Quantum Test of the Universality of Free Fall

We simultaneously measure the gravitationally induced phase shift in two Raman-type matter-wave interferometers operated with laser-cooled ensembles of ${}^{87}\mathrm{Rb}$ and ${}^{39}\mathrm{K}$ atoms. Our measurement yields an Eötvös ratio of ${\eta }_{\mathrm{Rb},\mathrm{K}}=(0.3\pm 5.4)\times {10}^{-7}$. We briefly estimate possible bias effects and present strategies for future improvements.

Figure 1

Space-time evolution of the rubidium and potassium atoms in a Mach-Zehnder-type interferometer positioned in a constant gravitational field pointing downwards. Coherent Raman processes at $t=0,T$ and $2T$ between the states $|F_i=1,p⟩$ and $|F_i=2,p+\hslash k_{\mathrm{eff},i}⟩$, where $i$ is either Rb or K resulting from the $\pi /2-$, $\pi -$, and $\pi /2$ pulses allow for momentum transfer in the downward (thick lines) and the upward (thin lines) directions. The difference in the velocity change between rubidium (black lines) and potassium (red lines) is not to scale.

Figure 2

Determination of the gravitational accelerations $g_{\mathrm{Rb}}$ and $g_{\mathrm{K}}$ of rubidium and potassium from two orientations of the interferometer determined by the upward ($+$) and downward ($-$) directions of the momentum transfer indicated in Fig. 1. We display the corresponding signals (individual data points) and the respective sinusoidal least squares fit functions (curves) for three pulse separation times $T=8\mathrm{ms}$ (blue squares, dash-dotted blue line), $T=15\mathrm{ms}$ (red circles, dashed red line), and $T=20\mathrm{ms}$ (black diamonds, solid black line). Because of bias contributions that are independent of the direction of the momentum transfer, the values $a_i^{(\pm )}(g)$ are offset symmetrically around their half difference $[a_i^{(+)}(g)-a_i^{(-)}(g)]/2$ marked for rubidium (solid vertical line). The displayed data sets have been adjusted in their signal offsets. Moreover, we have subtracted a linear fitting function from the data sets to correct for slow offset drifts in the detection caused by variations in the arrival time.

Figure 3

Normalized Allan deviations ${\sigma }_{\mathrm{Rb}}$, ${\sigma }_{\mathrm{K}}$, and ${\sigma }_{\eta }$ of the signals providing us with the accelerations $g_{\mathrm{Rb}}$ and $g_{\mathrm{K}}$ of rubidium (black circles) potassium (red squares), and of the Eötvös ratio ${\eta }_{\mathrm{Rb},\mathrm{K}}$ (blue diamonds) and its asymptotic behavior (dashed line), respectively, in their dependence on the integration time $\tau$. We achieve a statistical uncertainty of the Eötvös ratio of ${\sigma }_{\eta }=5.4\times {10}^{-7}$ after 4096 s integration. The measurement is solely limited by the stability of the potassium signal. The first data point at $\tau =32\mathrm{s}$ is given by the time required to obtain a single data point with the scheme described in detail in the text.