# Hybrid Electrostatic-Atomic Accelerometer for Future Space Gravity Missions
## 2 Materials and Methods
### Hybrid Instrument Simulator
#### 2.3.3 Cold Atom Interferometer  
In a cold atom interferometer, the test-mass is a gas of cold atoms obtained by laser cooling and trapping techniques. This cloud of atoms is released from a trap, and its acceleration is measured by an atom interferometry technique (see Section 1.2 for more details). The simulator models the motion of each individual atom of the cloud along the three axes. Due to computation time, the number of atoms is limited to a few hundred. Each one is assigned with a random initial position and velocity, corresponding to the atomic cloud spatial dispersion and the atomic cloud temperature. Contrary to the electrostatic accelerometer with its high measurement rate of 1 kHz, the atomic interferometer presents a low sample frequency, limited by the interferometer cycle duration \(T_{c}=4\) s in our simulations. The cycle begins with the trapping of the atomic cloud. The atoms are then released, and the interferometer phase runs for 2 s, with the atoms interacting with the laser beams. Finally, the proportion of atoms in the two atomic states is detected. The simulator therefore delivers the measurement for the atoms acceleration once per cycle.  
For the CAI, a Mach-Zehnder-type interferometer is simulated, consisting of a sequence of three laser pulses. A very simple model was used, where the measurement of the atom interferometer can be seen as a combination of three distance measurements between the atoms and the reference mirror. The simulator therefore models the dynamics of the atoms relative to the mirror. As developed in the next part, this mirror can either be a fixed mirror or the proof-mass of the electrostatic accelerometer.  
For the sake of simplification, the laser pulse durations of the CAI are considered infinitely short compared to the interrogation time \(T\), and the intrinsic effects that modify the energy level of the two involved atomic states between the first and third laser pulse are not taken into consideration for the moment in our simulation. The interferometer laser beams are considered with a uniform intensity distribution, perfectly retro-reflected on the reference mirror and directed along the \(X\) axis. It is possible to add in the simulation a phase noise and a phase bias.  
The probability for one atom to be in one particular atomic state is deduced from the phase considering an ideal contrast \(C=1\) and offset \(P_{0}=\frac{1}{2}\), and the proportion of atoms measured in each atomic state is computed as the mean value of this probability for all the atoms composing the cloud, in addition to a white noise corresponding to the quantum projection noise (see Section of Performance Aspects). The extraction of the measured acceleration is computed at each cycle \(i\) from this mean probability \(\overline{P}_{i}\) according to:  
\[a_{CAI,i}=\frac{\epsilon_{i}}{k_{eff}T^{2}}\arccos\left[\frac{2(\overline{P}_{ 0,i}-\overline{P}_{i})}{\overline{C}_{i}}\right]+n_{i}\frac{2\pi}{k_{eff}T^{2}} \tag{7}\]where \(e_{i}=\pm 1\) and \(n_{i}\in\mathbb{Z}\)  
\(e_{i}\) and \(n_{i}\) result from the fringe indetermination due to the cosine function linking the interferometric phase and the acceleration. In the hybridization scheme, they are determined thanks to the EA. At this stage of simulator development, \(\overline{C}_{i}\) and \(\overline{P}_{0,i}\), resulting from the averaging of the signal over all the atoms, are considered perfectly known.