Measurement of the Gravity-Field Curvature by Atom Interferometry

We present the first direct measurement of the gravity-field curvature based on three conjugated atom interferometers. Three atomic clouds launched in the vertical direction are simultaneously interrogated by the same atom interferometry sequence and used to probe the gravity field at three equally spaced positions. The vertical component of the gravity-field curvature generated by nearby source masses is measured from the difference between adjacent gravity gradient values. Curvature measurements are of interest in geodesy studies and for the validation of gravitational models of the surrounding environment. The possibility of using such a scheme for a new determination of the Newtonian constant of gravity is also discussed.

Figure 1

(a) Scheme of the experiment. ${}^{87}\mathrm{Rb}$ atoms are trapped and cooled in a MOT. Three atomic clouds are launched in rapid sequence along the vertical direction with a moving optical molasses. Near the apogees of the atomic trajectories, a measurement of the vertical acceleration sensed by the three clouds is performed by Raman interferometry. External source masses are positioned in order to maximize the average gravity curvature at the three clouds’ positions. (b) Gravitational acceleration along the symmetry axis ($a_z$) produced by the source masses and the Earth’s gravity gradient; a constant value accounting for the Earth’s gravitational acceleration was subtracted. The spatial regions of the three atom interferometers are indicated by the thick red lines.

Figure 2

Bias error in the differential phase from the 2D elliptical fit (red circles), the new 3D fit routine (black squares), and the Bayesian method for different ${\phi }_2$ angle values; ${\phi }_1$ is kept fixed at $\pi /2$. Synthetic data are generated with a Gaussian detection noise (${\sigma }_d=0.01$) and a fringes contrast of 0.3. In the Bayesian analysis, we feed the algorithm both with the exact Gaussian noise as used in the simulation (blue triangles) and with the value obtained after introducing a 10% error on ${\sigma }_d$ (green rhombi). For ${\phi }_2<0.1\mathrm{rad}$, the ${\chi }^2$ numerical minimization fails in the 2D fit. The knowledge of the noise model affecting the data becomes critical in the Bayesian analysis when the phase angle approaches zero.

Figure 3

Typical three-dimensional Lissajous figure obtained by plotting the output signal of the upper atom interferometer as a function of the lower and central one (red circles) and ellipse in 3D best fitting the data (black line). Orthogonal projections on the three Cartesian planes are also shown.