Sensitivity to New Physics of Isotope Shift Studies using the Coronal Lines of Highly Charged Calcium Ions
Nils-Holger Rehbehn, Michael K. Rosner, Hendrik Bekker, Julian C. Berengut, Piet O. Schmidt, Steven A. King, Peter Micke, Ming Feng Gu, Robert Müller, Andrey Surzhykov, José R. Crespo López-Urrutia
SSensitivity to New Physics of Isotope Shift Studies using the Coronal Lines of HighlyCharged Calcium Ions
Nils-Holger Rehbehn ID , ∗ Michael K. Rosner ID , Julian C. Berengut ID ,
2, 1
PietO. Schmidt ID ,
3, 4
Steven A. King ID , Peter Micke ID ,
3, 1
Ming Feng Gu, RobertM¨uller ID ,
3, 6
Andrey Surzhykov ID ,
3, 6 and Jos´e R. Crespo L´opez-Urrutia ID † Max-Planck-Institut f¨ur Kernphysik, D–69117 Heidelberg, Germany School of Physics, University of New South Wales, Sydney, New South Wales 2052, Australia Physikalisch–Technische Bundesanstalt, D–38116 Braunschweig, Germany Leibniz Universit¨at Hannover, D–30167 Hannover, Germany Space Science Laboratory, University of California, Berkeley, CA 94720, USA Technische Universit¨at Braunschweig, D–38106 Braunschweig, Germany (Dated: February 5, 2021)Promising searches for new physics beyond the current Standard Model (SM) of particle physicsare feasible through isotope-shift spectroscopy, which is sensitive to a hypothetical fifth force betweenthe neutrons of the nucleus and the electrons of the shell. Such an interaction would be mediatedby a new particle which could in principle be associated with dark matter. In so-called King plots,the mass-scaled frequency shifts of two optical transitions are plotted against each other for a seriesof isotopes. Subtle deviations from the expected linearity could reveal such a fifth force. Here, westudy experimentally and theoretically six transitions in highly charged ions of Ca, an element withfive stable isotopes of zero nuclear spin. Some of the transitions are suitable for upcoming high-precision coherent laser spectroscopy. Combining them with clock transitions in singly charged Ca + ions enables application of a generalized King plot scheme that allows for eliminating higher-orderSM-related nonlinearities, thereby increasing the sensitivity of ongoing searches for unknown forcesand particles. Over the last century, all the essential predictions ofthe Standard Model (SM) of particle physics were ex-perimentally confirmed; finally, the missing Higgs bosonwas found in the year 2013 [1–3]. Several fundamentalquestions remain unanswered, however: What are darkmatter and dark energy? How can we explain neutrinooscillations? What causes the apparent asymmetry be-tween matter and antimatter content in the universe?Consequently, the SM is considered to be incomplete,and new physics (NP) is sought after.By exploiting the unrivalled accuracy of laser spec-troscopy, atomic physics offers unique sensitivity forsearches for physics beyond the SM (for a review see[4]). Feeble non-gravitational interactions between nor-mal matter and, e. g. , dark matter, would result inchanges of atomic and molecular energy levels. Depend-ing on the dark matter candidate and the properties ofthe field representing it, oscillations [5, 6], drifts [7] ortransient changes [8] in energy levels can occur. Opticalatomic clocks [9] with up to 18 digits of accuracy have al-ready been employed to put bounds on the mass of darkmatter candidates [10–13].Recently, a complementary approach [14, 15] proposesprobing the existence of a hypothetical fifth force cou-pling electrons and neutrons by means of isotope-shift(IS) spectroscopy [16, 17]. Isotopic perturbations of op-tical transitions are dominated by two effects: (i) the nu-clear recoil (mass shift, MS), and (ii) the modification of ∗ [email protected] † [email protected] the electron-nucleus interaction potential by the nuclearcharge distribution (field shift, FS). The dependence onthe mostly poorly known nuclear charge distribution iseliminated by measuring [18–23] two different transitionsand using a so-called King-plot [24]. This yields a lin-ear relationship in first order between the two transitionfrequencies. A fifth force coupling electrons and neu-trons would break this linearity [14, 15, 25–27]. Cau-tion is, however, needed, since nonlinearity can also arisefrom higher-order SM effects [15, 26], which then cloudthe NP effects. To separate them, high precision atomicand nuclear structure calculations for the former are re-quired [26, 28–31], which are feasible for few-electron sys-tems such as highly charged ions (HCI). Alternatively, ageneralized King plot (GKP) proposed by Mikami et al. [31] and further developed by Berengut et al. [32] em-ploys measurements of additional electronic transitionsfor eliminating the impact of higher-order SM effects onsuch NP searches.Recently, optical-clock-like spectroscopy of HCI wasdemonstrated by means of sympathetic laser cooling andquantum logic operations in a linear Paul trap [36]. Rel-ative fractional uncertainties as low as 10 − can beachieved through absolute frequency measurements. Foroptical transition frequencies of the order of 500 THz(2 eV) this corresponds to an absolute uncertainty of theorder of 100 mHz (0 . a r X i v : . [ phy s i c s . a t o m - ph ] F e b I n t e n s i t y ( a r b . un i t s ) Ca I n t e n s i t y ( a r b . un i t s ) Ca Total momentum J025 E n e r g y ( e V ) P P Total momentum J025 E n e r g y ( e V ) P P Total momentum J020 E n e r g y ( e V ) P P Total momentum J020 E n e r g y ( e V ) P P Total momentum J020 E n e r g y ( e V ) P P Total momentum J0.02.5 E n e r g y ( e V ) P P FIG. 1. Fits to the Zeeman components of the studied transitions. Magenta arrows mark their positions and relative intensities.The level diagrams in the insets were calculated with fac [33].TABLE I. Forbidden optical fine-structure transitions in highly charged Ca ions: Measured energies, vacuum wavelengthshave been observed, transition probabilities A ki and theoretical energy values were calculated in this work with fac [33]and ratip/grasp [34]. Their respective SM electronic sensitivity coefficients K and F have been calculated with fac . Forcomparison, the 729 nm Ca + transition has been included with values from the NIST database [35].Observed fac ratip/grasp Elec. coeff.Ion Transition Energy (eV) Wavel. (nm) En. (eV) A ki (s − ) En. (eV) A ki (s − ) K (eV u) F (eV/fm )Ca + s p P ○ / - P ○ / . × − -104 × − Ca + s p P - P . × − -123 × − Ca + s p P - P . × − -196 × − Ca + s p P - P . × − -21 × − Ca + s p P ○ / - P ○ / . × − -577 × − Ca + s s p P ○ - P ○ . × − -134 × − Ca + p d -4 s D / - S / . × − -1 . × − In this Letter, we experimentally determine accuratewavelengths for six magnetic dipole (M1) forbidden linesof Ca ions in the charge stages 11 + through 16 + , andidentify which ones transition directly to the electronicground state and thus are suitable for coherent laser spec-troscopy [44] IS measurements. We perform calculationsof their IS and use them to construct King plots (KP) aswell as generalized King plots (GKP) to suppress higher-order SM terms, to see their effects and find the mostsuitable combination of transitions for the search of NPinfluences.Calcium ions are particularly suitable for King-plotstudies since this element has five stable isotopes withzero nuclear spin. Additionally, Ca has narrow tran-sitions in various charge states which can be accu-rately measured by high-precision laser spectroscopy us- ing common optical frequency metrology methods. Inthe past, the isotope shifts of broad dipole allowed [45–50] as well as narrow transitions [16, 18, 51] in singlycharged Ca have been investigated. In HCI, the stronglybound outer electrons can undergo fine-structure [52–59]and hyperfine-structure [60–64] transitions in the opticalrange. Some of them contribute to the corona spectrumand are therefore known as coronal lines. Optical tran-sitions in HCI can also arise from level crossings [65–68].These involve orbitals of very different character, andthus promise a high sensitivity to NP (see, e.g. [69–76]),making them strong candidates for such studies [77].To produce Ca HCI, we used the Heidelberg electronbeam ion trap (EBIT) [78–80]. There, an electron beamemitted by a thermionic cathode is accelerated to en-ergies between 600 eV and 1200 eV and strongly com-pressed by a coaxial 8 T magnetic field. Depending onits energy, electron impact ionization brings atoms cross-ing the beam to the desired charge state. Resulting HCIare radially trapped by the negative space charge poten-tial generated by the electron beam and axially by biaseddrift tubes. A small oven is used to evaporate a substancecontaining Ca. The resulting molecular beam crosses theelectron beam and is dissociated there, thus releasing Caatoms for ionization. The trap is periodically dumped ev-ery few seconds in order to avoid a slow accumulation ofundesired ions of barium and tungsten, elements whichare constituents of the thermoionic cathode and slowlyevaporate from there.In the trap, the forbidden optical transitions of inter-est are excited by electron impact. Using four lensesand several mirrors, we project a rotated image of thehorizontal ion cloud onto the vertical entrance slit of a2-meter Czerny-Turner spectrometer [59, 68] equippedwith a cryogenically cooled CCD camera. We calibratethe spectral dispersion function through a polynomial fitof the line positions of known transitions [35], emitted bysuitable hollow cathode lamps. Spectra of such lamps areautomatically taken before and after each of the approx-imately 30-minute-long exposures needed for the HCIlines. The two-dimensional spectral images are cleanedfrom cosmic events, corrected from optical aberrationsand calibrated, as explained in detail in Refs. [59, 67, 68].Transition wavelengths are determined by fitting Gaus-sians to the resolved Zeeman components, taking calcu-lated Clebsch-Gordan coefficients as initial parametersfor the g -factors. Results are shown in figure 1 with therespective level diagrams of the transitions, and summa-rized in Table I.For the observed p j → p j transitions in the chargestates from Ca + to Ca + , as well as the Ca + s j → d j transition, we calculated theoretical transition energiesand their dependence on hypothetical NP with the Flex-ible Atomic Code ( fac ) version 1.1.5 [33] and crosscheckwith ratip/grasp [34]. For this, a fictitious mediatorΦ is modeled by a Yukawa central potential V Φ ( r ) andintroduced as a perturbation to the electromagnetic fieldacting upon the electrons: V Φ ( r ) = y e y n ( A − Z ) ̵ h c4 πr exp (− c ̵ h ⋅ m Φ ⋅ r ) . (1)Here, ̵ h is the reduced Planck constant and c the speed oflight. The coupling strength is defined as y e y n , where y e and y n are the couplings of Φ to electrons and neutrons,respectively. A is the mass number and Z is the nuclearcharge. For the present estimates, we set a value of y e y n = × − , which is below the limits of current exclusionplots, such as shown in Ref. [32]. The Yukawa range isset by the mediator mass m Φ : a lighter particle has alonger-range effect than a heavier one.In first-order perturbation theory, the IS is defined fortransition i as the sum of the SM mass and field shift, aswell as the shift induced by the NP mediator:IS = IS SM + IS NP ≡ δν ai = K i µ a + F i δ ⟨ r a ⟩+ y e y n X i γ a , (2) Mediator mass m (eV/c )101234 I S N P ( k H z )
40 44
CaFACCa nm Ca nm Ca nm Ca nm Ca nm Ca nm Ca +729.3 nm RATIP/GRASPCa nm Ca nm Ca nm Ca nm Ca nm Ca nm Ca +729.3 nm IS SM -1.51 GHz-2.72 GHz-1.00 GHz-1.04 GHz-1.34 GHz-1.57 GHz-1.72 GHz FIG. 2. Effect of the NP particle on the isotope shift (IS NP )predicted using fac and ratip/grasp for the Ca isotope pair(40,44) as a function of the mediator mass for a couplingstrength y e y n = × − . The dominant SM contribution(IS SM ) was subtracted from the total IS to make the smallerIS NP contributions visible.TABLE II. NP electronic coefficients X( m Φ ) in eV calculatedwith fac and ratip/grasp for different mediator masses m Φ . m Φ eV / c eV / c eV / c eV / c Trans. fac grasp fac grasp fac grasp fac grasp Ca + + + + + + + -31.9 -35.3 9.37 14.8 2.51 4.52 0.119 0.223 where a is the isotope pair ( A, A r ) , thus δν ai = ν Ai − ν A r i isthe difference of transition i between isotope A and refer-ence isotope A r ; K i , F i and X i are electronic constants ofthe shift terms, µ a = / m A − / m A r is the difference of theinverse isotope masses and δ ⟨ r a ⟩ = ⟨ r A ⟩ − ⟨ r A r ⟩ is the dif-ference of the mean square nuclear charge radii. The NPpart depends on the isotopes with γ a = ( A − Z )−( A r − Z ) = A − A r and y e y n ∝ V Φ (see equation 1) defining the cou-pling strength with regards to the coupling range.Figure 2 shows the IS of the studied transitions inducedby NP for the Ca isotope pair (40,44); the associated elec-tronic coefficients are listed in table II. To highlight theeffects of the NP particle, the dominating SM contribu-tion was subtracted from the total IS. For comparison,we also included the Ca + s j → d j transition. It showsa larger shift than the p j → p j transitions due to thedifference in l between the initial and final state.To analyze the IS without an accurate knowledge ofthe change of the nuclear charge radius ⟨ r ⟩ , one can usethe approach of King [24], where the isotope shifts oftwo transitions in different isotope pairs a , divided bythe mass parameter µ a (denoted as m for modified), are Ca nm (THz u)0.441220.441200.441180.441160.441140.441120.44110 m C a + . n m ( T H z u ) k H z u (42,40)0.05 0.00 0.05kHz u 0.300.150.00 k H z u (46,40) 0.08 0.00 0.08kHz u0.080.000.080.16 k H z u (44,40)0.04 0.00 0.04kHz u 0.000.080.160.24 k H z u (48,40) FIG. 3. King’s plot of fac -calculated transitions in Ca + . and Ca + . . It assumes a mediator with mass m Φ = eV / c and a coupling strength y e y n = × − . The er-ror bars depict a 100 mHz measurement uncertainty, the massuncertainty is neglected. plotted against each other. A linear behavior along allpoints is expected from first-order perturbation theory,while nonlinearities would arise from either higher ordereffects from the SM, or from NP parts:m δν a = F F ⋅ m δν a + ( K − F F K )+ y e y n ⋅ ( X − F F X ) ⋅ m γ a . (3)Here, the first two terms represent the linear behavior be-tween the isotope shifts of the two transitions i = i =
1, from equation 2, for different isotope pairs a . Thethird term, with the variable m γ a = γ a µ a , induces a nonlin-earity when X i / F i varies for the transitions and mediatormasses. The King plot can be constructed from the fac results, where the SM and NP electronic coefficients canbe taken from table I and table II, respectively. This isshown for the mediator mass of m Φ = eV / c and acoupling strength of y e y n = × − in figure 3.To quantify the chances of detecting NP, we can dis-play the nonlinearity over its error, using the equationsfrom Berengut et al. [32]. This allows us to plot the low-est possible coupling strength where the nonlinearity canstill be resolved with a given measurement uncertainty.Figure 4 plots this measurement boundary for differentmediator masses with an expected measurement uncer-tainty of 100 mHz for future coherent laser spectroscopymeasurements. The Ca + . , Ca + . (violet) curvecorresponds to the transitions used in Solaro et al. [16]with a measurement uncertainty of 20 Hz. The browncurve corresponds to this transition pair with the un-certainty level from the present paper. Two out of thefour transitions ground-state transitions studied here areshown in a King plot. The most promising candidateis the Ca + . , Ca + . pair, but the HCI-only pairCa + . , Ca + . shows also a greater chance of find- Mediator mass (eV/c )10 C o u p li n g c o n s t a n t y e y n = 100mHzSolaro et al .( = 20Hz) GKPCa +729.3nm , Ca , Ca KP w/ SM nonlin.Ca , Ca
KP w/o SM nonlin.Ca +729.3nm , Ca +732.6nm Ca +729.3nm , Ca +732.6nm Ca +729.3nm , Ca Ca , Ca FIG. 4. Projected bounds on Yukawa interactions excluded bydifferent King plots with measurement errors of ∆ ν = ratip/grasp [34]. ing NP than the Ca + -only pair. To show potential higherorder SM effects, a quadratic mass shift in order of 3 Hz[26] is added to the HCI pair Ca + . , Ca + . . Thisterm causes additional SM nonlinearities, which greatlylimit the bounds that can be placed in the couplingstrength (orange). However, it is expected that ongo-ing calculations of the quadratic mass shift terms in Ca + and the different Ca HCI will reduce their contributionsto the general uncertainty. Furthermore, the problem canbe bypassed altogether by using the generalized King plot(blue), as the additional transition is used to separate thehigher-order SM effects from the NP effects, similarly towhat has been done to obtain equation 3. It is importantto mention that specific knowledge of the higher ordersof the SM contributions is not needed here [31, 32].Presently, the fractional accuracy of absolute frequencymeasurements is limited to the low 10 − range by thedefinition of the second in the SI system of units (see,for example, Weyers et al. [81]). This leads to this pa-per’s 100 mHz uncertainty for an individual transition. Itwould be a factor of √ + ions, the number ofcombinations suitable for NP searches grows [14, 15, 25–27, 87] and opens up the use of the generalized King-plotanalysis [32] for this type of experiment. ACKNOWLEDGEMENTS
Financial support was provided by the Max - Planck- Gesellschaft and the Physikalisch - Technische Bun- desanstalt. We acknowledge support from the MaxPlanck - Riken - PTB Center for Time, Constants andFundamental Symmetries, the Deutsche Forschungsge-meinschaft through SCHM2678/5-1, the collaborativeresearch centers “SFB 1225 (ISOQUANT)”and “SFB1227 (DQ-mat)”, and Germany’s Excellence Strategy- EXC-2123/1 QuantumFrontiers - 390837967. Thisproject also received funding from the European Metrol-ogy Programme for Innovation and Research (EMPIR)cofinanced by the Participating 5 States and from theEuropean Union’s Horizon 2020 research and innova-tion programme (Project No. 17FUN07 CC4C). JCBwas supported in this work by the Alexander von Hum-boldt Foundation and the Australian Research Council(DP190100974). [1] P. W. Higgs, Phys. Rev. Lett. , 508 (1964).[2] G. Aad, T. Abajyan, B. Abbott, J. Abdallah, S. AbdelKhalek, A. Abdelalim, and O. Abdinov, Physics LettersB , 1 (2012).[3] S. Chatrchyan, V. Khachatryan, A. Sirunyan, A. Tu-masyan, W. Adam, E. Aguilo, and T. Bergauer, PhysicsLetters B , 30 (2012).[4] M. S. Safronova, D. Budker, D. DeMille, D. F. J. Kimball,A. Derevianko, and C. W. Clark, Rev. Mod. Phys. ,025008 (2018).[5] A. Arvanitaki, J. Huang, and K. Van Tilburg, Phys.Rev. D , 015015 (2015).[6] Y. V. Stadnik and V. V. Flambaum, Phys. Rev. Lett. , 161301 (2015).[7] Y. V. Stadnik and V. V. Flambaum, Phys. Rev. Lett. , 201301 (2015).[8] A. Derevianko and M. Pospelov, Nat Phys , 933(2014).[9] A. D. Ludlow, M. M. Boyd, J. Ye, E. Peik, and P. O.Schmidt, Reviews of Modern Physics , 637 (2015).[10] C. J. Kennedy, E. Oelker, J. M. Robinson, T. Bothwell,D. Kedar, W. R. Milner, G. E. Marti, A. Derevianko,and J. Ye, Phys. Rev. Lett. , 201302 (2020).[11] P. Wcis(cid:32)lo, P. Morzy´nski, M. Bober, A. Cygan, D. Lisak,R. Ciury(cid:32)lo, and M. Zawada, Nat. Astron. , 0009 (2016).[12] P. Wcis(cid:32)lo, P. Ablewski, K. Beloy, S. Bilicki, M. Bober,R. Brown, R. Fasano, R. Ciury(cid:32)lo, H. Hachisu, T. Ido,J. Lodewyck, A. Ludlow, W. McGrew, P. Morzy´nski,D. Nicolodi, M. Schioppo, M. Sekido, R. L. Targat,P. Wolf, X. Zhang, B. Zjawin, and M. Zawada, Sci. Adv. , eaau4869 (2018).[13] B. M. Roberts, P. Delva, A. Al-Masoudi, A. Amy-Klein,C. Baerentsen, C. Baynham, E. Benkler, S. Bilicki,S. Bize, W. Bowden, J. Calvert, V. Cambier, E. Cantin, E. A. Curtis, S. D¨orscher, M. Favier, F. Frank, P. Gill,R. M. Godun, G. Grosche, C. Guo, A. Hees, I. R.Hill, R. Hobson, N. Huntemann, J. Kronjaeger, S. Koke,A. Kuhl, R. Lange, T. Legero, B. Lipphardt, C. Lis-dat, J. Lodewyck, O. Lopez, H. S. Margolis, H. Alvarez-Martinez, F. Meynadier, F. Ozimek, E. Peik, P.-E. Pot-tie, N. Quintin, C. Sanner, L. D. Sarlo, M. Schioppo,R. Schwarz, A. Silva, U. Sterr, C. Tamm, R. L. Targat,P. Tuckey, G. Vallet, T. Waterholter, D. Xu, and P. Wolf,New J. Phys. (2020), 10.1088/1367-2630/abaace.[14] C. Delaunay, R. Ozeri, G. Perez, and Y. Soreq, Phys.Rev. D , 093001 (2017).[15] J. C. Berengut, D. Budker, C. Delaunay, V. V. Flam-baum, C. Frugiuele, E. Fuchs, C. Grojean, R. Harnik,R. Ozeri, G. Perez, and Y. Soreq, Phys. Rev. Lett. ,091801 (2018).[16] C. Solaro, S. Meyer, K. Fisher, J. C. Berengut, E. Fuchs,and M. Drewsen, Phys. Rev. Lett. , 123003 (2020).[17] I. Counts, J. Hur, D. P. L. A. Craik, H. Jeon, C. Le-ung, J. Berengut, A. Geddes, A. Kawasaki, W. Jhe, andV. Vuleti´c, arXiv:2004.11383 (2020), arXiv:2004.11383.[18] F. W. Knollmann, A. N. Patel, and S. C. Doret, Phys.Rev. A , 022514 (2019).[19] T. Manovitz, R. Shaniv, Y. Shapira, R. Ozeri, andN. Akerman, Phys. Rev. Lett. , 203001 (2019).[20] H. Miyake, N. C. Pisenti, P. K. Elgee, A. Sitaram, andG. K. Campbell, Phys. Rev. Research , 033113 (2019).[21] F. Gebert, Y. Wan, F. Wolf, C. N. Angstmann, J. C.Berengut, and P. O. Schmidt, Physical Review Letters , 053003 (2015).[22] C. Shi, F. Gebert, C. Gorges, S. Kaufmann,W. N¨ortersh¨auser, B. K. Sahoo, A. Surzhykov, V. A.Yerokhin, J. C. Berengut, F. Wolf, J. C. Heip, and P. O.Schmidt, Applied Physics B , 2 (2017). [23] P. M¨uller, K. K¨onig, P. Imgram, J. Kr¨amer, andW. N¨ortersh¨auser, Physical Review Research , 043351(2020).[24] W. H. King, J. Opt. Soc. Am. , 638 (1963).[25] C. Frugiuele, E. Fuchs, G. Perez, and M. Schlaffer, Phys.Rev. D , 015011 (2017).[26] V. V. Flambaum, A. J. Geddes, and A. V. Viatkina,Phys. Rev. A , 032510 (2018).[27] S. Fichet, Phys. Rev. Lett. , 131801 (2018).[28] V. A. Yerokhin, R. A. M¨uller, A. Surzhykov, P. Micke,and P. O. Schmidt, Phys. Rev. A , 012502 (2020).[29] P.-G. Reinhard, W. Nazarewicz, and R. F. Garcia Ruiz,Phys. Rev. C , 021301(R) (2020).[30] M. Tanaka and Y. Yamamoto, Progress of The-oretical and Experimental Physics (2020),10.1093/ptep/ptaa121.[31] K. Mikami, M. Tanaka, and Y. Yamamoto, The Euro-pean Physical Journal C , 896 (2017).[32] J. C. Berengut, C. Delaunay, A. Geddes, and Y. Soreq,Phys. Rev. Research , 043444 (2020).[33] M. F. Gu, Canadian Journal of Physics , 675 (2008).[34] S. Fritzsche, Computer Physics Communications ,1525 (2012).[35] A. Kramida, Y. Ralchenko, J. Reader, and NIST ASDTeam, “NIST Atomic Spectra Database (ver. 5.7),”(2020), accessed: 2019-07-27.[36] P. Micke, T. Leopold, S. A. King, E. Benkler, L. J. Spieß,L. Schm¨oger, M. Schwarz, J. R. Crespo L´opez-Urrutia,and P. O. Schmidt, Nature , 60 (2020).[37] R. Barbieri and T. Ericson, Physics Letters B , 270(1975).[38] H. Leeb and J. Schmiedmayer, Phys. Rev. Lett. , 1472(1992).[39] V. V. Nesvizhevsky, G. Pignol, and K. V. Protasov,Phys. Rev. D , 034020 (2008).[40] Y. N. Pokotilovski, Physics of Atomic Nuclei , 924(2006).[41] S. L. Adler, R. F. Dashen, and S. B. Treiman, Phys.Rev. D , 3728 (1974).[42] M. Bordag, U. Mohideen, and V. Mostepanenko, PhysicsReports , 1 (2001).[43] M. Bordag, G. L. Klimchitskaya, U. Mohideen, andV. M. Mostepanenko, Advances in the Casimir effect , Vol.145 (OUP Oxford, 2009).[44] P. O. Schmidt, T. Rosenband, C. Langer, W. M. Itano,J. C. Bergquist, and D. J. Wineland, Science , 749(2005).[45] W. N¨ortersh¨auser, K. Blaum, K. Icker, P. M¨uller,A. Schmitt, K. Wendt, and B. Wiche, The EuropeanPhysical Journal D - Atomic, Molecular, Optical andPlasma Physics , 33 (1998).[46] A. Mortensen, J. J. T. Lindballe, I. S. Jensen,P. Staanum, D. Voigt, and M. Drewsen, Phys. Rev. A , 042502 (2004).[47] C. Gorges, K. Blaum, N. Fr¨ommgen, C. Geppert,M. Hammen, S. Kaufmann, J. Kr¨amer, A. Krieger,R. Neugart, R. S´anchez, and W. N¨ortersh¨auser, Jour-nal of Physics B: Atomic, Molecular and Optical Physics , 245008 (2015).[48] Y. Hashimoto, M. Kitaoka, T. Yoshida, andS. Hasegawa, Applied Physics B , 339 (2011).[49] C. Shi, F. Gebert, C. Gorges, S. Kaufmann,W. N¨ortersh¨auser, B. K. Sahoo, A. Surzhykov, V. A.Yerokhin, J. C. Berengut, F. Wolf, J. C. Heip, and P. O. Schmidt, Applied Physics B , 2 (2016).[50] F. Gebert, Y. Wan, F. Wolf, C. N. Angstmann, J. C.Berengut, and P. O. Schmidt, Phys. Rev. Lett. ,053003 (2015).[51] J. Benhelm, G. Kirchmair, U. Rapol, T. K¨orber, C. F.Roos, and R. Blatt, Phys. Rev. A , 032506 (2007).[52] C. A. Morgan, F. G. Serpa, E. Tak´acs, E. S. Meyer, J. D.Gillaspy, J. Sugar, J. R. Roberts, C. M. Brown, andU. Feldman, Phys. Rev. Lett. , 1716 (1995).[53] I. Dragani´c, J. R. Crespo L´opez-Urrutia, R. DuBois,S. Fritzsche, V. M. Shabaev, R. S. Orts, I. I. Tupit-syn, Y. Zou, and J. Ullrich, Phys. Rev. Lett. , 183001(2003).[54] R. S. Orts, Z. Harman, J. R. C. L´opez-Urrutia, A. N.Artemyev, H. Bruhns, A. J. G. Mart´ınez, U. D.Jentschura, C. H. Keitel, A. Lapierre, V. Mironov, V. M.Shabaev, H. Tawara, I. I. Tupitsyn, J. Ullrich, and A. V.Volotka, Phys. Rev. Lett. , 103002 (2006).[55] J. R. Crespo L´opez-Urrutia, Canadian Journal of Physics , 111 (2008).[56] J. R. Crespo L´opez-Urrutia and Z. Harman, in Fun-damental Physics in Particle Traps , Springer Tracts inModern Physics No. 256, edited by W. Quint and M. Vo-gel (Springer Berlin Heidelberg, 2014) pp. 315–373, dOI:10.1007/978-3-642-45201-7 10.[57] S. Murata, T. Nakajima, M. S. Safronova, U. I.Safronova, and N. Nakamura, Phys. Rev. A , 062506(2017).[58] A. Windberger, F. Torretti, A. Borschevsky, A. Ryabt-sev, S. Dobrodey, H. Bekker, E. Eliav, U. Kaldor,W. Ubachs, R. Hoekstra, J. R. Crespo L´opez-Urrutia,and O. O. Versolato, Phys. Rev. A , 012506 (2016).[59] H. Bekker, C. Hensel, A. Daniel, A. Windberger,T. Pfeifer, and J. R. Crespo L´opez-Urrutia, Phys. Rev.A , 062514 (2018).[60] I. Klaft, S. Borneis, T. Engel, B. Fricke, R. Grieser,G. Huber, T. K¨uhl, D. Marx, R. Neumann, S. Schr¨oder,P. Seelig, and L. V¨olker, Phys. Rev. Lett. , 2425(1994).[61] J. R. Crespo L´opez-Urrutia, P. Beiersdorfer, D. W.Savin, and K. Widmann, Physical Review Letters ,826 (1996).[62] P. Seelig, S. Borneis, A. Dax, T. Engel, S. Faber, M. Ger-lach, C. Holbrow, G. Huber, T. K¨uhl, D. Marx, K. Meier,P. Merz, W. Quint, F. Schmitt, M. Tomaselli, L. V¨olker,H. Winter, M. W¨urtz, K. Beckert, B. Franzke, F. Nolden,H. Reich, M. Steck, and T. Winkler, Phys. Rev. Lett. , 4824 (1998).[63] J. R. Crespo L´opez-Urrutia, P. Beiersdorfer, K. Wid-mann, B. B. Birkett, A.-M. M˚artensson-Pendrill, andM. G. H. Gustavsson, Phys. Rev. A , 879 (1998).[64] P. Beiersdorfer, S. B. Utter, K. L. Wong, J. R. CrespoL´opez-Urrutia, J. A. Britten, H. Chen, C. L. Harris, R. S.Thoe, D. B. Thorn, E. Tr¨abert, M. G. H. Gustavsson,C. Forss´en, and A.-M. M˚artensson-Pendrill, Phys. Rev.A , 032506 (2001).[65] J. C. Berengut, V. A. Dzuba, and V. V. Flambaum,Phys. Rev. Lett. , 120801 (2010).[66] J. C. Berengut, V. A. Dzuba, V. V. Flambaum, andA. Ong, Phys. Rev. A , 022517 (2012).[67] A. Windberger, J. R. Crespo L´opez-Urrutia,H. Bekker, N. S. Oreshkina, J. C. Berengut, V. Bock,A. Borschevsky, V. A. Dzuba, E. Eliav, Z. Harman,U. Kaldor, S. Kaul, U. I. Safronova, V. V. Flambaum, C. H. Keitel, P. O. Schmidt, J. Ullrich, and O. O.Versolato, Phys. Rev. Lett. , 150801 (2015).[68] H. Bekker, A. Borschevsky, Z. Harman, C. H. Keitel,T. Pfeifer, P. O. Schmidt, J. R. Crespo L´opez-Urrutia,and J. C. Berengut, Nature Communications , 5651(2019).[69] A. Ong, J. C. Berengut, and V. V. Flambaum, in Fun-damental Physics in Particle Traps (Springer, 2014) pp.293–314.[70] J. C. Berengut, V. A. Dzuba, V. V. Flambaum, andA. Ong, Physical Review Letters , 210802 (2011).[71] A. Derevianko, V. A. Dzuba, and V. V. Flambaum,Physical Review Letters , 180801 (2012).[72] V. A. Dzuba and V. V. Flambaum, Hyperfine Interact. , 79 (2015).[73] M. S. Safronova, V. A. Dzuba, V. V. Flambaum, U. I.Safronova, S. G. Porsev, and M. G. Kozlov, Phys. Rev.A , 052509 (2014).[74] S. G. Porsev, U. I. Safronova, M. S. Safronova, P. O.Schmidt, A. I. Bondarev, M. G. Kozlov, I. I. Tupitsyn,and C. Cheung, Phys. Rev. A , 012802 (2020).[75] C. Cheung, M. S. Safronova, S. G. Porsev, M. G. Kozlov,I. I. Tupitsyn, and A. I. Bondarev, Phys. Rev. Lett. ,163001 (2020).[76] N. Kimura, R. Kodama, K. Suzuki, S. Oishi, M. Wada,K. Okada, N. Ohmae, H. Katori, and N. Nakamura,Phys. Rev. A , 052508 (2019).[77] M. G. Kozlov, M. S. Safronova, J. R. Crespo L´opez-Urrutia, and P. O. Schmidt, Rev. Mod. Phys. , 045005(2018).[78] M. A. Levine, R. E. Marrs, J. R. Henderson, D. A.Knapp, and M. B. Schneider, Physica Scripta , 157(1988).[79] M. A. Levine, R. E. Marrs, J. N. Bardsley, P. Beiersdor-fer, C. L. Bennett, M. H. Chen, T. Cowan, D. Dietrich,J. R. Henderson, D. A. Knapp, A. Osterheld, B. M. Pen-etrante, M. B. Schneider, and J. H. Scofield, NuclearInstruments and Methods in Physics Research SectionB: Beam Interactions with Materials and Atoms , 431 (1989).[80] J. R. Crespo L´opez-Urrutia, A. Dorn, R. Moshammer,and J. Ullrich, Physica Scripta T80 , 502 (1999).[81] S. Weyers, V. Gerginov, M. Kazda, J. Rahm, B. Lip-phardt, G. Dobrev, and K. Gibble, Metrologia , 789(2018).[82] T. Rosenband, D. B. Hume, P. O. Schmidt, C. W.Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E.Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Did-dams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J.Wineland, and J. C. Bergquist, Science , 1808 (2008),https://science.sciencemag.org/content/319/5871/1808.full.pdf.[83] R. M. Godun, P. B. R. Nisbet-Jones, J. M. Jones, S. A.King, L. A. M. Johnson, H. S. Margolis, K. Szymaniec,S. N. Lea, K. Bongs, and P. Gill, Phys. Rev. Lett. ,210801 (2014).[84] B. A. C. O. Network, Collaboration, :, K. Beloy, M. I.Bodine, T. Bothwell, S. M. Brewer, S. L. Bromley, J.-S. Chen, J.-D. Deschˆenes, S. A. Diddams, R. J. Fasano,T. M. Fortier, Y. S. Hassan, D. B. Hume, D. Kedar,C. J. Kennedy, I. Khader, A. Koepke, D. R. Leibrandt,H. Leopardi, A. D. Ludlow, W. F. McGrew, W. R.Milner, N. R. Newbury, D. Nicolodi, E. Oelker, T. E.Parker, J. M. Robinson, S. Romisch, S. A. Sch¨affer, J. A.Sherman, L. C. Sinclair, L. Sonderhouse, W. C. Swann,J. Yao, J. Ye, and X. Zhang, “Frequency ratio measure-ments with 18-digit accuracy using a network of opticalclocks,” (2020), arXiv:2005.14694 [physics.atom-ph].[85] S. D¨orscher, N. Huntemann, R. Schwarz, R. Lange,E. Benkler, B. Lipphardt, U. Sterr, E. Peik, and C. Lis-dat, “Optical frequency ratio of a Yb + single-ion clockand a Sr lattice clock,” (2020), arXiv:2009.05470[physics.atom-ph].[86] T. Pruttivarasin, M. Ramm, S. G. Porsev, I. I. Tupitsyn,M. S. Safronova, M. A. Hohensee, and H. H¨affner, Nature , 592 (2015).[87] J. C. Berengut, V. V. Flambaum, and E. M. Kava, Phys.Rev. A84