# Hybrid Electrostatic-Atomic Accelerometer for Future Space Gravity Missions
## 2 Materials and Methods
### Rotation Issues for the CAI  
Characterization of the CAI contrast behavior is of great importance since it impacts directly the sensitivity of the accelerometer. Several effects can impact the contrast of the interferometer, and we can classify these effects into two categories, those that make imperfect the Raman beamsplitters/mirrors and those that impact the atomic phase and that it position- and/or velocity-dependent. For this second category of effects, we can understand their impact on the contrast as phenomena that render less efficient the spa  
Figure 8: Representation of the experimental setup ((**a**) side view and (**b**) top view) used to study rotation impact on the CAI and to demonstrate rotation compensation through EA proof-mass actuation. The dimensions are not to scale. The distance between PZT A–PZT B and PZT B–PZT C is 56 cm. tial/momentum overlap of the two wave packets related to the two interferometer arms at the output of the interferometer. Another vision is to consider that these phenomena impact differently each atom, according to their position or velocity, conducting an averaged signal, which is the contribution of dephased sine functions with a global contrast lower than the one of the individual atom signal. These two visions lead to the same evaluation of the contrast. Note that in our experimental setup, the detection scheme does not allow us to discriminate position-/velocity-dependent phases [69]. The main issue of this kind of instrument in orbit that impacts the interferometer contrast is the satellite rotation, and we report here some experimental demonstrations dealing with this problem. The hybrid prototype, composed of the cold atom gravimeter and the electrostatic accelerometer (see Figure 7), is mounted on a passive isolation platform (PIP), which is placed on a homemade excitation table driven by three linear piezoelectric actuators (PZT A, B, C). For rotation experiments, the isolation platform is blocked so that its top plate stays fixed relative to the bottom. Consequently, no ground vibration isolation is provided. The dynamic excitation is made through actuation of only PZT B, generating a rotation around the \(Z\) axis. A two-axis gyroscope is used to measure the generated rotation \(\Omega_{z}\) and unwanted residual rotation \(\Omega_{y}\). Parasitic linear accelerations can also be measured thanks to 3 classical linear accelerometers. To generate the rotation of the whole setup, the PZT is driven with a sine function to reduce parasitic excitations coming from numerous resonances of the mechanical structure. The same kind of excitation signals are provided to the secondary entries of the EA to control the EA's proof-mass for rotation compensation or for studying the impact of mirror rotation on the CAI contrast. These types of sine excitations are also useful for the proof-mass control to ensure a smooth and accurate control. The phase \(\phi_{exc}\), frequency \(v_{exc}\), and amplitude \(\theta^{0}_{exc}\) of the excitation are controllable. Since the PZT excitation is a sine function, the hybrid instrument is submitted to angle variations \(\theta(t)\), angular velocities \(\Omega(t)\), and angular accelerations \(\dot{\Omega}(t)\) with the respective amplitude \(\theta^{0}_{exc}\), \(\Omega^{0}_{exc}=\theta^{0}_{exc}\cdot(2\pi v_{exc})\), and \(\dot{\Omega}^{0}_{exc}=\theta^{0}_{exc}\cdot(2\pi v_{exc})^{2}\). The resulting acceleration coming from rotation can be written as:  
\[\vec{a}_{rot}(t)=-\vec{\Omega}(t)\times\left[\vec{\Omega}(t)\times\vec{r}_{at }(t)\right]-2\vec{\Omega}(t)\times\vec{v}_{at}(t)-\vec{\Omega}(t)\times\vec{r} _{at}(t) \tag{5}\]  
where \(\vec{r}_{at}(t)\) and \(\vec{v}_{at}(t)\) are, respectively, the time-dependent position, respectively velocity, of the considered atom relative to the reference mirror frame. The first term corresponds to the centrifugal acceleration, the second one to the Coriolis acceleration, and the third one to the angular acceleration term.