# Hybrid Electrostatic-Atomic Accelerometer for Future Space Gravity Missions
### Cold Atom Interferometry  
This new generation of instruments, relying on the manipulation of matter waves through atom interferometry [19], appears currently as one of the most promising candidates for highly precise and accurate inertial measurements [20]. Cold atom interferometers have indeed proven on the ground to be very high-performance sensors with the development in recent decades of cold atom gravimeters [21; 22; 23; 24; 25], gravity gradiometers [26; 27], and gyroscopes [28; 29]. In addition to the undeniable contribution they could bring for practical applications such as inertial navigation [30; 31], they also appear very promising for exploring fundamental physics such as for the determination of the fine-structure constant [32; 33], the gravitational constant [34; 35], and for testing [PERSON]'s theory of general relativity with quantum objects [36]. In that field, atom interferometers seem notably promising for detecting gravitational waves [37] and testing the weak equivalence principle [38; 39]. This technology has demonstrated impressive results in the laboratory environment and is beginning also to prove its interest for on-board applications with the recent demonstrations of gravity field mapping during shipborne [40] and airborne [41] campaigns with a cold atom gravimeter. Progressively, proposals for using cold atom interferometers in space have emerged targeting at the beginning more particularly fundamental physics applications [6; 37], but recently opened up to Earth observation objectives [8; 11; 13; 14; 15; 42]. Space offers indeed a micro-gravity environment that should allow in principle increasing the interrogation time and, thus, the scale factor of the atom interferometer compared to ground-based atomic instruments, where the size of the instrument limits the duration of atoms' free-fall and, thus, the sensitivity of the instrument. In the rest of the document, the study deals with an atom interferometer dedicated only to acceleration measurements, which is the objective of the proposed space instrument. In a cold atom accelerometer, the test mass is a gas of cold atoms obtained by laser cooling and trapping techniques [43]. This cloud of cold atoms is released from a trap, evolving in free-fall in a vacuum chamber, and its acceleration is measured by an atom interferometry technique. Typically, a Mach-Zehnder-type interferometer is realized, consisting of a sequence of three equally spaced laser pulses, which drive transitions between two atomic stable states. During each light pulse, the atoms interact with the laser beam and have a probability to absorb a photon out of the laser beam. In this way, the recoil of a photon can be transferred to the atom. This absorption probability can be adjusted by controlling the duration and intensity of the light pulses such that the atomic wave can be either equally split, deflected, or re-combined so as to form an atom interferometer. At the end of the interferometer, the proportion \(P\) of atoms in one of the two stable states depends sinusoidally on the phase \(\Delta\Phi\) of the interferometer (see Figure 1), which is proportional to the acceleration \(\vec{a}\) of the atoms, assumed constant here, along the laser direction of propagation defined by its wave vector \(\vec{k}_{\text{eff}}\). The expression of the output signal is given at first order by:  
\[P = P_{0}-\frac{C}{2}\cos(\Delta\Phi)\] (1) with \[\Delta\Phi \approx \vec{k}_{\text{eff}}\cdot\vec{a}\;T^{2}. \tag{2}\]  
Here, \(T\) is the time between the three laser pulses constituting the Mach-Zehnder interferometer, \(P_{0}\) is the fringe offset, and \(C\) is the fringe contrast. The output signal \(P\) of the atomic sensor, corresponding to the probability for the atom to be in one of the two output ports of the interferometer, could be measured typically thanks to fluorescence or absorption imaging techniques.