Measurement of the Differential Static Scalar Polarizability of the $^\mathbf{88}$Sr$^{+}$ Clock Transition

A. E. Wallin; K. J. Hanhij\"arvi; T. Fordell; T. Lindvall

Measurement of the Differential Static Scalar Polarizability of the $^\mathbf{88}$Sr$^{+}$ Clock Transition

T0 review · 0 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper reports that the differential static scalar polarizability of the 88Sr+ clock transition is −4.8314(20)×10^−40 J m^2/V^2 (−29.303(12) au), 3.5 times more precise than the previous value and disagreeing with it by 5 standard…

desk verdict A careful, well-cross-checked measurement of the 88Sr+ polarizability with a genuinely new analysis method; the 5 sigma tension with the previous value needs independent confirmation, but the paper's internal consistency is strong. read the letter →

arxiv 2507.02603 v1 pith:GIAHTGEN submitted 2025-07-03 physics.atom-ph

T. Lindvall , K. J. Hanhijärvi , T. Fordell , A. E. Wallin This is my paper · ORCID

classification physics.atom-ph PACS 32.10.Dk37.10.Ty06.30.Ft

keywords differentialstaticscalarpolarizability88Sr+opticalclockmicromotionmagictrapdrivefrequencyMathieuequationblackbodyradiationshiftsecond-orderDopplerion

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 sets out to pin down the differential static scalar polarizability of the $^{88}$Sr$^{+}$ optical clock transition, the quantity that sets how strongly ambient blackbody radiation shifts the clock frequency. It exploits the 'magic' ion-trap drive frequency at which micromotion-induced second-order Doppler and quadratic Stark shifts cancel, measuring this zero crossing in a single ion while switching between minimized and large excess micromotion. By repeating the measurement at several Mathieu $q_z$ values, the polarizability is extracted without knowing the angle between the rf electric field and the trap axis, which previously dominated the systematic error. The result is $\Delta\alpha_0 = -4.8314(20)\times 10^{-40}\,\mathrm{J\,m^2/V^2} = -29.303(12)\,\mathrm{au}$, a factor of 3.5 improvement in uncertainty and a 5-$\sigma$ discrepancy with the earlier value. If correct, room-temperature $^{88}$Sr$^{+}$ clocks that used the old value need a fractional frequency correction of $-4\times 10^{-18}$, and the polarizability-related uncertainty drops to $2.2\times 10^{-19}$ at 295 K.

What carries the argument

The load-bearing object is the 'magic' trap drive frequency $\Omega_0$, the frequency at which the micromotion-induced second-order Doppler shift and the quadratic Stark shift cancel for an ion with negative $\Delta\alpha_0$. The relation between the measured zero-crossing frequency and the polarizability is carried by the nonhomogeneous Mathieu equation solution for excess micromotion: the mean rf-field harmonic ratios $\bar{c}_2$ and $\bar{c}_3$ determine how $\Omega_0$ depends on the Mathieu parameter $q_z$ and the angle $\theta_\Omega$ through Eq. (4). Measuring $\Omega_0$ over a range of $q_z$ values and fitting Eq. (4) yields both $\Delta\alpha_0$ and $\theta_\Omega$ without a separate angle determination, converting the dominant systematic into a fit parameter.

What would settle it

A decisive check would be an independent measurement of $\Delta\alpha_0$ for the $^{88}$Sr$^{+}$ clock transition by a different technique, such as a calibrated ac Stark shift measurement or a cryogenic-versus-room-temperature clock comparison, that disagrees with $-29.303(12)$ au by more than the combined uncertainties. A more targeted falsifier: if the fitted zero-crossing frequencies show a systematic dependence on the applied excess-micromotion direction or amplitude that persists beyond the model's corrections, the $q_z$-fitting model is biased.

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Extended reading notes

Core claim

The central claim is that the differential static scalar polarizability of the $^{88}$Sr$^{+}$ $^2S_{1/2} \rightarrow {}^2D_{5/2}$ clock transition is $\Delta\alpha_0 = -4.8314(20)\times 10^{-40}\,\mathrm{J\,m^2/V^2} = -29.303(12)\,\mathrm{au}$, a fractional uncertainty of $4.1\times 10^{-4}$. The value is derived from a single trapped ion in an interleaved clock scheme: the clock is alternately locked with minimized micromotion and with large applied excess micromotion, and the frequency difference is measured as a function of the rf drive frequency near the magic zero crossing. Fitting those shifts with a two-parameter expression from the Mathieu equation yields the zero-crossing frequency $\Omega_0$, and repeating the fits at $q_z = 0.34\ldots 0.71$ lets both $\Delta\alpha_0$ and the previously problematic angle $\theta_\Omega$ between the rf field and the trap axis be obtained from the same fit. Measurements at different excess-micromotion levels and with opposite ion displacement agree, and the uncertainty budget is dominated by statistics. The paper further shows that the new value disagrees with the previous measurement by 5 standard deviations while remaining close to theory, and that it implies a $-4\times 10^{-18}$ fractional frequency correction for room-temperature $^{88}$Sr$^{+}$ clocks that used the older value.

Load-bearing premise

The result stands on the assumption that the measured micromotion-induced frequency shifts follow exactly the theoretical Mathieu-equation relation between the drive frequency and the relative strengths of the rf-field harmonics, so that the fitted zero-crossing frequency is a clean measure of the polarizability; if unmodeled trap distortions change those harmonic ratios in a frequency-dependent way, the extracted value would be biased.

Editorial extensions

If this is right

Room-temperature $^{88}$Sr$^{+}$ clocks that used the previous polarizability value must apply a fractional frequency correction of $-4\times 10^{-18}$, a shift comparable to the $10^{-18}$-level targets for a redefined SI second.
The polarizability-related blackbody-radiation uncertainty of the $^{88}$Sr$^{+}$ clock falls to $2.2\times 10^{-19}$ at 295 K, well below the $1\times 10^{-18}$ total-uncertainty target.
The new value, close to theory but roughly 80 times more precise, becomes a benchmark for atomic-structure calculations of Sr$^{+}$.
Via existing polarizability-transfer schemes, the improved accuracy propagates to other ion species, including $^{171}$Yb$^{+}$ clocks.
The $q_z$-scan method extends to other ions with negative differential polarizability, such as $^{138}$Ba$^{+}$, $^{226}$Ra$^{+}$, and the secondary transition in $^{176}$Lu$^{+}$.

Reading between the lines

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

If the 5-sigma discrepancy with the earlier measurement is real, published $^{88}$Sr$^{+}$ clock frequencies and optical frequency ratios that relied on the old polarizability value may need re-evaluation; the paper identifies one published ratio, and comparisons between clocks on the same transition would not reveal the common offset.
The success of turning the unknown rf-field angle into a fit parameter suggests a general strategy for trapped-ion precision measurements: scan a dimensionless trap parameter and fit the geometric angle out, provided the underlying theory relation is known.
A direct extension would be to run the same $q_z$-scan at cryogenic temperatures, where the blackbody-radiation shift is much smaller; consistency of the extracted $\Delta\alpha_0$ would test the assumed temperature scaling, while any residual would expose temperature-dependent systematics.
Because the uncertainty is statistics-dominated, longer averaging or a two-clock configuration could push the polarizability uncertainty toward the $10^{-5}$ level, where the Mathieu-model corrections (harmonic ratios, trap anharmonicity) would become the limiting systematics and could be tested by comparing scans at different excess-micromotion amplitudes.

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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

The headline is simple: this is a careful ion-clock measurement that gives a new Δα0 for the 88Sr+ clock transition with 3.5× lower uncertainty than the previous value, and a 5σ discrepancy to go with it. The genuinely new piece is the qz-fitting method—measuring the magic drive frequency at several Mathieu qz values turns the rf-field angle θΩ from a dominant systematic into a fitted nuisance parameter. Data are openly available, and the uncertainty budget is statistics-dominated at 2.0×10⁻⁴³ of 2.0×10⁻⁴³.

What is most convincing is internal consistency. The second-order, third-order, and second-order-in-qz Mathieu solutions give Ω0^0 values within 0.7 kHz of each other across a 14.39 MHz value. The fitted θΩ = 31.2(3.1)° hits the independent geometric estimate of 32(1)°. The qz fit suppresses anharmonicity errors by an order of magnitude versus a single-qz measurement, shown explicitly in the supplement. Results agree across EMM levels, qz values from 0.34 to 0.71, and opposite displacement directions. This is solid, careful metrology.

The soft spot is the one you flagged: Eq. (4) is load-bearing, and no measurement directly determines the harmonic field ratios c̄2 and c̄3 at the ion. All of the validation shares the same Mathieu model, so an unmodeled qz-dependent distortion—spectral impurity, electrode asymmetry—could in principle bias Ω0^0. But I read the paper as reducing this to a minor concern, not a load-bearing flaw. If the model were significantly wrong, the three solution orders would not converge to 5×10⁻⁵ relative, and the angle would not absorb the fit-function errors while Ω0^0 stays put. The agreement is indirect, and a referee could reasonably ask for a direct probe of the 2Ω/3Ω field content, but nothing here suggests the central value is fragile.

The 5σ discrepancy with Dubé et al. is unexplained, and the paper says so plainly. The critique of the earlier measurement—a poorly characterized high-EMM clock, single beam, no common reference—is plausible, and the new result sits close to the theory value. An independent measurement would settle which value is right. That is exactly the kind of situation where this paper deserves to be in the literature.

This is a yes for peer review. I would cite it if I worked on ion clocks or BBR shifts, and I would bring it to a reading group for the discussion of model dependence in precision measurements.

Referee Report

0 major / 3 minor

Summary. The manuscript reports a precision measurement of the differential static scalar polarizability Δα0 of the 88Sr+ 2S1/2 → 2D5/2 clock transition, using the cancellation of the micromotion-induced second-order Doppler and quadratic Stark shifts at a 'magic' trap drive frequency. A single ion is interleaved between a reference clock with minimized excess micromotion and a high-EMM clock, and the zero-crossing frequency is measured as a function of the Mathieu parameter qz. A fit of the zero-crossing frequencies to the Mathieu-derived expression, Eq. (4), yields the zero-field crossing frequency Ω0^0 and the angle θΩ as a free parameter, avoiding the need for a priori knowledge of the rf-field direction. The result is Δα0 = -4.8314(20)×10^-40 J m^2/V^2 = -29.303(12) au, with an uncertainty dominated by statistics (2.0×10^-43 J m^2/V^2 out of a 2.0×10^-43 total), a factor of 3.5 improvement over Ref. [15], and a 5σ discrepancy with that value. The paper reports cross-checks at different EMM levels, different qz values, opposite ion displacements, and with three different orders of the Mathieu approximation.

Significance. If the result holds, it is an important step for 88Sr+ optical clocks: it reduces the polarizability-related BBR shift uncertainty to 2.2×10^-19 at 295 K and implies a -4×10^-18 fractional-frequency correction for clocks that used the previous value. The qz-fitting method is a genuine methodological advance: it removes the need to know the angle θΩ and converts it into a fit parameter, with the fitted value 31.2(3.1)° consistent with the geometric estimate 32(1)°. The uncertainty budget (Table I) is dominated by statistics (2.0×10^-43 of 2.0×10^-43 J m^2/V^2), with the largest systematic contribution (ion temperature) at 0.22×10^-43. The paper's strengths include interleaved common-mode rejection, multiple cross-checks (EMM levels, opposite displacement directions, qz values, three orders of Mathieu approximation), and openly available data. The principal residual risk is that Eq. (4) is not directly validated by an experimental measurement of the harmonic field-intensity ratios c̄2 and c̄3.

minor comments (3)

[Header] The DOI '10.1103/52by-28mr' in the header appears to be a placeholder or malformed; please verify the final DOI before publication.
[Theory, Eq. (3)] The sentence 'For a constant applied dc field, ⟨E^2⟩ depends on the Mathieu parameters but not on Ω' is central to the method but could be expanded by one clause explaining why the product of the rf gradient and the dc-induced displacement is independent of Ω; this would help readers unfamiliar with Paul-trap scaling.
[References] Reference [8] is cited with an arXiv DOI (10.48550/ARXIV.1801.10134); if a journal-published version now exists, it would be preferable to cite the published version.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Δα0 is solved from the measured zero-crossing frequency using known constants and independently measured Mathieu parameters; the fitted angle is cross-validated by geometry.

full rationale

The central quantity Δα0 is not an input to any fit but is solved from the fitted zero-crossing frequency Ω0^0 via the closed condition Ω0^0 = (e/mc)√(−hν0/Δα0), using only known constants. In the first stage, the measured EMM-induced shift versus drive frequency is fitted with Eq. (5), whose zero crossing Ω0 is a measured line-shape feature; Eq. (5) is a second-order expansion of the theory shift Eq. (3) and does not contain Δα0 as a fit parameter. In the second stage, the Ω0(qz) curve is fitted to Eq. (4) with free parameters Ω0^0 and θΩ; the ratios c̄2 and c̄3 entering Eq. (4) are computed from separately measured Mathieu parameters through the Mathieu solution, not fitted to the polarizability. The fitted angle θΩ = 31.2(3.1)° agrees with an independent geometric estimate 32(1)°, and the robustness checks using second-order and qz²-expanded Mathieu solutions show that the extracted Ω0^0 is stable. Self-citations to the group's prior trap characterization [27] and to the Supplemental Material [29] provide an independent trap-geometry measurement and a mathematical derivation, respectively; they do not import the target Δα0 value. The skeptic's concern that c̄2 and c̄3 are not directly measured is a model-validity and correctness risk, not a circular reduction, because no fitted or assumed quantity is re-identified as the prediction. The 5σ disagreement with the independent experimental value [15] further shows that the result carries independent information. Therefore no circular step is present.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

No new physical entities are introduced. The central measurement rests on two fitted parameters (Ω0^0 and θΩ), standard Mathieu theory, and several domain assumptions about the trap and EMM geometry that are validated by cross-checks. The most significant assumption is the validity of Eq. (4) in describing the qz dependence of the zero crossing.

free parameters (3)

Ω0^0 (zero-crossing frequency at qz approaching zero) = 14.3916(30) MHz (third-order fit)
Fitted parameter in the qz fit of Eq. (4); the central Δα0 is computed from this frequency using Ω0^0 = (e/(mc)) sqrt(-hν0/Δα0).
θΩ (angle between rf field direction and trap z-axis) = 31.1(3.1)°
Nuisance fit parameter in the same qz fit; agrees with the geometric estimate of 32(1)° from laser beam directions, supporting the model.
Per-scan zero crossing Ω0 and slope k = varied per run
Fit parameters in Eq. (5) for each EMM shift vs drive frequency scan; the individual DSSP values in Fig. 4 are derived from the slopes, providing validation.

assumptions (6)

domain assumption The total scalar micromotion shift is the sum of the second-order Doppler shift and the quadratic Stark shift (Eq. 1).
Standard physical model for the ion's motion in a Paul trap; assumed without independent derivation in the main text.
standard math The ion velocity spectrum is related to the rf electric field spectrum by the nonhomogeneous Mathieu equation solution, giving the harmonic intensity ratios in Eq. (2).
Derived in the Supplemental Material using the Mathieu equation and recurrence relations; relies on a harmonic trap potential.
domain assumption The rf field at the drive frequency is orthogonal to the cooling and probe beams A and B.
Ensured by minimizing photon-correlation contrast at Ω for beams A and B; if this failed, a first-order Doppler shift would bias the measured zero crossing.
domain assumption The trap rf gradient is radially symmetric, so the mean harmonic ratios can be written as c̄_n = c_{n,z} cos^2 θΩ + c_{n,r} sin^2 θΩ.
Based on the authors' previous characterization [27]; the small a_i parameters are first neglected and later corrected.
domain assumption The total mean-square rf field ⟨E^2⟩ at the ion remains constant when qz is held constant while the drive frequency Ω is varied.
Critical for reducing Eq. (5) to a two-parameter fit; the DDS amplitude register and step attenuator keep qz constant to about 10^-3.
domain assumption Trap anharmonicity coefficients are bounded by worst-case estimates from FEM simulation and secular-frequency measurements, inflated by about a factor of two.
Used to correct the harmonic ratios c̄2 and c̄3; the paper shows the qz fitting suppresses the resulting DSSP error by an order of magnitude.

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Cite this review

Pith. "Pith review of Measurement of the Differential Static Scalar Polarizability of the $^\mathbf{88}$Sr$^{+}$ Clock Transition." pith.science (2026). https://pith.science/paper/GIAHTGEN

@misc{pith2026250702603,
  author       = {Pith},
  title        = {Pith review of: Measurement of the Differential Static Scalar Polarizability of the $^\mathbf88$Sr$^+$ Clock Transition},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GIAHTGEN}},
  note         = {Machine review of arXiv:2507.02603}
}

abstract

We report on a precision measurement of the differential static scalar polarizability $\Delta\alpha_0$ of the ${}^{2}\!S_{1/2} \rightarrow {}^{2}\!D_{5/2}$ optical clock transition in the $^{88}$Sr$^{+}$ ion. The polarizability was determined from the 'magic' ion-trap drive frequency where the micromotion-induced second-order Doppler and quadratic Stark shifts cancel, using a single clock in an interleaved scheme by switching between minimized and large micromotion. By measuring at different Mathieu $q_z$ parameters, $\Delta\alpha_0$ can be obtained without prior knowledge of the angle between the rf electric field and the trap axis, which would otherwise dominate the systematic uncertainty. For validation, measurements were carried out at different micromotion levels and by displacing the ion in opposite directions. The results show excellent consistency and our value, $\Delta\alpha_0 = -4.8314(20)\times 10^{-40}\;\mathrm{J\, m^2/V^2} = -29.303(12)\;\mathrm{au}$, reduces the uncertainty by a factor of 3.5 compared to a previous measurement, while showing a discrepancy of $5 \sigma$. Our measurement reduces the polarizability-related uncertainty of the $^{88}$Sr$^{+}$ clock to $2.2\times 10^{-19}$ for a blackbody radiation temperature of 295 K -- a significant step towards total uncertainties ${<}1\times10^{-18}$. Using existing polarizability transfer schemes, the result can reduce the uncertainty also for other ion species.

Figures

Figures reproduced from arXiv: 2507.02603 by the authors.

**Figure 2.** FIG. 2. (a) Measured micromotion-induced scalar frequency [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 4.** FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

Figures from the paper (3 more)

**Figure 3.** Figure 3: FIG. 3. (a) Measured zero crossing frequencies with statist [PITH_FULL_IMAGE:figures/full_fig_p004_3.png]

**Figure 5.** Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p004_5.png]

**Figure 6.** Figure 6: FIG. 6. (a) Interleaved clock sequence for measuring the DSS [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]

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K. J. Arnold, R. Kaewuam, S. R. Chanu, T. R. Tan, Z. Zhang,and M. D. Barrett, Precision measurements of the 138Ba+ 6/u1D4602/u1D4461/2 − 5/u1D4512 /u1D4375/2 clock transition, Phys. Rev. Lett. 124, 193001 (2020)

2020
[31]

C. A. Holliman, M. Fan, A. Contractor, S. M. Brewer, and A. M. Jayich, Radium ion optical clock, Phys. Rev. Lett. 128, 033202 (2022)

2022
[32]

K. J. Arnold, S. Bustabad, Z. Qi, Q. Qichen, Z. Zhang, Z. Z hao, and M. D. Barrett, Validating a lutetium frequency referenc e., J. Phys. Conf. Ser. 2889, 012040 (2024)

2024
[33]

Lindvall, K

T. Lindvall, K. J. Hanhij ¨arvi, T. Fordell, and A. E. Wallin, High-accuracy determination of Paul-trap stability parameters for 6 electric-quadrupole-shift prediction, J. Appl. Phys. 132, 124401 (2022)

2022
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Keller, H

J. Keller, H. L. Partner, T. Burgermeister, and T. E. Mehlst¨aubler, Precise determination of micromotion for trapped-ion opti cal clocks, J. Appl. Phys. 118, 104501 (2015)

2015
[35]

[ 15, 27, 30, 31], for derivation of equations and treatment of trap anharmonicity

See Supplemental Material [URL will be inserted by publ isher], which includes Ref. [ 15, 27, 30, 31], for derivation of equations and treatment of trap anharmonicity
[37]

Bentine, C

E. Bentine, C. Foot, and D. Trypogeorgos, (py)LIon: A pa ckage for simulating trapped ion trajectories, Comput. Phys. Commun. 253, 107187 (2020)

2020
[38]

Note that in [ 15, 16], the angle /u1D6FDis the angle between the total rf electric ﬁeld E and the trap /u1D467axis
[39]

Dub ´e, A

P. Dub ´e, A. A. Madej, J. E. Bernard, L. Marmet, J.-S. Boulanger, and S. Cundy, Electric Quadrupole Shift Cancellation in Sin gle- Ion Optical Frequency Standards, Phys. Rev. Lett. 95, 033001 (2005)

2005
[40]

This requires the azimuthal angle of the rf direction, /u1D711Ω, whose uncertainty of ≈5◦ has a completely negligible eﬀect
[41]

Jiang, B

D. Jiang, B. Arora, M. S. Safronova, and C. W. Clark, Blackbody-radiation shift in a 88Sr+ ion optical frequency stan- dard, J. Phys. B: At. Mol. Opt. Phys. 42, 154020 (2009)

2009
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A. A. Madej, J. E. Bernard, P. Dub ´e, L. Marmet, and R. S. Windeler, Absolute frequency of the 88Sr+ 5/u1D4602/u1D4461/2 – 4/u1D4512/u1D4375/2 reference transition at 445 THz and evaluation of systemati c shifts, Phys. Rev. A 70, 012507 (2004)

2004
[43]

A. D. Shiner, A. A. Madej, P. Dub ´e, and J. E. Bernard, Absolute optical frequency measurement of saturated absorption lin es in Rb near 422 nm, Appl. Phys. B 89, 595 (2007)

2007
[44]

Lindvall, T

T. Lindvall, T. Fordell, I. Tittonen, and M. Merimaa, Un polar- ized, incoherent repumping light for prevention of dark sta tes in a trapped and laser-cooled single ion, Phys. Rev. A 87, 013439 (2013)

2013
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Fordell, T

T. Fordell, T. Lindvall, P. Dub ´e, A. A. Madej, A. E. Wallin, and M. Merimaa, Broadband, unpolarized repumping and clearout light sources for Sr+ single-ion clocks, Opt. Lett. 40, 1822 (2015)

2015
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H ¨afner, S

S. H ¨afner, S. Falke, C. Grebing, S. Vogt, T. Legero, M. Merimaa, C. Lisdat, and U. Sterr, 8 × 10−17 fractional laser frequency instability with a long room-temperature cavity, Opt. Lett. 40, 2112 (2015)

2015
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Lindvall, A

T. Lindvall, A. E. Wallin, K. J. Hanhij¨arvi, and T. Fordell, Noise- induced servo errors in optical clocks utilizing Rabi interrogation, Metrologia 60, 045008 (2023)

2023
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Lindvall, K

T. Lindvall, K. J. Hanhij ¨arvi, T. Fordell, and A. E. Wallin, Zenodo, 2025, http://doi.org/10.5281/zenodo.15793152. END MATTER Laser system and interrogation sequence —The ion is Doppler cooled using a commercial external-cavity diode laser at 422 nm, frequency stabilized to ...

2025 doi
[49]

4) measured with opposite ion displacement also conﬁrms that the 300-ms dwell time is suﬃcient

and Δ/u1D6FC0 values (Fig. 4) measured with opposite ion displacement also conﬁrms that the 300-ms dwell time is suﬃcient. Secular frequencies and Mathieu parameters—Secular fre- quencies were measured using the tickler voltage and photon correlation method [ 27]. For the high...

2025
[50]

Dub ´e, A

P. Dub ´e, A. A. Madej, M. Tibbo, and J. E. Bernard, High- Accuracy Measurement of the Diﬀerential Scalar Polarizabi lity of a 88Sr+ Clock Using the Time-Dilation Eﬀect, Phys. Rev. Lett. 112, 173002 (2014)

2014
[51]

Lindvall, K

T. Lindvall, K. J. Hanhij ¨arvi, T. Fordell, and A. E. Wallin, High-accuracy determination of Paul-trap stability parameters for electric-quadrupole-shift prediction, J. Appl. Phys. 132, 124401 (2022)

2022
[52]

C. A. Schrama, E. Peik, W. W. Smith, and H. Walther, Novel miniature ion traps, Opt. Commun. 101, 32 (1993)

1993
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Bentine, C

E. Bentine, C. Foot, and D. Trypogeorgos, (py)LIon: A pac kage for simulating trapped ion trajectories, Comput. Phys. Commun. 253, 107187 (2020)

2020

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [15]

K. J. Arnold, R. Kaewuam, A. Roy, T. R. Tan, and M. D. Barret t, Blackbody radiation shift assessment for a lutetium ion clo ck, Nat. Commun. 9, 1650 (2018)

work page 2018

[2] [1]

The measurements utilized the full tuning range, 14.22–14.61 MHz, of the heli cal resonator, see End Matter for details

Thus even the maximum ion displacement, 8 µm, causes no signiﬁcant change in the cooling/probe beam intensity at the ion. The measurements utilized the full tuning range, 14.22–14.61 MHz, of the heli cal resonator, see End Matter for details. Theory—The scalar shift caused by EMM is the sum of the second-order Doppler shift, Δ/u1D708D2 = −( /u1D7080/2)⟨/u...

work page

[3] [2]

For a constant applied dc ﬁeld, ⟨/u1D4382⟩ depends on the Mathieu parameters but not on Ω

as Δ/u1D708s ≈ − Δ/u1D6FC0 2ℎ       1 − ( Ω0 0 Ω )2 1 + ¯/u1D70C2/4 + ¯/u1D70C3/9 1 + ¯/u1D70C2 + ¯/u1D70C3       ⟨/u1D4382⟩, (3) where ⟨/u1D4382⟩ ≈ ( 1 + ¯/u1D70C2 + ¯/u1D70C3)⟨/u1D4382 (Ω)⟩ is the total mean-square rf ﬁeld. For a constant applied dc ﬁeld, ⟨/u1D4382⟩ depends on the Mathieu parameters but not on Ω. In [ 27], the rf gradient of...

work page

[4] [3]

Deﬁning the zero crossing of Eq

[32]. Deﬁning the zero crossing of Eq. ( 3) as Ω0 = Ω 0( ¯/u1D70C2, ¯/u1D70C3) = Ω 0 0 √ 1 + ¯/u1D70C2/4 + ¯/u1D70C3/9 1 + ¯/u1D70C2 + ¯/u1D70C3 , (4) the expression in square brackets in (3) becomes 1 − (Ω0/Ω)2. This function is not well suited for ﬁtting to data over a narrow frequency range, so we use a second-order series expansion i n /u1D6FF/Ω0, whe...

work page

[5] [4]

(kHz) FIG

evaluated 4 (a) 14.25 14.3 14.35 14.4 0/2 (MHz) (b) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 qz -5 0 5 10Res. (kHz) FIG. 3. (a) Measured zero crossing frequencies with statist ical un- certainties (error bars) as function of /u1D45E/u1D467. Each point is the result of a ﬁt like in Fig. 2. The uncertainties of the /u1D45E/u1D467values are ≲3 × 10−4 and the correspond...

work page 2009

[6] [5]

has only two ﬁt parameters, the zero crossing Ω0 and the slope at the zero crossing, /u1D458. By contrast, in [15, 16] the ion secular frequencies were kept con- stant, which causes ⟨/u1D4382⟩ to have a complicated dependence on Ω and necessitates using a physically less meaningful ﬁt function with three free parameters. 3 Diﬀerential frequency shifts —Al...

work page

[7] [6]

This gives Ω0 0/2/u1D70B= 14.3915(30) MHz and /u1D703Ω = 31.2(3.1)◦, in agreement with the nominal angle given above. To demonstrate the robustness of the /u1D45E/u1D467 ﬁtting method, we repeat the ﬁtting using (i) the second-order Mathieu so- lution, which yields Ω0 0/2/u1D70B= 14.3916(30) MHz and /u1D703Ω = 30.9(3.1)◦, and (ii) a series expansion to or...

work page

[8] [7]

A. D. Ludlow, M. M. Boyd, J. Y e, E. Peik, and P. O. Schmidt, Optical atomic clocks, Rev. Mod. Phys. 87, 637 (2015)

work page 2015