Observation of Nonlinearity of Generalized King Plot in the Search for New Boson

168

518 297 652 280 285.8(3.7)

This work

170

171

518 296 293 795 809.6(3.0)

518 295 836 590 863.6(0.3)

This work

[26]

172

173

518 295 018 023 803.6(3.1)

518 294 576 845 268(10)

This work

[27]

174

518 294 025 309 216.9(2.3)

518 294 025 309 217.8(0.9)

This work

[25]

176

518 293 078 387 442.1(3.7)

This work

D. Absolute frequency of clock transition

In addition, by referencing the reported absolute frequency of the 1S0 − 3P0 transition for 171Yb [26], our IS

measurements provide the absolute frequencies for five

bosonic isotopes, which is summarized in Table V.

V. CONCLUSIONS AND PROSPECTS

In conclusion, we measure ISs for neutral Yb isotopes on

an ultranarrow optical clock transition 1S0 − 3P0 with an

accuracy of a few hertz. The determined ISs are combined

with the recently reported IS measurements for two optical

transitions of Ybþ , enabling us to construct the two King

plots. Both of them show very large nonlinearity, demonstrating the high sensitivity to the higher-order effect in the

IS. We also carry out the generalized King plot using the

three optical transitions. Our analysis shows a deviation

from linearity at the 3σ uncertainty level and jye yn j=ðℏcÞ <

1.2 × 10−10 for m < 1 keV with the 95% C.L.

APPENDIX A: NUCLEAR MASS AND NUCLEAR

CHARGE RADIUS

As the inverse mass differences, we use the masses

of the nuclei. They are calculated by the formula

mA ¼ mnucleus ¼ matom ðAÞ − Zme þ Eb ðZÞ with the atomic

masses matom ðAÞ given in Table VI, where the electron

mass me ¼ 5.485 799 090 65ð16Þ × 10−4 amu, where

amu represents the atomic mass unit [53,54]. We

evaluate the binding energy with Eb ðZÞ ¼ 14.4381Z2.39 þ

1.55468 × 10−6 Z5.35 eV with the uncertainty of 1.1 keV

given by Ref. [55]. In the case of Yb, this is Eb ð70Þ ¼

4.106ð12Þ × 10−4 amu with the conversion factor

of 1 eV ¼ 1.073 544 102 33ð32Þ × 10−9 amu.

The ionic masses are used in Ref. [12]. As shown in

Appendix D, our 2D analysis of the Ybþ data with the

nuclear masses is consistent with Ref. [12]. In our other 2D

021033-11

A Self-archived copy in

Kyoto University Research Information Repository

https://repository.kulib.kyoto-u.ac.jp

KOKI ONO et al.

PHYS. REV. X 12, 021033 (2022)

TABLE VI. Atomic masses and differences in mean-square

nuclear charge radii. These values are respectively given by

Ref. [54] and Ref. [24].

matom ðAÞ (amu)

168

170

172

174

176

167.933 891 30(10)

169.934 767 243(11)

171.936 386 654(15)

173.938 867 546(12)

175.942 574 706(16)

ðA0 ; AÞ

APPENDIX C: STATISTICAL TEST OF

LINEARITIES

δhr2 iA A (fm2 )

(170,168)

(172,170)

(174,172)

(176,174)

transition is applied to pump the 173Yb atoms into the

j↑ð↓Þi ¼ jmF ¼ þð−Þ5=2i states. It is noted that the PA

light is not applied to the 173Yb atoms since the clock

transition frequency associated with the multiply occupied

sites is well separated from that of the singly occupied

sites [57,58].

0.1348(6)

0.1266(6)

0.0989(6)

0.0944(5)

and 3D analyses, the changes in χ 2 are a few percent when

the atomic masses are employed instead of the nuclear

masses. Thus, the precise definition of masses to describe

the first-order MS is not relevant in the present work. In

particular, the bound on the new particle and its future

prospect given in Fig. 8 vary only about 1% even if we use

the atomic masses.

To evaluate the isotope dependence of the QFS, we use

the mean-square charge radii given by Ref. [24]. These

values are summarized in Table VI.

The nonlinearity of the King plot is analyzed with

the χ 2 function. We measured νAA

for the isotope pairs

Aγ ¼ fð168; 170Þ; ð170; 174Þ; ð172; 174Þ; ð174; 176Þg. In

Ref. [12], for the transitions α and β, the isotope pairs Aα ¼

Aβ ¼ fð168; 170Þ; ð170; 172Þ; ð172; 174Þ; ð174; 176Þg are

measured. In addition to these pairs, the ν170;174

and ν170;174

are measured in Ref. [12], and we perform the χ 2 analysis

including these redundant data. Although it is possible to

quantify nonlinearities by introducing a geometrical measure like areas of triangles as discussed in Ref. [22], the χ 2

analysis is more straightforward to handle the redundant

data and multiple sources of nonlinearities.

1. 2D case

We consider the following model for the 2D King plot

with the ðλ1 ; λ2 Þ transitions:

AA

þ cλ1 νAA

νAA

λ2 ¼ cμ δμ

λ1 ;

APPENDIX B: ISOTOPE-DEPENDENT

EXPERIMENTAL PARAMETERS

Table VII summarizes the experimental parameters

which depend on the isotopic species. The PA lines are

identified experimentally, whose resonance frequencies are

consistent with the theoretical calculation in Ref. [56].

Instead of the PA, the spin-polarized 171Yb atoms are

prepared using the optical pumping associated with the

S0 ðF ¼ 1=2Þ − 3P1 ðF0 ¼ 1=2Þ transition. As well as 171Yb,

the resonant light with 1S0 ðF ¼ 5=2Þ − 3P1 ðF0 ¼ 5=2Þ

ðC1Þ

where cμ and cλ1 are the model parameters associated with

the electronic factors. The corresponding χ 2 function is

given by

X  X νAA0 − ν˜ AA0 2

νAA

λ¼λ1 ;λ2 ðA;A0 Þ∈Aλ

 170;172

172;174

170;174 2 

þ νλ

− ν˜ λ

þ λ

σ ν170;174

χ 2 ¼ χ 2mass þ

ðC2Þ

TABLE VII. Summary of isotope-dependent experimental parameters. The parameters ΔA and τA correspond to the detuning of the PA

line from the 1S0 − 3P1 transition and the PA time for the isotope A, respectively. In addition, N A and T A correspond to the number of the

atom before the interrogation and the atom temperature after the evaporative cooling, respectively. It is noted that the 174Yb atoms are

cooled below the transition temperature of the Bose-Einstein condensation (BEC) with no discernable thermal components.

ðA0 ; AÞ

(170,168)

(174,170)

(174,172)

(171,174)

(174,176)

ð173↑; 173↓Þ

τ0 (s)

τ (s)

64.6

258.4

40.5

162

28.6

114.4

39.7

238.2

36

144

34

136

ΔA =2π (MHz)

τA (ms)

N A (×103 )

T A (μK)

168

170

−2072

10

10

0.5

−6213

20

15

0.3

171

172

10

0.5

−1143

10

15

0.7

021033-12

173

174

176

20

0.1

−3687

25

BEC

−598

25

0.2

A Self-archived copy in

Kyoto University Research Information Repository

https://repository.kulib.kyoto-u.ac.jp

OBSERVATION OF NONLINEARITY OF GENERALIZED KING …

where the tilded quantities and σ ð·Þ correspond to the

measured quantities and their experimental uncertainties,

respectively. The first term χ 2mass contains the contributions

from the parameters related to nuclear masses described in

Appendix A. Note that the last term is only included for the

transitions α and β. For example, the χ 2½α;β , the χ 2 function for

ðλ1 ; λ2 Þ ¼ ðα; βÞ, has the following parameters: four inde0

pendent νAA

α ’s and two model parameters (cμ and cα ). Thus,

we have 4 þ 2 ¼ 6 fitting parameters. Additionally, we have

the following measurements with errors: five ν˜ AA

α ’s and five

ν˜ AA

β ’s. Thus, we have 5 þ 5 ¼ 10 experimental constraints.

Hence, the d.o.f. is 10 − 6 ¼ 4. It is noted that the d.o.f. of the

χ 2½α;β in Ref. [12] is 2 since the redundant measurements

PHYS. REV. X 12, 021033 (2022)

χ 2½α;β ¼ 15.37

2. 3D case

As well as the 2D case, the linear model for the 3D King

plot with the ðλ1 ; λ2 ; λ3 Þ transitions is considered:

AA

AA

þ cλ1 νAA

νAA

λ3 ¼ cμ δμ

λ1 þ cλ2 νλ2 :

ðC3Þ

The corresponding χ 2 function has the same form as

Eq. (C2) where the summation for λ now includes λ3 .

The d.o.f. of χ 2½γ;α;β is 3 ¼ 14 (observations for α, β, and γ)

−3 (fitting parameters) −8 (independent ISs on α and γ).

APPENDIX D: 2D KING RELATION

WITH ðα; βÞ TRANSITIONS

χ 2½α;β ðQFSÞ ¼ 4.3 ðp ¼ 0.23Þ;

TABLE VIII.

Sources

QFS

PS

QFS and PS

χ 2½α;β ðPSÞ ¼ 5.4

ðD2Þ

ðp ¼ 0.15Þ:

ðD3Þ

Following Ref. [12], we minimize χ 2 including both of the

higher-order ISs,

χ 2½α;β ðQFS; PSÞ ¼ 4.0

ðp ¼ 0.13Þ;

ðD4Þ

where the p value is calculated with the d.o.f. of 2. From

the PS fit, we obtain the bound on the new physics

coupling 3.8 × 10−11 ≤ ð−1Þsþ1 ye yn =ðℏcÞ ≤ 1.7 × 10−10

for the new particle mass of 10 eV at 95% C.L. The

summary of the above fit results and the minimum χ 2 are

shown in Table VIII.

Here we consider the origin of the minimum values of χ 2

obtained in the above analysis. The ISs should satisfy the

transitive consistency condition νλA A ¼ νAλ A0 − νAA

for any

transitions by definition. With this condition, the shifts of

some isotope pairs can be given by combinations of

other pairs. In the case of Ref. [12], three isotope pairs

(170,172), (172,174), and (170,174) are in this situation.

The minimum contributions from the condition to χ 2 are

given by

χ 2Cλ ðA0 ; A1 ; A2 Þ

The combination ðα; βÞ is the same as that investigated

in Ref. [12]. Here we perform the analysis for this

combination to check the consistency of our analysis

with that in Ref. [12]. The minimum χ 2 and the corresponding p value are

ðD1Þ

The best-fit parameters are Fβα ¼ 1.011 410 06ð86Þ and

K βα ¼ 120.160ð23Þ GHz amu. They are consistent with

the results shown by Ref. [12], F0βα ¼ 1.011 410 24ð86Þ

and K 0βα ¼ 120.208ð23Þ GHz amu. Note that the central

values are slightly different from each other because

we have used different values for the inverse mass

differences. We then introduce the QFS or the PS in

Eq. (8) as higher-order ISs. Including an additional source

of the ISs, one of four d.o.f. is consumed by the additional

fit parameter Hβα. Their χ 2 minima are

are used to improve the precision of ν˜ 170;172

and

ν˜ 170;174

αðβÞ

αðβÞ

˜ 170;172

˜ 172;174

˜ 170;174

are

ν˜ 172;174

αðβÞ , meaning that ν

αðβÞ , ν

αðβÞ , and ν

αðβÞ

not independent of each other. From the same argument,

the d.o.f. of the χ 2½α;γ and χ 2½β;γ is 3 in our analysis.

When the QFS and PS are considered as the nonlinear

source of the King relation, the fitting function is modified

by adding cq ½δhr2 i2 AA and cp ðA − A0 Þ to the right-hand

side of Eq. (C1), respectively. Here cq and cp are a fitting

parameters for the QFS and PS, respectively, and the χ 2

function has the d.o.f. of 3 for the case of ðα; βÞ and 2 for

both of ðα; γÞ and ðβ; γÞ.

ðp ¼ 4.0 × 10−3 Þ:

ðνAλ 1 A0 − νAλ 2 A0 − νAλ 1 A2 Þ2

ðσ Aλ 1 A0 Þ2 þ ðσ Aλ 2 A0 Þ2 þ ðσ Aλ 1 A2 Þ2

ðD5Þ

Using the results of Ref. [12], we find the lower limit of χ 2

when we include the transitions α or β:

χ 2Cα ð172; 170; 174Þ ¼ 0.1262;

ðD6Þ

Best-fit parameters with transition pairs ðλ1 ; λ2 Þ ¼ ðα; βÞ. The error of each fit parameter is evaluated as 1σ.

χ 2 (p value)

Fβα

K βα (GHz amu)

Hβα;QFS (kHz=fm4 )

Hβα;PS (kHz)

4.3 (0.23)

5.4 (0.15)

4.0 (0.13)

1.011 401 6(27)

1.011 401 8(27)

1.011 401 2(28)

120.381(70)

121.55(44)

120.77(81)

71(21)

52(45)

40(13)

13(26)

021033-13

A Self-archived copy in

Kyoto University Research Information Repository

https://repository.kulib.kyoto-u.ac.jp

KOKI ONO et al.

PHYS. REV. X 12, 021033 (2022)

χ 2Cβ ð172; 170; 174Þ ¼ 3.911;

ðD7Þ

χ 2Cα ð172; 170; 174Þ þ χ 2Cβ ð172; 170; 174Þ ¼ 4.037:

ðD8Þ

These are the theoretical minima of χ 2 in our analyses with

α and/or β included.

APPENDIX E: ANALYSIS OF THE

NONLINEARITY OF GENERALIZED

KING PLOT ASSUMING QFS AND PS

Here we suppose that the QFS and PS are the two distinct

higher-order ISs involved in the data, as assumed in

Refs. [12,22]. Furthermore, we consider the particular case

where the QFS is eliminated in the 3D King plot construction and the PS remains as the origin of the nonlinearity in the 3D King relation. This corresponds to the

case of Eq. (11) with the PS-origin nonlinearity term given

by hPS. From this analysis, we find that the χ 2 minimum

reaches the theoretical minimum, and we can determine the

best-fit values for hPS as well as for the coefficients f α , f γ ,

and kμ given in Table IX. The hPS is expressed as

hPS ¼ αNP ðXβ − f α Xα − f γ Xγ Þ. See Appendix F for the

details. We employed the electronic factor of the β

transition Xβ which is calculated by the configuration

interaction method in Ref. [12]. The other factors Xα;γ

are reconstructed so as to reproduce the results of Ref. [22].

We evaluated f α;γ also with other electronic factors given

by Ref. [22]. Thus, from the best-fit value of hPS , we can

determine the favored region of the coupling and mass of

the new boson, as shown as the black shaded region in

Fig. 9. For instance, the favored region is 1.1 × 10−11 ≤

ð−1Þs ye yn =ðℏcÞ ≤ 4.5 × 10−11 in the 95% C.L. when the

mass of the new boson is 10 eV. Note that the product of the

couplings ye yn is positive (negative) for s ¼ 0 (s ¼ 1) in

the smaller mass side of the peak structure, namely,

m ≲ 10 keV. This peak structure is attributed to the

cancellation of the electronic factors, and the sign of

ye yn changes across the peak. The suggested favored

region, however, conflicts with the exclusion limit set by

the other terrestrial experiment, obtained from the product

of the individual bounds on the couplings with electron and

neutron, as shown as the orange line in Fig. 9. Thus, we

conclude that the QFS plus PS assumption is not valid to

describe the observed nonlinearities in the Yb=Ybþ system.

FIG. 9. Product of couplings jye yn j of a new boson as a

function of the mass m. The black shaded region represents

the 95% confidence interval of the new physics coupling obtained

from the fit using Eq. (11) with h ¼ hPS . For comparison, the

constraint from electron anomalous magnetic moment (g − 2)

measurements combined with neutron scattering measurements is

shown as the orange shaded region, as in Fig. 8. In addition, the

favored regions and constraints from Ybþ analysis [12] are

shown as red and blue lines and shaded regions, in exactly the

same manners as in Ref. [12].

APPENDIX F: EXPLICIT FORMS

OF hPS AND hQFS

Here we consider two distinct higher-order sources,

defined as I λ δηA A and Jλ δζA A , and the IS is expressed as

0.993(16)

kμ (GHz amu)

hPS (kHz)

0.030(26)

111.1(9.0)

50(15)

ðF1Þ

f λ1 ¼

F λ2 I λ3 − F λ3 I λ2

F λ2 I λ1 − F λ1 I λ2

ðF2Þ

f λ2 ¼

F λ3 I λ1 − F λ1 I λ3

F λ2 I λ1 − F λ1 I λ2

ðF3Þ

kμ ¼ K λ3 − f λ1 K λ1 − f λ2 K λ2 ;

ðF4Þ

h ¼ Jλ3 − f λ1 Jλ1 − f λ2 J λ2 :

ðF5Þ

If Jλ δζA A corresponds to the QFS or PS, the nonlinear

terms hQFS and hPS are expressed as

ð2Þ

fγ

When we eliminate δhr2 iA A and δηA A by combining the

three transitions ðλ1 ; λ2 ; λ3 Þ, the coefficients of the 3D King

relation Eq. (11) are expressed as

TABLE IX. Best-fit parameters for 3D King relation with the

PS term as an origin of the nonlinearity. The error of each fit

parameter is evaluated as 1σ.

fα

νλA A ¼ K λ δμA A þ Fλ δhr2 iA A þ I λ δηA A þ Jλ δζA A :

ð2Þ

ð2Þ

hQFS ¼ Gλ3 − f λ1 Gλ1 − f λ2 Gλ2 ;

ðF6Þ

hPS ¼ αNP ðXλ3 − f λ1 Xλ1 − f λ2 Xλ2 Þ:

ðF7Þ

Note that the coefficients in hPS and hQFS , shown in

Eqs. (F2) and (F3), depend on the electronic factors of

the eliminated terms I λ1 , I λ2 , and I λ3 .

021033-14

A Self-archived copy in

Kyoto University Research Information Repository

https://repository.kulib.kyoto-u.ac.jp

OBSERVATION OF NONLINEARITY OF GENERALIZED KING …

TABLE X. Best-fit parameters for the 3D King relation with the

QFS term as the origin of the nonlinearity. In the QFS fit, the error

of each fit parameter is evaluated as 1σ. The second line is the

coefficients at m ¼ 10 eV calculated by the electronic factors

given by Ref. [22].

QFS fit

[22]

fα

fγ

kμ

(GHz amu)

hQFS

(kHz=fm4 )

1.018(13)

1.063

−0.010ð21Þ

−0.069

124.1(7.7)

240.1

70(21)

−57

APPENDIX G: POSSIBLE CONSISTENCY CHECK

IN THE NONLINEARITY FIT

In Appendix E, following Refs. [12,22], we describe the

3D King plot analysis under the assumption that the QFS

and PS are two distinct higher-order ISs involved in the

data. This picture is not plausible to explain the observed

deviations from the linearities because the given favored

region is excluded by the other experiments. In general,

even if the dataset is fit by some higher-order ISs well, they

do not have to be the origins of the observed nonlinearity.

Here, we discuss a method to test the origins of the higherorder ISs.

Different from the argument in Appendix E where we

attribute the PS to the source of the nonlinearity and the

QFS is eliminated by the construction of 3D King plot, we

here attribute the QFS, instead of the PS, to the source to

explain the leftover nonlinearity and the PS is eliminated.

These two constructions should be treated on an equal

footing, in principle. This fit gives us the χ 2 of the

theoretical minimum; see Table X for the fit result. In this

case, the given fit coefficients can be calculated with the

electronic factors Fλ , K λ , and Xλ ðλ ∈ fα; β; γgÞ using the

formulas shown in Appendix F. The coefficients calculated

by the electronic factors given in Ref. [12] do not match the

fit result for m < 1 keV [59]. Table X shows the QFS fit

result and the theoretical coefficients at m ¼ 10 eV. This

means, as long as we use the numerical results given by

Ref. [22], the original assumption to include only the QFS

and the PS (m < 1 keV) as the higher-order ISs is inconsistent with the data. In the generalized King relation, this

method helps us to test the consistency of some higherorder ISs with experimental results independent of other

experimental bounds.

[1] P. A. Zyla et al. (Particle Data Group), The Review of

Particle Physics, Prog. Theor. Exp. Phys. 2020, 083C01

(2020).

[2] G. Bertone, D. Hooper, and J. Silk, Particle Dark Matter:

Evidence, Candidates and Constraints, Phys. Rep. 405, 279

(2005).

[3] M. Dine and A. Kusenko, Origin of the Matter-Antimatter

Asymmetry, Rev. Mod. Phys. 76, 1 (2003).

PHYS. REV. X 12, 021033 (2022)

[4] J. E. Kim and G. Carosi, Axions and the Strong CP

Problem, Rev. Mod. Phys. 82, 557 (2010).

[5] M. S. Safronova, D. Budker, D. DeMille, Derek F. Jackson

Kimball, A. Derevianko, and C. W. Clark, Search for New

Physics with Atoms and Molecules, Rev. Mod. Phys. 90,

025008 (2018).

[6] C. Delaunay, R. Ozeri, G. Perez, and Y. Soreq, Probing

Atomic Higgs-like Forces at the Precision Frontier, Phys.

Rev. D 96, 093001 (2017).

[7] J. C. Berengut, D. Budker, C. Delaunay, V. V. Flambaum, C.

Frugiuele, E. Fuchs, C. Grojean, R. Harnik, R. Ozeri, G.

Perez, and Y. Soreq, Probing New Long-Range Interactions

by Isotope Shift Spectroscopy, Phys. Rev. Lett. 120, 091801

(2018).

[8] W. H. King, Comments on the Article “Peculiarities of the

Isotope Shift in the Samarium Spectrum,” J. Opt. Soc. Am.

53, 638 (1963).

[9] W. H. King, Isotope Shifts in Atomic Spectra (Plenum,

New York, 1984).

[10] F. W. Knollmann, A. N. Patel, and S. C. Doret, Part-perBillion mMasurement of the 42S1=2 → 32D5=2 ElectricQuadrupole-Transition Isotope Shifts between 42;44;48Caþ

and 40Caþ, Phys. Rev. A 100, 022514 (2019).

[11] C. Solaro, S. Meyer, K. Fisher, J. C. Berengut, E. Fuchs, and

M. Drewsen, Improved Isotope-Shift-Based Bounds on

Bosons beyond the Standard Model through Measurements

of the 2D3=2 –2D5=2 Interval in Caþ , Phys. Rev. Lett. 125,

123003 (2020).

[12] I. Counts, J. Hur, D. P. L. Aude Craik, H. Jeon, C. Leung,

J. C. Berengut, A. Geddes, A. Kawasaki, W. Jhe, and V.

Vuletić, Evidence for Nonlinear Isotope Shift in Ybþ Search

for New Boson, Phys. Rev. Lett. 125, 123002 (2020).

[13] T. Manovitz, R. Shaniv, Y. Shapira, R. Ozeri, and N.

Akerman, Precision Measurement of Atomic Isotope Shifts

Using a Two-Isotope Entangled State, Phys. Rev. Lett. 123,

203001 (2019).

[14] T. Takano, R. Mizushima, and H. Katori, Precise Determination of the Isotope Shift of 88Sr − 87Sr Optical Lattice

Clock by Sharing Perturbations, Appl. Phys. Express 10,

072801 (2017).

[15] H. Miyake, N. C. Pisenti, P. K. Elgee, A. Sitaram, and G. K.

Campbell, Isotope-Shift Spectroscopy of the 1S0 → 3P1 and

S0 → 3P0 Transitions in Strontium, Phys. Rev. Research 1,

033113 (2019).

[16] S. A. Blundell, P. E. G. Baird, C. W. P. Palmer, D. N. Stacey,

and G. K. Woodgate, A Reformulation of the Theory of Field

Isotope Shift in Atoms, J. Phys. B 20, 3663 (1987).

[17] K. Mikami, M. Tanaka, and Y. Yamamoto, Probing New

Intra-atomic Force with Isotope Shifts, Eur. Phys. J. C 77,

896 (2017).

[18] V. V. Flambaum, A. J. Geddes, and A. V. Viatkina, Isotope

Shift, Nonlinearity of King Plots, and the Search for New

Particles, Phys. Rev. A 97, 032510 (2018).

[19] M. Tanaka and Y. Yamamoto, Relativistic Effects in the

Search for New Intra-atomic Force with Isotope Shifts,

Prog. Theor. Exp. Phys. 2020, 103B02 (2020).

[20] C. W. P. Palmer, Reformulation of the Theory of the Mass

Shift, J. Phys. B 20, 5987 (1987).

[21] S. O. Allehabi, V. A. Dzuba, V. V. Flambaum, and A. V.

Afanasjev, Nuclear Deformation as a Source of the

021033-15

A Self-archived copy in

Kyoto University Research Information Repository

https://repository.kulib.kyoto-u.ac.jp

KOKI ONO et al.

[22]

[23]

[24]

[25]

[26]

[27]

[28]

[29]

[30]

[31]

[32]

[33]

[34]

[35]

[36]

PHYS. REV. X 12, 021033 (2022)

Nonlinearity of the King Plot in the Ybþ Ion, Phys. Rev. A

103, L030801 (2021).

J. C. Berengut, C. Delaunay, A. Geddes, and Y. Soreq,

Generalized King Linearity and New Physics Searches with

Isotope Shifts, Phys. Rev. Research 2, 043444 (2020).

V. A. Dzuba, V. V. Flambaum, and S. Schiller, Testing

Physics beyond the Standard Model through Additional

Clock Transitions in Neutral Ytterbium, Phys. Rev. A 98,

022501 (2018).

J. S. Schelfhout and J. J. McFerran, Isotope Shifts for

1S –3Po Yb Lines from Multiconfiguration Dirac-Hartree0

0;1

Fock Calculations, Phys. Rev. A 104, 022806 (2021).

N. Poli, Z. W. Barber, N. D. Lemke, C. W. Oates, L. S. Ma,

J. E. Stalnaker, T. M. Fortier, S. A. Diddams, L. Hollberg,

J. C. Bergquist, A. Brusch, S. Jefferts, T. Heavner, and T.

Parker, Frequency Evaluation of the Doubly Forbidden

S0 → 3P0 Transition in Bosonic 174Yb, Phys. Rev. A 77,

050501(R) (2008).

F. Riehle, P. Gill, F. Arias, and L. Robertsson, The CIPM

List of Recommended Frequency Standard Values: Guidelines and Procedures, Metrologia 55, 188 (2018).

C. Clivati, G. Cappellini, L. F. Livi, F. Poggiali, M. S. de

Cumis, M. Mancini, G. Pagano, M. Frittelli, A. Mura, G. A.

Costanzo, F. Levi, D. Calonico, L. Fallani, J. Catani, and M.

Inguscio, Measuring Absolute Frequencies beyond the GPS

Limit via Long-Haul Optical Frequency Dissemination,

Opt. Express 24, 11865 (2016).

H. Katori, M. Takamoto, V. G. Pal’chikov, and V. D.

Ovsiannikov, Ultrastable Optical Clock with Neutral Atoms

in an Engineered Light Shift Trap, Phys. Rev. Lett. 91,

173005 (2003).

E. C. Seltzer, K X-Ray Isotope Shifts, Phys. Rev. 188, 1916

(1969).

M. Kitagawa, K. Enomoto, K. Kasa, Y. Takahashi, R.

Ciuryło, P. Naidon, and P. S. Julienne, Two-Color Photoassociation Spectroscopy of Ytterbium Atoms and the

Precise Determinations of s-Wave Scattering Lengths,

Phys. Rev. A 77, 012719 (2008).

Z. W. Barber, J. E. Stalnaker, N. D. Lemke, N. Poli, C. W.

Oates, T. M. Fortier, S. A. Diddams, L. Hollberg, C. W.

Hoyt, A. V. Taichenachev, and V. I. Yudin, Optical Lattice

Induced Light Shifts in an Yb Atomic Clock, Phys. Rev. Lett.

100, 103002 (2008).

R. J. Fasano, Y. J. Chen, W. F. McGrew, W. J. Brand, R. W.

Fox, and A. D. Ludlow, Characterization and Suppression

of Background Light Shifts in an Optical Lattice Clock,

Phys. Rev. Applied 15, 044016 (2021).

J. H. Denschlag, J. E. Simsarian, H. Häffner, C. McKenzie,

A. Browaeys, D. Cho, K. Helmerson, S. L. Rolston, and

W. D. Phillips, A Bose-Einstein Condensate in an Optical

Lattice, J. Phys. B 35, 3095 (2002).

T. Akatsuka, M. Takamoto, and H. Katori, Optical Lattice

Clocks with Non-Interacting Bosons and Fermions, Nat.

Phys. 4, 954 (2008).

S. G. Porsev, Y. G. Rakhlina, and M. G. Kozlov, ElectricDipole Amplitudes, Lifetimes, and Polarizabilities of the

Low-Lying Levels of Atomic Ytterbium, Phys. Rev. A 60,

2781 (1999).

A. V. Taichenachev, V. I. Yudin, C. W. Oates, C. W. Hoyt,

Z. W. Barber, and L. Hollberg, Magnetic Field-Induced

[37]

[38]

[39]

[40]

[41]

[42]

[43]

[44]

[45]

[46]

[47]

[48]

[49]

[50]

[51]

[52]

021033-16

Spectroscopy of Forbidden Optical Transitions with Application to Lattice-Based Optical Atomic Clocks, Phys. Rev.

Lett. 96, 083001 (2006).

Y. Takata, S. Nakajima, J. Kobayashi, K. Ono, Y. Amano,

and Y. Takahashi, Current-Feedback-Stabilized Laser System for Quantum Simulation Experiments Using Yb Clock

Transition at 578 nm, Rev. Sci. Instrum. 90, 083002 (2019).

L.-S. Ma, P. Jungner, J. Ye, and J. L. Hall, Delivering the

Same Optical Frequency at Two Places: Accurate Cancellation of Phase Noise Introduced by an Optical Fiber or

Other Time-Varying Path, Opt. Lett. 19, 1777 (1994).

N. Nemitz, T. Ohkubo, M. Takamoto, I. Ushijima, M. Das,

N. Ohmae, and H. Katori, Frequency Ratio of Yb and Sr

Clocks with 5 × 10−17 Uncertainty at 150 Seconds Averaging Time, Nat. Photonics 10, 258 (2016).

I. Ushijima, M. Takamoto, and H. Katori, Operational

Magic Intensity for Sr Optical Lattice Clocks, Phys. Rev.

Lett. 121, 263202 (2018).

N. Darkwah Oppong, L. Riegger, O. Bettermann, M. Höfer,

J. Levinsen, M. M. Parish, I. Bloch, and S. Fölling,

Observation of Coherent Multiorbital Polarons in a TwoDimensional Fermi Gas, Phys. Rev. Lett. 122, 193604

(2019).

K. Beloy, J. A. Sherman, N. D. Lemke, N. Hinkley, C. W.

Oates, and A. D. Ludlow, Determination of the 5d6s 3D1

State Lifetime and Blackbody-Radiation Clock Shift in Yb,

Phys. Rev. A 86, 051404(R) (2012).

K. Ono, J. Kobayashi, Y. Amano, K. Sato, and Y. Takahashi,

Antiferromagnetic Interorbital Spin-Exchange Interaction

of 171Yb, Phys. Rev. A 99, 032707 (2019).

D. Hanneke, S. Fogwell, and G. Gabrielse, New Measurement of the Electron Magnetic Moment and the Fine

Structure Constant, Phys. Rev. Lett. 100, 120801 (2008).

L. Morel, Z. Yao, P. Clad´e, and S. Guellati-Kh´elifa,

Determination of the Fine-Structure Constant with an

Accuracy of 81 Parts per Trillion, Nature (London) 588,

61 (2020).

H. Leeb and J. Schmiedmayer, Constraint on Hypothetical

Light Interacting Bosons from Low-Energy Neutron Experiments, Phys. Rev. Lett. 68, 1472 (1992).

Y. N. Pokotilovski, Constraints on New Interactions from

Neutron Scattering Experiments, Phys. At. Nucl. 69, 924

(2006).

V. V. Nesvizhevsky, G. Pignol, and K. V. Protasov, Neutron

Scattering and Extra-Short-Range Interactions, Phys. Rev.

D 77, 034020 (2008).

M. S. Safronova, S. G. Porsev, C. Sanner, and J. Ye, Two

Clock Transitions in Neutral Yb for the Highest Sensitivity

to Variations of the Fine-Structure Constant, Phys. Rev.

Lett. 120, 173001 (2018).

A. Yamaguchi, S. Uetake, D. Hashimoto, J. M. Doyle, and

Y. Takahashi, Inelastic Collisions in Optically Trapped

Ultracold Metastable Ytterbium, Phys. Rev. Lett. 101,

233002 (2008).

S. Kato, K. Inaba, S. Sugawa, K. Shibata, R. Yamamoto, M.

Yamashita, and Y. Takahashi, Laser Spectroscopic Probing

of Coexisting Superfluid and Insulating States of an Atomic

Bose-Hubbard System, Nat. Commun. 7, 11341 (2016).

Y. Nakamura, Y. Takasu, J. Kobayashi, H. Asaka, Y.

Fukushima, K. Inaba, M. Yamashita, and Y. Takahashi,

A Self-archived copy in

Kyoto University Research Information Repository

https://repository.kulib.kyoto-u.ac.jp

OBSERVATION OF NONLINEARITY OF GENERALIZED KING …

[53]

[54]

[55]

[56]

Experimental Determination of Bose-Hubbard Energies,

Phys. Rev. A 99, 033609 (2019).

W. Huang, M. Wang, F. Kondev, G. Audi, and S. Naimi, The

AME 2020 Atomic Mass Evaluation (I). Evaluation of Input

Data, and Adjustment Procedures, Chin. Phys. C 45,

030002 (2021).

M. Wang, W. Huang, F. Kondev, G. Audi, and S. Naimi, The

AME 2020 Atomic Mass Evaluation (II). Tables, Graphs

and References, Chin. Phys. C 45, 030003 (2021).

D. Lunney, J. M. Pearson, and C. Thibault, Recent Trends in

the Determination of Nuclear Masses, Rev. Mod. Phys. 75,

1021 (2003).

M. Borkowski, R. Ciuryło, P. S. Julienne, S. Tojo, K.

Enomoto, and Y. Takahashi, Line Shapes of Optical

PHYS. REV. X 12, 021033 (2022)

Feshbach Resonances Near the Intercombination Transition of Bosonic Ytterbium, Phys. Rev. A 80, 012715 (2009).

[57] F. Scazza, C. Hofrichter, M. Höfer, P. C. De Groot, I. Bloch,

and S. Fölling, Observation of Two-Orbital Spin-Exchange

Interactions with Ultracold SUðNÞ-Symmetric Fermions,

Nat. Phys. 10, 779 (2014).

[58] G. Cappellini, M. Mancini, G. Pagano, P. Lombardi, L. Livi,

M. Siciliani de Cumis, P. Cancio, M. Pizzocaro, D.

Calonico, F. Levi, C. Sias, J. Catani, M. Inguscio, and L.

Fallani, Direct Observation of Coherent Interorbital SpinExchange Dynamics, Phys. Rev. Lett. 113, 120402 (2014).

[59] This is in good contrast with the agreement between the

calculations and experiments for Fβα within 0.4%, reported

in Ref. [12].

021033-17

...