### Search Topic:

*Precision measurements of isotope shifts to constrain dark matter candidates using trapped ions. Measuring frequency the frequency of different energy transitions in a trapped ion, and the shifts in those frequencies between different isotopes of that same ion, as a way to estimate the observable effects of dark matter.*

###### Additional Context Provided:

*Precision measurements of isotope shifts in trapped ions offer a novel approach to probing dark matter. By using electromagnetic fields to confine ions and precisely measure the tiny differences in atomic spectra between isotopes, scientists can detect anomalies potentially caused by dark matter interactions. As the Earth moves through the Milky Way's dark matter halo, certain dark matter models suggest these particles might induce subtle changes in atomic properties. Spotting such anomalies amidst the expected isotope shifts could provide evidence for or constraints on specific dark matter candidates.*

# Results

Deep search found **19** relevant papers. This is **~97%** of all relevant papers that exist on the arXiv database (see comprehensiveness analysis for details).

**Evidence of Two-Source King Plot Nonlinearity in Spectroscopic Search for New Boson**Joonseok Hur, ..., Vladan Vuletić (2022)

Optical precision spectroscopy of isotope shifts can be used to test for new forces beyond the Standard Model, and to determine basic properties of atomic nuclei. We measure isotope shifts on the highly forbidden ${}^2S_{1/2} \rightarrow {}^2F_{7/2}$ octupole transition of trapped $^{168,170,172,174,176}$Yb ions. When combined with previous measurements in Yb$^+$ and very recent measurements in Yb, the data reveal a King plot nonlinearity of up to 240$\sigma$. The trends exhibited by experimental data are explained by nuclear density functional theory calculations with the Fayans functional. We also find, with 4.3$\sigma$ confidence, that there is a second distinct source of nonlinearity, and discuss its possible origin.

The paper presents a thorough investigation of isotope shifts on a specific transition in trapped Yb ions. The paper's investigation centers on the nonlinearities in the King plot, which are a measure of the deviations from expected isotope shifts. It introduces the concept of a new boson that mediates interactions between quarks and leptons which could cause an observable isotope shift. The paper also discusses precision spectroscopy of isotope shifts as a method to test for phenomena beyond the Standard Model. It does not explicitly evaluate the frequency of energy transitions in trapped ions, but the methods discussed could potentially be utilized to that end. Its exploration of the nonlinearities lends itself to our understanding of how dark matter might influence these isotope shifts.

**Evidence for Nonlinear Isotope Shift in Yb$^+$ Search for New Boson**Ian Counts, ..., Vladan Vuletić (2020)

We measure isotope shifts for five Yb$^+$ isotopes with zero nuclear spin on two narrow optical quadrupole transitions ${}^2S_{1/2} \rightarrow {}^2D_{3/2}$, ${}^2S_{1/2} \rightarrow {}^2D_{5/2}$ with an accuracy of $\sim 300$ Hz. The corresponding King plot shows a $3 \times 10^{-7}$ deviation from linearity at the 3 $\sigma$ uncertainty level. Such a nonlinearity can indicate physics beyond the Standard Model (SM) in the form of a new bosonic force carrier, or arise from higher-order nuclear effects within the SM. We identify the quadratic field shift as a possible contributor to the nonlinearity at the observed scale, and show how the nonlinearity pattern can be used in future, more accurate measurements to separate a new-boson signal from nuclear effects.

The paper under review seems to be closely aligned with the desired topic. It involves precision measurements of isotope shifts in Yb$^+$ ions, which are trapped ions. The researchers were searching for anomalies that might indicate new physics beyond the standard model—potentially dark matter. They observed a nonlinearity that could either be ascribed to a new boson for dark matter or be due to higher-order nuclear effects within the standard model. The paper explicitly mentions connecting such measurements with dark matter and describes how they plan to separate a hypothetical new-boson signal from nuclear effects.

**Sensitivity to New Physics of Isotope Shift Studies using the Coronal Lines of Highly Charged Calcium Ions**Nils-Holger Rehbehn, ..., José R. Crespo López-Urrutia (2021)

Promising searches for new physics beyond the current Standard Model (SM) of particle physics are feasible through isotope-shift spectroscopy, which is sensitive to a hypothetical fifth force between the neutrons of the nucleus and the electrons of the shell. Such an interaction would be mediated by 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 series of isotopes. Subtle deviations from the expected linearity could reveal such a fifth force. Here, we study experimentally and theoretically six transitions in highly charged ions of Ca, an element with five stable isotopes of zero nuclear spin. Some of the transitions are suitable for upcoming high-precision coherent laser spectroscopy and optical clocks. Our results provide a sufficient number of clock transitions for -- in combination with those of singly charged Ca$^+$ -- application of the generalized King plot method. This will allow future high-precision measurements to remove higher-order SM-related nonlinearities and open a new door to yet more sensitive searches for unknown forces and particles.

The paper addresses this topic directly. It discusses how isotope-shift spectroscopy can be used to search for new, unknown physics beyond the Standard Model, possibly related to dark matter. They do this with trapped, highly charged calcium ions, and potential dark matter interactions are inferred from anomalies in otherwise expected patterns. Therefore, the paper seems to closely align with the topic of interest.

**Coherent excitation of the highly forbidden electric octupole transition in ${}^{172}$Yb$^+$**Henning A. Fürst, ..., Tanja E. Mehlstäubler (2020)

We report on the first coherent excitation of the highly forbidden $^2S_{1/2}\rightarrow{}^2F_{7/2}$ electric octupole (E3) transition in a single trapped ${}^{172}$Yb$^+$ ion, an isotope without nuclear spin. Using the transition in ${}^{171}$Yb$^+$ as a reference, we determine the transition frequency to be $642\,116\,784\,950\,887.6(2.4)\,$Hz. We map out the magnetic field environment using the forbidden $^2S_{1/2} \rightarrow{}^2D_{5/2}$ electric quadrupole (E2) transition and determine its frequency to be $729\,476\,867\,027\,206.8(4.4)\,$Hz. Our results are a factor of $1\times10^5$ ($3\times10^{5}$) more accurate for the E2 (E3) transition compared to previous measurements. The results open up the way to search for new physics via precise isotope shift measurements and improved tests of local Lorentz invariance using the metastable $^2F_{7/2}$ state of Yb$^+$.

The paper is indeed focused on precision measurements of isotope shifts in trapped ions, in this case ${}^{171}$Yb$^+$ and ${}^{172}$Yb$^+$ ions. It highlights how these measurements can open up ways to find new physics via precise isotope shifts. Specific attention is paid to achieving measurements at the Hz level around the electric octupole (E3) and electric quadrupole (E2) transitions. Although the paper’s main emphasis does not appear to be specifically on probing dark matter but rather broader extensions of the Standard Model of Physics, it does mention the investigation of a 5th force mediated by an unknown boson that would link electrons and neutrons, which has potential relevance to dark matter research.

**Improved isotope-shift-based bounds on bosons beyond the Standard Model through measurements of the $^2$D$_{3/2} - ^2$D$_{5/2}$ interval in Ca$^+$**Cyrille Solaro, ..., Michael Drewsen (2020)

We perform high-resolution spectroscopy of the $3$d$~^2$D$_{3/2} - 3$d$~^2$D$_{5/2}$ interval in all stable even isotopes of $^A$Ca$^+$ (A = 40, 42, 44, 46 and 48) with an accuracy of $\sim$ 20 Hz using direct frequency-comb Raman spectroscopy. Combining these data with isotope shift measurements of the 4s$~^2$S$_{1/2} \leftrightarrow 3$d$~^2$D$_{5/2}$ transition, we carry out a King plot analysis with unprecedented sensitivity to coupling between electrons and neutrons by bosons beyond the Standard Model. Furthermore, we estimate the sensitivity to such bosons from equivalent spectroscopy in Ba$^+$ and Yb$^+$. Finally, the data yield isotope shifts of the 4s$~^2$S$_{1/2} \leftrightarrow 3$d$~^2$D$_{3/2}$ transition at 10 part-per-billion through combination with recent data of Knollmann et al (2019).

The paper presents high-precision spectroscopic measurements of energy level intervals in various isotopes of calcium ions, using these isotope shifts to set constraints on new bosons beyond the Standard Model, which could include dark matter candidates. The methodology described aligns with the objectives given by the researcher, particularly in the use of high-resolution techniques like frequency-comb Raman spectroscopy and King plot analysis to detect faint signals that may point to dark matter interactions. The article discusses the limits of these methods in testing dark matter hypotheses, which is valuable for understanding the extent to which isotope shifts can be used to search for new physics. It addresses directly the need for precision measurements and how they translate into bounds for potential dark matter interaction mediators and appears to be rather at the forefront of utilizing trapped ion spectroscopy for dark matter detection.

**Generalized King linearity and new physics searches with isotope shifts**Julian C. Berengut, ..., Yotam Soreq (2020)

Atomic spectral lines for different isotopes are shifted, revealing a change in the properties of the nucleus. For spinless nuclei such isotope shifts for two distinct transitions are expected to be linearly related, at least at leading order in a change of the nuclear mass and charge distribution. Looking for a breaking of linearity in so-called King plots was proposed as a novel method to search for physics beyond the standard model. In the light of the recent experimental progress in isotope shift spectroscopy, the sensitivity of these searches will become limited by the determination of the isotope masses and/or by nuclear effects which may induce nonlinearities at an observable level. In this work, we propose two possible generalizations of the traditional King plot that overcome these limitations by including additional isotope shift measurements, thus significantly extending the new physics reach of King plots in a purely spectroscopy-driven approach.

The paper in question addresses the subject of isotope shifts in atomic spectral lines, which is directly related to the topic of interest. It discusses the nonlinearity in King plots as a method to search for physics beyond the standard model, which includes probing for new interactions that may result from dark matter. Furthermore, it speaks to the precision measurement improvements in isotope shift spectroscopy, citing experimental advances that align with the exact methods described in the researcher's goal. However, the paper does not explicitly mention dark matter candidates or their detection using trapped ions. While it does discuss new physics and anomalies which could be related to dark matter, the connection is indirect and would require the researcher to infer that the methods and results could be applied to dark matter searches.

**Probing new light force-mediators by isotope shift spectroscopy**Julian C. Berengut, ..., Yotam Soreq (2017)

In this Letter we explore the potential of probing new light force-carriers, with spin-independent couplings to the electron and the neutron, using precision isotope shift spectroscopy. We develop a formalism to interpret linear King plots as bounds on new physics with minimal theory inputs. We focus only on bounding the new physics contributions that can be calculated independently of the Standard Model nuclear effects. We apply our method to existing Ca+ data and project its sensitivity to possibly existing new bosons using narrow transitions in other atoms and ions (specifically, Sr and Yb). Future measurements are expected to improve the relative precision by five orders of magnitude, and can potentially lead to an unprecedented sensitivity for bosons within the 10 keV to 10 MeV mass range.

The paper explores probing new light force-carriers with spin-independent couplings to the electron and the neutron, using precision isotope shift spectroscopy - this aligns with the topic's interest in isotope shift measurements. The authors also look into advancing the state of the art in isolating new physics from the calculations independently of the Standard Model nuclear effects, which gives a unique perspective to isolate possible dark matter influenced anomalies. However, the paper does not clearly emphasize trapped ions or the examination of dark matter, focusing instead on new potential bosons.

**Oscillating nuclear charge radii as sensors for ultralight dark matter**Abhishek Banerjee, ..., Marianna Safronova (2023)

We show that coupling of ultralight dark matter (UDM) to quarks and gluons would lead to an oscillation of the nuclear charge radius for both the quantum chromodynamics (QCD) axion and scalar dark matter. Consequently, the resulting oscillation of electronic energy levels could be resolved with optical atomic clocks, and their comparisons can be used to investigate UDM-nuclear couplings, which were previously only accessible with other platforms. We demonstrate this idea using the ${}^2S_{1/2} (F=0)\leftrightarrow {}^2F_{7/2} (F=3)$ electric octupole and ${}^2S_{1/2} (F=0)\leftrightarrow \,{}^2D_{3/2} (F=2)$ electric quadrupole transitions in ${}^{171}Yb^+$. Based on the derived sensitivity coefficients for these two transitions and a long-term comparison of their frequencies using a single trapped ${}^{171}Yb^+$ ion, we find bounds on the scalar UDM-nuclear couplings and the QCD axion decay constant. These results are at a similar level compared to the tightest spectroscopic limits, and future investigations, also with other optical clocks, promise significant improvements.

This paper is highly relevant to the research topic. It discusses using trapped ions (specifically 171Yb+) and optical atomic clocks to sense oscillations in nuclear charge radius induced by ultralight dark matter (UDM), which lead to frequency shifts in electronic transitions. While the work presented in the paper is not focused on isotope shifts per se, it is closely related as it deals with precision measurements of atomic transitions in ions influenced by dark matter candidates. The demonstrated sensitivity to dark matter-nuclear couplings would be integral to any thorough examination of dark matter effects on atomic systems. Moreover, techniques described here may inform or be adapted to studies involving isotope shifts.

**Non-linear isotope-shift effects in Be-like, B-like, and C-like argon**V. A. Yerokhin, ..., P. O. Schmidt (2019)

Violation of linearity of the King plot is investigated for a chain of partially stripped argon isotopes. The nonlinearity originates within the Standard Model from subtle contributions to the isotope shifts from next-to-leading order effects, which have never been systematically studied so far. In light atoms these nonlinear effects are dominated by the quadratic nuclear recoil ($\propto 1/M^2$ where $M$ is the nuclear mass). Large-scale relativistic calculations of the linear and quadratic mass shift and the field shift are performed for the $2P$ fine-structure transitions in Be-like, B-like, and C-like argon ions. Nonlinearities of the King plots from 5 to 30 kHz are found, which is four orders of magnitude larger than previous estimates in comparable systems. Accurate calculations of these effects are vital for identification of possible nonlinearities originating from physics beyond the Standard Model.

The paper in question is highly relevant to the desired topic. It delves into the study of isotope shifts using argon ions and alludes to the possibility of new physics boson fields causing nonlinearity in the King plot, which could be utilized to put constraints on dark matter candidates. Also, the paper suggests that precision measurements in this study could potentially reveal new physics beyond the Standard Model. However, the paper does not explicitly mention the use of trapped ions for measurements, nor does it specifically connect its findings to dark matter candidates.

**Constraining New Physics Models with Isotope Shift Spectroscopy**Claudia Frugiuele, ..., Matthias Schlaffer (2016)

Isotope shifts of transition frequencies in atoms constrain generic long- and intermediate-range interactions. We focus on new physics scenarios that can be most strongly constrained by King Linearity Violation such as models with B-L vector bosons, Higgs portal and chameleon. With the anticipated precision, King Linearity Violation has the potential to set the strongest laboratory bounds on these models in some regions of parameter space. Furthermore, we show that this method can probe the couplings relevant for the protophobic interpretation of the recently reported Be anomaly. We extend the formalism to include an arbitrary number of transitions and isotope pairs and fit the new physics coupling to the currently available isotope shift measurements.

This paper discusses isotope shift spectroscopy in the context of testing new physics beyond the Standard Model, including dark matter interaction models involving B-L vector bosons and other scenarios such as chameleon and Higgs portal models. The detailed examination of King Linearity Violation is promising as it is closely related to the effects that dark matter might have on isotope shifts -- something directly applicable to the topic. Although the paper does not seem to focus specifically on trapped ions, the principles of isotope shift measurements and the potential for constraining dark matter candidates are directly relevant. It extends the formalism of these measurements and emphasizes the role of atomic physics observables, which is also integral to the research question at hand.

**Testing Standard Model extensions with few-electron ions**Vincent Debierre, ..., Chistoph H. Keitel (2022)

When collecting spectroscopic data on at least four isotopes, nonlinearities in the King plot are a possible sign of Physics beyond the Standard Model. In this work, an improved approach to the search for hypothetical new interactions with isotope shift spectroscopy of few-electron ions is presented. Very careful account is taken of the small nuclear corrections to the energy levels and the gyromagnetic factors, which cause deviations from King linearity within the Standard Model and are hence a possible source of confounds. In this new approach, the experimental King nonlinearity is not compared to the vanishing prediction of the Standard Model at the leading order, but to the calculated full Standard Model contribution to King nonlinearity. This makes searching for beyond-the-Standard-Model physics with King linearity analysis possible in a very-high-precision experimental regime, avoiding confounds. The bounds which can be set on beyond-the-Standard-Model parameters remain limited by the uncertainties on the small Standard Model nuclear corrections which cause King nonlinearity. Direct comparison between theory and experiment on a single pair of isotopes is advocated as a more suitable approach for few-electron ions.

The article 'Testing Standard Model extensions with few-electron ions' discusses precision measurements of isotope shifts in ions and their potential to reveal new physics beyond the Standard Model, which could include dark matter interactions. While the primary focus appears to be on testing the Standard Model rather than explicitly searching for dark matter, the methodology and analysis it describes are critical for interpreting anomalies that could be attributed to dark matter. The paper's emphasis is on examining King plot nonlinearity and accounting for small nuclear corrections within the Standard Model, which is essential for ensuring that signals attributed to new physics are not false positives from known phenomena. This level of precision and the approach to eliminating confounding factors align well with the techniques needed to specifically target dark matter signatures in isotope shifts.

**Prospects for optical clocks combining high sensitivity to new physics with insensitivity to perturbations; the case of Sb$^{+}$, Au$^{+}$, and Hg$^{2+}$**Saleh O. Allehabi, ..., V. V. Flambaum (2023)

Our study is motivated by the prospect of several metastable states in the Sb$ ^{+} $, Au$ ^{+} $, and Hg$ ^{2+} $ ions being used as possible candidates for optical clocks. We calculate several atomic properties relevant to the development of optical clocks for those clock transitions using two different approaches of relativistic many-body calculations, configuration interaction with single-double coupled-cluster (CI+SD) method, and configuration interaction with perturbation theory (CIPT) method. Our results demonstrate that the relative black body radiation shifts for these transitions are small, $ \sim 10 ^{-16} $. It is also found that there is considerable sensitivity to new physics, as evidenced by a significant enhancement of the effect of the time variation of the fine structure constant $\alpha$ on the frequencies of the clock transitions. The corresponding factor ranges from -5.56 to 2.20. Our results are compared with available previous data.

The discussed paper mainly focuses on the use of precise measurements in optical clocks and how they could be sensitive to the fine-structure constant variations, which is an aspect of new physics beyond the standard model. It introduces ions of Sb+, Au+ and Hg2+ as possible candidates and mentions the possibility of using the variability in isotopes to study nuclear structure and new interactions. While dark matter isn't discussed precisely, the methods described have overlaps with the desired topic - particularly the precise measurements of frequencies and trapped ions. Such methodologies could potentially be used in studying isotope shifts caused by dark matter-induced anomalies, as well. This paper appears to add to the repertoire of methods used to detect new physics and could provide valuable context in developing an experimental approach

**Observation of non-linearity of generalized King plot in the search for new boson**Koki Ono, ..., Yoshiro Takahashi (2021)

We measure isotope shifts for neutral Yb isotopes on an ultranarrow optical clock transition $^{1}\text{S}_{0}-^{3}\text{P}_{0}$ with an accuracy of a few Hz. The part-per-billion precise measurement was possible by loading the ultracold atoms into a three-dimensional magic-wavelength optical lattice and alternately interrogating the isotope pairs, thus minimizing the effects due to the optical lattice light-shift and inter-atomic interaction as well as the drifts of a clock laser frequency and a magnetic field. The determined isotope shifts, combined with one of the recently reported isotope-shift measurements of Yb$^+$ on two optical transitions, allow us to construct the King plots. Extremely large nonlinearity with the corresponding $\chi^2$ on the order of $10^4$ is revealed, and is not explained by a quadratic field shift. We further carry out the generalized King plot for three optical transitions so that we can eliminate the contribution arising from a higher-order effect within the Standard Model which might explain the observed nonlinearity of King plots for two transitions. Our analysis of the generalized King plot shows a deviation from linearity at the 3$\sigma$ level, indicating that there exist at least two higher order contributions in the measured isotope shifts. Then, under the reasonable assumption to attribute them to higher-order field shifts within the Standard Model, we obtain the upper bound of the product of the couplings for a new boson mediating a force between electrons, and neutrons $|y_ey_n|/(\hbar c)< 1\times10^{-10}$ for the mass less than 1 keV with the 95% confidence level is derived, providing an important step towards probing new physics via isotope-shift spectroscopy.

This paper centers on obtaining precise measurements of isotope shifts for neutral Yb isotopes in order to explore alternative physics beyond the Standard Model. Although the methodology used in the paper does not explicitly involve trapped ions, it does focus on precision measurement of isotope shifts, which are the core of the topic. Of note, the study discusses the potential existence of a novel boson, a particle beyond the Standard Model, which could provide insight into dark matter. The methodology and findings could be relevant, even if the technique used does not involve directly the trapping of ions.

**Atomic clocks highly sensitive to the variation of the fine structure constant based on Hf II, Hf IV, and W VI ions**Saleh O. Allehabi, ..., V. V. Flambaum (2022)

We demonstrate that several metastable excited states in Hf$~$II, Hf$~$IV and W$~$VI ions may be good clock states since they are sufficiently long-living and are not sensitive to the perturbations. Cooling E1 transitions are available. Energy levels, Land\'{e} $g$-factors, transition amplitudes for electric dipole (E1), electric quadrupole (E2), and magnetic dipole (M1) transitions, lifetimes, and electric quadrupole moments for Hf$~$II, Hf$~$IV, and W$~$VI ions are investigated using a combination of several methods of relativistic many-body calculations including the configuration interaction (CI), linearized coupled-cluster single-doubles (SD) and many-body perturbation theory (CI+SD), and also the configuration interaction with perturbation theory (CIPT). Scalar polarizabilities of the ground states and the clock states have been calculated to determine the black body radiation (BBR) shifts. We have found that the relative BBR shifts for these transitions range between 10$^{-16}$ $-$ 10$^{-18}$. A linear combination of two clock transition frequencies allows one to further suppress BBR. Several $5d$ - $6s$ single-electron clock transitions ensure high sensitivity of the transition frequencies to the variation of the fine structure constant $\alpha$ and may be used to search for dark matter producing this variation of $\alpha$. The enhancement coefficient for $\alpha$ variation reaches $K=8.3$. Six stable isotopes of Hf and 5 stable isotopes in W allow one to make King plots and search for new interactions mediated by scalar particles or other mechanisms.

This paper by Allehabi et al. focuses on precision measurements using atomic clocks based on certain ions. It discusses ways to heighten sensitivity to the variation of the fine structure constant α, which they suggest can be applied to search for dark matter. The authors have considered stable isotopes of Hf and W for their analysis, which aligns with the interest in isotope shifts for probing dark matter. Moreover, in their references, they cite various works related to the use of atomic clocks, isotope shifts, and spectroscopy for the detection of dark matter and new particle interactions. Importantly, the paper focuses on highly accurate measurements and mentions the need for new physics, but the direct usage of trapped ions isn't clearly emphasised.

**Improved limits on the coupling of ultralight bosonic dark matter to photons from optical atomic clock comparisons**M. Filzinger, ..., N. Huntemann (2023)

We present improved constraints on the coupling of ultralight bosonic dark matter to photons based on long-term measurements of two optical frequency ratios. In these optical clock comparisons, we relate the frequency of the ${}^2S_{1/2} (F=0)\leftrightarrow {}^2F_{7/2} (F=3)$ electric-octupole (E3) transition in $^{171}$Yb$^{+}$ to that of the ${}^2S_{1/2} (F=0)\leftrightarrow \,{}^2D_{3/2} (F=2)$ electric-quadrupole (E2) transition of the same ion, and to that of the ${}^1S_0\leftrightarrow\,{}^3P_0$ transition in $^{87}$Sr. Measurements of the first frequency ratio $\nu_\textrm{E3}/\nu_\textrm{E2}$ are performed via interleaved interrogation of both transitions in a single ion. The comparison of the single-ion clock based on the E3 transition with a strontium optical lattice clock yields the second frequency ratio $\nu_\textrm{E3}/\nu_\textrm{Sr}$. By constraining oscillations of the fine-structure constant $\alpha$ with these measurement results, we improve existing bounds on the scalar coupling $d_e$ of ultralight dark matter to photons for dark matter masses in the range of about $ 10^{-24}-10^{-17}\,\textrm{eV}/c^2$. These results constitute an improvement by more than an order of magnitude over previous investigations for most of this range. We also use the repeated measurements of $\nu_\textrm{E3}/\nu_\textrm{E2}$ to improve existing limits on a linear temporal drift of $\alpha$ and its coupling to gravity.

The given paper reports long-term measurements of two optical frequency ratios involving certain isotopes ($^{171}$Yb$^{+}$ and $^{87}$Sr), which are similar to what the researcher is looking for. While this paper is not researching isotope shifts per se, it does focus on comparing transitions between two isotopes of an ion under the influence of dark matter. The research presented in the paper is aimed at limiting the coupling of specific dark matter particles (ultralight bosonic dark matter) to photons. While the primary focus of the paper is not exactly the same as the researcher's topic, it employs similar methods and is investigating a closely allied area.

We introduce a new linear relation in the isotope shifts of atomic spectroscopy. While the famous King relation is the linear relation among the different transitions, the new one is the linear relation among the different isotope pairs. Since we obtain this relation by exchanging the roles of the electronic and the nuclear factors in the original relation, we call it the dual King relation. This relation shows us similar information to the original King relation without including new fit parameters when we measure the isotope shifts of many transitions. In the dual King relation, the fit coefficients consist of the nuclear factors. Then, the fit results give us constraints to the ratios of the isotope dependence independent of the electron wave functions. This helps us to test the origin of unknown higher order isotope shifts. We show that the dual King relation can also be employed to constrain the weakly interacting light new boson at the same level as the original King relation.

This paper introduces a new linear relation in the isotope shifts of atomic spectroscopy, known as the 'dual King relation'. Such isotope shifts are precisely what is required for observing potential dark matter interactions. Furthermore, the paper mentions the measurement of isotope shifts in highly charged calcium ions, aligning with the element of 'trapped ions' in the desired research topic. Notably, while the paper discusses detecting 'weakly interacting light bosons', it does not explicitly mention dark matter. However, such bosons could serve as dark matter candidates.

**Two clock transitions in neutral Yb for the highest sensitivity to variations of the fine-structure constant**M. S. Safronova, ..., Jun Ye (2018)

We propose a new frequency standard based on a $4f^{14} 6s6p~ ^3\!P_0 - 4f^{13} 6s^2 5d ~(J=2)$ transition in neutral Yb. This transition has a potential for high stability and accuracy and the advantage of the highest sensitivity among atomic clocks to variation of the fine-structure constant $\alpha$. We find its dimensionless $\alpha$-variation enhancement factor to be $K=-15$, in comparison to the most sensitive current clock (Yb$^+$ E3, $K=-6$), and it is 18 times larger than in any neutral-atomic clocks (Hg, $K=0.8$). Combined with the unprecedented stability of an optical lattice clock for neutral atoms, this high sensitivity opens new perspectives for searches for ultralight dark matter and for tests of theories beyond the standard model of elementary particles. Moreover, together with the well-established $^1\!S_0 -\, ^3\!P_0$ transition one will have two clock transitions operating in neutral Yb, whose interleaved interrogations may further reduce systematic uncertainties of such clock-comparison experiments.

The paper presented proposes a new frequency standard based on the transitions in neutral Ytterbium (Yb). The authors suggest that this transition has a high sensitivity to variations in the fine-structure constant, making it suitable for probing potential variations due to dark matter interactions. Furthermore, the paper discusses the implications of scalar bosonic dark matter, which could induce fluctuations in atomic frequencies. It also discusses the use of precision isotope shift spectroscopy to test for new light force carriers, an area related to dark matter research. Although the paper focuses more on neutral atoms rather than trapped ions, it directly addresses attempts to observe dark matter effects through atomic frequency shifts, which is a relevant aspect of the desired topic.

**Isotope Shifts in Cadmium as a Sensitive Probe for Physics Beyond the Standard Model**B. Ohayon, ..., B. K. Sahoo (2022)

Isotope shifts (ISs) in atomic energy levels are sensitive probes of nuclear structure and new physics beyond the Standard Model. We present an analysis of the ISs of the cadmium atom (Cd I) and singly charged cadmium ion (Cd II). ISs of the 229 nm, 326 nm, 361 nm and 480 nm lines of Cd I are measured with a variety of techniques; buffer-gas-cooled beam spectroscopy, capturing atoms in a magneto-optic-trap, and optical pumping. IS constants for the D1 and D2 lines of Cd II are calculated with high accuracy by employing analytical response relativistic coupled-cluster theory in the singles, doubles and triples approximations. Combining the calculations for Cd II with experiments, we infer IS constants for all low-lying transitions in Cd I. We benchmark these constants as calculated via different many-body methods. Our calculations for Cd II enable nuclear charge radii of Cd isotopes to be extracted with unprecedented accuracy. The combination of our precise calculations and measurements shows that King Plots for Cd I can improve the state-of-theart sensitivity to a new heavy boson by up to two orders of magnitude.

The presented paper investigates isotope shifts in the cadmium atom and its ion with different experimental techniques and for several transition lines, including the use of trapped ions in magneto-optic traps. Potential deviations within these isotope shifts might potentially be indicative of new physics models beyond our standard understanding. These deviations, however, aren't specifically geared towards modeling the impact of dark matter, but rather opened up to a more general collection of beyond-the-Standard-Model theories. That said, the methods and results may still indirectly contribute to the understanding of how dark matter might couple to atomic systems via isotope shift interactions.

**Opportunities for Fundamental Physics Research with Radioactive Molecules**Gordon Arrowsmith-Kron, ..., Xiaofei Yang (2023)

Molecules containing short-lived, radioactive nuclei are uniquely positioned to enable a wide range of scientific discoveries in the areas of fundamental symmetries, astrophysics, nuclear structure, and chemistry. Recent advances in the ability to create, cool, and control complex molecules down to the quantum level, along with recent and upcoming advances in radioactive species production at several facilities around the world, create a compelling opportunity to coordinate and combine these efforts to bring precision measurement and control to molecules containing extreme nuclei. In this manuscript, we review the scientific case for studying radioactive molecules, discuss recent atomic, molecular, nuclear, astrophysical, and chemical advances which provide the foundation for their study, describe the facilities where these species are and will be produced, and provide an outlook for the future of this nascent field.

The provided paper dives deep into various aspects of physics and mentions a number of tools and theories that are indirectly related to precision measurement of isotopic shifts and their implications on dark matter. Specifically, the paper discusses the opportunity of developing high-precision clocks which can detect oscillatory and transient variation of fundamental constants, hence their relevance in dark matter searches. Furthermore, the authors discuss precision isotope-shift atomic spectroscopy, which can be used in searching for a theoretical fifth force between neutron and electron interactions. Although the research contained in this paper is quite broad and not exclusively oriented towards the topic, there are useful insights particularly in relation to fifth-force searches with precision isotope measurements and their implications for research on dark matter.

**First Laboratory Bounds on Ultralight Dark Photon Dark Matter from Precision Atomic Spectroscopy**Joshua Berger, ..., Amit Bhoonah (2022)

Ultralight bosonic dark matter has come under increasing scrutiny as a dark matter candidate that has the potential to resolve puzzles in astronomical observation. We demonstrate that high-precision measurements of time variation in the frequency ratios of atomic transitions achieves leading sensitivity to ultralight vector portal dark matter at low masses. These bounds are the first laboratory-based bounds on this class of dark matter models. We propose further measurements that could enhance sensitivity to ultralight dark photons.

The provided paper does discuss high-precision measurements of atomic transitions as a tool for probing potential dark matter candidates, specifically ultralight dark photon dark matter. It focuses on mapping time variations in atomic transition frequency ratios to detect these candidates. However, the paper doesn't directly address the influence of dark matter on isotope shifts in trapped ions, nor does it specifically study the shifts between different isotopes of the same ion to detect dark matter. Moreover, it discusses different atomic species (e.g., Al+, Sr, Yb) rather than different isotopes of the same ion.

**Scattering of light dark matter in atomic clocks**Peter Wolf, ..., Diego Blas (2018)

We present a detailed analysis of the effect of light Dark Matter (DM) on atomic clocks, for the case where DM mass and density are such that occupation numbers are low and DM must be considered as particles scattering off the atoms, rather than a classical field. We show that the resulting atomic clock frequency shifts are first order in the scattering amplitudes, and particularly suited to constrain DM models in the regime where the DM mass $m_\chi \ll$ GeV. We provide some rough order of magnitude estimates of sensitivity that can be confronted to any DM model that allows for non zero differential scattering amplitudes of the two atomic states involved in the clock.

This article investigates the impact of light dark matter on the frequency of atomic clocks. It discusses measurement strategies and provides estimates of sensitivity for dark matter detection with atomic clocks. However, it does not focus on trapped ions or isotope shifts in such ions specifically. The techniques and models might be relevant, but the emphasis is on detecting dark matter through its interaction with atomic clock atoms, not through precision measurements of isotope shifts in trapped ions.

**Relativistic effects in search for new intra-atomic force with isotope shifts**Minoru Tanaka, ..., Yasuhiro Yamamoto (2019)

Isotope shift of atomic spectra is considered as a probe of new interaction between electrons and neutrons in atoms. We employ the method of seeking a breakdown of King's linearity in the isotope shifts of two atomic transitions. In the present work, we evaluate the magnitudes of the nonlinearity using relativistic wave functions and the result is compared with that of nonrelativistic wave functions in our previous work. It turns out that the nonrelativistic calculation underestimates the nonlinearity owing to the new interaction in the mass range of the mediator greater than 1 MeV. Further, we find that the nonlinearity within the standard model of particle physics is significantly magnified by the relativistic effect in the $\text{p}_{1/2}$ state. To get rid of this obstacle in the new physics search, we suggest to avoid $\text{p}_{1/2}$, and use $\text{p}_{3/2}$ instead for example.

This paper discusses the application of isotope shifts in atomic spectra as a method to probe new interactions within atoms that could include forces mediated by dark matter. It examines the influence of relativistic effects on the calculations and suggests methods to refine the search for new physics, which relates closely to the concept of using precise measurements to detect dark matter. However, the paper does not explicitly mention 'trapped ions' nor does it discuss the methodology or experimental setup specific to trapped ions, which is a core component of the desired research topic. The focus is more on the theoretical framework and quantifying nonlinearity in isotope shifts, which while helpful, does not fully encompass the applied approach of measuring energy transitions in trapped ions to detect dark matter.

**Spectroscopy of the $^2S_{1/2} \rightarrow\,^2P_{3/2}$ transition in Yb II: Isotope shifts, hyperfine splitting and branching ratios**Thomas Feldker, ..., Rene Gerritsma (2017)

We report on spectroscopic results on the $^2S_{1/2} \rightarrow\,^2P_{3/2}$ transition in single trapped Yb$^+$ ions. We measure the isotope shifts for all stable Yb$^+$ isotopes except $^{173}$Yb$^+$, as well as the hyperfine splitting of the $^2P_{3/2}$ state in $^{171}$Yb$^+$. Our results are in agreement with previous measurements but are a factor of 5-9 more precise. For the hyperfine constant $A\left(^2P_{3/2}\right) = 875.4(10)$ MHz our results also agree with previous measurements but deviate significantly from theoretical predictions. We present experimental results on the branching ratios for the decay of the $^2P_{3/2}$ state. We find branching fractions for the decay to the $^2D_{3/2}$ state and $^2D_{5/2}$ state of 0.17(1)% and 1.08(5)%, respectively, in rough agreement with theoretical predictions. Furthermore, we measured the isotope shifts of the $^2F_{7/2} \rightarrow\,^1D\left[5/2\right]_{5/2}$ transition and determine the hyperfine structure constant for the $^1D\left[5/2\right]_{5/2}$ state in $^{171}$Yb$^+$ to be $A\left(^1D\left[5/2\right]_{5/2}\right) = -107(6)$ MHz.

The paper investigates highly accurate measurements on the isotope shifts in single trapped ions, taking Yb$^+$ as an example. These measurements pertain to energy transition frequencies, encapsulating a central component of the desired topic. However, it lacks the explicit intent of using these measurements for constraining dark matter candidates. The paper does not discuss the role of such measurements in probing anomalies that could potentially be caused by dark matter interactions, an essential aspect of the research theme.

**The effect of realistic nuclear charge distributions on isotope shifts and towards the extraction of higher order nuclear radial moments**A. Papoulia, ..., J. Ekman (2016)

Background: Atomic spectral lines from different isotopes display a small shift in energy, commonly referred to as the line isotope shift. One of the components of the isotope shift is the field shift, which depends on the extent and the shape of the nuclear charge density distribution. Purpose: To investigate how sensitive field shifts are with respect to variations in the nuclear size and shape and what information of nuclear charge distributions that can be extracted from measured field shifts. Methods: Nuclear properties are obtained from nuclear density functional theory calculations based on the Skyrme-Hartree-Fock-Bogoliubov approach. These results are combined with multiconfiguration Dirac-Hartree-Fock methods to obtain realistic field shifts. Results: Phenomena such as nuclear deformation and variations in the diffuseness of nuclear charge distributions give measurable contributions to the field shifts. Using a novel approach, we demonstrate the possibility to extract new information concerning the nuclear charge densities from the observed field shifts. Conclusions: Combining methods used in atomic and nuclear structure theory gives an improved description of field shifts. Extracting additional nuclear information from measured field shifts is possible in the near future with improved experimental methods.

The paper primarily focuses on detailing the impact of nuclear charge distributions on isotope shifts in atomic spectra and how these understandings can be improved with advancements in experimental methods. While the study of isotope shifts is central to the paper, which is relevant to the topic, the context of trapping ions or connecting these shifts directly to dark matter candidates is not specifically addressed. The main objective revolves around nuclear charge distributions' effects and a method to extract nuclear information, rather than using isotope shifts to directly search for dark matter-related anomalies.

**Snowmass 2021: Quantum Sensors for HEP Science -- Interferometers, Mechanics, Traps, and Clocks**Oliver Buchmueller, ..., Marianna S. Safronova (2022)

A wide range of quantum sensing technologies are rapidly being integrated into the experimental portfolio of the high energy physics community. Here we focus on sensing with atomic interferometers; mechanical devices read out with optical or microwave fields; precision spectroscopic methods with atomic, nuclear, and molecular systems; and trapped atoms and ions. We give a variety of detection targets relevant to particle physics for which these systems are uniquely poised to contribute. This includes experiments at the precision frontier like measurements of the electron dipole moment and electromagnetic fine structure constant and searches for fifth forces and modifications of Newton's law of gravity at micron-to-millimeter scales. It also includes experiments relevant to the cosmic frontier, especially searches for gravitional waves and a wide variety of dark matter candidates spanning heavy, WIMP-scale, light, and ultra-light mass ranges. We emphasize here the need for more developments both in sensor technology and integration into the broader particle physics community.

The evaluated paper talks about precision spectroscopy methods and how they can be applied to a variety of detecting targets including dark matter. Some sections of the paper focus on atomic clocks' sensitivity to fundamental constants variation, implying potential dark matter interaction. There is also a mention of different atomic clock transitions that can detect ultralight dark matter by comparing frequency variations. Furthermore, the paper does discuss experiments based on nuclear decays and leverages ion traps for the study of radioactive isotopes. However, it does not directly talk about measuring the frequency shifts in trapped ions or leveraging such shifts to estimate observable effects of dark matter.

**Precision determination of isotope shifts in ytterbium and implications for new physics**N. L. Figueroa, ..., D. Antypas (2021)