# Hybrid Electrostatic-Atomic Accelerometer for Future Space Gravity Missions
### Cold Atom Interferometry
Note that the above-mentioned acceleration \(\vec{a}\) is more rigorously the differential acceleration of the atoms relative to a mirror acting as a reference in the instrument. This acceleration value is similar to what is measured by an inertial accelerometer. Figure 1: Output signal \(P\) of a cold atom accelerometer. One measurement \(P_{i}\) corresponds to several possible acceleration values \(a_{i}^{n}\), \(a_{i}^{sn}\). This ambiguity can be resolved thanks to hybridization with a classical accelerometer. Here, \(P_{0}=0.5\) and \(C=1\). In the case where the acceleration to be measured is not subjected to large acceleration variations, i.e., \(\Delta a\ll\frac{T}{2k_{\mathrm{eff}}t^{2}}\), it is possible thanks to a specific technique [44] to identify the fringe index corresponding to the measurement and to retrieve unambiguously the true acceleration value. Otherwise, for large shot-to-shot acceleration variations (\(\Delta a>\frac{\pi}{2k_{\mathrm{eff}}t^{2}}\)), the true acceleration value can no longer be recovered. This ambiguity concerning the acceleration determination reduces consequently the shot-to-shot measurement range of cold atom sensors. In regard to this limitation, it is typical to associate a classical accelerometer or seismometer with the cold atom instrument [40; 41; 45; 46; 47; 48]. This classical accelerometer allows the identification of the fringe index corresponding to the atomic acceleration measurement (see Figure 1). Moreover, the interferometer phase duration (\(2T\)) being smaller than the cycling time (\(T_{c}\)), mainly due to the loading time of the cold atomic cloud (cooling and trapping phase) and to the detection period, the atomic instrument presents measurement dead times (see Figure 2) leading to potential errors [49; 50]. The atom interferometer is thus sensitive to acceleration only during the interferometer phase. Note that the measurement rate is typically a few Hz for this kind of instrument on the ground. The interferometer temporal acceleration response function \(f_{\mathrm{a}}(t)\)[48; 51] follows a triangle shape (see Figure 2) and highlights that the CAI, for each cycle, gives an average of the acceleration between the first and third pulse with a peak sensitivity during the second laser pulse. In a first approximation, this acceleration response function \(f_{\mathrm{a}}(t)\) is related to the output interferometer phase \(\Delta\Phi\) as follows [48; 51]:  
\[\Delta\Phi = \vec{k}_{\mathrm{eff}}\cdot\int f_{\mathrm{a}}(t)\;\vec{a}(t)dt \tag{3}\]  
\[\mathrm{with}\;f_{\mathrm{a}}(t) \approx (t+T)\cdot\Theta(t+T)-2t\cdot\Theta(t)+(t-T)\cdot\Theta(t-T) \tag{4}\]  
where \(\Theta(t)\) is the Heaviside function:  
\[\left\{ \begin{array}{l l l}\Theta(t)=0&\mathrm{for}&t<0\\ \Theta(t)=1&\mathrm{for}&t\geq 0\end{array} \right.\]  
This acceleration response function corresponds, in the frequency domain, to a low-pass filter transfer function of a \(1/2T\) cut-off frequency [51]. Figure 2: Typical atom interferometer sequence. After cooling and trapping the atoms, the interferometer is realized with a total duration time \(2T\). The phase at the output of the interferometer is measured during the subsequent detection phase. The measurement is given with a cycling time \(T_{c}\). The atomic instrument is only sensitive to acceleration during the interferometer period.