Improved Isotope-Shift-Based Bounds on Bosons beyond the Standard Model through Measurements of the ${{}^2\mathrm{D}}_{3/2}-{{}^2\mathrm{D}}_{5/2}$ Interval in ${\mathrm{Ca}}^{+}$

We perform high-resolution spectroscopy of the $3d{{}^2\mathrm{D}}_{3/2}-3d{{}^2\mathrm{D}}_{5/2}$ interval in all stable even isotopes of ${{}^A\mathrm{Ca}}^{+}$ ($A=40$, 42, 44, 46, and 48) with an accuracy of $\sim 20\mathrm{Hz}$ using direct frequency-comb Raman spectroscopy. Combining these data with isotope shift measurements of the $4s{{}^2\mathrm{S}}_{1/2}\leftrightarrow 3d{{}^2\mathrm{D}}_{5/2}$ 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 ${\mathrm{Ba}}^{+}$ and ${\mathrm{Yb}}^{+}$. Finally, the data yield isotope shifts of the $4s{{}^2\mathrm{S}}_{1/2}\leftrightarrow 3d{{}^2\mathrm{D}}_{3/2}$ transition at 10 parts per billion through combination with recent data of Knollmann, Patel, and Doret [Phys. Rev. A 100, 022514 (2019)].

Figure 1

Schematic of the experimental setup and the relevant electronic levels of the ${\mathrm{Ca}}^{+}$ ion. The isotope shift of the 732-nm transition can be deduced from the isotope shifts of the 729-nm transition and of the D-fine-structure splitting. The D-fine-structure splitting is measured successively on the different calcium isotopes by direct frequency-comb Raman spectroscopy [18]. The transition frequency is deduced from the measurement of the comb repetition rate on a frequency counter referenced to a GPS-disciplined rubidium standard. The 729-nm laser used to probe the $4s{{}^2\mathrm{S}}_{1/2}\leftrightarrow 3d{{}^2\mathrm{D}}_{5/2}$ transition is locked to an ultrastable high-finesse cavity providing a short-term linewidth $<1\mathrm{kHz}$. The absolute laser frequency is deduced from a measurement on the frequency counter of the beating between the laser and the frequency comb with the latter locked to an ultrastable laser at $1.5\mu \mathrm{m}$.

Figure 2

Two-dimensional King plot of the 732- and 729-nm transitions. The line is a fit to our data using a weighted orthogonal distance regression. The extracted fit parameters are given in the text. We point out that the isotope shift of the 732-nm transition is deduced from measurements of the isotope shift of the 729-nm transition $\delta {\nu }_{729}^{A,40}$ and of the D-splitting isotope shift $\delta {\nu }_{\mathrm{DSIS}}^{A,40}$. Hence, the measurement accuracy on $\delta {\nu }_{729}^{A,40}$ ($\delta {\nu }_{\mathrm{DSIS}}^{A,40}$) translates into an error bar parallel (perpendicular) to the fitted line, emphasizing that the analysis is limited by the achieved fractional accuracy on $\delta {\nu }_{\mathrm{DSIS}}^{A,40}$.

Figure 3

Current and projected constraints ($2\sigma$) on the coupling strength $|y_e{y}_n|/(\hslash c)$ of electrons and neutrons to a new boson $\varphi$ of mass $m_{\varphi }$. Existing bound [8] from measurements of the ${\mathrm{S}}_{1/2}\text{−}{\mathrm{P}}_{1/2}$ and ${\mathrm{D}}_{3/2}\text{−}{\mathrm{P}}_{1/2}$ transition isotope shifts in ${\mathrm{Ca}}^{+}$ with an accuracy of $\mathcal{O}(100\mathrm{kHz})$ [13] (black line, labeled as P). Constraint imposed by this work (red, solid line), limited by the $\sim 20\text{−}\mathrm{Hz}$ measurement uncertainty of the DSIS. Projection for a 10-mHz uncertainty on the DSIS (red, dotted line). Projected constraints from measurements in ${\mathrm{Ba}}^{+}$ (DSIS at 20-Hz level, green dashed line; 10 mHz, green dotted line) and in ${\mathrm{Yb}}^{+}$ isotopes (dark blue line, also for 20 Hz and 10 mHz) (for details, see [32]). The curves end at masses $m_{\varphi }$ that correspond to the inverse nuclear radii. For comparison, constraints from other experiments are shown as shaded areas [8]: fifth force [60, 61] (dark orange), $(g-2{)}_e$ measurements [49, 50] combined with neutron scattering data [51, 52, 53, 54] (light blue), or SN 1987A (light orange), and star cooling in globular clusters [44, 45, 46, 47, 48] (orange). The gray bar represents the range of $y_e{y}_n$ to explain the Be anomaly at $m_{\varphi }=17\mathrm{MeV}$ [8, 9, 10, 11, 12], while the part within the horizontal gray lines is the remaining allowed range with $y_e$ between the NA64 [55] and $\mathrm{NA}48/2$ [57] limits and $y_n$ from Ref. [58] including isospin mixing and breaking.