High-accuracy multi-ion spectroscopy with mixed-species Coulomb crystals

B. Lipphardt; C. Lisdat; E. Benkler; E. Peik; H. Liu; H. N. Hausser; I. M. Richter; J. Keller; J. Kiethe; J. Klose

REVIEW 3 major objections 4 minor 18 references

High-accuracy multi-ion spectroscopy with mixed-species Coulomb crystals

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper reports simultaneous spectroscopy of up to four indium-115 clock ions in a mixed-species Coulomb crystal, showing that a decay-dead-time model explains the observed instability scaling and predicts full 1/sqrt(N)…

desk verdict Solid first demonstration of a four-ion In+ clock with a clean dead-time model; the N=2/N=4 verification is thin and the central number has a typo, but the main claims hold up. read the letter →

arxiv 2501.09615 v1 pith:2PD2IGNF submitted 2025-01-16 physics.atom-ph quant-ph

J. Keller , H. N. Hausser , I. M. Richter , T. Nordmann , N. M. Bhatt , J. Kiethe , H. Liu , E. Benkler

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B. Lipphardt S. Dörscher K. Stahl J. Klose C. Lisdat M. Filzinger N. Huntemann E. Peik T. E. Mehlstäubler

This is my paper · ORCID

classification physics.atom-phquant-ph

keywords multi-ionopticalclocksCoulombcrystalsindium-115ionquantumprojectionnoisesympatheticcoolingclockinstabilitydead-timemodelfrequencycomparison

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to show that an optical clock built from several ions can beat the single-ion noise limit without sacrificing accuracy. The authors run simultaneous clock spectroscopy on up to four $^{115}\mathrm{In}^+$ ions held in a mixed-species Coulomb crystal (an ordered chain of trapped ions) with $^{172}\mathrm{Yb}^+$ cooling ions, and find agreement with the single-ion In$^+$ clock below 1 part in $10^{17}$. With four clock ions the measured instability drops from $1.6\times10^{-15}/\sqrt{t/(1\,\mathrm{s})}$ to $9.2\times10^{-16}/\sqrt{t/(1\,\mathrm{s})}$. They explain the less-than-ideal scaling with a model for dead time spent waiting for excited ions to decay, and the model predicts that adding a repump laser would restore the full $1/\sqrt{N}$ quantum-projection-noise advantage. If correct, this gives a practical path to shorter averaging times in the most accurate class of optical clocks.

What carries the argument

The load-bearing object is a binomial dead-time model for spontaneous-decay-based state initialization. Given $n_{\mathrm{exc}}$ excited ions among $N$, the average time until the $n_{\mathrm{min}}$-th ion decays is computed from the binomial distribution, and averaging over the number of excited ions (each independently excited with probability $p_{\mathrm{exc}} = 0.26$) yields the mean dead time $T_{\mathrm{decay}}(n_{\mathrm{min}}, N, p_{\mathrm{exc}})$. This enters the instability formula $\sigma_y(t) = \sigma_{y,1}(t)/\sqrt{n_{\mathrm{min}}} \cdot \sqrt{(T_c + T_{\mathrm{decay}})/T_c}$, where $T_c$ is the cycle time without decay dead time. The model quantitatively matches the measured instability scaling with clock-ion number $N$, and identifies the removal of $T_{\mathrm{decay}}$ as the condition for reaching $1/\sqrt{N}$ scaling.

What would settle it

Measure site-resolved excitation probabilities and decay-time histograms for $N=2$ and $N=4$ crystals; if $p_{\mathrm{exc}}$ is not uniform, the model's match is coincidental. The sharpest test: add the $482\,\mathrm{nm}$ repump and measure the $N=4$ instability; if it does not approach $0.8\times10^{-15}/\sqrt{t/(1\,\mathrm{s})}$ (half the single-ion value) within the stated uncertainty, the central prediction fails.

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Extended reading notes

Core claim

The central claim is that multi-ion $^{115}\mathrm{In}^+$ clock spectroscopy in a mixed-species Coulomb crystal is accurate and its instability behaviour is quantitatively understood. First frequency comparisons of the two- and four-ion clock against $^{87}\mathrm{Sr}$ and $^{171}\mathrm{Yb}^+$ agree with the single-ion In$^+$ clock below $1\times10^{-17}$, while the four-ion configuration cuts the instability from $1.6\times10^{-15}$ to $9.2(4)\times10^{-16}$ per $\sqrt{t/(1\,\mathrm{s})}$. The paper derives a binomial model for the average dead time incurred when state preparation waits for spontaneous decay (lifetime $195\,\mathrm{ms}$), showing that this dead time erodes the ideal $1/\sqrt{N}$ improvement. The same model indicates that deterministic state initialization via a $482\,\mathrm{nm}$ repump would eliminate this dead time and reach the quantum-projection-noise limit, scaling as $1/\sqrt{N}$.

Load-bearing premise

The model assumes every clock ion is excited with the same probability $p_{\mathrm{exc}}=0.26$ and that the $195\,\mathrm{ms}$ spontaneous decay is the only dead time that grows with ion number; if excitation probability varies across crystal sites or with $N$, the match to the measured instability scaling could be coincidental and the predicted $1/\sqrt{N}$ recovery after repumping would need revision.

Editorial extensions

If this is right

With a $482\,\mathrm{nm}$ repump laser, the same apparatus is expected to reach the quantum-projection-noise limit, giving $\sigma_y \propto 1/\sqrt{N}$ and roughly halving the $N=4$ instability relative to single-ion operation.
Choosing $n_{\mathrm{min}} < N$ (continuing once a subset of clock ions has decayed) can reduce dead time enough to lower the instability, and the paper provides upper-bound estimates for those operating points.
First comparisons show that multi-ion operation preserves agreement with the single-ion In$^+$ clock at the $1\times10^{-17}$ level, so the accuracy advantage of In$^+$ is not lost when scaling to four ions.
The decay-time formulas apply to any multi-ion clock whose state preparation relies on spontaneous decay, not just In$^+$, so the analysis can guide species and crystal-size choices in future designs.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: the model's assumption of a uniform excitation probability $p_{\mathrm{exc}}$ across crystal sites should be tested directly by recording site-resolved Rabi contrast and decay histograms; if $p_{\mathrm{exc}}$ varies with position, the $N$-dependence of the dead time would shift and the predicted repump gain would need recomputation.
Editorial inference: the same dead-time analysis likely applies to other ion-clock species that lack a repump transition, where the waiting-time penalty would enter the same formula with a different lifetime.
Editorial inference: the ad hoc $0.9$ contrast reduction in the Rabi model suggests that state preparation itself is not perfectly understood; if that contrast loss is fixed, $p_{\mathrm{exc}}$ could increase above $0.26$, which would improve the no-repump instability and slightly change the model curves.
Editorial inference: a natural testable extension is to record the full probability distribution of the $n_{\mathrm{min}}$-th decay time rather than just its mean, since the model's binomial prediction could be validated histogram-by-histogram.

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Formalized claims in Lean

Claim #1: The central claim is that multi-ion $^{115}\mathrm{In}^+$ clock spectroscopy in a mixed-species Coulomb crystal is accurate and its instability behaviour is quantitatively understood. First frequency comparisons of the two- and four-ion clock against $^{87}\mathrm{Sr}$ and $^{171}\mathrm{Yb}^+$ agree with the single-ion In$^+$ clock below $1\times10^{-17}$, while the four-ion configuration cuts th

/-- @claim 1 The central claim is that multi-ion $^{115}\mathrm{In}^+$ clock spectroscopy in a mixed-species Coulomb crystal is accurate and its instability behaviour is quantitatively understood. First frequency comparisons of the two- and four-ion clock against $^{87}\mathrm{Sr}$ and $^{171}\mathrm{Yb}^+$ agree with the single-ion In$^+$ clock below $1\times10^{-17}$, while the four-ion configuration cuts th -/
def central_claim : Prop :=

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a solid experimental letter from PTB showing that four In+ ions in a mixed Coulomb crystal can be spectroscopically resolved and used as an optical clock, with a measured instability improvement from 1.6e-15 to 9.2e-16/sqrt(t/1s) and agreement with single-ion results below 1e-17. The dead-time model in Eqs. (1)-(8) is a clean analytic extension of binomial decay statistics to the "wait for nmin decays" protocol, and it should be useful to other crystal-clock groups.

What is genuinely new is the first simultaneous clock operation with up to four In+ ions and the derivation of the decay-related dead-time scaling. The Rabi oscillation data and site-resolved spectroscopy in Fig. 1 support the claim that all four ions can be addressed with roughly the single-ion contrast. The comparison campaign against Sr and Yb+ is competently done, and the 1e-17 consistency check is a meaningful sanity signal.

The soft spots are real but not damaging. The model is calibrated by normalizing to the N=1 instability and then compared with only two derived instability points (N=2 and N=4), and the N=2/N=4 numbers are extracted by subtracting a Yb+ clock noise contribution, not from a direct In+ vs Sr comparison. The paper does not report a measurement of pexc or cycle-time components at N=2/N=4; it assumes the single-ion pexc=0.26 holds at all sites. My own estimate, however, does not support the stress-test worry that a pexc shift from 0.26 to 0.20 would move the predicted N=4 instability by an amount comparable to the measured improvement: using Eq. (8) with Tc=330ms, the shift is a few percent, not a factor of 1.7. So the qualitative conclusion--that decay dead time explains the observed scaling and that a repump should restore 1/sqrt(N)--is not brittle to that assumption. The larger caveat is that the dead-time model only treats decay-related dead time; other N-dependent dead times are assumed absent, and the 0.9 contrast factor in Fig. 1c hints that excitation/state-preparation may not be perfectly understood. Also, the paper contains a numerical typo: the abstract says 9.2e-16, while Sec. 3.4 writes 9.2e-15. That needs fixing.

Bottom line: this is a useful and honest experimental step, not a breakthrough. It deserves a proper referee. If I were citing multi-ion In+ clock work in the next year, I would cite it, but I would not treat the model's N=4 prediction as quantitatively confirmed until a direct pexc/N-cycle-time calibration is published.

Referee Report

3 major / 4 minor

Summary. This paper reports simultaneous spectroscopy and first clock operation with up to four 115In+ clock ions sympathetically cooled by 172Yb+ ions in a linear Coulomb crystal. The authors present site-resolved spectroscopy and Rabi oscillations for a 4In+-8Yb+ crystal, frequency comparisons against a 87Sr lattice clock and a 171Yb+ E3 single-ion clock, and an instability analysis. The central quantitative claims are that the measured instability is reduced from 1.6e-15/sqrt(t/1s) with one In+ ion to 9.2e-16/sqrt(t/1s) with four ions, that a derived model for spontaneous-decay-related dead time during state preparation explains the observed scaling of instability with clock-ion number N, and that adding a 482 nm repump laser should restore 1/sqrt(N) scaling. The dead-time model is derived analytically in Eqs. (1)-(8).

Significance. If the central claims hold, the paper provides a credible route to multi-ion optical clocks with reduced quantum projection noise while retaining the low systematic uncertainties of a single-ion clock. The analytic dead-time model is a useful contribution and is internally consistent. The paper also demonstrates agreement between multi-ion and single-ion In+ clock frequencies below 1e-17, and quotes a single-ion systematic uncertainty of 2.5e-18. These are notable experimental achievements. The main limitations are that the experimental test of the scaling model rests on only two derived instability points and on an unmeasured assumption that the excitation probability pexc is independent of N and crystal site. The repump prediction is plausible but depends on all other dead-time contributions being N-independent, which is not demonstrated.

major comments (3)

[Section 3.4, Eq. (8), and Fig. 4b] The central claim that the dead-time model 'matches the observed scaling' rests on Eq. (8) with pexc = 0.26 and Tc = 330 ms taken from single-ion operation only. No measurement of pexc, of the cycle-time components, or of the decay-related dead time is reported for the N = 2 and N = 4 crystals. Since Tdecay in Eq. (6) depends sensitively on pexc for N > 1, and since Fig. 1c requires an ad hoc 0.9 contrast reduction, the excitation/preparation model is not exact. If pexc varied with N or with crystal site, the agreement in Fig. 4b could be fortuitous. The authors should either measure pexc and the relevant dead-time components for each crystal size, or provide a sensitivity analysis showing that the conclusion is robust to the plausible range of pexc and Tc variations.
[Fig. 4b and Section 3.4] The N = 2 and N = 4 instability points are not directly measured against the Sr lattice clock; they are inferred from In+/Yb+ (E3) ratio measurements by subtracting the Yb+ clock instability. The method of subtraction, the treatment of correlated noise between the two clocks, and the corresponding uncertainty propagation are not described. Because the model curve is normalized to the directly measured N = 1 point, the comparison to only two derived points is a weak test of the scaling prediction. The authors should state the subtraction procedure explicitly and, if possible, add direct measurements against the Sr lattice for N = 2 and N = 4.
[Sections 3.3, 3.5, and Fig. 4] The prediction that a repump laser will restore 1/sqrt(N) scaling assumes that all dead-time contributions other than spontaneous decay are N-independent. However, sympathetic cooling and state-detection durations could in principle depend on the number and arrangement of ions in the crystal. The footnote to Section 3.3 acknowledges that the treatment overestimates instabilities for nmin < N, but the claim for nmin = N also relies on the implicit assumption that Tc is independent of N. The authors should state this assumption explicitly and provide evidence, for example by reporting the measured cycle times for each crystal composition, that no significant N-dependent dead time remains.

minor comments (4)

[Abstract and Section 3.4] The abstract quotes an instability of 9.2e-16/sqrt(t/1s) for N = 4, while Section 3.4 states sigma_y = 9.2(4)e-15/sqrt(t/1s); one of these is a typo and should be corrected.
[Figure 1c caption] The caption states 'all 2N = 24 radial modes', but N is elsewhere used for the number of clock ions. Here the total number of ions in the 4In+-8Yb+ crystal gives 24 radial modes, so the notation is ambiguous and should be clarified.
[Reference [18]] Reference [18] is given only as 'Safronova M priv. comm.'; if this is the sole support for the feasibility of the 482 nm repump transition, a more complete reference or a published source should be provided.
[Section 3.2, Eq. (5)] The summation notation in Eq. (5) is unconventional and would be clearer if written with explicit indicator functions or with a unified sum over j and i, rather than the conditional subscripts on the inner sum.

Circularity Check

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No significant circularity: the dead-time scaling model is calibrated on the N=1 point and then tested against independently measured N=2 and N=4 instabilities.

full rationale

The paper's central derivation is the dead-time model in Eqs. (5)-(8), which computes the average waiting time for n_min spontaneous decays from the measured lifetime tau = 195 ms and a binomial excitation distribution, and then combines it with QPN using Eq. (8). The only free normalization is the single-ion instability scale, which is explicitly set to the observed N=1 instability (Sec. 3.4: 'adjusted to the observed instability at N=1'), so the N=1 point is a calibration, not a prediction. The scaling claim rests on the N=2 and N=4 instabilities, which are measured in independent frequency comparisons against Sr and Yb+ clocks (Fig. 2) and are not used to determine p_exc = 0.26 or T_c = 330 ms; these parameters are taken from single-ion operation. The 0.9 contrast scaling in Fig. 1c is an acknowledged imperfection in the Rabi-oscillation model, not an input to the instability model. The self-citation to [9] supplies the spectroscopy sequence, the single-ion systematic uncertainty, and the N=1 normalization, but it is a published, experimentally falsifiable data source and does not by construction force the N=2/N=4 agreement. The footnote even states that the n_min < N curves are upper bounds, showing that the model is not rigged to match. Thus no circular step reduces the prediction to its inputs.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central experimental claims rest on a small set of physical assumptions about ion excitation, decay, and systematic uncertainty, plus the standard statistical model. No new entities are introduced. The main free parameters are the Rabi contrast reduction factor (0.9) and the N=1 normalization of the instability model.

free parameters (2)

Rabi contrast reduction factor = 0.9
Figure 1c scales the theoretical Rabi oscillation model by 0.9 to match the observed contrast; the paper attributes the missing contrast to imperfect state preparation.
Instability model normalization = set to N=1 observed instability
Figure 4b says the theoretical curves are 'adjusted to the observed instability at N=1' before comparing to N=2 and N=4, so one overall scale is fitted.

assumptions (5)

domain assumption The 3P0 clock state of 115In+ decays spontaneously with lifetime tau = 195 ms.
Used in the dead-time model, Eqs. (5)-(8); value taken from reference [15].
domain assumption Each clock ion is excited independently with equal probability pexc per probe, with pexc = 0.26 measured for a single ion and assumed for N=2 and N=4.
Used in Eq. (7) and the model curves of Figure 4; nonuniform excitation would break the binomial model.
domain assumption The radial motional modes are at 1.3 times the Doppler temperature for the Rabi oscillation model.
Figure 1c uses this to estimate contrast and infer that thermal time dilation shifts are at the low 1e-18 level; it is stated as a coarse estimate.
domain assumption Larger crystals have systematic uncertainties dominated by effects independent of ion number.
Section 2 states that a full systematic evaluation has not been performed for the larger crystals and that most contributions are expected to be ion-number-independent.
domain assumption The noise of the Yb+ E3 clock and the In+ clock are independent, allowing quadrature subtraction to infer In+ instability from ratio measurements.
Used to derive the N=2 and N=4 instability points in Section 3.4; correlated noise would bias the inferred values.

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Cite this review

Pith. "Pith review of High-accuracy multi-ion spectroscopy with mixed-species Coulomb crystals." pith.science (2026). https://pith.science/paper/2PD2IGNF

@misc{pith2026250109615,
  author       = {Pith},
  title        = {Pith review of: High-accuracy multi-ion spectroscopy with mixed-species Coulomb crystals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2PD2IGNF}},
  note         = {Machine review of arXiv:2501.09615}
}

abstract

Multi-ion optical clocks offer the possibility of overcoming the low signal-to-noise ratio of single-ion clocks, while still providing low systematic uncertainties. We present simultaneous spectroscopy of up to four ${}^{115}$In${}^+$ clock ions in a linear Coulomb crystal, sympathetically cooled with ${}^{172}$Yb${}^+$ ions. In first clock comparisons, we see agreement below $1\times10^{-17}$ with results obtained using a single In${}^+$ ion, for which we have evaluated the systematic uncertainty to be $2.5\times10^{-18}$. Operation with four clock ions reduces the instability from $1.6\times10^{-15}/\sqrt{t/(1\;\mathrm{s})}$ to $9.2\times10^{-16}/\sqrt{t/(1\;\mathrm{s})}$. We derive a model for decay-related dead time during state preparation, which matches the observed scaling of instability with clock ion number $N$, and indicates that $1/\sqrt{N}$ scaling can be achieved with the addition of a repump laser.

Figures

Figures reproduced from arXiv: 2501.09615 by the authors.

**Figure 1.** Spectroscopy with four In+ ions. a) Mixed-species crystal consisting of four In+ clock ions and eight Yb+ cooling ions. b) Site-resolved frequency scan of the clock transition (mF = mF ′ = +9/2 component). The frequency axis offset is the same for all plots. c) Site-resolved Rabi oscillations compared to the theoretical expectation at 1.3 times the Doppler temperature for all 2N = 24 radial modes, scaled by 0.9 to m… view at source ↗

**Figure 2.** Optical frequency comparisons to 87Sr (top) and 171Yb+ E3 (bottom) using different numbers of clock ions. Vertical shaded areas show the durations of the respective datasets. The green lines and shaded areas depict the weighted means and corresponding uncertainties. All shown uncertainties are purely statistical, based on the observed white frequency noise. suggests that shifts are at the low 10−18 level, but more d… view at source ↗

**Figure 3.** Decay times in ensembles of ions. a) Temporal evolution of t [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Multi-ion clock instabilities. a) Expected quantum projectio [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Arnold K, Hajiyev E, Paez E, Lee C H, Barrett M D and Bollinger J 201 5 Phys. Rev. A 92(3) 032108

work page

[2] [2]

Aharon N, Spethmann N, Leroux I D, Schmidt P O and Retzker A 20 19 New J. Phys. 21 083040

work page

[3] [3]

Cui K, Valencia J, Boyce K T, Clements E R, Leibrandt D R and Hume D B 2022 Phys. Rev. Lett. 129(19) 193603

work page 2022

[4] [4]

Leibrandt D R, Porsev S G, Cheung C and Safronova M S 2024 Nat. Commun. 15 5663

work page 2024

[5] [5]

Steinel M, Shao H, Filzinger M, Lipphardt B, Brinkmann M, Didier A, M ehlst¨ aubler T E, Lindvall T, Peik E and Huntemann N 2023 Phys. Rev. Lett. 131(8) 083002

work page 2023

[6] [6]

Itano W M, Bergquist J C, Bollinger J J, Gilligan J M, Heinzen D J, Moore F L, Raizen M G and Wineland D J 1993 Phys. Rev. A 47(5) 3554–3570

work page 1993

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Herschbach N, Pyka K, Keller J and Mehlst¨ aubler T E 2012 Appl. Phys. B 107 891–906

work page 2012

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Keller J, Burgermeister T, Kalincev D, Didier A, Kulosa A P, Nordman n T, Kiethe J and Mehlst¨ aubler T E 2019Phys. Rev. A 99 013405

work page