Atom interferometry in an Einstein Elevator (2025)

Overview of microgravity platforms for cold-atom physics

Our platform, the EE in Bordeaux, provides the longest daily microgravity availability among all ground-based facilities worldwide. Its exceptional repeatability and reliability have enabled extensive studies of all-optical cooling techniques, achieving quantum degeneracy under microgravity conditions³⁰. We successfully demonstrated acceleration-sensitive atom interferometry with interrogation times reaching up to 2T = 200 ms. Our results are competitive with current space-based experiments while avoiding the significant technical complexities and stringent size, weight, and power (SWaP) constraints inherent in space missions.

To place these achievements in context, we compare our platform with other microgravity facilities (see Table1).

Low-orbit platforms

The CAL aboard the ISS has achieved Bose-Einstein condensation³⁴, produced dual-species ultracold gases²², and demonstrated bubble-shaped traps³⁹. Atom interferometry experiments using Bragg pulses were conducted, but interrogation times were limited (2T = 16 ms) due to strong vibrations caused by onboard equipment, human activity, and accelerations associated with orbital dynamics²⁴. Similarly, atom interferometry on the Chinese Tiangong Space Station (TSS) measured rotation rates using thermal atoms, achieving longer interrogation times (2T = 150 ms) compared to the ISS, yet limited by reduced signal-to-noise ratios resulting from the higher thermal cloud temperatures (~2 μK)²⁵. Despite offering excellent microgravity conditions (10⁻⁷ g), the TSS faces strict SWaP restrictions.

Parabolic flight platforms

The MAIUS sounding rocket missions offer brief (~6-min) parabolic flights reaching altitudes above 240 km. These missions successfully conducted numerous short-duration experiments with Bose-Einstein condensates²⁰. Recent flights demonstrated atom interferometry with spinor condensates at short interrogation times (2T = 4 ms)²³. Although these missions proved capable of remote operation under challenging conditions, their potential is limited by infrequent launch schedules.

Novespace’s Zero-G aircraft performs around 30 parabolas per flight, each providing roughly 22 s of weightlessness at altitudes below 10 km. Atom interferometry on this aircraft platform has successfully measured accelerations²⁸ and tested the WEP²⁹, though strong vibrations and rotation rates restrict the achievable interrogation time to 2T = 20 ms.

Ground-based platforms

The ZARM drop tower in Bremen enables atom interferometry experiments with quantum degenerate gases and notably longer interrogation times (up to 2T = 700 ms)^26,40. However, operational constraints, specifically the need to evacuate the vacuum tube between experiments, significantly limit its repetition rate (only 2–3 drops per day). To address this limitation, the recently introduced ZARM GraviTower employs an airtight slider mechanism, allowing up to 2.5 s of microgravity and substantially higher repetition rates (960 drops/day)²⁷.

The HITec/LUH EE in Hannover, operational since 2020, offers an optimal balance between microgravity quality and experiment repetition rate, providing up to 4 s of controlled free fall within a 40-m vertical path. While several cold-atom experiments are planned for this platform^38,41, experimental results have not yet been published.

Experimental setup and measurement sequence

The sensor head consists of a titanium vacuum chamber, ion pump, magnetic gradient and bias compensation coils, a mu-metal magnetic shield, and all optics required for the 3D magneto-optical trap (MOT) and Raman excitation. The vertical Raman beam is retro-reflected by a mirror that acts as an inertial reference frame. A three-axis mechanical accelerometer (Colibrys SF3000L) is fixed to the rear of this mirror to monitor its motion. Other critical hardware, including power supplies, RF synthesizers, analog and digital electronics, and fibered laser systems, reside in racks near the EE. They are connected to the sensor head through a series of 15-m-long coaxial cables and optical fibers. The two Raman frequencies are generated using an electro-optic phase modulator driven by an ultra-low phase noise microwave synthesizer. The two optical frequencies propagate in the same optical fiber, providing excellent common-mode rejection that mitigates the impact of laser phase noise on the AI. Additional details can be found in^29,30,32,36.

In this work, the sensor head is fixed to the EE platform. The EE was designed and built by the French company Symétrie. Its operational principle relies on mimicking the free-fall trajectory of an object in Earth’s gravity. The sensor head, fixed to its moving platform, experiences up to half a second of weightlessness by launching it upward and following a pre-programmed parabolic trajectory using a precisely-controlled motorized position servo. Measurements of the moving platform’s position are provided by two incremental grating rulers. Vertical accelerations are achieved thanks to two linear motors located on both sides of the platform. The cycling rate of the EE is limited by the 12-s cool-down period required by the linear motors.

We evaluated the quality of the EE trajectory by recording the acceleration of the reference mirror and the rotation rate of the platform during its motion. Figure5a shows the typical acceleration profiles obtained throughout one full cycle. The vertical acceleration profile is comprised of standard gravity (1g), hypergravity (2g), and microgravity (0g) phases. Figure5c shows residual vibrations measured by the mechanical accelerometer during the 0g phase. The peak-to-peak residual acceleration is below 50 mg on the vertical Z-axis and below 100 mg on the horizontal X- and Y-axes (not shown). Rotation rates about the horizontal axes (Ω_X and Ω_Y) are monitored using two fiber optic gyroscopes (KVH DSP-1750) fixed directly to the elevator platform. Figure5d shows a typical rotation rate profile during the 0g phase, which features peak-to-peak values less than 10 mrad/s.

The experimental sequence is illustrated in Fig.5b. A six-beam MOT of ⁸⁷Rb is loaded directly from background vapor for typically 4 s with an optical power of about 100 mW and an intensity of 20 I_sat. The cooling beams are red-detuned by −18 MHz (−3Γ) from the ⁸⁷Rb cycling transition, and the ratio of cooling to repump light intensity is I_rep/I_cool = 0.1. About 5 × 10⁷ atoms are loaded in this configuration. Atoms are fully loaded into the MOT during the dead time between parabolas, and they are maintained during the pull-up and injection phases. When the EE platform reaches the 0g phase, it triggers our control system to initiate the sub-Doppler cooling stage. This stage is performed during the first 10 ms of the 0g phase, where the MOT gradient coils are turned off, the detuning of the cooling beams is linearly ramped to −24Γ, and the overall cooling power is reduced to ~1I_sat. At this point the cloud temperature reaches ~7 μK with all ground-state magnetic sub-levels populated. The atoms are then prepared in the \(\left\vert F=1,{m}_{F}=0\right\rangle\) state via a purification sequence. First, a “repump” pulse resonance with the \(\left\vert F=1\right\rangle \to \left\vert {F}^{{\prime} }=2\right\rangle\) transition pumps atoms to the \(\left\vert F=2\right\rangle\) state. Then, a magnetic field bias of 130 mG is applied along the vertical direction to split the magnetic sub-levels. This is followed by a microwave π-pulse, resonant with the clock transition at 6.834 GHz, 500 μs in duration, that transfers population from \(\left\vert F=2,{m}_{F}=0\right\rangle\) to \(\left\vert F=1,{m}_{F}=0\right\rangle\). Finally, we remove atoms remaining in the \(\left\vert F=2\right\rangle\) manifold with a 600 μs push pulse resonant with the \(\left\vert F=2\right\rangle \to \left\vert {F}^{{\prime} }=3\right\rangle\) cycling transition. Only atoms in the \(\left\vert F=1,{m}_{F}=0\right\rangle\) remain. This state preparation sequence is followed by a sequence of Raman pulses that form the AI. A π/2−π/2 sequence is used to perform optical Ramsey spectroscopy to determine the coherence of the sample at large free-fall times (see below). We use a π/2−π−π/2 pulse sequence to construct the quantum accelerometer. The detuning of the Raman beam from the \(\left\vert {F}^{{\prime} }=2\right\rangle\) excited state is Δ ≃ −830 MHz, and the duration of the Raman π-pulse is 2τ ≃ 20 μs. State detection is performed by collecting atomic fluorescence on a photodiode. We apply two light pulses to the sample after the interferometer along the vertical direction—one resonant with the \(\left\vert F=2\right\rangle \to \left\vert {F}^{{\prime} }=3\right\rangle\) transition to measure the number of atoms in \(\left\vert F=2\right\rangle\) (N₂), and one that includes an additional repump sideband, giving the total number of atoms in both ground states (N_T). The peak intensity of these pulses is ~37I_sat.

To reconstruct the blurred atomic fringes due to the residual vibrations of the EE platform, we make use of the mechanical accelerometer fixed to the rear of the reference mirror, measuring the acceleration perpendicular to the mirror’s surface. During the interferometer, this time-varying acceleration is weighted by the response function of the AI⁴², and integrated over the total interrogation time—providing an estimate of the vibration-induced phase shift. AI fringes are reconstructed from noise by plotting the measured population ratio as a function of this estimated phase shift³². The mechanical sensor has a significant bias (a few 10⁻² m/s²) that typically drifts by about 10⁻³ m/s² over 10 min due to its inherent temperature dependence. In practice, this drift can be corrected by tracking the phase shift of the reconstructed atomic fringe, the AI being much more stable by several orders of magnitude³⁶. In the absence of contrast loss mechanisms, the self-noise of the mechanical accelerometer limits our ability to resolve the atomic fringes at large interrogation times. Replacing the Colibrys SF3000L with a more sensitive accelerometer (e.g., a Nanometrics Titan) will help extend this range.

Ramsey interferometry

To assess our ability to manipulate atoms throughout the entirety of the zero-gravity phase provided by our simulator, we conduct Ramsey fringes. Ramsey interferometry is widely used for high-resolution spectroscopy, primary frequency standards in cesium fountain clocks⁴³, and tests of fundamental physics with optical atomic clocks⁴⁴. The frequency sensitivity of the Ramsey technique scales inversely with the interrogation time T_R in a π/2−π/2 sequence of pulses. Recent experiments have pushed the limits of Ramsey interrogation in reduced gravity environments⁴⁵, achieving up to T_R = 400 ms on a 0g aircraft⁴⁶. Here, we probe the ⁸⁷Rb clock transition on the EE using co-propagating optical Raman beams with interrogation times up to 380 ms.

Figure6 and 6b show Ramsey fringes obtained for T_R = 100 ms and 380 msrespectively. Fits to these data yield a fringe half-width at half-maximum of Δν = 1/2T_R as low as 1 Hz at T_R = 380 ms. The uncertainty on the fit of the fringes is less than 1 Hz, and the light shift is not corrected in these datasets. This measurement is not conducted for metrological purposes, but it illustrates the potential for exploring long-time interference effects in our EE. Figure6d shows measurements of the short-term frequency sensitivity σ_ν as a function of the interrogation time. Analytically, σ_ν is given by the following expression:

where \({{{\rm{SNR}}}}=C/2{\sigma }_{{{{\rm{res}}}}}\) is the signal-to-noise ratio of the Ramsey fringes determined by the fringe contrast C and the standard deviation of fit residuals \({\sigma }_{{{{\rm{res}}}}}\). Note that each point corresponds to a single shot and \({\sigma }_{{{{\rm{res}}}}}\) is extracted from each dataset with no time averaging. The measured fringe contrast decreases with the interrogation time, as shown in Fig.6c. This is due to the finite temperature of the atoms: as the cloud expands, it samples the lower-intensity regions of the Gaussian Raman beam, which reduces the effective Rabi frequency Ω_co and therefore the efficiency of the final beamsplitter pulse. As a consequence, the short-term sensitivity of the measurement is limited to σ_ν = 8 × 10⁻¹² for T_R = 380 ms. Our best sensitivity σ_ν = 5.4 × 10⁻¹² is reached for T_R = 300 ms. This corresponds to a trade-off between the gain with interrogation time and the reduction of fringe contrast.

Double diffraction and double single diffraction

The two optical frequencies required to drive a two-photon Raman transition are derived from an optical phase modulator. They are therefore phase-locked, spatially overlapped, and feature the same optical polarization. We inject this light into a single-mode polarization-maintaining fiber connected to a commercial beam collimator that expands the Raman beam diameter to ~20 mm. This beam is aligned vertically through the atoms, with one λ/4-plate on either side of the vacuum system, before being retro-reflected by the reference mirror. With a suitable choice of the first λ/4-plate angle, the polarization of the two Raman beams will be lin⊥lin, where they drive velocity-sensitive transitions (counter-propagating configuration), or σ⁺/σ⁻, where they drive velocity-insensitive transitions (co-propagating configuration). In the latter case, the two Raman transitions are degenerate due to the absence of a Doppler shift. This transition is visible near zero detuning in Fig.1b due to imperfect lin⊥lin polarization. Under gravity, accelerated atoms experience a Doppler shift that lifts the degeneracy of the two counter-propagating transitions. In weightlessness, all of these Raman transitions are degenerate and occur simultaneously.

Because our thermal cloud has a relatively large range of velocities (≫ℏk_eff/M where M is the mass of a rubidium-87 atom), two regimes of atomic diffraction are possible. For the zero-momentum class, \(\left\vert F=1,p=0\right\rangle\) is simultaneously coupled with both states \(\left\vert F=2,p=\pm \hslash {k}_{{{{\rm{eff}}}}}\right\rangle\) leading to double diffraction⁴⁷. For a momentum class p ≠ 0, the laser pulse couples simultaneously \(\left\vert F=1,p\right\rangle\) with \(\left\vert F=2,p+\hslash {k}_{{{{\rm{eff}}}}}\right\rangle\) on the one hand, and \(\left\vert F=1,-p\right\rangle\) with \(\left\vert F=2,-p-\hslash {k}_{{{{\rm{eff}}}}}\right\rangle\) on the other hand—leading to DSD²⁹. The double diffraction regime is avoided by choosinga Rabi frequency Ω_eff/2π ≃ ν_R, which determines the pulse bandwidth, and a Raman detuning ∣δ∣ > ν_R, where ν_R ≃ 15 kHz is the two-photon recoil frequency [see Fig. 3a]. Typically, we use δ = 50 kHz. Under these conditions, the first Raman pulse selects a narrow range of atomic velocities symmetrically about p = 0. The DSD process leads to two symmetric AIs as discussed above [see Fig.3b]. Only a fraction of the total number of atoms participates in the upper (+) or the lower (−) interferometers.

After the interferometer sequence, position-velocity correlation in the two diffracted wavepackets leads to two spatially separated clouds. For 2T = 200 ms, the two interferometer ports are separated by 8 mm, which is small compared to the width of each cloud (~20 mm) due to the finite temperature of the sample. We detect the fluorescence of both clouds simultaneously by imaging the light on a single photodiode. This provides a measurement of the population ratio R, and hence the acceleration Φ_a through Eq. (2).

Model of Raman spectroscopy in microgravity

We perform a numerical simulation to model the Raman spectrum in microgravity using measured experimental parameters. The atom-light interaction leads to a four-photon process that drives Raman transitions with effective wavevectors ±k_eff = ± (k₁−k₂) ≃ ± 2ℏk₁, since k₂ ≃ −k₁. The two optical frequency components of the incident beam (ω₁, k₁ and ω₂, k₂) have the same linear polarization. Generally, the polarization also contains a small residual circular component that allows co-propagating Raman transitions to contribute to the Raman spectrum. We consider a one-dimensional system of atoms with two long-lived electronic ground states \(\left\vert e\right\rangle\) and \(\left\vert g\right\rangle\) separated in frequency by \({\omega }_{\left\vert e\right\rangle }-{\omega }_{\left\vert g\right\rangle }\). This splitting is much larger than the Doppler width of the sample. Each atom has an arbitrary momentum class p and can be excited by integer multiples of the photon momentum ℏk_eff—creating a ladder of coupled momentum states. We truncate this state space to ±2ℏk_eff in both ground states, resulting in a 10-level model of the atom. This truncation is justified because we operate with Raman π-pulses that have a Fourier width similar to \(\omega_R = 2\pi \nu_R\). As a result, higher-order momentum states (±Nℏk_eff with N > 2) are not significantly populated due to their larger detunings. We label these bare states by \(\left\vert g,j\right\rangle\) and \(\left\vert e,j\right\rangle\), where j = 0,±1,±2 is an integer representing momentum state p + jℏk_eff. The full atom-light system is represented by an energy–momentum diagram shown in Fig.7. This complex system includes residual co-propagating transitions and two opposite velocity classes in the case of DSD. Consequently, a number of additional near-resonant momentum states can be accessed through two- and four-photon processes. These transitions are indicated by solid and dashed arrows in Fig.7b–d.

We express the time-dependent atomic wavefunction \(\left\vert \psi (t)\right\rangle\) as the following superposition of bare states \(\left\vert n\right\rangle\) with corresponding amplitudes \({c}_{\left\vert n\right\rangle }(t)\):

Here, the wavefunction is written in the interaction representation, where the basis states rotate at a frequency corresponding to their internal energies \(\hslash {\omega }_{\left\vert n\right\rangle }\). Following the calculation in ref. ⁴⁸, we modify the basis to the field-interaction representation. Here, the states rotate at an additional frequency corresponding to the detuning relative to the two-photon resonance between the central states \(\left\vert g,0\right\rangle\) and \(\left\vert e,0\right\rangle\):

where the detunings are as follows:

Here, ω_D = k_effp/M is the Doppler frequency and \({\omega }_{{{{\rm{R}}}}}=\hslash {k}_{{{{\rm{eff}}}}}^{2}/2M\) the two-photon recoil frequency.

The wavefunction in the field-interaction representation is \(\left\vert \tilde{\psi }(t)\right\rangle={\sum }_{n}{a}_{\left\vert n\right\rangle }(t)\left\vert n\right\rangle\). The benefit of this representation is that the effective Hamiltonian becomes time-independent, and the dynamics are less computationally costly to determine. The time evolution of the system is computed by numerically solving the Schrödinger equation^47,49:

by computing the matrix exponential of the effective Hamiltonian \({{\mathbb{H}}}_{{{{\rm{eff}}}}}\):

The effective Hamiltonian can be written as the sum of two matrices:

where \({\mathbb{D}}\) is a diagonal matrix containing the state-dependent detunings:

and \({\mathbb{A}}\) is a matrix with only off-diagonal elements that describe the coupling between states:

where χ_eff ≡ Ω_eff/2 and χ_co ≡ Ω_co/2 are half-Rabi frequencies. The effective two-photon Rabi frequency is given by \({\Omega }_{{{{\rm{eff}}}}}=\frac{{\Omega }_{gi}^{*}{\Omega }_{ei}}{2\Delta }\), where Ω_gi and Ω_ei are Rabi frequencies associated with the one-photon transitions between \(\left\vert g\right\rangle\) and \(\left\vert i\right\rangle\), and between \(\left\vert e\right\rangle\) and \(\left\vert i\right\rangle\), respectively. We account for residual co-propagating transitions with the Rabi frequency Ω_co = ϵΩ_eff, where ϵ ≪ 1 is a factor that describes the defect in linear polarization. These transitions are insensitive to the Doppler effect and transfer negligible recoil to the atoms since k₁ + k₂ ≃ 0. Consequently, co-propagating transitions occur at a fixed detuning δ₀ for all momentum states under consideration.

The simulation takes into account the velocity dispersion of the atomic cloud through the Doppler shift ω_D(v) = k_effv of velocity-dependent transitions. We assume a 1D Maxwell-Boltzmann velocity distribution at temperature \({{{\mathcal{T}}}}\) with probability density:

where \({\sigma }_{v}=\sqrt{2{k}_{{{{\rm{B}}}}}{{{\mathcal{T}}}}/M}\). To compute the population \({P}_{\left\vert n\right\rangle }(\delta,t)\) of the state \(\left\vert n\right\rangle\) as a function of laser detuning δ, we integrate our numerical solution (9) over this velocity distribution:

The total population in the upper ground state \(\left\vert e\right\rangle\) is obtained by summing the populations of the five associated momentum states: \(\left\vert e,-2\right\rangle,\left\vert e,-1\right\rangle,\left\vert e,0\right\rangle,\left\vert e,+1\right\rangle,\left\vert e,+2\right\rangle\). This model of Raman spectra is shown in Fig.2 for our experimental parameters, with a velocity spread of the order σ_v ≈ 3v_R (three times the two-photon recoil velocity).

Contrast loss due to rotation noise

The sensitivity of the AI at long interrogation times (T > 50 ms) is primarily limited by two effects: (1) reduction of the Rabi frequency due to cloud expansion and spatial averaging of the Raman beam, and (2) the rotation rate of the EE platform and its associated noise. The relatively high temperature of our thermal samples exacerbates both of these effects. The former effect is well understood and has been studied elsewhere⁵⁰. In this section, we describe the latter effect since it has unique features on the EE platform. Rotations of the Raman wavevector cause a separation of atomic trajectories at the final π/2 pulse. This “open interferometer” effect can be quantified by an overlap integral between the wavepackets at either output port of the interferometer. Following previous work^29,51, we introduce the displacement vector at time t:

Here, δR(t) and δP(t) are the position and momentum displacement vectors between two atomic wavepackets at time t, and Δt = t−t₀ is the free-expansion time of the wavefunction relative to release time t₀, which we take to be t₀ = 0 for simplicity. The contrast loss at the time of the final beamsplitter pulse t = t₃ is well approximated by the following overlap integral

provided the expansion time \(\Delta t\gg \hslash /M{\sigma }_{v}^{2}\), where σ_v is the velocity spread of the wavepacket. Assuming a thermal velocity distribution, the spatial probability density ∣ψ(r,t)∣² can be written:

where \({\sigma }_{r}^{2}(t)={\sigma }_{0}^{2}+{\sigma }_{v}^{2}{t}^{2}\) is the e⁻¹ spatial width of the distribution at time t, σ₀ is the initial wavepacket spread, and \({\sigma }_{v}=\sqrt{2{k}_{{{{\rm{B}}}}}{{{\mathcal{T}}}}/M}\) is the e⁻¹ velocity width. For our temperature of \({{{\mathcal{T}}}}=7.5\,\mu\)K, this width is σ_v ≃ 3.8 cm/s. Thus, the calculations that follow are valid for expansion times Δt ≫ 500 ns.

We consider the two wavepacket trajectories associated with a vertically-oriented, single diffraction Mach-Zehnder interferometer (k_eff = k_zz) in a reference frame undergoing a constant rotation with rotation vector Ω = (Ω_x,Ω_y,Ω_z). The displacement at the time of the final beamsplitter pulse t₃ = Δt = 2T can be shown to be (up to order T⁴ and Ω²):

Evaluating the contrast loss, we find:

where we made the approximation σ_r(2T) ≈ 2σ_vT ≫ σ₀ and we truncated terms in the exponential larger than T⁴ and Ω².

For the more general case of a time-varying rotation rate Ω(t), we numerically compute the orientation of the Raman beam using a two-axis gyroscope mounted directly on the EE platform. We apply a 3D rotation matrix that orients the effective wavevector \({{{{\bf{k}}}}}_{{{{\rm{eff}}}}}^{(i)}\) at the time of each Raman pulse (i = 1,2,3). We then calculate the displacement vectors at the final π/2 pulse as follows:

where we took an expansion time of Δt = 2T in the last expression. The contrast loss is then computed numerically using Eq. (15).

Figure8 shows our model of the contrast loss factor as a function of interrogation time T at sample temperatures of \({{{\mathcal{T}}}}=7.5\,\mu\)K and 30 nK. Our model combines the “open interferometer” effects of the time-varying rotation rate of the EE platform and the thermal expansion of the cloud. Our present temperature of 7 μK leads to ~40% contrast after only T ≃ 30 ms. The second major effect is the residual rotation of the EE platform. We also predict significant fluctuations in the contrast due to the measured rotation rate noise. These effects result from mechanical vibration modes of the EE platform that are excited during its motion. The amplitude of these modes varies during the motion, and thus, the contrast loss is strongly dependent on when the Raman pulses occur. A comparison with measured contrast loss shows reasonable agreement with our model predictions from EE platform rotation measurements. We attribute the discrepancies to differences in the amplitude and timing of the reference mirror motion compared to the EE platform. The sensitivity to reference mirror rotations can be largely mitigated using ultracold atoms in an optical dipole trap³⁰, where temperatures of 30 nK are routinely achieved, or by compensating the residual rotation in real time with gyroscopes and a tip-tilt mirror⁵².

Bayesian estimation of the acceleration sensitivity

In this section, we describe a robust Bayesian method to estimate interference fringe parameters (amplitude A, offset μ_B, and offset noise σ_B) from a set of atomic population measurements. From these estimates, we accurately determine the AI signal-to-noise ratio and single-shot acceleration sensitivity. This analysis method does not rely on methods such as non-linear least-squares fitting, nor does it require large amounts of data to estimate sample distributions from histograms. Furthermore, it converges at an optimal rate (\(\sim 1/\sqrt{N}\)) toward unbiased estimates of these fringe parameters.

Fringe noise model

The total population ratio measured at the AI output can be expressed as

where A is the fringe amplitude, B is the fringe offset, and ϕ is the interferometer phase. Considering the residual vibration levels experienced on the EE platform and interrogation times of T ≫ 1 ms, the phase ϕ is randomly scanned over a large interval, covering many sinusoidal periods. Therefore, after being wrapped on the ϕ ∈ [0, 2π] range, it can be safely assumed that the random variable ϕ is described by a uniform probability distribution. Moreover, the fringe amplitude A and offset B can be considered as random variables affected by Gaussian noise distributions with mean values μ_A and μ_B and standard deviations σ_A and σ_B, respectively.

The objective is to estimate the AI sensitivity \({\mathbb{S}}\), which can be written as:

where σ_ϕ = σ_B/μ_A is the phase noise at the mid-fringe location. It is therefore important that we obtain good estimates of the fringe amplitude μ_A and the offset noise σ_B. We note that the contrast noise σ_A does not enter directly into the AI sensitivity because it produces zero phase noise at the mid-fringe. However, it can indirectly influence the estimate of both μ_A and σ_B under specific conditions. In the following simplified treatment, we consider the case where the offset noise is the dominant source of noise (σ_B/μ_B ≫ σ_A/μ_A), allowing us to treat the amplitude A as a constant. The more general case follows naturally from this treatment, but is significantly more costly due to one additional integration.

The measured population ratio R is the sum of two independent random variables R = X + Y, with \(X=A\cos (\phi )\) and Y = B, whose PDFs are respectively given by:

Here, f_X(u) is the usual PDF of a sinusoidal function, which features a characteristic bi-modal shape with singularities at u = ±A. The PDF of the sum R is given by the convolution f_R = (f_X * f_Y), which can be written as:

The effect of the offset noise is illustrated in Fig.9. Generally, the larger the noise parameter σ_B, the more the smearing of the bi-modal distribution.

Bayesian estimation algorithm

Once the sensor noise model has been derived, it can be used in the frame of a Bayesian analysis in order to optimally estimate the various parameters of interest (μ_B, A, and σ_B in this case). The Bayesian algorithm is similar to that presented in refs. ^32,53. It consists of updating the probability distribution for the parameter state V = {μ_B,A,σ_B} after each new measurement of R. An iteration k of the algorithm can be decomposed into three main steps that are looped over the set of measurements {R}:

1. 1.

   The prior distribution for the current iteration P(V)_k is set equal to the conditional distribution from the previous iteration \(P{(V| {R}_{k-1})}_{k-1}\).

2. 2.

   A new measurement R_k is added, and the likelihood distribution P(R_k∣V) is computed directly from the noise model [Eq. (23)].

3. 3.

   The updated conditional distribution after the kth measurement is derived from Bayes’ rule: \(P{(V| {R}_{k})}_{k}=P{(V)}_{k}\,P({R}_{k}| V)/N\), where N is a normalizing factor.

The application of this Bayesian analysis in a 3D parameter space is computationally demanding. We found that the execution time can be significantly reduced by using pre-computed look-up tables for the likelihood distribution P(R_k∣V). Starting from a uniform prior distribution P(V)₀ in this 3D space, the algorithm quickly converges to a much narrower PDF, from which the parameters estimates and confidence intervals can be accurately retrieved (see Fig.4). These estimates and uncertainties can finally propagate to the AI signal-to-noise ratio SNR = A/σ_B and the shot-to-shot sensitivity:

Drift tracking

The approach presented above implicitly assumes that fringe parameters {μ_B,A,σ_B} are constant quantities over time. However, in practice, instabilities in some experimental parameters, such as the detection pulse intensity, can result in slow temporal variations of the fringe parameters.

Figure9 illustrates how drifts in the offset μ_B can lead to underestimates of the SNR if they are not taken into account. Since these variations are slow with respect to the cycling time of our experiment, an effective mitigation strategy consists of dynamically tracking the offset drift by applying the same Bayesian approach on a “sliding window,” i.e., a moving subset of successive experimental data points. Thanks to its rapid convergence, the Bayesian algorithm is capable of providing a reliable estimate of the “instantaneous” offset value after only a few tens of measurements, making it possible to track the drift accurately, as demonstrated in Fig.9. This drift can then be removed from the experimental measurements before a final Bayesian analysis can be applied on the full (corrected) dataset. The number of points is chosen empirically to optimize the offset drift estimation—leading to the best histogram. It must be long enough to enable a good convergence of the Bayesian algorithm, but small enough to avoid any “averaging effect” of the drift.

The efficiency of this offset tracking method was also benchmarked on simulated data and has produced better results than alternative methods based on a moving average or low-pass filtering, highlighting once again the optimality of the Bayesian approach.