Joint mass-and-energy test of the equivalence principle at the ${10}^{-10}$ level using atoms with specified mass and internal energy

We use rubidium atoms with specified mass and internal energy to carry out a joint mass-energy test of the equivalence principle (EP). We improve the four-wave double-diffraction Raman transition method (4WDR) we proposed before to select atoms with a certain mass and angular momentum state, and form a dual-species atom interferometer. By using the extended 4WDR to ${}^{85}\mathrm{Rb}$ and ${}^{87}\mathrm{Rb}$ atoms with different angular momenta, we measure their differential gravitational acceleration, and we determine the value of the Eötvös parameter, $\eta$, which measures the strength of the violation of EP. The Eötvös parameters of the four paired combinations ${}^{85}\mathrm{Rb}|F=2\rangle \text{−}{}^{87}\mathrm{Rb}|F=1\rangle$, ${}^{85}\mathrm{Rb}|F=2\rangle \text{−}{}^{87}\mathrm{Rb}|F=2\rangle$, ${}^{85}\mathrm{Rb}|F=3\rangle \text{−}{}^{87}\mathrm{Rb}|F=1\rangle$, and ${}^{85}\mathrm{Rb}|F=3\rangle \text{−}{}^{87}\mathrm{Rb}|F=2\rangle$ were measured to be ${\eta }_1=(1.5\pm 3.2)\times {10}^{-10}$, ${\eta }_2=(-0.6\pm 3.7)\times {10}^{-10}$, ${\eta }_3=(-2.5\pm 4.1)\times {10}^{-10}$, and ${\eta }_4=(-2.7\pm 3.6)\times {10}^{-10}$, respectively. The violation parameter of mass is constrained to ${\eta }_0=(-0.8\pm 1.4)\times {10}^{-10}$, and that of internal energy to ${\eta }_E=(0.0\pm 0.4)\times {10}^{-10}$ per reduced energy ratio $a$ ($a=h{\nu }_0/m_i^{85}{c}^2$, and ${\nu }_0=1$ GHz). This work opens a door for joint tests of two attributes beyond the traditional pure mass or energy tests of EP with quantum systems.

Figure 1

Schematic diagram of 4WDR-e ${}^{85}\mathrm{Rb}\text{−}{}^{87}\mathrm{Rb}$ dual-species AI. (a) Relevant sublevels covering the energy interval from micro-eV to giga-eV. Raman lasers with frequencies of ${\omega }_1$, ${\omega }_2$, and ${\omega }_3$ are used for ${}^{85}\mathrm{Rb}$ atoms, while those with frequencies ${\omega }_1$, ${\omega }_2$, and ${\omega }_4$ are for ${}^{87}\mathrm{Rb}$ atoms; ${\delta }_1$ is the detuning of ${\omega }_1$, ${\delta }_2$ is the detuning of ${\omega }_2$. ${\omega }_1$ and ${\omega }_2$ are detuned to the blue side of transitions ${}^{85}\mathrm{Rb}|F=3\rangle$ to $|F^{\prime }=4\rangle$ with a detuning of ${\mathrm{\Delta }}_1$, and ${}^{87}\mathrm{Rb}|F=2\rangle$ to $|F^{\prime }=3\rangle$ with a detuning of ${\mathrm{\Delta }}_2$. (b) The 4WDR-e configuration for ${}^{85}\mathrm{Rb}\text{−}{}^{87}\mathrm{Rb}$ dual-species AI. The blue dashed lines represent LGS atoms, and red solid lines represent UGS atoms.

Figure 2

Schematic diagram of dual-species AI and experimental setup. Experimental setup and process. PM: polarization maintaining; C field coil: constant-field coil.

Figure 3

Schematic diagram of 4WDR dual-species AIs with specified mass and internal energy. (a) ${}^{85}\mathrm{Rb}|F=2\rangle \text{−}{}^{87}\mathrm{Rb}|F=1\rangle$ dual-species AI. (b) ${}^{85}\mathrm{Rb}|F=2\rangle \text{−}{}^{87}\mathrm{Rb}|F=2\rangle$ dual-species AI. (c) ${}^{85}\mathrm{Rb}|F=3\rangle \text{−}{}^{87}\mathrm{Rb}|F=1\rangle$ dual-species AI. (d) ${}^{85}\mathrm{Rb}|F=3\rangle \text{−}{}^{87}\mathrm{Rb}|F=2\rangle$ dual-species AI.

Figure 4

Experimental data. (a) Experimentally measured $\eta$ values, where the error corresponding to the effective wave vector is corrected. Parts (a1), (a2), (a3), and (a4) are measurements for ${\eta }_1$, ${\eta }_2$, ${\eta }_3$, and ${\eta }_4$, respectively. (b) Allan deviation of ${\eta }_1$ (red squares), ${\eta }_2$ (black dots), ${\eta }_3$ (green boxes), and ${\eta }_4$ (blue triangles). (c) Dependence of $\eta$ values on energy, the intercept value of the fitted straight line ${\eta }_0=(-0.8\pm 1.4)\times {10}^{-10}$, and the slope value ${\eta }_E=(0.0\pm 0.4)\times {10}^{-10}$. $a=h{\nu }_0/m_i^{85}{c}^2$ and ${\nu }_0=1$ GHz.

Figure 5

Dependence of the ac Stark shift on the magic intensity ratio of the Raman lasers. $\mathrm{MIR}=I_1:I_2:I_3:I_4$, which is magic intensity ratio of four Raman beams with frequencies of ${\omega }_1$, ${\omega }_2$, ${\omega }_3$, and ${\omega }_4$. The red dots are data before improvement when changing the Raman laser intensity from $10\%$ to $100\%$, the maximum residual frequency shift is 6 kHz. The blue squares are data after improvement, and the frequency shift is less than 500 Hz. The error bars are obtained using Gaussian fitting.

Figure 6

Dependence of ${\eta }_1$ measurements and their residuals $R$ on the duration of blow away pulse. The blue squares are experimental data, and the red curve is the quadratic polynomial fitting. The error bars are got by the Allan deviation with an integration time of 4480 s or 8960 s. The corresponding value of $\mathrm{\Delta }{\eta }_1$ is fitted as $0.1\times {10}^{-10}$ at pulse duration of 2 ms. The red dots are residuals, and the black line is the quadratic polynomial fit, the uncertainty of $\mathrm{\Delta }{\eta }_1$ is obtained as $0.2\times {10}^{-10}$ by the deviation of residuals.

Figure 7

Dependence of $\mathrm{\Delta }{\eta }_4$ measurements on the position difference of two atom clouds. The velocity difference is 24 mm/s and the interval between velocity-selection pulses for ${}^{85}\mathrm{Rb}$ and ${}^{87}\mathrm{Rb}$ is 140 ms. The position difference is 3.36 mm, and the fitting value is $(3.6\pm 0.5)\times {10}^{-10}$/mm. The error bars are obtained by the Allan deviation with an integration time of 4480 or 8960 s.

Figure 8

Dependence of uncertainty of ${\eta }_1$ measurements and their residuals $R$ on the current C-field. The blue squares are measurements using $I_c=50$, 150, 200, 300 mA. The error bars are obtained by the Allan deviation with an integration time of 4480 or 8960 s. The red line is a fitting curve. The corresponding value of $\mathrm{\Delta }{\eta }_1$ is fitted as $0.5\times {10}^{-10}$ when the $I_c$ is extrapolated to 320 mA. The red dots are residuals, and the black line is the quadratic polynomial fit. The uncertainty of $\mathrm{\Delta }{\eta }_1$ is obtained as $0.3\times {10}^{-10}$ by the deviation of residuals.

Figure 9

Dependence of $\eta$ measurements and their residuals $R$ on the size of detection beams. The blue squares are measurements using a detection beam with a diameter of 5, 10, 15, and 20 mm. The uncertainties of four measurements are all within $1.0\times {10}^{-10}$. The error bars are obtained by the Allan deviation with an integration time of 4480 or 8960 s. The corresponding value of $\mathrm{\Delta }\eta$ is fitted as $0.5\times {10}^{-10}$ at a beam diameter of 15 mm. The red dots are residuals, and the black line is the quadratic polynomial fit. The uncertainty of $\mathrm{\Delta }{\eta }_1$ is obtained as $0.5\times {10}^{-10}$ by the deviation of residuals.