# Statistical Analysis of the Rotation Induced Decay of the Contrast in an Onboard Atom Interferometer

Benoît Kaczmarczuk, Joel Gomes Baptista, Sébastien Merlet, Leonid A

Sidorenkov, Quentin Beaufils, Franck Pereira dos Santos

► To cite this version:

Benoît Kaczmarczuk, Joel Gomes Baptista, Sébastien Merlet, Leonid A Sidorenkov, Quentin Beaufils, et al.. Statistical Analysis of the Rotation Induced Decay of the Contrast in an Onboard Atom Interferometer. IEEE Sensors Journal, 2025, 25 (16), pp.30889-30896. 10.1109/JSEN.2025.3586409. hal-05363851

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# Statistical analysis of the rotation induced decay of the contrast in an onboard atom interferometer

Benoît Kaczmarczuk, Joel Gomes Baptista, Sébastien Merlet, Leonid A. Sidorenkov, Quentin Beaufils, Franck Pereira dos Santos

**Abstract**— We present a statistical analysis of the measurement noise of an atom interferometer under large phase and contrast noise caused by rotation fluctuations. The extraction of the standard deviation of the contrast fluctuations out of transition probability measurements allows us to determine the level of rotation noise applied to the onboard instrument. We include an experimental validation of the method.

**Index Terms**— Atom Interferometry (AI), Inertial Sensors, Navigation

## I. INTRODUCTION

QUANTUM sensors based on cold-atom interferometry show performances that have or will have a significant impact in numerous fields including geophysics [1], [2] and geodesy [3]–[10], gravitational wave detection [11], tests of fundamental physics [12]–[16] and inertial navigation [17]. The variety of potential applications stimulated an effort to deploy this technology in diverse environments. Initially limited to quiet and controlled laboratory experiments [20], atom interferometers (AI) can be adapted to operate under critical conditions on ground [18], [19], onboard a vehicle [21], [22] (boat, aircraft) or in space [23].

Whether they are intended for gravity field recovery or inertial navigation, onboard applications are particularly demanding since they impose high levels of acceleration and angular velocity fluctuations. AIs have an inherent $2\pi$ phase ambiguity that limits their dynamic range and their ability to fully benefit from their high sensitivity in a noisy environments. Besides, they are sensitive to inertial effects such as Coriolis and centrifugal accelerations, which can cause measurement noise and reduce the contrast [24], limiting further their sensitivity. Methods to overcome those limitations have been developed, relying for instance on the hybridization with classical inertial sensors [19], [25].

Calibrations of the instrument in noisy environments rely on a statistical analysis of signal fluctuations, which allow to extract key parameters such as the contrast and the detection noise, which are assumed constant [26]. However, in many

This work was supported in part by a government grant managed by the Agence Nationale de la Recherche under the Plan France 2030 with the reference "ANR-22-PETQ-0005", and by the European Defence Fund (EDF) under grant agreement 101103417 EDF-2021-DIS-RDIS-ADEQUADE.

Benoît Kaczmarczuk, Joël Gomes, Sebastien Merlet, Leonid Sidorenkov, Quentin Beaufils, Franck Pereira dos Santos are members of LTE, Observatoire de Paris, Université PSL, Sorbonne Université, Université de Lille, LNE, CNRS, 61 avenue de l'Observatoire, 75014 Paris, France.

situations, the angular velocity of the instrument varies which leads to contrast fluctuations. In this article, we build on the previous approach and propose two different methods to extend the analysis of the statistical distribution of AI measurements to the case of large phase and contrast fluctuations caused by acceleration and rotation noise. In comparison to previous works, our methods allow to estimate both contrast and phase fluctuations with the same data set. The next section is dedicated to the description of these methods, and the last section presents experimental tests of the methods that were performed with a cold atom interferometer.

## II. ANALYSIS OF THE TRANSITION PROBABILITY DISTRIBUTION

### A. Context

We consider a three-pulse atom interferometer similar to the one described in [26]. The measured quantity at the output of the interferometer is the transition probability between the two $^{87}$Rb hyperfine levels of the ground state:

$$P_i = P_0 + \frac{C_i}{2} \cos(\phi_i) + \delta P_i \quad (1)$$

where $P_0$ is the offset of the interference fringes, $C_i$ is the contrast defined as the fringe amplitude in transition probability and $\delta P_i$ is an additive white Gaussian detection noise of standard deviation $\sigma_P$. Here the index $i$ enumerates the successive measurements. The phase difference accumulated between the two arms of the interferometer $\phi_i$ is proportional to the acceleration along the measurement axis as well as rotation induced inertial pseudo-accelerations (Coriolis, centrifugal and Euler). In the presence of acceleration fluctuations, the variation of the phase between successive measurements can exceed $2\pi$, causing an ambiguity on the corresponding transition probability. When the contrast is constant ($C_i = C_0$), an analysis of the transition probability distribution over a large number of measurements allows to demonstrate the
<figure><img src="image_2.png" /><figcaption>Fig. 1. In this work we consider a three pulse atom interferometer in a configuration analogous to an optical Mach-Zehnder interferometer. The atomic wavefunction undergoes a series of splitting, mirror, and recombination laser pulses. The transition probability between the two velocity states is measured after recombination. Under random acceleration phase, the transition probability density function shows a characteristic shape (green histogram). Comparatively to previous works, we include the effect of angular velocity fluctuations (blue histogram).</figcaption></figure>

coherence of the interferometer and estimate $C_0$, $P_0$ and $\sigma_P$ [26]. Figure 2a shows the corresponding distribution, the Probability Density Function (PDF), for a random phase considered as a white noise of Gaussian standard deviation $\sigma_\phi \gg 2\pi$.

For onboard applications, rotations of the instrument can induce a loss of contrast via inhomogeneous Coriolis phase shifts: the finite temperature of the cloud is associated with a spread of velocity at the release, resulting in a phase inhomogeneity at the output of the interferometer. In the absence of a spatially resolved output state detection, this leads to a loss of contrast [28] that can be expressed as:

$$C_i = C_0 \exp(-2\sigma_v^2 k_{eff}^2 \Omega_i^2 T^4) \quad (2)$$

where $\Omega_i$ is the angular velocity in the plane perpendicular to the measurement direction, $\sigma_v = \sqrt{k_0 T/m}$ is the velocity dispersion for a temperature $T$ and mass $m$, $k_{eff} = 4\pi/780$ nm is the effective wavevector of the interferometer beamsplitting pulse, and $T$ is the interferometer separation time. A fluctuating angular velocity leads to a change of the PDF illustrated in figure 2b, c and d where $\Omega_i$ was modeled by a Gaussian white noise of increasing standard deviations $\sigma_\Omega$.

## B. First method : least squares fit

The first method we present is based on a least square fit of the histogram with four free parameters: $P_0$, $C_0$, $\sigma_p$ and $\sigma_\Omega$, respectively the central transition probability, the unperturbed contrast, the detection noise standard deviation (assumed here independent of $P_i$) and the angular velocity noise standard deviation.

<figure><img src="image_3.png" /><figcaption>Fig. 2. Probability Density Functions (PDF) of the transition probability. 10 000 successive measurements were simulated with a random phase ($\sigma_\phi \gg 2\pi$) and with a random angular velocity fluctuations $\Omega_i$ of increasing standard deviations $\sigma_\Omega$. a: $\sigma_\Omega = 0$, b: $\sigma_\Omega = 0.5\Omega_c$, c: $\sigma_\Omega = \Omega_c$, d: $\sigma_\Omega = 2\Omega_c$, $\sigma_P = 5 \times 10^{-3}$, $P_0 = 0.3$ and $C_0 = 0.2$. $\Omega_c$ is defined in equation (3). Solid lines are the results of the fits, following the procedure described in section II-B</figcaption></figure>

We introduce the characteristic angular velocity $\Omega_c$:

$$\Omega_c = \frac{1}{2\sigma_v k_{eff} T^2} \quad (3)$$

and express the contrast $C_i$ as:

$$C_i = C_0 \exp \left( -\frac{1}{2} \frac{\Omega_i^2}{\Omega_c^2} \right) \quad (4)$$

The transition probability at each measurement $i$ is thus given by:

$$P_i = P_0 + \delta P_i^{\text{det}} + \frac{C_0}{2} e^{-\frac{1}{2} \frac{\Omega_i^2}{\Omega_c^2}} \cos(\phi_i) \quad (5)$$

That we rewrite as:

$$P_i = P_0 + \delta P_i^{\text{det}} + \delta P^{\text{phase}}(\phi_i, \Omega_i) \quad (6)$$

The transition probability is a function of three random variables and the sum of two random functions. Its density function in the continuous domain can be written as the convolution of the density functions of $P^{\text{det}}$ and $P^{\text{phase}}$:

$$f_P(x) = (f_{\delta P^{\text{det}}} * f_{\delta P^{\text{phase}}})(x - P_0) \quad (7)$$

In order to adjust the histograms we use the cumulative distribution function $F_P(x)$ defined as the integral of the density function:

$$F_P(x) = (f_{\delta P^{\text{det}}} * F_{\delta P^{\text{phase}}})(x - P_0) \quad (8)$$
We can express the PDF of the detection noise as a Gaussian distribution :

$$f_{\delta P_{det}}(x) = \frac{1}{\sigma_P \sqrt{2\pi}} e^{-\frac{1}{2} \frac{x^2}{\sigma_P^2}} \quad (9)$$

The detail of the calculation of the cumulative distribution function of $\delta P^{phase}$ is given in the Appendix I. We obtain :

$$F_{\delta P^{phase}}(x) = \int_{0}^{1} \frac{\Omega_c}{\sigma_{\Omega} t \sqrt{-\pi \ln(t)}} \left(1 - \frac{1}{\pi} \cos^{-1}\left(\frac{2x}{C_0 t}\right)\right) dt \quad (10)$$

Note that this function is only valid for a normal distribution of the random parameters. In order to adjust the histogram with this function, we perform a least-squares fit with a statistical approach :

* the histogram gives the measured probability for the transition probability $\mathcal{P}$ to be in the interval $[x_{j-1}, x_j]$ as $N_j/N$ where $N$ is the total number of measurements, and $N_j$ is the number of measurements that fall in that interval.

* the cumulative distribution gives the theoretical probability for $\mathcal{P}$ to be in the interval $[x_{j-1}, x_j]$ as $F_{\mathcal{P}}(x_j) - F_{\mathcal{P}}(x_{j-1})$.

The best adjustment of the parameters $[\sigma_p, \sigma_{\Omega}, C_0, P_0]$ is the one that minimizes the quantity :

$$\sum_{j=1}^{N_{bin}} \left( F_{\mathcal{P}}(x_j) - F_{\mathcal{P}}(x_{j-1}) - \frac{N_j}{N} \right)^2 \quad (11)$$

When constructing the histograms, we adjust the binning according to the Terrell-Scott rule [27]. To ensure better convergence, we apply zero-padding equal to $2 \times 10\%$ of the data range before constructing the histogram :

$$\sqrt[3]{2N} \times (1 + 20\%) \approx 32 \quad (12)$$

### C. Second method : equation of moments

The second method is based on the calculation of the moments of the distribution. The four unknown parameters ($P_0, C_0, \sigma_p$ and $\sigma_{\Omega}$) can be determined by a set of four equations based on the moments of different order. Since the distribution is an even function, only the even order moments are non null.

The first moment is the mean :

$$\mathbb{E}[\mathcal{P}] = P_0 \quad (13)$$

The second central moment, the variance, is given by:

$$\mathbb{E}[(\mathcal{P} - P_0)^2] = \sigma_p^2 + \frac{1}{8} \frac{C_0^2}{\sqrt{1 + 2 \frac{\sigma_{\Omega}^2}{\Omega_c^2}}} \quad (14)$$

The fourth central moment, the kurtosis, is given by:

$$\mathbb{E}[(\mathcal{P} - P_0)^4] = 3\sigma_p^4 + \frac{3}{4} \frac{\sigma_p^2 C_0^2}{\sqrt{1 + 2 \frac{\sigma_{\Omega}^2}{\Omega_c^2}}} + \frac{3}{128} \frac{C_0^4}{\sqrt{1 + 4 \frac{\sigma_{\Omega}^2}{\Omega_c^2}}} \quad (15)$$

The sixth central moment, is given by:

$$\mathbb{E}[(\mathcal{P} - P_0)^6] = 15\sigma_p^6 + \frac{45}{8} \frac{\sigma_p^4 C_0^2}{\sqrt{1 + 2 \frac{\sigma_{\Omega}^2}{\Omega_c^2}}} + \frac{45}{128} \frac{\sigma_p^2 C_0^4}{\sqrt{1 + 4 \frac{\sigma_{\Omega}^2}{\Omega_c^2}}} + \frac{5}{1024} \frac{C_0^6}{\sqrt{1 + 6 \frac{\sigma_{\Omega}^2}{\Omega_c^2}}} \quad (16)$$

We estimate $(\sigma_p, \sigma_{\Omega}, C_0, P_0)$ by equating theoretical moments with empirical moments [29] : $\mathbb{E}[\mathcal{P}] = \sum_i P_i$; $\mathbb{E}[(\mathcal{P} - P_0)^2] = \sum_i (P_i - P_0)^2$, $\mathbb{E}[(\mathcal{P} - P_0)^4] = \sum_i (P_i - P_0)^4$ and $\mathbb{E}[(\mathcal{P} - P_0)^6] = \sum_i (P_i - P_0)^6$. Combining these equations, we derive an equation with one unknown, $\sigma_p$, which we solve numerically, and with which we then calculate analytically the other unknowns (see details in the Supplementary Material).

### D. Performance evaluation

Figure 2 shows transition probability histograms fitted with the method described in section II-B for different values of $\sigma_{\Omega}$. Each histogram has been produced with the same values of $\sigma_p, C_0$, and $P_0$. All four parameters have been left free for the fit. The results are listed in table I, With uncertainties throughout the article defined at one $\sigma$.

To evaluate the ability of the adjustment procedure to retrieve and discriminate successfully $\sigma_{\Omega}$ from $\sigma_p$, we realized series of 100 measurement session simulations for different values of $\sigma_{\Omega}$ and $\sigma_p$. Each session consists in 10 000 successive simulated interferometer transition probability measurements. The 100 resulting histograms are fitted and the fit results are averaged and compared in figure 3 (left hand side) to the values used for the simulation of the histograms. Figure 4 displays the standard deviations of the fitted values.

TABLE I

<table><caption>RESULTS FROM THE FITS OF FIGURE 2. THE NUMBERS IN BRACKETS ARE THE VALUES USED FOR THE SIMULATION OF THE HISTOGRAMS. THE RESULTS OF THE FIT ARE LISTED ALONG WITH THE STANDARD DEVIATION ERRORS DEDUCED FROM THE COVARIANCE MATRICES.</caption><thead><tr><th rowspan="2">Data set<br>(σ<sub>Ω</sub>/Ω<sub>c</sub>)</th><th colspan="4">Results</th></tr><tr><th>σ<sub>Ω</sub>/Ω<sub>c</sub></th><th>σ<sub>P</sub> (5 × 10<sup>-3</sup>)</th><th>P<sub>0</sub> (0.3)</th><th>C<sub>0</sub> (0.2)</th></tr></thead><tbody><tr><td>a (0)</td><td>0.20<br>±0.01</td><td>4.6 × 10<sup>-3</sup><br>±0.2 × 10<sup>-3</sup></td><td>0.3001<br>±0.0001</td><td>0.2035<br>±0.0004</td></tr><tr><td>b (0.5)</td><td>0.51<br>±0.01</td><td>5.0 × 10<sup>-3</sup><br>±0.4 × 10<sup>-3</sup></td><td>0.3001<br>±0.0003</td><td>0.2010<br>±0.0006</td></tr><tr><td>c (1)</td><td>1.07<br>±0.01</td><td>5.1 × 10<sup>-3</sup><br>±0.6 × 10<sup>-3</sup></td><td>0.3001<br>±0.0004</td><td>0.2006<br>±0.0009</td></tr><tr><td>d (2)</td><td>1.97<br>±0.01</td><td>5.1 × 10<sup>-3</sup><br>±0.1 × 10<sup>-3</sup></td><td>0.2998<br>±0.0001</td><td>0.2017<br>±0.0014</td></tr></tbody></table>

These results can be compared to the results of the moments equation method, listed in table II and plotted on the right hand side of figure 3 and figure 4.

The two methods give qualitatively similar results. However, we identify three areas where the methods are less effective :

* The moments method shows a large dispersion and bias on the estimated $\sigma_{\Omega}$ for low detection noise ($\sigma_p < \sigma_{\Omega}$).
<caption>TABLE II</caption>

RESULTS OF THE MOMENTS METHOD. THE NUMBERS IN BRACKETS ARE
THE VALUES USED FOR THE DATA GENERATION.

<table><thead><tr><th rowspan="2">Data set<br>(σ<sub>Ω</sub>/Ω<sub>c</sub>)</th><th colspan="4">Results</th></tr><tr><th>σ<sub>Ω</sub>/Ω<sub>c</sub></th><th>σ<sub>P</sub> (5 × 10<sup>-3</sup>)</th><th>P<sub>0</sub> (0.3)</th><th>C<sub>0</sub> (0.2)</th></tr></thead><tbody><tr><td>a (0)</td><td>0.28</td><td>1.5 × 10<sup>-3</sup></td><td>0.3011</td><td>0.2072</td></tr><tr><td>b (0.5)</td><td>0.51</td><td>4.8 × 10<sup>-3</sup></td><td>0.2995</td><td>0.2009</td></tr><tr><td>c (1)</td><td>0.95</td><td>4.9 × 10<sup>-3</sup></td><td>0.3002</td><td>0.2003</td></tr><tr><td>d (2)</td><td>2.00</td><td>3.8 × 10<sup>-3</sup></td><td>0.2992</td><td>0.2019</td></tr></tbody></table>

<figure><img src="image_4.png" /><figcaption>Fig. 3. Relative bias on the estimates of $\sigma_P$ (top) and $\sigma_\Omega$ (bottom) as a function of $\sigma_P$ on the y-axis and $\sigma_\Omega$ on the x-axis. Left: fits of the PDF, Right: method of moments. The bias is represented by color intensity. Each pixel is the average of 100 draws.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 4. Relative standard deviations on the estimates of $\sigma_P$ (top) and $\sigma_\Omega$ (bottom) as a function of $\sigma_P$ on the y-axis and $\sigma_\Omega$ on the x-axis. On the right are the results from the method of moments, and on the left are the results from the fits method. The relative standard deviation is represented by color intensity. Each pixel is the average of 100 successive simulations and the corresponding relative standard deviation is plotted as an error bar.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 5. Relative biases of the estimations of detection (on the left) and rotation velocity noise (on the right) using the method of moments (in orange) and the fit method (in blue), as a function of the number of draws. The bars represent the relative dispersion given by $\sigma(\sigma_{p,fit/mom})/\sigma_P$ for the detection noise estimation and $\sigma(\sigma_{\Omega,fit/mom})/\sigma_\Omega$ for the standard deviation of the rotational speed estimation. The parameters set for the simulation are $\sigma_\Omega = 2\Omega_c$, $\sigma_P = 0.02$ et $C_0 = 0.2$</figcaption></figure>

0.003). This is correlated with the absence of a real solution for $\sigma_P$ (see supplementary material).

* Both methods show a large dispersion and bias on the estimated $\sigma_\Omega$ for low rotational noise ($\sigma_\Omega < 0.1$). This increase in dispersion is explained by the smaller variations in the PDF and in the moments with respect to $\sigma_\Omega$, compared to the statistical fluctuations resulting from the finite number of draws. This increased dispersion also results in a bias. Indeed, $\sigma_\Omega$ being bound to positive values, the average of its fluctuating values scales with their dispersion.

* Both methods fail to discriminate $\sigma_P$ from $\sigma_\Omega$ when both parameters are large.

According to Figure 5, the dispersion and biases for both methods decrease down to zero as the number of draws increases. We note though for the larger number of draws a resolved bias in the estimation of the detection noise using the fit method, which tends to be systematically underestimated. A convergence analysis indicates that this bias originates from the numerical method used to estimate the PDF.

### III. EXPERIMENTAL TESTS

We tested experimentally our ability to evaluate the amplitude of the rotation velocity noise using the above methods with a cold-atom gravity gradiometer. The experimental setup, described in [30], consists of two cold atom clouds in free fall,

vertically separated by a distance $d = 1$ m, and interrogated by the same pair of counter-propagating Bragg laser beams, thus realizing two simultaneous interferometers. Their common reference mirror is mounted on a tip-tilt platform and can rotate with a controlled constant angular velocity $\Omega_M$ during the interferometer sequence, which allows us to mimic a rotation of the whole system (see for example [24]). The phase of one interferometer is:

$$ \phi = k_{eff} a T^2 \quad (17) $$

where $k_{eff}$ is the effective wave vector of the Bragg lasers, $a$ is the acceleration of the atoms with respect to the reference mirror and $T$ is the interrogation time.

For the interferometer duration used for this demonstration, of $2T = 120$ ms, mirror vibrations induced by ground seismic noise are not large enough to completely scramble the phase.
We thus artificially increase the phase noise by applying a random frequency difference between the Bragg lasers, so as to add a contribution to the interferometer phase spanning across several radians, ensuring that $\sigma_{\phi} \gg 2\pi$.

The transition probability is measured with a velocity-selective fluorescence detection system [31], where the atom clouds traverse a resonant laser light-sheet and their fluorescence is detected on a photodiode. The detection system has a limited spatial area that allows to measure only the atoms with a limited velocity in the direction of interest. This truncation of the detected atom cloud thus imposes an effective velocity distribution $\sigma_{veff}$ smaller than the one deduced from the cloud temperature, and different for the two clouds falling from different heights. We calibrated the resulting contrast loss in the experiment as a function of $\Omega_M$ (figure 6). This allowed to determine experimentally the value of $\sigma_{veff}$ and $\Omega_c$ for each cloud. We find $\Omega_c = 1.79 \pm 0.06$ mrad s$^{-1}$ for the upper cloud and $\Omega_c = 1.34 \pm 0.03$ mrad s$^{-1}$ for the lower cloud.

<figure><img src="image_7.png" /><figcaption>Fig. 6. Normalized contrast reduction due to the rotation of the interferometer versus the angular velocity of the mirror, for the upper (red) and lower (blue) atom clouds. Dashed lines are Gaussian fits yielding the values of $\Omega_c$ from which we can deduce $\sigma_{veff}$. The solid line is the expected contrast, calculated with equation 2 using the experimental value of the atomic cloud's temperature.</figcaption></figure>

We then applied random fluctuations to the mirror's angular velocity $\Omega_M$, with centered Gaussian distributions of standard deviations $\sigma_{\Omega}$ ranging between 0 and $2$ mrad.s$^{-1}$, each to a set of ($\simeq 10$ 000) successive interferometer cycles. Typical results of the corresponding transition probabilities are displayed in Figure 7, for $\sigma_{\Omega} = 0$ and $\sigma_{\Omega} = 2$ mrad.s$^{-1}$. The resulting histograms were fitted using the procedure described in section II-B and the fit results ($\sigma_{\Omega}^{\text{fit}}$, $\sigma_{P}^{\text{fit}}$, $P^{\text{fit}}$, $C_{0}^{\text{fit}}$) are plotted in figure 8, along with the results given by the moments method applied to the same data sets. We performed a linear fit of the extracted value of $\sigma_{\Omega}^{\text{fit}}$ as a function of the experimentally applied value $\sigma_{\Omega}$ and obtained $\sigma_{\Omega}^{\text{fit}}/\Omega_c = (0.908 \pm 0.008) \times \sigma_{\Omega}/\Omega_c$ for the lower cloud and $\sigma_{\Omega}^{\text{fit}}/\Omega_c = (0.976 \pm 0.009) \times \sigma_{\Omega}/\Omega_c$ for the upper cloud concerning the fit method; and $\sigma_{\Omega}^{\text{mom}}/\Omega_c = (0.932 \pm 0.008) \times \sigma_{\Omega}/\Omega_c$ for the lower cloud and $\sigma_{\Omega}^{\text{mom}}/\Omega_c = (0.921 \pm 0.013) \times \sigma_{\Omega}/\Omega_c$ for the upper cloud concerning the moments method. We observe that the agreement is consistently close but smaller than one,

<figure><img src="image_8.png" /><figcaption>Fig. 7. Experimental PDFs with and without rotation noise, for the upper and lower atomic cloud. The solid lines are the results of the fit method. The number of measurements is 32 664 with $\sigma_{\Omega} = 0$ mrad s$^{-1}$ and 9 077 with $\sigma_{\Omega} = 2$ mrad s$^{-1}$</figcaption></figure>

which could be explained by an experimental underestimation of $\sigma_{veff}$.

The other parameters ($\sigma_P$, $P_0$, $C_0$) are not expected to change with the applied $\sigma_{\Omega}$. The significant variations in the noise value $\sigma_P$, uncorrelated to $\sigma_{\Omega}$, can be explained by variations in the number of detected atoms from one measurement sequence to another.

## IV. CONCLUSION

We presented two alternative statistical methods to calibrate a cold atom interferometer in a noisy environment. It allows to retrieve key instrumental parameters: the average transition probability $P_0$, the unperturbed contrast $C_0$, the transition probability standard deviation due to detection noise $\sigma_P$ and the angular velocity standard deviation $\sigma_{\Omega}$ that causes contrast noise. Those methods are suited for onboard applications where the noise levels can be non-stationary. Regular noise calibrations allow to optimize instrumental parameters such as the interrogation time of the interferometer or the gain in a fringe-lock scheme [19]. In the future, tests of these techniques could be performed on various onboard atom interferometry data, when available, from future envisioned missions onboard satellites, boats, planes or research balloons. The implementation of the techniques on hybridized systems will be also studied.

The two methods were tested with simulated and experimental data. The numerical simulations demonstrated the effectiveness of the two methods to separate detection noise from rotation induced contrast noise, in a large region of parameters. The experimental tests showed a good agreement between the angular velocity dispersion $\sigma_{\Omega}$ applied to the mirror and the value obtained with both methods, further validating their relevance. Both methods require a large number of sample points to work effectively, which can limit their
<figure><img src="image_9.png" /><figcaption>Fig. 8. Results of the fit method (left) and the moment method (right) on experimental results for the upper (red symbols) and lower clouds (green symbols).</figcaption></figure>

effectiveness when the noise parameters vary too fast. Possible improvements would be optimizing computation speed for real time applications, or taking into account variations in the amplitude of the detection noise. Also, the study of the shape of the PDF could be used to characterize detection noise beyond the uniform case considered here, such as when operating at the quantum projection noise limit or below, via for instance the use of spin squeezing protocols [32].

# APPENDIX I
## EXPRESSION OF THE CUMULATIVE DISTRIBUTION FUNCTION

The expression for the loss of contrast induced by rotation in an interferometer for a constant angular velocity during one cycle is given by:

$$C = \exp \left( - \frac{2k_b T}{m} k_{eff}^2 (\Omega_x^2 + \Omega_y^2) T^4 \right) \quad (18)$$

With:

* x and y the two directions transverse to the direction of the Raman pulses in z.

* Ωx and Ωy are the rotation velocities of the frame of reference in x and y.

If a rotation is applied only to the mirror with a rotation velocity Ω.

Introducing Ω_c = 1/(2√(k_b T/k_{eff} T^2)), we get:

$$C = \exp \left( -\frac{1}{2} \frac{\Omega^2}{\Omega_c^2} \right) \quad (19)$$

The transition probability finally writes as:

$$\mathcal{P} = P_0 + \delta P^{det} + \frac{C_0}{2} e^{-\frac{1}{2} \frac{\Omega_c^2}{\Omega_c^2} \cos(\phi)} \quad (20)$$

We assume that Ω follows a centered standard distribution with standard deviation σ_Ω. The distribution function of the mirror rotation speed is given by:

$$f_{\Omega}(x) = \frac{1}{\sqrt{2\pi}\sigma_{\Omega}} \exp\left(-\frac{1}{2}\frac{x^2}{\sigma_{\Omega}^2}\right) \quad (21)$$

We define the random variable X as:

$$X = \frac{C_0}{2} e^{-\frac{1}{2} \frac{\Omega_c^2}{\Omega_c^2}} = g_X(\Omega) \quad (22)$$

with $g_X(x) = \frac{C_0}{2} e^{-\frac{1}{2} \frac{x^2}{\Omega_c^2}}$

Expressing the derivative of g

$$g'_{X}(x) = -\frac{1}{2} \frac{2x}{\Omega_c^2} \frac{C_0}{2} e^{-\frac{1}{2} \frac{x^2}{\Omega_c^2}} = -\frac{x}{\Omega_c^2} g_X(x) \quad (23)$$

and introducing $g_{X,1}^{-1}(y)$ and $g_{X,2}^{-1}(y)$ the two solutions of the equation $g_X(x) = y$,

$$g_{X,1}^{-1}(y) = \Omega_c \sqrt{2 \ln \left( \frac{C_0}{2y} \right)} \quad (24)$$

$$g_{X,2}^{-1}(y) = -\Omega_c \sqrt{2 \ln \left( \frac{C_0}{2y} \right)} \quad (25)$$

The density function of the new random variable writes as:

$$\begin{aligned} f_X(y) &= \frac{f_\Omega(g_{X,1}^{-1}(y))}{|g'_X(g_{X,1}^{-1}(y))|} + \frac{f_\Omega(g_{X,2}^{-1}(y))}{|g'_X(g_{X,2}^{-1}(y))|} \\ &= \left| \frac{\Omega_c^2}{y\Omega_c\sqrt{2\ln\left(\frac{C_0}{2y}\right)}} \right| f_\Omega\left(\Omega_c\sqrt{2\ln\left(\frac{C_0}{2y}\right)}\right) \\ &\quad + \left| \frac{\Omega_c^2}{y\Omega_c\sqrt{2\ln\left(\frac{C_0}{2y}\right)}} \right| f_\Omega\left(-\Omega_c\sqrt{2\ln\left(\frac{C_0}{2y}\right)}\right) \quad (26) \\ &= \frac{\Omega_c^2}{y\Omega_c\sqrt{2\ln\left(\frac{C_0}{2y}\right)}} \frac{1}{\sigma_\Omega\sqrt{2\pi}} e^{-\frac{\Omega_c^2}{\sigma_\Omega^2} \ln\left(\frac{C_0}{2y}\right)} \\ &\quad + \frac{\Omega_c^2}{y\Omega_c\sqrt{2\ln\left(\frac{C_0}{2y}\right)}} \frac{1}{\sigma_\Omega\sqrt{2\pi}} e^{-\frac{\Omega_c^2}{\sigma_\Omega^2} \ln\left(\frac{C_0}{2y}\right)} \end{aligned}$$

$$\begin{aligned} f_X(y) &= \frac{\Omega_c}{\sigma_\Omega} \frac{1}{y\sqrt{\pi \ln\left(\frac{C_0}{2y}\right)}} \exp\left(-\frac{\Omega_c^2}{\sigma_\Omega^2} \ln\left(\frac{C_0}{2y}\right)\right) \\ &= \frac{\Omega_c}{\sigma_\Omega} \frac{1}{y\sqrt{\pi \ln\left(\frac{C_0}{2y}\right)}} \left(\frac{C_0}{2y}\right)^{-\frac{\Omega_c^2}{\sigma_\Omega^2}} \end{aligned} \quad (27)$$

Let Φ, the random variable associated with the interferometric phase for a totally random phase, that is to say that φ varies much more than 2π. The Cumulative distribution function of the cosine of φ is given by:
$$F_{\cos \phi}(x) = 1 - \frac{1}{\pi} \cos^{-1}(x) \quad (28)$$

Let $\delta P^{phase} = X \times \cos \Phi$. With the product formula, the Cumulative distribution function of Y is:

$$ 
\begin{aligned}
F_{\delta P^{phase}}(x) &= \int_{-\infty}^{+\infty} f_X(t) F_{\cos \phi}(x/t) dt \\
&= \int_0^{\frac{C_0}{2}} \frac{\Omega_c}{\sigma_\Omega} \frac{1 - \frac{1}{\pi} \cos^{-1}\left(\frac{x}{t}\right)}{t \sqrt{\pi \ln\left(\frac{C_0}{2t}\right)}} \left(\frac{C_0}{2t}\right)^{-\frac{\Omega_c^2}{\sigma_\Omega^2}} dt \\
&= \int_0^1 \frac{\Omega_c}{\sigma_\Omega} \frac{1 - \frac{1}{\pi} \cos^{-1}\left(\frac{2x}{C_0 t}\right)}{t \sqrt{-\pi \ln(t)}} t^{\frac{\Omega_c^2}{\sigma_\Omega^2}} dt
\end{aligned} 
\quad (29) $$

To obtain the transition probability, we add the random variable $\delta P^{phase}$ to a normal random variable, centered on $P_0$, with standard deviation $\sigma_p$ (the detection noise).

Finally, the Cumulative distribution function of the transition probability is the convolution between the Cumulative distribution function of $\delta P^{phase}$ and the above mentioned normal distribution, of standard deviation $\sigma_p$, centered on $P_0$:

$$F_P(x) = (F_{\delta P^{phase}} * f_{\delta P^{det}})(x - P_0) \quad (30)$$

$$ \text{with : } f_{\delta P^{det}}(x) = \frac{1}{\sigma_P \sqrt{2\pi}} e^{-\frac{1}{2} \frac{x^2}{\sigma_P^2}} $$

# APPENDIX II

## NUMERICAL RESOLUTION OF THE METHOD OF MOMENTS

The method of moments leads to a system of 4 equations with 4 unknowns, which can be reduced to finding the root of a degree-24 polynomial $P$. Due to the approximation of theoretical moments by empirical moments, it is possible that the polynomial has no root. In such a case, the root value is set to 0:

$$ 
\begin{aligned}
F(\sigma_P) = & \left(-\frac{9E_2^2 E_4}{4} + E_4^2\right) \\
& \left(-\frac{100E_2^2 E_4}{81} + \frac{1600E_4^2}{729} - E_2 E_6\right)^2 \\
& - \left(\frac{100E_2^2 E_4^2}{27} - \frac{1600E_4^3}{729} - 2E_2^3 E_6 + \frac{4E_2 E_4 E_6}{3}\right)^2
\end{aligned} 
\quad (31) $$

$$ \text{With : } \begin{cases} e_2 = \mathbb{E}[(P - P_0)^2] \\ e_4 = \mathbb{E}[(P - P_0)^4] \\ e_6 = \mathbb{E}[(P - P_0)^6] \end{cases} \quad (32) $$

And:

$$ 
\begin{cases} E_2(\sigma_p) = (e_2 - \sigma_p^2)^2 \\ E_4(\sigma_p) = (e_4 - 6e_2\sigma_p^2 + 3\sigma_p^4)^2 \\ E_6(\sigma_p) = (e_6 - 15e_4\sigma_p^2 + 45e_2\sigma_p^4 - 15\sigma_p^6)^2 \end{cases} 
\quad (33) $$

Solving for the roots of the polynomial $F$ we find the value of $\sigma_P$. $P_0$ is determined by averaging the data.

All that remains is to determine $C_0$ and $\sigma_\Omega$ using the two following equations:

$$e_2 = \sigma_p^2 + \frac{1}{8} \frac{C_0^2}{\sqrt{1 + 2 \frac{\sigma_\Omega^2}{\Omega_c^2}}} \quad (34)$$

$$e_4 = 3\sigma_p^4 + \frac{3}{4} \frac{\sigma_p^2 C_0^2}{\sqrt{1 + 2 \frac{\sigma_\Omega^2}{\Omega_c^2}}} + \frac{3}{128} \frac{C_0^4}{\sqrt{1 + 4 \frac{\sigma_\Omega^2}{\Omega_c^2}}} \quad (35)$$

$$ \sigma_\Omega = \Omega_c \sqrt{\frac{-\beta + \sqrt{\beta^2 - \beta}}{2}} \quad \text{with } \beta = 1 - \frac{4 E_4(\sigma_p)}{9 E_2(\sigma_p)^2} \quad (36) $$

Finally, using (34), we find $C_0$:

$$C_0 = \sqrt{8 \sqrt{1 + 2 \frac{\sigma_\Omega^2}{\Omega_c^2}} \cdot (e_2 - \sigma_p^2)} \quad (37)$$

The polynomial $F$ in (31) has multiple roots, which we numerically determine for $\sigma_P$ in the [0-0.01] range. However only one root is the actual solution to our problem. For each possible root $\sigma_P$, we calculate the corresponding $C_0$ and $\sigma_\Omega$. To find the triplet $(\sigma_P, C_0, \sigma_\Omega)$ that is the solution to our problem, we select the triplet $(\sigma_P, C_0, \sigma_\Omega)$ which minimizes the quantity:

$$ S(\sigma_P, C_0, \sigma_\Omega) = \left( \mathbb{E}[(P - P_0)^2] - \sum_i (P_i - P_0)^2 \right)^2 \\ + \left( \sqrt{\mathbb{E}[(P - P_0)^4]} - \sqrt{\sum_i (P_i - P_0)^4} \right)^2 \\ + \left( \sqrt[3]{\mathbb{E}[(P - P_0)^6]} - \sqrt[3]{\sum_i (P_i - P_0)^6} \right)^2 \quad (38) $$

Equation (36) imposes: $0 < \beta^2 - \beta$ which, after calculation, gives: $9E_2(\sigma_p)^2 < 4E_4(\sigma_p)$. The estimation of the theoretical moments by the empirical moments may lead to a violation of this condition. In such a case, we consider: $\sigma_\Omega = \Omega_c \sqrt{\frac{-\beta}{2}}$. If this is not sufficient, then $\sigma_\Omega = 0$. As for the equation (37), it is necessary that $e_2 - \sigma_p^2 < 0$. If this is not the case, we set arbitrarily $C_0 = 0$.

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