Micromotion-Synchronized Pulsed Doppler Cooling of Trapped Ions Alexander Kato1Andrei Nomerotski2and Boris B. Blinov1 1University of Washington Department of Physics Seattle Washington USA 98195

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Micromotion-Synchronized Pulsed Doppler Cooling of Trapped Ions

Alexander Kato,1Andrei Nomerotski,2and Boris B. Blinov1

1University of Washington, Department of Physics, Seattle, Washington, USA, 98195

2Physics Department, Brookhaven National Laboratory, Upton, New York, USA, 11973

(Dated: February 10, 2023)

We propose and demonstrate a new method for Doppler cooling trapped-ion crystals where the distribution

of micromotion amplitudes may be large and uneven. The technique uses pulses of Doppler cooling light

synchronized with the trap RF that selectively target ions when their velocity is near a node, leading to more

uniform cooling across a crystal by a single tone of cooling light. We lay out a theoretical framework that

describes where this technique is practical, and provide a simple experimental demonstration.

I. INTRODUCTION

Micromotion is a time-dependent, driven motion that is

present in all radiofrequency (RF) ion traps. This may push

ions beyond the Lamb-Dicke regime [1, 2], leading to fre-

quency and amplitude modulation in addressing beams [3]

and presenting a signiﬁcant obstacle to high-ﬁdelity quantum

logic operations. Signiﬁcant micromotion may even cause dif-

ﬁculties in Doppler cooling–typically a straightforward pro-

cess for trapped ion crystals [2, 4]. For linear ion traps, this

RF-driven motion can be minimized [5, 6] for all ions by plac-

ing them at the nodal line where the RF electric ﬁeld ampli-

tude is zero, making this type of traps the most widely used

by the experimenters.

However, several recent proposals suggest 2D and 3D crys-

tals may be used for quantum computing and simulations [3,

7–10]. These proposals allude to the possibility of scaling up

the number of qubits in a given area, opening up the potential

for a wide range of quantum simulations that are more suited

to a native 2D geometry and leading to new options for error

correction that may improve the threshold for fault-tolerant

quantum computing. To overcome the adverse effect of mi-

cromotion on gate ﬁdelity, it has been shown that segmented

pulses may be used [3, 7]. Moreover, quantum simulations

with 2D crystals may be achieved by making use of trans-

verse motional modes to generate entanglement [3, 8, 9]. To-

wards this goal, recent experiments have demonstrated good

isolation between transverse motion [11, 12], where micro-

motion may be minimized and ground state cooling has been

achieved, and radial motion, where excess micromotion is

present, conﬁnement is weaker, and efﬁcient cooling may be

difﬁcult.

To reach the low temperatures required for quantum in-

formation experiments, one must effectively cool along each

trap axis. This necessitates having at least some component

of the cooling beams point in the direction of excess micro-

motion, which may have a dramatic effect on the steady-state

Doppler cooling [2, 4].To avoid these adverse effects one can

cool largely along an axis where no micromotion is present

[7, 13], such as with lateral 2D crystals. However, at least

some of the cooling beam’s k-vector must point in a direc-

tion where micromotion is present, since otherwise thermal

motion in all directions is not cooled. As the size of crystals

scales up, this effect may become signiﬁcant. Moreover, for

radial 2D crystals [11, 14, 15], and for 3D crystals, it is gener-

ally not possible to isolate a direction in space with no excess

micromotion. Therefore, for large trapped ion crystal, it may

be necessary to directly take into account and overcome the

detrimental effects of micromotion on Doppler cooling.

Due to micromotion, the cooling laser frequency in each

ion’s rest frame is continuously Doppler-shifted, by varying

amounts across a crystal. This causes the absorption spec-

trum and range of frequencies for which steady state cool-

ing is efﬁcient to vary for different ions. It was suggested to

use power-broadened and far-detuned cooling beam to allow

more even cooling across a wide range of micromotion ampli-

tudes [4]. In recent experiments, we implemented a two-tone

Doppler cooling [15] and were able to stabilize larger radial

2D crystals covering a broad range of micromotion. Multi-

tone Doppler cooling may be an avenue to stabilizing even

larger crystals. However, effective Doppler cooling of crys-

tals where ions have both large and differing amplitudes of

micromotion parallel to the cooling beam’s k-vector remains

an outstanding challenge.

In this paper, we propose using pulses of Doppler cooling

light synchronized with the trap RF phase to cool ions under-

going micromotion, as illustrated in Fig. 1. There exist two

points per RF period Twhere the velocity v=0 for all ions

simultaneously. We propose to use ∼ns laser pulses synchro-

nized with the nodes in the micromotion velocity (dashed line

in Figure 1.a) in order to narrow the range of ion speeds which

need to be addressed by the cooling beam. We then show that

this technique can be useful when cooling multiple ions with

differing amounts of micromotion using a single-tone laser

beam.

Pulsed Doppler cooling has been used before with the in-

tention of broadband cooling [16] or generating frequency

comb teeth deep into the ultraviolet (UV) range, offering to

reduce the complication associated with harmonic generation

of light [17, 18]. However, laser pulses have not been used

to cool trapped ions synchronized with the trap RF. In ad-

dition, micromotion-synchronized frequency modulation has

been used previously to compensate for the effect of micro-

motion on Raman transitions in surface traps [19–21]. The

frequency modulation scheme may be effective in cooling

trapped ions with the same amplitude and phase of micromo-

tion, but breaks down for 2D crystals, where ions have dif-

ferent amplitudes of micromotion with a phase ﬂip at the trap

center.

arXiv:2210.03835v2 [physics.atom-ph] 9 Feb 2023

FIG. 1: Trapped ion excess micromotion effect on laser absorption. a) An ion oscillating back and forth at frequency Ωin the

direction of the wave vector~

kof a cooling beam leads to a time dependent Doppler shift. The transparency of each dot

represents the magnitude of it’s velocity ~v, also indicated by the arrows. By pulsing a laser on (laser intensity indicated by the

dashed line) during only a small portion of the ion’s trajectory when the velocity is minimal (darkest points), one can

selectively cool a smaller velocity class. Time is measured in units of the RF period T. b) Doppler shift of a 493 nm light in the

reference frame of a Ba+ion undergoing micromotion at 8 MHz with 3 µm amplitude. Instantaneous Doppler shifts exceed

300 MHz, which is large compared to the 15 MHz natural linewidth of the transition.

II. PULSED COOLING

Previous approaches to modelling Doppler cooling under

micromotion have relied on time averaging steady state solu-

tions to the Schrodinger’s equation or the optical Bloch equa-

tions by sampling velocities over a period of micromotion

to produce an atomic absorption spectrum [2, 4, 15]. Yet in

the presence of signiﬁcant micromotion, the cooling is not

steady state since ions may be experiencing Doppler shifts

much larger then the linewidth of the atomic transition Γand

rapidly changing on a timescale similar to the excited state

lifetime τ. Moreover, pulsed lasers cause frequency combing

effects and fast intensity changes that cannot be captured in

the steady state. Therefore, to understand how pulsed Doppler

cooling works, we numerically solve the time-dependent op-

tical Bloch equations (see appendix for details).

Consider a crystal of 138Ba+ions undergoing micromotion

at a frequency Ω=2π×8 MHz (period T=2π/Ω=125 ns),

interaction with the 493-nm Doppler-cooling laser (transition

linewidth Γ31 =2π×15 MHz) and 650-nm repump laser

(transition linewidth Γ32 =2π×5 MHz), see appendix, Fig. 5.

These values are similar to what our experiment is capable of,

and are well representative of the regime where Ω<Γ31. Each

ion experiences oscillations around a ﬁxed point described by

~r=Acos(Ωt)where A=qrs/2 is the micromotion amplitude.

Here, rsis the displacement of the equilibrium position of the

ion from the trap center, and qis the relevant Matthieu pa-

rameter. Hence, the instantaneous velocity is v=AΩsin(Ωt),

and the instantaneous Doppler shift in the ion’s rest frame is

k·~v=kAΩsin(Ωt), where~

kis the laser wave vector. An exam-

ple of this is shown in Figure 1.b for an ion with micromotion

amplitude A=3µm.

We model the behaviour of the excited state population,

ρ33, as shown in Fig. 2. In order to compute the absorption

spectra, we time-average the solutions over many periods of

micromotion. In Figure 2.a we show the beneﬁts that can be

obtained by pulsing the cooling beams, as opposed to contin-

uous cooling, for different pulse widths. The absorption spec-

trum for continuous cooling obtained via this method (solid

line in Fig. 2.a) is similar to those found using the steady-

state model [15], yet the adverse effects due to coherent pop-

ulation trapping (CPT) are far more pronounced. As the am-

plitude of micromotion becomes large, multiple dips appear

at multiples of Ω. This is in contrast to the steady-state solu-

tions, where these CPT features smooth out as the spectrum

becomes power broadened [15]. In Figure 2.b we plot the ab-

sorption spectra for the ﬁxed laser pulse width of 5 ns and

various micromotion amplitudes. We note that the absorption

line width remains essentially unchanged as the micromotion

amplitude increases from 0 to 3 µm. In Fig. 2.c we show the

effect of power broadening in pulsed cooling, which shows

the expected behavior of the line width increasing at higher

laser intensities. The derivatives of the absorption curves in

Fig. 2 (a-c), which are proportional to the cooling rates, can

be found in panels (d-f).

In order to cool the ions effectively we must take into

consideration the level of velocity selection due to the pulse

width, the frequency combing effects from the pulse train, the

inﬂuence on CPT dips, and the saturation effects. Each has a

substantial impact on cooling efﬁciency and must be consid-

ered individually and with respect to each other.

First, we consider the effects of CPT when cooling using

pulsed lasers. For a 3-level λ-system such as 138Ba+, the

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Micromotion-SynchronizedPulsedDopplerCoolingofTrappedIonsAlexanderKato,1AndreiNomerotski,2andBorisB.Blinov11UniversityofWashington,DepartmentofPhysics,Seattle,Washington,USA,981952PhysicsDepartment,BrookhavenNationalLaboratory,Upton,NewYork,USA,11973(Dated:February10,2023)Weproposeanddemonstrateanew...

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Micromotion-Synchronized Pulsed Doppler Cooling of Trapped Ions Alexander Kato1Andrei Nomerotski2and Boris B. Blinov1 1University of Washington Department of Physics Seattle Washington USA 98195

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