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Thorium-229 in its Highest Charge States: Single-Ion Nuclear Clocks for Tests of Fundamental Interactions

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Pith's one-line read This paper argues that thorium-229 stripped to a bare nucleus or left with a single electron can serve as a single-ion nuclear clock read out by quantum logic spectroscopy, with the natural linewidth of the clock transition tuned by more…

desk verdict A credible design study for a highly charged Th-229 nuclear clock; the bare-nucleus case holds up, the H-like case needs error bars or softer claims. read the letter →

arxiv 2608.08133 v1 pith:MMJPW5VA submitted 2026-08-08 physics.atom-ph

C. Brandau , P. Micke , I. Hartl , C. M. Heyl , P. Indelicato , Yu. A. Litvinov , R. X. Schüssler , M. Vogel
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Th. Stöhlker
This is my paper · ORCID
classification physics.atom-ph
keywords nuclearclockthorium-229highlychargedionsquantumlogicspectroscopyhyperfinemixingisomericstatefundamentalconstantvariationlinearPaultrap
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

The paper makes the case that the two simplest forms of thorium-229 — the bare nucleus $^{229}$Th$^{90+}$ and the hydrogenlike ion $^{229}$Th$^{89+}$ with one $1s$ electron — are realistic platforms for single-ion nuclear clocks. It shows that quantum logic spectroscopy in a cryogenic linear Paul trap, using one co-trapped $^9$Be$^+$ ion for sympathetic cooling and readout, can coherently excite and detect the vacuum-ultraviolet nuclear transition with few-nanowatt lasers. The enabling mechanism is nuclear hyperfine mixing, which shortens the isomer lifetime from roughly 42 minutes to tens of milliseconds in hydrogenlike thorium and thereby tunes the natural linewidth over more than five orders of magnitude across charge states. If the scheme works, a clock based on the fully stripped nucleus would be the first timekeeper with no electrons at all, and comparisons across charge states would give unusually clean tests of fundamental interactions.

What carries the argument

The load-bearing mechanism is nuclear hyperfine mixing (NHM): for ions with an unpaired $j=1/2$ electron, the magnetic hyperfine interaction between the electron and the nucleus mixes the $F=2$ state of the nuclear ground state with the $F=2$ state of the isomeric state, repelling the two levels and opening a fast decay path that shortens the isomer lifetime dramatically. The second mechanism is quantum logic spectroscopy (QLS), in which the thorium ion and a $^9$Be$^+$ logic ion form a two-ion Coulomb crystal; a sequence of laser pulses maps the thorium nuclear state onto the beryllium qubit, which is read out by resonance fluorescence. The Rabi frequency $\Omega$ and the optical-Bloch-equation excitation probability, with the isomer lifetime entering through the decay rate $\Gamma=1/\tau$, turn the nuclear parameters and laser parameters into predicted excitation curves.

What would settle it

Measure the isomer lifetime of a single trapped $^{229}$Th$^{89+}$ ion: nuclear hyperfine mixing predicts a decay in tens of milliseconds, while the unmixed bare-nucleus lifetime is about 42 minutes. A measured lifetime close to 42 minutes, or a VUV resonance found far from the calculated hydrogenlike transition energy, would falsify the feasibility claim.

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Extended reading notes

Core claim

The central claim is that highly charged $^{229}$Th$^{q+}$ ions with $q=90$ to $87$ are not exotic obstacles but practical clock systems. For the hydrogenlike ion, the $1s$ electron creates a strong magnetic field at the nucleus; nuclear hyperfine mixing couples the $F=2$ hyperfine levels of the ground and isomeric nuclear states, shifting them by $\pm 0.019$ eV and accelerating the isomeric decay by about five orders of magnitude. The bare nucleus, by contrast, has no electron cloud at all, making it the prototype of a pure nuclear frequency standard. The paper evaluates the quantum logic spectroscopy sequence for both cases, including coherent Rabi flopping on the carrier and first-order motional sidebands, and concludes that a two-ion crystal of one thorium ion and one beryllium ion can cool, drive, and read out the nuclear transition with currently demonstrated or near-term VUV laser technology.

Load-bearing premise

The plan stands on published values for how the thorium nucleus changes its size and magnetism between the ground and isomeric state, and for how fast the isomeric state decays; if any of those is significantly wrong, the predicted clock frequencies and linewidths shift.

Editorial extensions

If this is right

  • A single trap holding one thorium ion and one $^9$Be$^+$ logic ion can perform coherent spectroscopy on the nuclear carrier transition and on motional sidebands at a few nanowatts of VUV power.
  • The same platform can compare different charge states and different transitions in situ, separating nuclear from electronic contributions and isolating sensitivities to variations of $\alpha$, $m_e/m_p$, and the quark-mass parameter $X_q$.
  • A clock on fully ionized $^{229}$Th$^{90+}$ would have no electronic many-body shifts, promising fractional uncertainties at the $10^{-19}$ level and beyond.
  • Nuclear hyperfine mixing shortens the isomer lifetime to milliseconds, so a VUV frequency comb with roughly 500 Hz comb-tooth linewidth suffices to drive the transition, relaxing the laser requirements for early demonstrations.
  • Comparing $K_\alpha\approx6000$ for the nuclear transition with $K_\alpha\approx6$ for the hyperfine atomic transition in the same ion allows fundamental-constant variations to be disentangled.

Reading between the lines

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

  • A natural next experiment would validate the Rabi-flopping curves with a non-radioactive highly charged ion whose optical transition has a similar lifetime and Lamb-Dicke parameter, before any thorium crystal is loaded.
  • If in situ comparisons of $^{229}$Th$^{90+}$ and $^{229}$Th$^{88+}$ are realized, the difference of their isomer shifts would give a nearly model-independent measurement of the nuclear charge-radius change that currently dominates the energy uncertainty.
  • The same quantum logic spectroscopy platform could use the two measured VUV transition frequencies and the ground-state hyperfine frequency to predict the fifth transition, giving an internal consistency check that needs no absolute laser calibration.
  • If the technology matures, transportable cryogenic ion traps could deliver pre-stripped $^{229}$Th$^{90+}$ ions to metrology laboratories, turning accelerator-based production into a one-time supply step.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper proposes single-ion nuclear clocks based on 229Th in its highest charge states, specifically H-like 229Th89+ and bare 229Th90+, implemented via quantum logic spectroscopy in a cryogenic linear Paul trap with a co-trapped 9Be+ logic ion. It presents Rabi-frequency and excitation-probability calculations for carrier and first-order motional sidebands using the optical Bloch equations, with assumed VUV frequency-comb and cw laser parameters. The paper also emphasizes nuclear hyperfine mixing as a mechanism that shortens the isomeric lifetime in few-electron ions and thereby tunes the clock-transition linewidth over several orders of magnitude by changing the charge state. It reports MDFGME calculations of the isomer shift ΔE_iso and discusses the potential of these systems for tests of fundamental interactions.

Significance. If the quantitative feasibility claims survive scrutiny, this proposal would be valuable: highly charged 229Th ions combine the advantages of nuclear clocks with the well-controlled environment of single trapped ions, and the quantum logic approach offers a universal readout for comparing different charge states and transitions in the same apparatus. The idea of exploiting nuclear hyperfine mixing to tune the natural linewidth is inventive and could substantially ease the demands on VUV laser systems. The paper is transparent in its use of standard optical Bloch equations and in its specification of the assumed laser parameters, and the reported ΔE_iso calculation is a concrete original contribution. The main weakness is that the central feasibility curves rest on nuclear input parameters whose uncertainties are not propagated into the quoted lifetimes, linewidths, or Rabi rates.

major comments (2)
  1. [Fig. 1 and the NHM paragraph] The H-like isomer lifetime (74 ms) and the NHM shifts (ΔE_NHM = ±0.019 eV) that determine the Rabi frequencies and linewidths used in Fig. 2 are taken from [63] as point values, without propagating the uncertainties in μ^(g)=0.366(6) μ_N, μ^(m)=−0.378(8) μ_N, B(M1)=0.022 W.u., and δ⟨r²⟩=0.0103 fm². Since Ω in Eq. (1) scales as √Γ, a factor-of-several change in the NHM rate moves the H-like Rabi curves and the sideband contrast in Fig. 2, and the text itself states that the transition energies of highly ionized 229Th still have comparatively large uncertainties. The authors should provide a sensitivity analysis over the allowed ranges of these nuclear inputs and state whether the quoted 74 ms lifetime and the 'more than five orders of magnitude' linewidth-tunability claim survive within those ranges. Without that, the H-like feasibility claim is not quantitatively grounded.
  2. [Fig. 2 and Eq. (1)] The two feasibility curves are computed for assumed laser parameters: P_L = 3 nW with Δν_L = 500 Hz for the H-like case and P_L = 10 nW with Δν_L = 1 Hz for the bare-nucleus case, together with a 5 µm focus and C²_ge G² = 0.5. The cw parameters are supported by a recent demonstration [42], but no comparable evidence is cited for a frequency-comb tooth with 3 nW power and 500 Hz linewidth, and C²_ge G² is set by hand rather than derived from the level structure and beam geometry. Because Fig. 2 is the central demonstration of QLS feasibility, the authors should either justify these values quantitatively or show how the excitation probability degrades when they are varied over realistic ranges.
minor comments (5)
  1. [Eq. (1) and Fig. 2] The text does not state how P_L is converted to I_L for the assumed 5 µm focus; please give the beam area and intensity definition used in Eq. (1).
  2. [Fig. 1] Figure 1 is visually crowded, and the numerical energy and lifetime labels are difficult to read; please increase font sizes and separate the panels more clearly.
  3. [Isomer-shift paragraph] The two values of ΔE_iso quoted in the text (0.086 eV for the bare nucleus relative to the crystal and 0.036 eV for H-like relative to the bare nucleus) are easy to confuse; please make the reference frames explicit at each occurrence.
  4. [Introduction] The sentence defining the quark-mass parameter, 'm_q = m_u + m_d/2', is ambiguous; please write the light-quark average mass explicitly.
  5. [References] The reference list contains several arXiv identifiers and preprint DOIs; please update to published versions where available.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the feasibility estimate is a forward calculation from externally published nuclear parameters and standard QLS/optical-Bloch equations, not a fitted or self-defined prediction.

full rationale

The paper's central claims are forward calculations. The Rabi frequency in Eq. (1) and the excitation probability in Eq. (2) are standard results applied with assumed laser parameters and with decay rates taken from prior published work. The H-like and Li-like lifetimes and hyperfine-mixing shifts are imported from the independently published PRL [63], whose authors overlap with the present paper; however, [63] is a peer-reviewed calculation based on stated nuclear inputs (magnetic moments and B(M1)) and is not re-fitted or tuned to match any target quantity in this paper. The isomer shift DeltaE_iso is computed with the MDFGME code from a literature value delta<r^2>=0.0103 fm^2, and the paper explicitly states agreement with independent calculations [64,65]; this is a calculated input, not a fitted output. The 'linewidth tunability over more than five orders of magnitude' is a direct consequence of the NHM lifetimes from [63] and earlier work [58-62], not a prediction that reduces by construction to this paper's own inputs. The acknowledged large uncertainties in transition energies for highly ionized 229Th are an accuracy and robustness caveat, not evidence of circular reasoning. No equation is defined in terms of the claimed result, no fitted parameter is renamed as a prediction, and the feasibility curves in Fig. 2 are obtained by inserting stated parameters into an explicit model. Accordingly, the circularity score is 0.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central proposal is built from imported nuclear parameters, standard two-level quantum optics, prior NHM results, and assumed beam and trap performance. The paper's genuine contribution is the synthesis and the quantitative Rabi feasibility estimates; no new particles, forces, or fields are introduced.

free parameters (4)
  • Coupling product C^2_ge G^2 = 0.5
    Hand-set in Eq. (1) for both the 2g to 2m transition in 229Th89+ and the 3/2+ to 5/2+ transition in 229Th90+; it scales the Rabi frequency and therefore every excitation probability shown in Fig. 2.
  • Assumed VUV laser power and linewidth = 3 nW at 500 Hz for 229Th89+; 10 nW at 1 Hz for 229Th90+
    Chosen as achievable or near-future values; the paper does not show that these powers and linewidths can actually be delivered to a single ion inside the Paul trap.
  • Motional mode parameters = omega_u = 2*pi*1.3 MHz; axial eigenvector component 0.86; eta about 0.15
    Chosen for a 9Be+-229Thq+ two-ion crystal; these set the sideband Rabi frequencies and the Lamb-Dicke suppression.
  • Laser focus diameter = 5 micrometers
    Assumed for computing intensity and Rabi frequency in Eq. (1); smaller or larger focus changes the excitation rate.
assumptions (6)
  • standard math Optical Bloch equations with relaxation (Torrey solution) describe the nuclear excitation
    Used without re-derivation to obtain Eq. (2) and Fig. 2; standard for a two-level system but neglects multilevel structure and unresolvable decoherence.
  • domain assumption QLS transfers the nuclear state to a 9Be+ qubit with high fidelity
    The paper assumes the universal QLS protocol works for the VUV nuclear transition; no pulse sequence or state-dependent force calculation is provided.
  • domain assumption NHM lifetimes, shifts, and formulas from Ref. [63] apply to the charge states considered
    The central linewidth-tuning asset is imported from prior work by overlapping authors and is not re-derived in this paper.
  • domain assumption Nuclear input values from literature (mu, B(M1), delta<r^2>) are correct within quoted uncertainties
    mu_g=0.366(6) mu_N, mu_m=-0.378(8) mu_N, B(M1)=0.022 W.u., delta<r^2>=0.0103 fm^2; the text admits large uncertainties propagate to transition energies.
  • domain assumption MDFGME multiconfiguration Dirac-Fock with a two-parameter Fermi nuclear model gives reliable DeltaE_iso
    Computational method is standard but no numerical inputs, grid parameters, or uncertainty estimates are provided in the paper.
  • domain assumption Ion production and storage capabilities at GSI/HITRAP are sufficient
    Claims about about 10^5 229Th89+ per accumulation and months-long storage at 4 K are stated without independent data or error analysis in this paper.

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Cite this review

Pith. "Pith review of Thorium-229 in its Highest Charge States: Single-Ion Nuclear Clocks for Tests of Fundamental Interactions." pith.science (2026). https://pith.science/paper/MMJPW5VA

@misc{pith2026260808133,
  author       = {Pith},
  title        = {Pith review of: Thorium-229 in its Highest Charge States: Single-Ion Nuclear Clocks for Tests of Fundamental Interactions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MMJPW5VA}},
  note         = {Machine review of arXiv:2608.08133}
}
abstract

The prospects and the implementation of single-ion nuclear clocks of $^{229}$Th$^{q+}$ ions in their highest charge states $q=90, \ldots, 87$ are discussed. Highly-ionized-thorium clocks are ideal for tests of fundamental interactions since the ions are elementary quantum systems composed of only a few building blocks. Two cases of $^{229}$Th$^{q+}$ clocks excel: a) one-electron $^{229}$Th$^{89+}$ that combines two nuclear-clock transitions in the VUV with hyperfine IR atomic-clock transitions, and, b) fully ionized $^{229}$Th$^{90+}$ which constitutes the prototype of a nuclear clock, one without any electrons. We evaluate the feasibility of such clocks by means of quantum logic spectroscopy (QLS) in linear Paul traps. Due to its universal nature, the QLS approach allows for systematic clock comparisons using different charge states as well as different spectroscopy transitions on the same experimental platform. A valuable asset towards single-ion $^{229}$Th$^{q+}$ clocks is the process of nuclear hyperfine mixing that enables the tunability of the natural linewidth of the clock transition over more than five orders of magnitude by changing the charge state.

Figures

Figures reproduced from arXiv: 2608.08133 by the authors.

Figure 1
Figure 1. Nuclear clock transitions between the I π = 5/2 + ground state and the 3/2 + isomeric state for the bare nucleus, 229Th90+ (left), and for H-like 229Th89+ (center and right). For 229Th89+ the density of the 1s-electron shifts the energy by ∆Eiso = 0.036 eV. (Center) Ordinary HFS without NHM. (Right) HFS+NHM including ∆ENHM = ±0.019 eV for the F = 2 states (red). The VUV transitions between ground and isomeric state … view at source ↗
Figure 2
Figure 2. (Top) Excitation probability of the isomer for [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

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Reference graph

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Pith tools

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