;scos?rt?/C138
× sin2?rt=2?;
pj↓↑ij1iw?0;
pj↑↓ij1iw?0;
pj↓↑ij0iw?4g2w;s
r4?2g2w;s?δ2?2g2w;scos?rt?/C138
× sin2/C18π???
3p/C19
sin2?rt=2?;
pj↑↓ij0iw?4g2w;s
r4?2g2w;s?δ2?2g2w;scos?rt?/C138
× sin2/C18π???
3p/C19
sin2?rt=2?;
pj↑↑ij0iw?8?2g2w;s?δ2?2g2w;scos2?rt?/C1382
9r4: ?C11?
The average number of dark ions is
Dw?8
9?pj↓↑ij1iw?pj↑↓ij1iw?pj↓↑ij0iw?pj↑↓ij0iw
?2pj↑↑ij0iw?
?2/C188
9/C192/C18
1?4g2w;s
r2sin2?rt=2?/C19
?1?dw?; ?C12?
with
dw?7?9cos?2π??
3p?
164g2w;s
r2sin2?rt=2?:
Since 4g2w;s≤r2, we have dw≤?7?9cos?2π??
3p?=16/C138≈0.06.
Therefore, we can neglect dwand approximate Dwas
Dw≈2/C188
9/C192/C18
1?4g2w;s
r2sin2?rt=2?/C19
: ?C13?
This expression has the same form as the fit function
Eq.(C1). Since dwis symmetric around resonance ( δ?0),
the neglected term is not expected to change the value of the
coupling frequency found by fitting to Eq. (C1).
Similarly, when a RSB pulse on mode sis applied,
we solve Eqs. (C10) with different initial conditions,
cj↓↑ij1is?0??c11andcj↓↓ij2is?0??c02, and the rest are
set to zero. In this case, the state j↓↓ij2iwj0isdoes notINDIRECT COOLING OF WEAKLY COUPLED TRAPPED-ION … PHYS. REV . X 14,021003 (2024)
021003-13evolve under the RSB interaction on mode s. The prob-
abilities pji;jijnis?t?are
pj↓↓ij2is?16g4w;ssin4?rt=2?
9r4;
pj↓↓ij1is?8g2w;s
r4cos2/C18π???
3p/C19
?2g2w;s?δ2?2g2w;scos?rt?/C138
× sin2?rt=2?;
pj↓↑ij1is?0;
pj↑↓ij1is?0;
pj↓↑ij0is?4g2w;s
r4sin2/C18π???
3p/C19
?2g2w;s?δ2?2g2w;scos?rt?/C138
× sin2?rt=2?;
pj↑↓ij0is?4g2w;s
r4sin2/C18π???
3p/C19
?2g2w;s?δ2?2g2w;scos?rt?/C138
× sin2?rt=2?;
pj↑↑ij0is?128g4w;ssin4?rt=2?
9r4: ?C14?
The average number of dark ions for mode sis
Ds?2/C188
9/C1924g2w;s
r2sin2?rt=2??1?ds?δ?/C138; ?C15?
with
ds?9sin2?π??
3p??8
8/C18
1?4g2w;s
r2sin2?rt=2?/C19
:
Since ds3p??8=8/C138≈0.06,Dscan be approxi-
mated as
Ds≈2/C188
9/C1924g2w;s
r2sin2?rt=2?; ?C16?
that is, in the form of Eq. (C1) as well.
APPENDIX D: CROSS-KERR COUPLING
EFFECTS IN SIDEBAND TRANSITIONS
In the experiments with a Be??Be?crystal, we exper-
imentally confirm the dephasing caused by cross-Kerrcoupling by comparing the sideband spectra of the zstretch
mode with and without cooling the xandyrocking modes
to near their ground states. Details and results are described
in the main text. Here, we develop a numerical model forthe sideband spectrum that takes into account the cross-Kerr coupling effects.
Assuming the zstretch, xrocking, and yrocking modes
are all in thermal states with respective mean occupationsof?n
zs,?nxr, and ?nyr, the average number of dark ions
Dzs?f?is
Dzs?f??XNzs
nzs?0XNxr
nxr?0XNyr
nyr?0p?nzs;nxr;nyr?
×czs?nzs;nxr;nyr;f?; ?D1?
with the probability of the three modes in a certain number
state
p?nzs;nxr;nyr??Y
i∈fzs;xr;yr g/C18?ni
1??ni/C19ni
and the transition amplitudes
czs?BΩ2
rsb;0?nzs?sin2?πΩrsb=?2Ωrsb;0?/C138
Ω2
rsb?D0:
We define the Rabi frequencies
Ωrsb?????????????????????????????????????????????????????????????????????????????????????????
Ω2
rsb;0??2π?f?frsb??χzs;xrnxr?χzs;yrnyr/C1382q
and Ωrsb;0?nzs??Ωexp?η2=2/C138?1=?nzs?1?1=2ηL1nzs?η2?/C138,
where L1nzs?η2?is the generalized Laguerre polynomial
andη?0.268. The cross-Kerr coupling strengths χzs;xr?
2π×75.86?5?Hz and χzs;yr?2π×95.4?7?Hz are esti-
mated using Eq. (16) in Ref. [46]. In the following analysis,
we considered number states up to Nzs?5andNxr?
Nyr?20in Eq. (D1), to sufficiently reflect mode occu-
pations after Doppler cooling in our data.
When both rocking modes are cooled to a mean
occupation of 0.05, the fits of the data to the modelyield B?1.78?24?,Ω?2π×0.86?10?kHz, f
0?
1201 .2124 ?1?MHz, and D0?0.05?5?. Using these
parameters, the data with only the axial modes sidebandcooled are fitted to determine the mean occupations of thetwo rocking modes, ?n
xr?2.4?1.5?and ?nxr?1.8?1.4?.
APPENDIX E: MINIMIZING ELECTRIC FIELDS
Gradients in the coupling potential (electric fields)
oscillating at frequency δw;scause driven motion of the
ions, which can be detrimental. In the Be?-Mg?-Be?
crystal, the driven motion of Mg?along the axial direction
phase modulates the Raman beams, reducing the Rabi
frequencies of the Mg?Raman transitions by a factor
jJ0?ΔkA?j, where J0is the zero-order Bessel function, Δk
is the magnitude of the wave vector difference of the
Raman beams, and Ais the amplitude of Mg?driven
motion along the direction of the Raman wave vectordifference.
We characterize the Mg
?driven motion induced by
applying U2?r;t?at variable strengths and measuring thePAN-YU HOU et al. PHYS. REV . X 14,021003 (2024)
021003-14Rabi frequency of the alternating-mode RSB transition
j↑iMjnial?j↓iMjn?1ial. To avoid coupling the alter-
nating mode to any other modes, we tune the parametric
drive frequency to 0.5 MHz, off resonance from any mode
frequency differences. We prepare the Mg?ion in j↑iMand
cool all axial modes to near their ground states. Then, a
RSB pulse and a parametric drive of a certain strength areapplied simultaneously on the ion, followed by fluores-cence detection. We repeat the experiment 100 times for
each RSB pulse duration and obtain the probability
P?j↑i
M?of Mg?being in j↑iMat this drive strength.
Such RSB oscillation traces are taken for various drive
amplitudes corresponding to on-resonance stretch-alternat-ing exchange rates ranging from r
0?0to 1.03 kHz. The
resulting oscillations are shown in Fig. 8(c) and fitted to
exponentially decaying sinusoidal functions to estimateRabi frequencies. In Fig. 8(d), the fractions of the Rabi
frequencies relative to the Rabi frequency with the drive off
(blue dots) are plotted versus r
0.Af i tt o jJ0?Δkβr0?j(red
line) is also shown, where β?101?2?nm=kHz relates the
amplitude Aof Mg?driven motion to the coupling rates
asA?βr0.
To reduce the driven motion in the zdirection, the
potential U2?r?needs to be adjusted. The zelectric field is
adjusted by changing the amplitudes of drives on electrodes3 and 4 in Fig. 8(b) byΔ
z, while the amplitude on
electrodes 7 and 8 is adjusted by ?Δz. We vary Δzuntil
the Raman Rabi frequency on the Mg?ion is maximized.
In addition, we change the amplitudes driving electrodes 5
and 6, above and below the Mg?,b y/C6Δxto minimize the x
electric field, which results in a smaller but noticeable
improvement in the Rabi frequency compared to onlyminimizing the zelectric field.
After minimizing the electric fields at the Mg
?position
due to the applied coupling potential, we repeat the
characterization of Mg?driven motion using the same
method. The resulting Rabi oscillations of the alternatingmode RSB transition are shown in Fig. 8(e)along with their
fits. In Fig. 8(f), we plot the relative Rabi rates as a function
ofr
0(blue dots) and fitted to jJ0?Δkβr0?j(red line), from
which we determine β?12.6?4?nm=kHz. This value is
nearly an order of magnitude smaller than prior to thecompensation. There are several technical limitations on
the suppression of the driven motion, such as drifting or
noisy stray fields, fluctuations in electrode potentials, anddiscretization errors of the digital-to-analog converters. It is
currently unclear which of these imperfections is the
dominant cause of the residual driven motion.
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