New Limits on Coupling of Fundamental Constants to Gravity Using ${}^{87}\mathrm{Sr}$ Optical Lattice Clocks

The ${}^{1}S_0$-${}^{3}P_0$ clock transition frequency ${\nu }_{\mathrm{Sr}}$ in neutral ${}^{87}\mathrm{Sr}$ has been measured relative to the Cs standard by three independent laboratories in Boulder, Paris, and Tokyo over the last three years. The agreement on the $1\times {10}^{-15}$ level makes ${\nu }_{\mathrm{Sr}}$ the best agreed-upon optical atomic frequency. We combine periodic variations in the ${}^{87}\mathrm{Sr}$ clock frequency with ${}^{199}\mathrm{Hg}^{+}$ and H-maser data to test local position invariance by obtaining the strongest limits to date on gravitational-coupling coefficients for the fine-structure constant $\alpha$, electron-proton mass ratio $\mu$, and light quark mass. Furthermore, after ${}^{199}\mathrm{Hg}^{+}$, ${}^{171}\mathrm{Yb}^{+}$, and H, we add ${}^{87}\mathrm{Sr}$ as the fourth optical atomic clock species to enhance constraints on yearly drifts of $\alpha$ and $\mu$.

Figure 1

(a) Spectrum of the ${}^{87}\mathrm{Sr}$ ${}^{1}S_0$-${}^{3}P_0$ clock transition with quality factor $2\times {10}^{14}$. (b) Measurements of the clock transition from JILA (circle), SYRTE (triangle), and U. Tokyo (square) over the last 3 years. Frequency data are shown relative to ${\nu }_0=429228004229800\mathrm{Hz}$. Weighted linear (dotted line) and sinusoidal (solid line) fits determine a yearly drift rate and an amplitude of annual variation. (c) Zoom into the four most recent measurements, showing agreement within $1.7\mathrm{Hz}$ and determining both drift and annual variation.

Figure 2

The upper panel shows fractional frequency drifts for ${}^{171}\mathrm{Yb}^{+}$, ${}^{87}\mathrm{Sr}$, H 1S-2S, and ${}^{199}\mathrm{Hg}^{+}$ versus their sensitivity to $\alpha$ variation relative to Cs. Sensitivity due to Cs is indicated as a dotted vertical line. A linear fit (solid line) determines yearly drift rates $\delta \alpha /\alpha$ and $\delta \mu /\mu$. The drift rate constraints from each species are shown in the lower panel as shaded bars. The fit determines a confidence ellipse (white) [3, 6, 7] with projections equal to the parameters’ 1-$\sigma$ uncertainties.

Figure 3

Earth orbiting around Sun (mass $m_{\odot }$) in gravitational potential $U$ on an orbit with semimajor axis $a$, eccentricity $ϵ$ (exaggerated to show geometry), and angular velocity $\Omega$. Earth is at radial distance $r$ from the Sun. The eccentric anomaly $E$ is the angle between the major axis and the orthogonal projection of Earth’s position onto a circle of radius $a$.

Figure 4

A fit to linear constraints on gravitational-coupling constants $k_{\alpha }$, $k_{\mu }$, and $k_q$ from three species determines a plane. Its value at $d_{\mu }=0$, $d_q=0$ is $k_{\alpha }$; its gradient along the $d_{\mu }$ ($d_q$) axis is $k_{\mu }$ ($k_q$). The table shows sensitivity constants and constraints for ${}^{87}\mathrm{Sr}$, ${}^{199}\mathrm{Hg}^{+}$, and the H maser.