Number-Resolved Detection of Dark Ions in Coulomb Crystals Fabian Schmid1Johannes Weitenberg1Jorge Moreno1 Theodor W. H ansch1 2Thomas Udem1 2and Akira Ozawa1

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Number-Resolved Detection of Dark Ions in Coulomb Crystals

Fabian Schmid,1, ∗Johannes Weitenberg,1Jorge Moreno,1

Theodor W. H¨ansch,1, 2 Thomas Udem,1, 2 and Akira Ozawa1, †

1Max-Planck-Institut f¨ur Quantenoptik, 85748 Garching, Germany

2Fakult¨at f¨ur Physik, Ludwig-Maximilians-Universit¨at M¨unchen, 80799 M¨unchen, Germany

(Dated: October 6, 2022)

While it is straightforward to count laser-cooled trapped ions by ﬂuorescence imaging, detecting

the number of dark ions embedded and sympathetically cooled in a mixed ion crystal is more

challenging. We demonstrate a method to track the number of dark ions in real time with single-

particle sensitivity. This is achieved by observing discrete steps in the amount of ﬂuorescence

emitted from the coolant ions while exciting secular motional resonances of dark ions. By counting

the number of ﬂuorescence steps, we can identify the number of dark ions without calibration and

without relying on any physical model of the motional excitation. We demonstrate the scheme by

detecting H+

2and H+

3ions embedded in a Be+ion Coulomb crystal in a linear radio frequency trap.

Our method allows observing the generation and destruction of individual ions simultaneously for

diﬀerent types of ions. Besides high-resolution spectroscopy of dark ions, another application is the

detection of chemical reactions in real time with single-particle sensitivity. This is demonstrated in

this work.

Trapped and laser-cooled ions have been used to in-

vestigate fundamental light-atom interactions [1–5], for

observation of ion-neutral chemical reactions [6–10], as

well as for optical frequency standards [11, 12] and quan-

tum computing [13–15].

Sympathetic cooling is a way for extending this tech-

nology to atomic or molecular ions that do not possess

suitable transitions for laser cooling. In this scheme, the

ions are trapped together with another ion species that

can be laser-cooled. Due to the mutual Coulomb inter-

action, the ions rapidly thermalize, indirectly cooling all

species. At suﬃciently low temperatures, the ions form

regular Coulomb crystals in the trap [16]. The technique

has found application in precision spectroscopy of HD+

and H+

2molecular ions [17–21], optical atomic clocks

based on quantum logic spectroscopy of Al+[22, 23],

spectroscopy of highly-charged ions [24, 25], and the

study of chemical reactions with molecular ions [7, 26–

29].

While the number of laser-cooled ions can be eas-

ily measured by ﬂuorescence imaging, identiﬁcation and

counting of the non-ﬂuorescing dark ions is more diﬃcult.

One method is to eject the ions from the trap and to ac-

celerate them onto a detector in an electric ﬁeld. The

diﬀerent ion species can be distinguished by their arrival

times [30–33]. While this method allows a quantitative

measurement of the number of ions of each species, it

has the disadvantage that it is destructive and a new ion

crystal has to be prepared after each measurement.

In linear Paul traps, lighter ions are more tightly con-

ﬁned than heavier ions. Lighter sympathetically cooled

ions therefore form a dark region in the center of the ﬂu-

orescence image of an ion crystal consisting of a heavier

coolant species. The number of dark ions can then be

∗fabian.schmid@mpq.mpg.de

†akira.ozawa@mpq.mpg.de

obtained by comparing experimental images with simu-

lated ones [17, 34, 35]. This method is non-destructive,

and the ion images can be acquired quickly and post-

processed later. However, diﬀerent dark ion species can-

not be distinguished.

Instead, secular excitation has been used for non-

destructive detection of trapped ions. In this method the

secular motion of the ions, i.e. the harmonic motion in the

time averaged trap potential, is excited resonantly by ap-

plying an additional oscillating electric ﬁeld. This trans-

fers energy into the motion of the surrounding coolant

ions and thereby increases their temperature. Due to the

temperature dependence of the Doppler broadening this

leads to a change in the amount of ﬂuorescence that can

be observed from the coolant ions. The secular motion of

the ions in three-dimensional Coulomb crystals has rich

dynamics that can complicate the analysis of the secular

excitation spectra. For example, the frequencies of the

secular resonances are inﬂuenced by space charge eﬀects

and the mechanical coupling between the ions [36–38].

The energy transfer to the coolant ions is expected to in-

crease with an increasing number of dark ions. Therefore,

the ﬂuorescence change induced by motional excitation

serves as a measure of the number of dark ions. However,

the relationship between the ﬂuorescence change and the

number of excited dark ions is in general non-linear and

is inﬂuenced by various experimental parameters, such

as the strength of the motional excitation and the geom-

etry of the mixed ion crystal. Therefore, evaluating the

number of dark ions quantitatively is challenging and of-

ten requires intricate modeling and calibration of the sig-

nal using molecular dynamics simulations [26, 39]. This

problem has been limiting the usage of the secular exci-

tation method for highly precise spectroscopy so far.

In this work, we show that by properly choosing ex-

perimental parameters, discrete steps in the secular ex-

citation signal can be observed that are identiﬁed with

individual dark ions leaving the trap or being generated

arXiv:2210.02112v1 [physics.atom-ph] 5 Oct 2022

within the trap. Hence such a signal is auto-calibrating

and counting the number of ions gives the ultimate ac-

curacy. The signal does not have to be calibrated using

a physical model of the secular excitation. Spurious sig-

nals at other frequencies that may arise from motional

coupling have no inﬂuence on the counting process and

can be safely ignored.

We experimentally demonstrate this method by res-

onantly exciting the radial motion of H+

2and H+

3ions

embedded in a laser-cooled Be+ion crystal. We observe

concomitant changes in the amount of ﬂuorescence from

the Be+ions when the number of trapped H+

2or H+

3ions

changes due to chemical reactions with neutral rest gas

molecules.

Spectroscopy of dark ions requires a scheme for de-

tecting that the target transition is being excited. This

can for example be achieved by monitoring that new

ion species are created by state-dependent photoioniza-

tion [40] or resonance-enhanced multiphoton dissocia-

tion [18–20, 39]. The reliable detection of single dark ions

demonstrates that this detection scheme can be single-

event sensitive and that the spectroscopy will be ulti-

mately limited only by quantum projection noise. Non-

linearities in the signal intensity may introduce a system-

atic frequency shift, especially when the spectrum con-

sists of multiple overlapping lines [39]. Accurate counting

of the dark ions gives rise to a spectroscopy signal with

negligible nonlinearity.

Chemical reactions at ultracold temperatures can be

investigated precisely for a small number of atoms or ions

after careful quantum-state preparation [41]. Our detec-

tion scheme can be employed to eﬃciently capture such

events with single-particle sensitivity.

We use a linear Paul trap to conﬁne the ions. As shown

in Fig. 1, it consists of four blade electrodes that are

spaced 0.45 mm from the trap axis and have an axial

length of 3.00 mm. Axial conﬁnement is provided by two

endcap electrodes that are located 3.50 mm from the trap

center.

The trap is driven with a radio frequency (RF) of

66.05 MHz with an amplitude of around 120 V, and a

static voltage of 400 V is applied to the endcaps. For

Be+this results in a radial secular frequency of around

1.6 MHz, corresponding to a Mathieu stability parameter

q≈0.07 [5], and an axial secular frequency of 645 kHz.

The single-particle radial secular frequency scales propor-

tional to the ion’s charge-to-mass ratio. The theoretical

values are 4.8 MHz and 7.2 MHz for H+

3and H+

2, respec-

tively.

The Be+ions are laser-cooled by driving the 2s 2S1/2

(F=2) →2p 2P3/2(F=3) cycling transition with circu-

larly polarized 313 nm light that is generated by sum-

frequency generation of two continuous wave ﬁber lasers

at 1051 nm and 1550 nm and subsequent frequency-

doubling [42]. The cooling transition has a natural

linewidth of Γ = 2π×18 MHz [43], and the cooling beam

contains two frequency components that are red-detuned

from the transition by 130 MHz and 460 MHz. We found

x y

RF DC

DC RF

PMT

Compensation

electrode

RF DC

zHorizontal

imaging

Vertical

imaging

Beam

splitter

EMCCD

FIG. 1. Geometry of the ion trap setup. Radial (top) and ax-

ial (bottom) view. The Coulomb crystal is not drawn to scale.

Radial conﬁnement is generated by the RF voltage Urf (t) that

is applied to one diagonal pair of blade electrodes. A static

voltage Uec is applied to the endcap electrodes for axial con-

ﬁnement. The cooling beam (purple) is directed along the

trap axis (z). The ﬂuorescence from the laser-cooled Be+ions

is imaged onto two EMCCD cameras. The static compensa-

tion voltage Ucand voltages applied to the two dc blade elec-

trodes are used to compensate stray ﬁelds in the trap caused

by patch potentials and electrode misalignment. A sinusoidal

voltage Uex(t) is added to the compensation electrode in order

to excite the secular motion of the trapped ions.

that adding the far-detuned component makes the sys-

tem more robust against losing the trapped ions when

strongly driving secular excitations. Both frequency com-

ponents can be switched on or oﬀ separately and have

similar intensities of around Isat, where Isat = 765 W/m2

is the saturation intensity of the cooling transition [43].

The cooling beam is aligned parallel to the trap axis and

propagates through holes in the end-cap electrodes. A

weak magnetic ﬁeld is applied parallel to the beam in

order to deﬁne the quantization axis. We use an electro-

optic modulator to create 1.25 GHz sidebands on the

cooling laser in order to re-pump the ions out of the

2s 2S1/2(F=1) dark state.

The trap is housed in an ultra-high vacuum chamber.

The background pressure measured with a cold-cathode

ion gauge is around 4 ×10−11 mbar.

We load Be+ions into the trap with the help of a

beam of beryllium atoms from an oven. The beam is sent

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Number-ResolvedDetectionofDarkIonsinCoulombCrystalsFabianSchmid,1,JohannesWeitenberg,1JorgeMoreno,1TheodorW.Hansch,1,2ThomasUdem,1,2andAkiraOzawa1,y1Max-Planck-InstitutfurQuantenoptik,85748Garching,Germany2FakultatfurPhysik,Ludwig-Maximilians-UniversitatMunchen,80799Munchen,Germany(Dated:Oct...

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Number-Resolved Detection of Dark Ions in Coulomb Crystals Fabian Schmid1Johannes Weitenberg1Jorge Moreno1 Theodor W. H ansch1 2Thomas Udem1 2and Akira Ozawa1

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