Characterization of a simultaneous dual-species atom interferometer for a quantum test of the weak equivalence principle

We present here the performance of a simultaneous dual-species matter-wave accelerometer for measuring the differential acceleration between two different atomic species (${}^{87}\mathrm{Rb}$ and ${}^{85}\mathrm{Rb}$). We study the expression and the extraction of the differential phase from the interferometer output. The differential accelerometer reaches a short-term sensitivity of $1.23\times {10}^{-7}g/\sqrt{\mathrm{Hz}}$ limited by the detection noise and a resolution of $2\times {10}^{-9}g$ after 11 000 s, a very high sensitivity using a dual-species atom interferometer. Thanks to the simultaneous measurement, such resolution levels can still be achieved even with vibration levels up to $3\times {10}^{-3}g$, corresponding to a common-mode vibration noise rejection ratio of 94 dB (rejection factor of 50 000). These results prove the ability of such atom sensors for realizing a quantum-based test of the weak equivalence principle at a level of $\eta \sim {10}^{-9}$ even with high vibration levels and a compact sensor.

Figure 1

Mach-Zehnder-type atom interferometer time diagram and associated acceleration response functions $f\left(t\right)$ and velocity response functions $g_s\left(t\right)$. The black dashed line curves correspond to ${\mathrm{\Omega }}_1={\mathrm{\Omega }}_2={\mathrm{\Omega }}_3=\frac{\pi }{2\tau }$, the red solid line curves correspond to ${\mathrm{\Omega }}_1\tau =0.85\frac{\pi }{2}, {\mathrm{\Omega }}_{2}2\tau =0.99\pi$, and ${\mathrm{\Omega }}_3\tau =1.08\frac{\pi }{2}$ similar to values of the Rabi frequencies associated with ${}^{85}\mathrm{Rb}$ in our experiment. Here, $T=3$ ms and $\tau =1$ ms; the small value of $T$ and high value of $\tau$ are arbitrarily chosen for a good readability of the curve but have no connection with the reality of the experiment.

Figure 2

Simultaneous interferometric signals for ${}^{87}\mathrm{Rb}$ and ${}^{85}\mathrm{Rb}$, with $T=47$ ms. From top to bottom, (a) interference fringes as a function of the microwave chirp $\alpha$, in black squares from ${}^{87}\mathrm{Rb}$ in red dots from ${}^{85}\mathrm{Rb}$. (b) Signal from ${}^{85}\mathrm{Rb}$ is plotted vs signal from ${}^{87}\mathrm{Rb}$ drawing the ellipse fitted by the solid blue line. (c) ${\varphi }_d$ is derived from the phase estimation likelihood function (red histogram). The fringes and the ellipse contain 2400 points; each point is acquired in 0.25 s corresponding to a global integration time of 600 s. The data were recorded with a single direction of the Raman wave vector $+{\vec{k}}_{\mathrm{eff}}$.

Figure 3

Schematic of the laser spectrum at 780 nm during the interferometric sequence with atomic level systems and Raman transitions for both isotopes. In solid lines the Raman pairs used for the atomic mirrors and beam splitters; in dashed lines the “parasite” Raman pairs are due to phase modulation generation, red color corresponding to ${}^{85}\mathrm{Rb}$ and blue to ${}^{87}\mathrm{Rb}$.

Figure 4

Allan deviations on the differential phase ${\varphi }_d$ expressed in $g$ units ${\varphi }_d/\left(k_{\mathrm{eff}}^{87}g{T}^2\right)$ with $g\simeq 9.81 \mathrm{m}{\mathrm{s}}^{-2}$. In black and red dots the Allan deviation corresponding to the two reversed directions of ${\vec{k}}_{\mathrm{eff}}$. The Allan deviation of the averaging over these two directions (blue dots) and its asymptotic behavior (blue line) is fitted by 295 $\mathrm{n}g/\sqrt{\tau }$. The best sensitivity achieved with our differential accelerometer (green dots, corresponding to data points from Fig. 2) and its asymptotic behavior (green line) is fitted by 123 $\mathrm{n}g/\sqrt{\tau }$. The first data point at $\tau =50$ s is given by the time required to obtain enough points to extract the phase from the ellipse (here 100 points).

Figure 5

Three ellipses of 2000 points plotted for three different vibration amplitudes: $3.2\times {10}^{-4}g, 7.6\times {10}^{-3}g$, and $2.4\times {10}^{-2}g$. The ellipses are progressively blurred because of additional inertial effects induced by the excitation (cf. Sec. 7d).

Figure 6

Resolution on the differential acceleration versus the amplitude of vibrations. The resolution is estimated from the ellipses, containing $N=2000$ points, for different amplitudes of vibrations at 2.08 Hz. The correlation with a mechanical accelerometer allows one to measure the vibrations, and to estimate the fringe amplitudes and the population offsets. These two parameters depend on inertial effects (cf. Sec. 7d); this dependency is (red dots) or is not (black squares) taken into account. In dashed lines, the resolution is plotted according to different values of the rejection ratio $r=88.2$ dB (black), $r=94$ dB (red, maximum achieved with our experiment), and $r=100$ dB (blue, theoretical limit with our experiment).

Figure 7

Normalized acceleration transfer function of a single species atom interferometer (black line, single species), a dual-species atom interferometer with wave-vector and Rabi frequency mismatches (differential 1, red line), and a dual-species atom interferometer with a wave-vector mismatch only (differential 2, blue line). The wave-vector mismatch is $\delta k/k_{\mathrm{eff}}=5\times {10}^{-6}$. The Rabi frequency mismatch between ${}^{87}\mathrm{Rb}$ and ${}^{85}\mathrm{Rb}$ is 0.1 (10%).

Figure 8

Rabi frequency mismatch during the first and the third pulse (top), and normalized difference in response functions (bottom) in the low frequency limit, between ${}^{87}\mathrm{Rb}$ and ${}^{85}\mathrm{Rb}$, as a function of the microwave power at 3.035 GHz. In the bottom curve, the experimental measurement (black squares) is compared to the theoretical value (red line) as expressed in Eq. (8). The horizontal uncertainty corresponds to the uncertainty on the determination of the Rabi frequency. The vertical uncertainty corresponds to the 1-$\sigma$ uncertainty coming from the determination of the scale factor by fitting the interferometric fringes.

Figure 9

Normalized ellipses obtained with a vibration level of $2.4\times {10}^{-2}g$. At the top, the ellipse is plotted with the raw data. At the bottom, the ellipse is shown with a better SNR resulting from the post-correction of spurious additional inertial effects. This correction is achievable thanks to the hybridization of the atom interferometer with classical inertial sensors.

Figure 10

Drop of contrast as a function of the amplitude of vibrations for ${}^{87}\mathrm{Rb}$ (black squares) and for ${}^{85}\mathrm{Rb}$ (red circles). The drop is induced by spurious inertial effects.