# Hybrid Electrostatic-Atomic Accelerometer for Future Space Gravity Missions
## 3 Results and Discussions
### Performance Aspects  
In the following, we give a rough estimation of some important effects that could limit the performance of the cold atom interferometer in space. This constitutes a non-exhaustive analysis with estimations given for a CAI based on a molasses source:  
* Fundamental limit: The fundamental limit in terms of sensitivity is given by the so-called quantum projection noise (QPN) [76; 77]. This noise per shot at the output of a double-diffraction atomic accelerometer is given by: \[\sigma_{a}^{QPN}=\frac{1}{2k_{eff}T^{2}}\frac{1}{C\sqrt{N_{det}}}\] (10) where \(N_{det}\) is the number of detected atoms contributing to the interferometer signal. This formula assumes working at mid-fringe. \(C\) refers to the interferometer contrast. \(k_{eff}\approx 4\pi/\lambda_{Rb}\approx 1.6\times 10^{7}\), where \(\lambda_{Rb}\) is the laser wavelength associated with the rubidium atomic transition. The QPN sensitivity of the CAI, relative to the interrogation time \(T\), is reported in Figure 26a. The contrast C is calculated taking into account an imperfect contrast without rotation of 0.5 coming, for instance, from imperfect Raman beamsplitters and mirrors. We can see that increasing the interrogation time does not improve continuously the sensitivity, which is a consequence of atom losses due to collisions during free-fall with the residual background gas, of contrast loss due to satellite rotation, and finally, of the spatial expansion of the atomic cloud relative to the finite size of the detection area, which conducts a loss of detected atoms and an increase of the quantum projection noise. This effect is illustrated in Figure 26b, which represents the evolution of the number of detected atoms according to the interrogation time. The input parameters lead to a number of 2.5 \(\times\) 10\({}^{8}\) atoms entering the interferometer. We can see, as explained, that the number of detected atoms decreases quickly with the interrogation time reaching for \(T=0.4\) s approximately 10\({}^{7}\) detected atoms. The assumption of an accelerometer limited by quantum projection noise should not stand for a number of detected atoms lower than 10\({}^{3}\)-10\({}^{4}\), where other types of noise should dominate in the detection. Therefore, typically, in our case, considering \(3\times 10^{5}\) detected atoms, a contrast \(C=0.34\) (beampsplitter efficiency + compensated satellite rotation), an interrogation time \(T=1\) s, and a cycling time \(T_{c}=3\) s, we found a quantum projection acceleration noise of \(\sigma_{a}^{QPN}=3\times 10^{-10}\) m\(\cdot\) s\({}^{-2}\). Hz\({}^{-1/2}\). * Detection noise: Considering a molasses atomic source allowing \(3\times 10^{5}\) detected atoms, a contrast \(C=0.34\) (beampsplitter efficiency + compensated satellite rotation), with an interrogation time \(T=1\) s and a cycling time \(T_{c}=3\) s, a near state-of-the-art detection noise of \(\sigma_{P}=10^{-3}\), and an interferometer operating at mid-fringe, we found an acceleration noise contribution of the detection of \(3\times 10^{-10}\) m\(\cdot\)s\({}^{-2}\)\(\cdot\)Hz\({}^{-1/2}\), similar to the quantum projection noise, constituting the fundamental limit of sensitivity performance. * CAI contrast: Any imperfections of the beamsplitters or mirrors forming the interferometer reduce the number of atoms involved in the interferometer and lead to parasitic paths, which could impact the contrast of the interferometer fringes. The finite position dispersion of the atomic cloud in conjunction with the inhomogeneous intensity profile of the Raman laser reduce also the global probability transition and induce a loss of contrast. For this contrast estimation, we considered a finite velocity distribution of the atoms with an effective atomic temperature of 0.1 \(\upmu\)K thanks to velocity selection and double-diffraction transition with an effective Rabi frequency of \(2\pi\times 100\) kHz, leading to an estimation of \(C\approx 0.5\).