Mach-Zehnder atom interferometry with non-interacting trapped Bose-Einstein condensates

Domain: neuroscience
Published: 2026-04-11
Editorial outcome: minor_revision

Abstract
The coherent manipulation of a quantum wave is at the core of quantum sensing. For instance, atom interferometers require splitting and recombination processes to map the accumulated phase shift into a measurable population signal. Although Bose Einstein condensates (BECs) are the archetype of coherent matter waves, their manipulation in double-well potentials has been limited by the strong interparticle collisions dominating over the tunneling energy. Here, we overcome this problem by using BECs with tunable interactions trapped in an innovative array of double-well potentials and exploiting quantum tunneling to realize coherent beam splitting. We operate several Mach-Zehnder interferometers in parallel, canceling common-mode potential instabilities by a differential analysis, thus demonstrating a trapped-atom gradiometer. Furthermore, by applying a spin-echo protocol, we suppress additional decoherence sources and approach unprecedented coherence times of one second. Our interferometer will find applications in precision measurements of forces with a sub-micron spatial resolution and in linear manipulation of quantum entangled states for sensing with sub shot-noise sensitivity.

Introduction
Bose-Einstein Condensates (BECs) of ultracold atomic gases [1,2] are recognized as powerful tools for both fundamental research [3,4] and emerging quantum technologies [5–7]. Often regarded as the matter-wave analog of lasers [8,9], BECs hold particular promise in the domain of high precision measurements [10]. Thanks to their remarkably low momentum spread [11], they are currently exploited in free-falling atom interferometers within drop towers [12–18] and earth-orbiting research laboratories [19]. Trapped atom interferometers [20–23] that exploit the unique properties of BECs promise to bring measurements of gravity, inertial forces and electromagnetic fields into a new realm. The large spatial coherence of a quantum degenerate gas enables splitting a BEC in two interferometric modes with large separation, allowing for high-sensitive measurements. Moreover, the quantum-limited size of a BEC provides high spatial resolution for sensing applications [10].

Among the several interferometric schemes with trapped BEC [24–28], much effort has been devoted to the realization of double-well systems [29–32] where the two spatial modes are separated by a potential barrier and coupled by tunneling [33–36]. These interferometers could enable force measurements with high spatial resolution in compact setups. However, collisions between atoms in high-density samples have caused dephasing of the interferometric signal [37, 38] and obstructed the realization of coherent beam splitters [39]. Previous valuable attempts to overcome these issues [36,40] have been limited by the short coherence times and a reduction of the interference contrast. Additionally, interatomic collisions prevent the linear rotations of the Bloch vector associated to the many-particle quantum entangled states, which are necessary for the operation of the interferometer below the shot-noise limit [41]. A possible solution involves tuning the collisional scattering length to zero via magnetic Feshbach resonances – as already employed to suppress interaction-induced dephasing in Bloch oscillation [42, 43]. However, non-interacting gases make the interferometer extremely sensitive to imperfections in the trapping potential. This occurs because even small energy mismatches between the wells directly translate into uncontrolled phase shifts, in the absence of the mean-field self-compensation whereby interactions generate a population imbalance that reduces the effective bias [44, 45]. Therefore, despite early optimistic expectations, atom interferometry based on BECs confined in a double-well trap has yet to fully realize its potential in the field of precision metrology.

In this work we overcome current roadblocks and demonstrate the operation of a full Mach-Zehnder Interferometer (MZI) with non-interacting BECs trapped in two spatially separated modes. The key ingredient is an innovative potential [46] made of an array of Double-Well (DW) traps where several interferometers operate simultaneously. In this way, residual instabilities of the trap, that could mask the coherence of a single sensor, are common mode and can be canceled using a differential analysis. Therefore, our system also realizes the first trapped atom gradiometer reported in the literature, extending the range of applicability of a technique already exploited in free falling atom interferometers [47]. Each MZI (see the scheme in Fig. 1) consists of a beam splitter, an interrogation time, and a final beam splitter that maps the accumulated relative phase into a population imbalance. The beam splitters operate in the linear regime with near-unit contrast by canceling inter-particle interactions via a magnetic Feshbach resonance and by tuning the height of the barrier separating each DW-changing the tunneling energy. By implementing a spin-echo protocol [48], we suppress residual technical noise and extend the interferometer’s coherence time to approximately 1 second nearly two orders of magnitude longer than previously reported values [36,40]. Our results unlock the full potential of trapped atom interferometry with Bose-Einstein condensates, establishing it as a powerful and promising platform for quantum-enhanced technologies.

Results
The experimental system consists of a BEC of 39K atoms manipulated by an innovative array of DWs [46], where interactions are tuned to zero using a broad magnetic Feshbach resonance [49,50]. During the final part of the evaporation stage, two collinear optical lattices aligned along the horizontal x directions are switched on adiabatically. Their wavelengths λ ≃ 1013 nm and λ ≃ 1064 nm create a Beat-Note superlattice (BNSL) [51] with a spacing equal to 1 λ1λ2 ∼ 10 µm. In this way, several independent BECs can be loaded in the minima of the potential, Fig. 1a. By using a third optical lattice superimposed to the others, and with wavelength λ 3 ≃ 1120 nm, we create an additional BNSL with spatial periodicity d = 1 λ1λ3 ∼ 5 µm. By changing the intensity of the optical potential, we form an array of DWs (see Supplemental Material) and realize the first beam splitter of each MZI, see Fig. 1b. This operation is followed by an interrogation time T, see Fig. 1c. An external homogeneous force provides a linear potential energy that affects each interferometer with an equal energy mismatch ε between the two wells. A spatial-dependent force realized in our experiment with an additional vertical dipole trap laser, instead, provides an additional non-common energy difference δ between the two modes of each MZI, see Fig. 1c. By performing a further beam splitter (Fig. 1d), we map the accumulated phase onto a population difference (Fig. 1e).

The second beam-splitter exploits the Josephson dynamics in the linear regime (Rabi [35]) of non-interacting atoms tunneling through the central barrier of each DW. In order to optimize the beam-splitting sequence, we have performed a dedicated experiment. We start with a BEC initially loaded in the left well. We reduce the barrier height with a linear ramp and then raise it back again in 20 ms. This is done by acting on the third lattice depth leaving unchanged the first two. The minimum height of the barrier is adjusted in order to achieve a balanced beam splitter. The corresponding optimal sequence is shown in Fig. 2a. In Fig. 2b, we show the distribution of the population imbalance z = (N L,j - N R,j )/(N L,j + N R,j ) after the splitting sequence (see histogram in Fig. 2b). Here, N R,j and N L,j are the number of atoms in the right and left side of the j-th DW (j = 1,2). Residual instabilities of the energy mismatch between the two modes prevent the achievement of shot noise limited splitting (black line in Fig. 2b). Shorter sequences with larger tunneling energies could help to approach such limit, but they could cause unwanted excitations of the atoms to higher lying modes. The possibility to operate simultaneously two or more interferometers requires that the tunneling energy of neighboring DWs is equal. This is demonstrated in Fig. 2c, where we report the Rabi oscillations of z j (for two neighboring DWs) as a function of tunneling time, at low barrier height. This measurement was performed by loading a Bose–Einstein condensate into the left site of an unbalanced double well. Over a period of 10 ms, the height of the central barrier was lowered while the energy offset between the two sites was compensated, there by initiating a Rabi dynamics with an amplitude slightly smaller than the initial imbalance z ∼ 1. The results show near-unity visibility of Rabi oscillations that would be prevented by particle-particle interactions [39].

In the following, we select two central DWs, each loaded with N ≈ 3000 atoms. An external harmonic confinement of frequency ω/2π, created by an additional vertical laser beam, causes a differential energy mismatch δ = mω2d2, where m is the atomic mass. This provides a phase difference between two interferometers equal to ∆ϕ = ϕ 1 - ϕ 2 = ℏ δT, where ϕ j is the interferometric phase acquired by the j-th interferometer. A plot of the two population imbalances, z 1 versus z 2 , is shown in Fig. 3a. For two perfectly correlated interferometers with sinusoidal signals z j = V j sin(ϕ + ϕ j ) + C j , where V j are visibilities, C j are oscillation offsets and ϕ is a common-mode phase noise uniformly distributed between 0 and 2π, the measurements scatter along an ellipse whose eccentricity depends on the relative phase ∆ϕ [52–54],

z˜2 1 + z˜2 2 - 2z˜ 1 z˜ 2 cos(∆ϕ) - sin2(∆ϕ) = 0, (1)

where z˜ j = (z j - C j )/V j . Values of V j smaller than 1 are mainly due to unwanted excitations of the atoms to higher-lying states, finite temperature effects, and detection noise. Values of C j different from 0 arise from unwanted energy mismatch between the two wells during the splitting sequences. Loss of correlations between the two interferometers, mainly due to the interactions and uncontrolled spatial inhomogeneities (see below), causes a spread of the experimental measurements out of the ellipse. We perform a multiparameter maximum likelihood analysis of the experimental data [55] and extract both the differential signal ∆ϕ and the uncorrelated noise σ∆ϕ (see Methods). To study ∆ϕ as a function of T, we collect three sets of measurements of ∆ϕ chosen in a -π + π interval around T = 20 ms, 45 ms and 85 ms (collected in three different days), see Fig. 3b. As expected, ∆ϕ increases linearly with T, with a slope proportional to δ. The interaction is set on the zero crossing of a broad Feshbach resonance around 350 Gauss for atoms in the absolute ground state |F = 1, m F = 1⟩, where F is the hyperfine quantum number [43]. The interferometric determination of the external trapping frequency from such slopes is in agreement, within the error bars, with the value determined using atoms sloshing in the external harmonic potential (see Fig. 3b and Supplemental Material).

To study the effect of interaction-induced dephasing, we fix T and analyze z 1 vs z 2 as a function of the atomic scattering length by tuning the magnetic field B. The corresponding values of σ∆ϕ are shown in Fig. 4 (points). As expected, we observe a minimum around B min = 350.45(6) Gauss, in agreement with the previously reported value [56]. We compare the experimental results with the model including the effect of interactions, imperfect splitting and technical decoherence. We predict (see Supplemental Material) that the uncorrelated noise for an initial coherent state follow the behavior

σ2 ∆ϕ (B) = 2 N (1 + σ2 BS) + 2N (1 + Nσ2 BS) (χ(B)T/ℏ)2 + σ2 tech . (2)

Here, N is the number of atoms in each interferometer (assumed equal) and χ(B) = χ el (B) + χ dd is the sum of the elastic particle-particle (χ el ) and the dipolar (χ dd ) interaction within each well and σ tech is the technical decoherence. We have χ(B) ∝ (B - B min ), where the interplay of the attractive χ dd and the repulsive χ el gives a specific value B min such that χ = 0. σ2 BS quantifies the classical fluctuations of the mean value of z due to uncontrolled energy mismatch between the two DWs after the first beam splitter. Such fluctuations enhance the spread of the quantum state on the Bloch sphere due to non linear dynamics (see inset in Fig. 4) that cause a rotation of the Bloch vector with a speed that is proportional to the distance from the equator. The experimental data are well reproduced by Eq. 2.

As shown by the previous analysis, near B min interactions are minimized by the precise canceling of the scattering length and the primary source of noise remains σ tech . Assuming that the technical decoherence is almost constant on the duration of the interferometer, but varying from shot to shot, it can be compensated by a spin-echo protocol. This consists of applying an additional π pulse at time T/2, midway through the interferometric sequence [57,58]. The spin echo is realized by a linear ramp of the barrier height, as in Fig. 2, but reaching a lower value (approximately 30% of the initial height). Such sequence lasts a time long enough to allow the exchange of the occupation of the two clouds between the two spatial modes. Note that this sequence is achievable due to the unique capability of our apparatus to perform linear rotations around the x-axis of the Bloch sphere. In Fig. 5, we compare the measurements of σ∆ϕ as a function of T for both the gradiometer and the spin-echo protocol with B = B min . Using the gradiometer scheme, we are able to measure a differential phase up to ≈ 200ms (blue triangles). Instead, with the spin-echo protocol we reach a significant decoherence only after 400ms (red circles). At this interrogation time, we further minimize the interaction-induced dephasing by finely tuning the Feshbach magnetic field. At the new optimum value B = 350.48(1) Gauss, using the spin-echo protocol, we extend the interrogation time up to T = 800ms (yellow squares). The origin of the residual noise is under investigation. Multiple π pulses can be used to cancel the effect of differential phase drifts occurring on the timescale T. This will allow to distinguish technical noise sources from fundamental limitations such as finite temperature or residual interparticle interactions.

Methods
Beat-note superlattice. The array of DWs is realized with three co-propagating lasers at wavelengths λ 1 , λ 2 and λ 3 retroreflected on a common mirror to form three optical lattices [46]. The beam waist of the lasers is ≈ 500 µm and their powers are 300/380/250 mW provide optical lattice depths of 370/400/240 nK, respectively. Using a polarizing cube we can superimpose the three lattices with an additional dipole trap beam with a wavelength of 1064 nm and with a 12 µm waist. This laser beam provides a radial confinement of the trapped modes along the y and the vertical z direction of ≈ 180 Hz. Its weak longitudinal confinement along the x direction of ≈ 1 Hz does not prevent the simultaneous balancing of many DWs. In addition it is used in combination with another dipole trap beam provided by an intense 100 W IPG laser to perform the final evaporative cooling stage to achieve condensation [50]. Once a quantum degenerate gas is produced, this second laser is switched off in 300 ms while, after a time τ, the intensity of the BNSL with a spacing of 10 µm is ramped up. By changing the value of τ we can control the number of DWs populated. We can achieve a maximum number of 8 DWs with ≈ 2·103 atoms each. The three optical lattice lasers are frequency locked to the same optical cavity (see Supplemental Material). In this way cavity length drifts, to the first order, cause only spatial translation of the DWs array.

Theoretical modeling. The single interferometer can be described within a standard two-mode approximation of the many-body order parameter: Ψ†(r) = ψ L (r)cˆ† L + ψ R (r)cˆ† R , where ψ L,R (r) are the wave function localized in the right and left well, respectively, and cˆ† L,R (cˆ L,R ) are bosonic mode creation (annihilation) operators, see e.g. Ref. [39] for details. The mode wave functions are computed as ψ L,R (r) = [ψ gs (r) ± ψ ex (r)]/ √ 2, where ψ gs (r) are the ground and first excited state of the non-interacting gas of energy E gs and E ex , respectively. We write the interaction potential as U(r - r′) = gδ(r - r′) + C dd U dd (r - r′), where the first term accounts for s-wave scattering contact interaction, with g = 4πℏ2a/m, and the second term describe dipole interaction U dd (r - r′) = 1 1 - 3cosθ 4π |r - r′|3 . By neglecting the dipolar interaction between nearby wells of the BNSL, we write the interaction contribution to the full Hamiltonian, H int = 1 2 ∫ dr ∫ dr′ Ψ†(r)Ψ†(r′)U(r - r′)Ψ(r)Ψ(r′) = (χ el + χ dd )J2, where J = (cˆ† L cˆ R + cˆ† R cˆ L )/2. For the parameters of our experiments, we estimate χ el = 0.46 a/a 0 Hz, where a 0 = 5.3 × 10-5, and χ dd = -0.01 Hz. To summarize:

Hˆ int = ℏ × (0.072 a/a 0 - 0.01) Hz Jˆ2. (3)

It implies that the interaction within each double well of the superlattice potential can be exactly canceled when a = 0.139 a 0 .

Multiparameter maximum likelihood estimation. We consider the set of m joint measurement results z 1 ≡ {z(1),...,z(m)} and z 2 ≡ {z(1),...,z(m)} obtained for the two interferometers and introduce the rescaled variable z˜(k) j = (z(k) j - C j )/V j . We assume that the output signal of each interferometer depends sinusoidally on the phase shift, with constant visibility V j and offset C j . The offset is estimated as C j = 1 m ∑ m k=1 z(k) j and, for our sets of data, where m ≈ 30, we find C j ≈ 0 (see e.g. caption of Fig. 3). The visibility is estimated as V j = max |z(k) j - C j |. The phase difference ∆ϕ and the uncorrelated noise σ∆ϕ are estimated jointly, as the values that maximize the likelihood function P(z˜ 1 , z˜ 2 |∆ϕ, σ∆ϕ ) = ∏ m j=1 P(z˜ 1 , z˜ 2 |∆ϕ, σ∆ϕ ). A semi-analytical expression for P(z˜ 1 , z˜ 2 |∆ϕ, σ∆ϕ ) is obtained by assuming a Gaussian distribution of width σ∆ϕ , giving

P(z˜ 1 , z˜ 2 |∆ϕ, σ∆ϕ ) = 1 (2π)3/2σ∆ϕ √ 1 - z˜2 1 √ 1 - z˜2 2 ∑ k=0,1 e - [θ + (-1)k(sin-1(z˜1) - sin-1(z˜2))]2 2σ2 ∆ϕ + e - [θ + (-1)k(sin-1(z˜1) + sin-1(z˜2) - π)]2 2σ2 ∆ϕ . (4)

Further details about the multiparameter maximum likelihood method are reported in Ref. [55]. To associate error bars to the estimated values ∆ϕ est and σ est ∆ϕ we use a Bootstrap method: we generate a large number of data z 1 and z 2 sampled according to P(z˜ 1 , z˜ 2 |∆ϕ est , σ est ∆ϕ ). Data are divided into samples of m values. For every sample we obtain new estimated values of ∆ϕ and σ∆ϕ . Finally, mean square fluctuations of the estimated values are computed.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The manuscript „Mach-Zehnder atom interferometry with non-interacting trapped Bose Einstein Condensates“ describes an experimental sequence that indeed implements a Mach-Zehnder-like sequence for space-wise separated, trapped matter wave packets of K-39. The outstanding achievement is the ability to adjust the atomic interactions via a Feshbach resonance throughout this sequence, in this case tuning them close to zero, realizing an ideal gas-like situation. This combination and level of experimental control has been a long-standing goal for over a decade and the work clearly merits publication in Nature Communications. This being said, there are several issues that prevent publication in the current form and need to be addressed. The work presented here is embedded in a global effort that started over 20 years ago, and many beautiful results have been obtained by authors, as well as the groups of Ketterle, Reichel, Schmiedmayer, Steinhauer, Oberthaler, Klempt, Treutlein, just to name a few. Some of the claims of novelty and “for the first time” should be checked carefully against this history, more specifics will be given below. Explicitly, the ability to perform x-axis rotation operations on the Bloch sphere is a necessary and demonstrated operation, the claimed uniqueness should be revised. We identify two main points of general criticism: • The role of interactions is somewhat mystified and the used terminology is ambiguous. The terms “decoherence” and “dephasing” are used (synonymously?) to describe the “harmful” effects of interactions. The authors should differentiate, especially in view of the fact that an echo sequence is applied. In the abstract as well in the main text, interactions are described as “strong” without any reference to other scales, like tunnel couplings or potential barrier heights or temperature. One of the groundbreaking works, highlighting the interplay of interactions and coherence time is Ketterle/Pritchard PRL 98, 030407 from 2007 (demonstrating 200 ms in an interacting system). This should be cited and the 200 ms be set as a benchmark, questioning the claim of an improvement by two orders of magnitudes. From an experimental perspective, the tuning of interactions is described poorly and indirectly via the applied magnetic fields. Only towards the end of the 3rd page, the reader learns about the used field values. The link between field and scattering properties is fully missing. A plot of \\Xi(B) = \\Xi_el(B) + \\Xi_dd over the relevant range should be provided in the Methods section. Plotting Fig. 4 in terms of scattering parameter instead of the rather technical magnetic field should be considered. We propose to cite New Journal of Physics 9, 223 (2007) D’Errico et al. additional to [41]. 2. The “linear beam splitter” is described actively avoiding the established terminology of the bosonic Josephson junction (here presumably in the Rabi regime), which actually has been established by some of the authors. It is difficult to understand the sequence used to produce Fig.2, especially panel c). Which time(s) shown in panel a) are changed to vary the time depicted in panel c)? The fluctuations of the initial population imbalance are huge. It would have been instructive to change the atomic interaction parameter in this sequence to separate technical from quantum effects. This is relevant, as ultimately, the echo-sequence is used to predominantly (?) reverse technical fluctuations, not a potential phase diffusion. Additionally, we would like to mention: • We don’t understand what is meant by „provides Heisenberg-limited spatial resolution”, please explain and provide a reference. • Why is a non-interacting gas „extremely sensitive to imperfections“? This probably connects to the non-mentioning of the relevant energy scales (tunnel couplings or potential barrier heights, temperature…) mentioned above • The left panel of Figure 1 is rather confusing. The vertical axis depicts time in the sense of a time-sequence, but also the “integrated profiles of the BEC density” and “trapping potential”. In at least two other publications, authors have given a more detailed description of an effective potential provided by a Beat-Note superlattice (https://doi.org/10.48550/arXiv.2504.12831 and 10.1103/PhysRevLett.127.020601). Especially in the PRL by Masi et al. the effective potential and the atomic density are calculated depending on the effect the fast optical modulation (depth) has on the shape of the BEC’s wavefunction and the tunnelling between the adjacent sites of the BNSL. Given that tunnelling plays an important role in the beam splitter and recombiner sequence, these details are of great interest to the reader. The depicted wavefunction in blue suggests that the modulation on top of the smooth effective potential is rather strong – is that correct? Ideally, the authors would provide a calculation of the time dependent optical potential through the splitting process, probably in the supplementary information. • The description „An external homogeneous force provides a linear potential energy that affects each interferometer with an equal energy mismatch ε between the two wells.„ Is very cryptic, reader needs to go to supplementary information to find out that there is an additional off-center beam applied. • The Fig. 4 caption refers to the methods section [43] to explain, but the histogram announced in the caption of figure 4b is actually missing – there is just one panel • The inset of Fig. 4 is not clear: is this a simulated (artificially broadened) wave function, or has any tomography been performed? • The title of this figure is “Interaction-induced decoherence” which sounds as if some fundamental effect is studied. However, the interactions merely turn the predominantly technical fluctuations into phase diffusion – which can be (partially) reversed using echo sequences? Should this really be called decoherence? • The sequences for the “splitter” and the “recombiner” seem to be different. What is the reason?", "reviewer_2": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.", "reviewer_3": "In this manuscript the authors report on the realization of several Mach-Zehnder atom interferometers based on Bose-Einstein condensates of 39K confined in an array of optical double-well potentials. The potentials are formed by a beat-note superlattice generated with three laser frequencies, which allow for tuning the barrier height between two individual wells. Additionally, interparticle interactions are suppressed by tuning the interaction strength with a broad magnetic Feshbach resonance, allowing the realization of a linear matter-wave beam splitter based on coherent tunneling between adjacent wells. Ultimately, the operation of multiple interferometers enables differential phase measurements and the demonstration of a trapped atom-gradiometer for a harmonic potential, created with an additional vertical laser beam. Furthermore, a spin-echo protocol is employed to mitigate technical dephasing and extend the coherence time of the interferometers to nearly one second. Phase shift and noise are analyzed using a multiparameter maximum likehood method and the experimental results are supported by a theoretical model including interaction-induced decoherence, beamsplitter imperfections and technical dephasing. The supplemental material provides a detailed derivation of the used noise model, the double-well potentials and a technical explanation of the magnetic field calibration, all of which are relevant to supporting the main results. The manuscript presents a compelling advance for quantum sensing with trapped atom interferometers. By combining coherent tunneling with interaction tuning and a spin-echo protocol the authors address a long-standing issue and achieve significant improvement in coherence time. The manuscript thereby presents a clear conceptual advancement and offers experimental validation of a previously unrealized interferometric architecture with potential applications in fundamental physics and applied sensing, which will be of interest to a broader community in those fields. The manuscript is globally well written, the methods are clearly described, and the results are supported by appropriate modeling and data analysis. However, I have the following remarks and questions, which I think can strengthen the manuscript: The authors state a 30-fold improvement in coherence time compared to other realizations of trapped atom interferometers. I think a quantitative comparison to these experiments, e.g. as a table, would strengthen the significance of the achieved results while also highlighting potential differences between the experiments conducted. The first beam splitter is performed by classically raising a potential barrier in the center of the Bose-Einstein condensate. How stable is this method over time? Are there significant shot-to-shot fluctuations in the splitting ratio or long term drifts which have to be taken into account? Could the authors please comment on the scalability of their approach? While here two double-wells were used in the gradiometer, is it possible to operate a larger number of wells and thereby increase the spacing between interferometers? What are the potential limitations? It is stated that the central double-wells are loaded with N = 3000 atoms for the interferometry scheme. Since interaction induced dephasing can effectively be controlled by magnetic Feshbach tuning, can the wells also be loaded with more atoms to enhance the sensitivity of the interferometer? The authors convincingly demonstrate differential phase measurements between two interferometers, consistent with a gradiometric setup. However, a sensitivity figure is not provided. Including one would greatly strengthen the manuscript as it would elevate the scheme from a proof-of-principle experiment to a functional sensor and allows to contextualize the performance relative to other interferometric schemes. It is stated that the interferometer operates in the absolute ground state |1,1>, which allows to presicly tune the scattering length around the Feshbach resonance at approximately 350 G. This implies a substantial sensitivity to magnetic fields. Since 39K features broad Feshbach resonance for all mF-states in F = 1, have the authors considered using a different mF-state with a reduced sensitivity to stray magnetic fields? Could environmental magnetic noise limit the application of the scheme to unknown gradients? Minor spelling remarks: Page 5: “In conclusion, we report the realization of Mach-Zehnder atom interferometry with trapped BEC, where the linear operations are enabled by the the precise canceling of the scattering length.“ Page 5 : “coopropagating“ -> co-propagating Page 5: “wavelenghts” -> wavelengths", "reviewer_4": "The manuscript describes an experimental realization of an atom interferometer using a Rb BEC. The beamsplitter operations are achieved through coherent tunneling through a double-well potential formed by optical lattices. The fidelity of the beamsplitter is quite good, but not perfect (~70%). The authors then describe a gradiometer that uses conjugate interferometers to suppress common mode noise and measure differences in phase accumulated between the interferometers. They use this interferometer to measure the harmonic potential of an additional laser beam. The results agree with expectation, with sensitivity at the part per ~100 level. The authors then describe a spin-echo sequence, which is used to extend measurement time and possibly increase sensitivity. Significance of scientific results. Atom interferometry using tunneling through double-well potentials has been somewhat dormant for a while, so this is a nice update for these type of methods. The experimental scheme used is clever and gets around some of the limitations of the method. The data and analysis look convincing and consistent. Overall, this manuscript describes a significant step forward for this type of atom interferometers, although I have some major reservations with regards to the possibility of this method becoming competitive with other state-of-the-art light-pulse atom interferometry techniques. 1. The measurement shown here has precision at the part per 100 level. It is unclear how could this be improved by more than 9 orders of magnitude to reach part per 10^11 precision of the best atom gravimeters (see fine structure constant and equivalence principle measurements). I suggest the authors clarify the application(s) where this method is superior, otherwise there is strong contradiction with the last sentence in the abstract: “Our interferometer will find applications in precision measurements of forces with a high spatial resolution and in linear manipulation of quantum entangled states for sensing with sub shot-noise sensitivity.” 2. Expanding on question 1, there are significant unknowns in the paper: the main residual noise (“extra stochastic noise”) is not understood so path to improvements are unclear. Entanglement is mentioned as a possibility in the system, but entanglement only becomes useful as a resource one the experiment is at shot-noise, which is not the case here. 3. How well will the magnetic field offset have to be stabilized and gradients controlled to be able to achieve competitive precision? 4. For the spin-echo sequence, it is as if the two interferometer arms are switched. Why does this not cancel the measured gradient along with the noise? Related, I ssugest showing the measured phase for the spin-echo experiment not only noise, same as in Figure 3b. Please compare to the expected harmonic potential values. In conclusion, I enjoyed reading the article and the experimental realizations, but I have some questions on the technical side and the presentation of this method as a precision sensing technique. Specific comments and suggestions The paper is quite well written in general, but there are a few points that could use some improvement. “Moreover, the quantum-limited size of a BEC provides Heisenberg-limited spatial resolution for sensing applications.” - It is unclear what “Heisenberg-limited” spatial resolution means. Could you give a number? Earlier sentences claim that a BEC has very narrow momentum spread, doesn’t that naively necessarily mean large position uncertainty per the uncertainty principle? “Each MZI (see the scheme in Fig. 1) consists of a beam splitter, an interrogation time, and a final beam splitter that maps the accumulated relative phase into a population imbalance.” - I would move this into a new paragraph. The way that paragraph is written is confusing, I suggest to finish the introduction and only then go into experiment description, “The energy difference δ can be then estimated by a differential analysis, as illustrated below.” – This is out of sequence, please move to where this is explained “The second beam-splitter exploits the linear evolution of non-interacting atoms tunneling” – it’s unclear to me what the authors mean by linear evolution? “the interferometric phase acquired by the the j-th interferometer.” – interferometer arm “As shown by the previous analysis, σtech is the major source of noise that limits the operation of our gradiometer close to Bmin, thanks to our high control of the scattering length.” - Strange wording, perhaps: “Near Bmin, interactions are minimized and the primary source of noise is σtech.” “by the the precise canceling of the scattering length.” “Our work demonstrates a long-sought achievement in the ultracold atom community – pursued for over two decades.” - This is a very subjective statement that I really doubt, I suggest removing it."}, "author_response": "1\nReviewer 1\nThe manuscript “Mach-Zehnder atom interferometry with non-interacting trapped Bose Einstein\nCondensates” describes an experimental sequence that indeed implements a Mach-Zehnder-like\nsequence for space-wise separated, trapped matter wave packets of K-39. The outstanding\nachievement is the ability to adjust the atomic interactions via a Feshbach resonance throughout\nthis sequence, in this case tuning them close to zero, realizing an ideal gas-like situation. This\ncombination and level of experimental control has been a long-standing goal for over a decade and\nthe work clearly merits publication in Nature Communications. This being said, there are several\nissues that prevent publication in the current form and need to be addressed.\nWe sincerely thank the Referee for recognizing the significance of our results, which they\ndescribe as outstanding, and for highlighting how the level of experimental control we have\nachieved has been the subject of research efforts for well over a decade. We are grateful for the\nReferee’sassessmentthatourmanuscript“clearly” deservespublicationinNatureCommunications.\nBelow, we address each comment raised by the Referee.\nThe work presented here is embedded in a global effort that started over 20 years ago, and\nmany beautiful results have been obtained by authors, as well as the groups of Ketterle, Reichel,\nSchmiedmayer, Steinhauer, Oberthaler, Klempt, Treutlein, just to name a few. Some of the claims\nof novelty and “for the first time” should be checked carefully against this history, more specifics\nwill be given below. Explicitly, the ability to perform x-axis rotation operations on the Bloch\nsphere is a necessary and demonstrated operation, the claimed uniqueness should be revised.\nWe thank the Referee for this comment. We have revised the manuscript, including some\nimportant missing references, and phrasing more precisely our claims of novelty.\nAmong the various types of interferometers based on trapped condensates, our work contributes\nto advancing the double-well scheme, in which atoms are confined in two spatial modes coupled\nvia tunneling through the separating barrier. This configuration has long faced the challenge of\nhaving the interaction energy NU much smaller to the tunneling energy J [see Spagnolli et al.\nPRL 118 230403 (2017) for definitions of the single-particle interaction energy U and J], where N\nis the number of atoms. As a result, although coherent Josephson dynamics (i.e., rotations around\nthe x-axis of the Bloch sphere) have been demonstrated, a precise mapping of the relative phase\n2\nonto a measurable observable has typically required switching off the double-well potential and\nanalyzing the interference pattern of the overlapping atomic clouds during time-of-flight expansion.\nThis scheme has been demonstrated in several works, e.g. by Ketterle, Hinds, and Oberthaler.\nAn alternative approach is through in-trap recombination of the two BECs [G. B. Jo et al., PRL\n98, 180401 (2007)], enabling phase-sensitive generation of collective excitations. However, both\nmethods do not rely on transferring the relative phase information to the population imbalance\nbetween the two modes, as occurs at the output of a Mach–Zehnder interferometer.\nThe only exception is the work by Schmiedmayer and collaborators [Berrada et al., Nat. Com-\nmun., 4, 2077, (2013)], which is properly cited in our manuscript. Our work belongs to this line\nof research, and to our knowledge, it is the first to achieve tunability of NU well below J, thus\nenabling fully linear operations—specifically, rotations of the Bloch vector of the N-particle system\naround the x-axis of the Bloch sphere, in an interferometric sequence.\nThe Referee mentions other scientists who have contributed to different types of interferometric\nschemes, which, however, differ from the interferometer discussed above. These include guided\nBECs coupled via Bragg or Raman transitions, and trapped BECs coupled by radiofrequency (rf)\nfields. In such systems, the strong Rabi coupling typically exceeds the interaction energy NU by a\nlarge margin, allowing for coherent rotations around the x-axis with high fidelity. We have added\nreferences to these valuable works and changed the abstract, the introduction and the conclusions\nto clarify our contribution in relation to the state of the art.\nThe role of interactions is somewhat mystified and the used terminology is ambiguous. The\nterms “decoherence” and “dephasing” are used (synonymously?) to describe the “harmful” effects of\ninteractions. The authors should differentiate, especially in view of the fact that an echo sequence\nis applied.\nWe agree with the Referee that the two terms have been used synonymously. As suggested we\nhave modified the main text using dephasing to describe the \"harmful\" effect of interactions and\ndecoherenceasanyothersourceofuncontrolledbroadeningoftheinterferometricphaseuncertainty.\nIn the abstract as well in the main text, interactions are described as “strong” without any\nreference to other scales, like tunnel couplings or potential barrier heights or temperature.\nWe agree with the Referee. In the previous version of the manuscript we have used the adjective\n3\n\"strong\" in order to stress the following considerations. During the beam splitting operation,\ninteraction energy has to be weaker than the tunneling energy to guarantee linear splitting. During\nthe interrogation time, the dephasing due to interaction must be smaller than the energy resolution\nwe want to achieve. In the new version of the manuscript we have removed any statement with\ngeneric \"strong\" interactions and we have clarified the energy scales.\nWhy is a non-interacting gas ’extremely sensitive to imperfections“? This probably connects\nto the non-mentioning of the relevant energy scales (tunnel couplings or potential barrier heights,\ntemperature) mentioned above\nOneofthemainlimitationsoftrappedatominterferometryistheinstabilityoftheenergiesofthe\nspatial modes. For instance, in a double-well potential created with optical tweezers, uncontrolled\nintensity fluctuations of the trapping laser can cause drifts in the energy offset ∆E between the\ntwo wells. As a result, the interferometric phase becomes affected by technical noise on the order\nof T∆E/¯h, where T is the interrogation time.\nAlthough interatomic collisions are typically a source of phase diffusion, they can also play a\nstabilizing role by compensating for slow drifts in ∆E that is, fluctuations that remain constant\nduring a single measurement run (over T), but vary from shot to shot. This compensation arises\nbecause, during the initial splitting of the condensate, a greater number of atoms tend to occupy\nthe lower-energy well, leading to an increase in interaction energy that reduces the effect of the\nenergy mismatch ∆E. Another way to describe this compensation mechanisms is to follow the\ndescription of Ref. [46] where it is stressed that an interacting BEC is a superfluid that keeps the\nchemical potential spatially constant. For this reason, when the BEC is split, the two condensates\nin the two wells have almost the same chemical potential and the average phase difference evolution\nis not influenced by ∆E.\nIn our case, where we work with non-interacting BECs, this self-compensation mechanism is\nabsent, and our system is therefore much more sensitive to fluctuations in ∆E. We have tried to\nclarify this point in the main text with a specif comment.\nOne of the groundbreaking works, highlighting the interplay of interactions and coherence\ntime is Ketterle/Pritchard PRL 98, 030407 from 2007 (demonstrating 200 ms in an interacting\nsystem). This should be cited and the 200 ms be set as a benchmark, questioning the claim of an\nimprovement by two orders of magnitudes.\ndemonstrates a significant coherence time of 200 ms, and we have now included this important\nreference in our manuscript. However, we would like to clarify the basis for our claim of a two-\norders-of-magnitude improvement over previous results. As discussed in the previous answer, the\nstronger the interaction, the more efficiently the technical noise affecting the energy imbalance is\ncompensated, effectively reducing the phase decoherence. However, with the interaction energy\nalso being much larger than the tunneling energy it is not possible to realize linear beam splitting\noperations.\nTo give a quantitative comparison, in G.-B. Jo et al., the reported chemical potential reaches\napproximately 6 kHz. In this regime, the relative phase cannot be directly read out in-trap. In-\nstead, the condensates must be released and allowed to expand at much lower densities, with the\nphase extracted from the resulting interference pattern. Contrarily, in Barreda et al., an in-trap\nrecombination was performed to complete a Mach-Zehnder sequence, but this required a significant\nreduction of the atom number to bring the final chemical potential down to about 500 Hz, close to\nthe tunneling energy.\nFor these reasons, we chose to compare our results to earlier experiments in which the entire\nMach-Zehnder sequence was performed with atoms fully trapped, and where coherence times were\non the order of 30 ms. Nonetheless, we agree with the Referee that such distinctions may be\ntoo specialized for a general readership, and that the work by G.-B. Jo et al. should indeed be\nconsidered a benchmark for double-well interferometry with trapped BECs. We have accordingly\nadded the citation and revised the text to reflect this broader perspective.\nFrom an experimental perspective, the tuning of interactions is described poorly and indirectly\nvia the applied magnetic fields. Only towards the end of the 3rd page, the reader learns about\nthe used field values. The link between field and scattering properties is fully missing. A plot of\nΞ(B) = Ξ (B)+Ξ over the relevant range should be provided in the Methods section. Plotting\nel dd\nFig. 4 in terms of scattering parameter instead of the rather technical magnetic field should be\nconsidered. We propose to cite New Journal of Physics 9, 223 (2007) D’Errico et al. additional to\n[41].\nWe have added the suggested plot in the Supplementary Material where we report the depen-\ndence of the scattering length as a function of the applied magnetic field (see Fig. 3). Furthermore,\n5\nwe now mention the tunability of the interaction via a Feshbach resonance already at the beginning\nof the manuscript, adding the suggested reference.\nIn Fig. 4 we have preferred to report the uncertainty of the differential phase as a function of\nthe magnetic field for the following reason. As explained to the Referee in one of the previous\nanswers, the lowest decoherence rate might occur for slightly positive scattering lengths that could\ncompensate for instabilities of the energy mismatches ∆E between the two wells during the first\nbeam splitter. This could shift the minimum of the decoherence with respect to the exact zero\ncrossing of the scattering length. Therefore we have preferred to leave the precise measurement\nof the dependence of the scattering length on the external magnetic field to future work. For\nexample, by using a tunneling beam splitter also at the beginning of the interferometric sequence\nmight avoid this interaction shielding effect working.\nThe “linear beam splitter” is described actively avoiding the established terminology of the\nbosonic Josephson junction (here presumably in the Rabi regime), which actually has been\nestablished by some of the authors\nWe agree that we are using a platform analogous to the one used in Spagnolli et al. [Phys. Rev.\nLett. 118, 230403 (2017)], where Josephson physics has been explored across the zero crossing of\nthe collisional scattering length for both positive and negative meanfiled energies. Unfortunately,\nduring revision, we inadvertently removed a citation to this work, which is highly relevant to our\nstudy. We have intentionally avoided using the terminology of Josephson junctions, which is more\nappropriate for tunnel-coupled superfluids, since our non-interacting condensates do not exhibit\nsuperfluid behavior. Moreover, our beam-splitting sequence has distinct characteristics: the barrier\nheight is rapidly lowered and then raised again, unlike in standard Josephson oscillations where\natom transfer occurs at a fixed barrier height.\nThat said, we agree with the Referee that highlighting the connection with Josephson junction\nsystems is valuable. We have therefore added a clarifying sentence, along with the suggested\nreference, in the section of the manuscript where the beam-splitting sequence is introduced.\nIt is difficult to understand the sequence used to produce Fig.2, especially panel c). Which\ntime(s) shown in panel a) are changed to vary the time depicted in panel c)? The fluctuations of\nthe initial population imbalance are huge.\ntunneling between the two, almost balanced, modes for a specific time such that the left (right)\nmode has a 50% probability to tunnel to the right (left) one (see Fig. 2a). From numerical\nsimulationsweknowthattheoverallbeamsplittingtimecannotbeshorterthan20msifexcitations\nto higher lying modes must be avoided. As a consequence, assuming linear ramps of the barrier\nheight (are easier to be implemented), the only adjustable parameter is the value of the minimum\nbarrier height V needed during the procedure. We experimentally determine such value with\nmin\nthe following procedure. We start with a condensate fully loaded in the left mode, and by applying\nthe aforementioned sequence, we vary V until the final distribution of the atomic imbalance is\nmin\ncentered around zero (see Fig. 2b). To stress the connection between Fig. 1 and Fig. 2a) it should\nbe clear now that the time 20 ms in Fig. 2a) corresponds to t in Fig. 1.\nBS2\nRegarding Fig.2 c) the goal of this measurement is to show that the tunneling energies are the\nsame for two neighbouring double-wells. That’s why we extend the dynamics as much as possible\nup to 240 ms. In this measurement the condensate has been initially loaded in the left well by\nrising the central barrier height and by applying an energy mismatch between the two modes. To\ntriggerthedynamics,atTime=0ms,westartalinearevolutiontowardszerooftheenergymismatch\nbetween the minima of the two wells and of the barrier height towards 70% of its maximum value.\nThe ramp lasts 10 ms and during this time the imbalance slightly reduces. For this reason our first\nmeasurement of the imbalance is recorded at Time=10 ms and such value set the amplitude of the\noscillations.\nRegarding the comment of the Referee on the \"huge fluctuations of the initial population im-\nbalance\" we do not really understand what they are referring to. The uncertainty is the one we\ntypically have in the measurement of the atomic imbalance. If they are mentioning the reduced\nvalue of ≈ 0.75, it should be clear from what explained above that this is due to the simultane-\nous ramping of the barrier height and energy mismatch. Finally we have preferred to look at the\ngradiometric measurement to estimate the visibility V of the interference (see Fig. 3).\nj\nThese aspects are clarified when discussing Fig. 2 in the main text as well as in the Supplemen-\ntary Material.\nIt would have been instructive to change the atomic interaction parameter in this sequence to\nseparate technical from quantum effects. This is relevant, as ultimately, the echo-sequence is used\nto predominantly (?) reverse technical fluctuations, not a potential phase diffusion.\nneeds J ≫ NU, so interactions are deleterious. When we optimize the beam splitting sequence we\nstart with all the atoms in one of the two modes and even a small value of the scattering length\ncauses macroscopic quantum self trapping [Spagnolli et al. Phys. Rev. Lett. 118, 230403 (2017)],\ni.e. the atoms don’t even reach the z = 0 balanced condition.\nIf the Referee mentions the tuning of the scattering length during the interferometric sequence\nwe remind that this is done later in the paper (see Fig. 4).\nOne possibility could be to exploit interaction during the first beam splitter. As already stressed\nthis could help to suppress the technical noise. Of course it should be canceled during the second\nbeam splitter. We have decided to leave this kind of investigation for future works because the\noptimal value of the scattering length would depend on the technical noise level of our apparatus\nand interaction phase diffusion. Then a precise modeling of the observed residual decoherence is\nnot straightforward.\nAdditionally, we would like to mention: We don’t understand what is meant by ’provides\nHeisenberg-limited spatial resolution’, please explain and provide a reference.\nThe term Heisenberg limit may be misleading in this context and we have revised the text\naccordingly. Our intention was to highlight that a pure quantum state (namely a macroscopically-\noccupied ground state of the harmonic oscillator), is generally more localized than a thermal state,\nunder the same external confinement. Since a pure state saturates the Heisenberg uncertainty\nrelation, we loosely referred to this condition as the Heisenberg limit, intending it as a synonym\nfor the system being in a pure state.\nThe left panel of Figure 1 is rather confusing. The vertical axis depicts time in the sense of\na time-sequence, but also the “integrated profiles of the BEC density” and “trapping potential”.\nIn at least two other publications, authors have given a more detailed description of an effective\npotential provided by a Beat-Note superlattice (https://doi.org/10.48550/arXiv.2504.12831\nand 10.1103/PhysRevLett.127.020601). Especially in the PRL by Masi et al. the effective potential\nand the atomic density are calculated depending on the effect the fast optical modulation (depth)\nhas on the shape of the BEC’s wavefunction and the tunnelling between the adjacent sites of\nthe BNSL. Given that tunnelling plays an important role in the beam splitter and recombiner\nsequence, these details are of great interest to the reader. The depicted wavefunction in blue\n8\nsuggests that the modulation on top of the smooth effective potential is rather strong – is that\ncorrect? Ideally, the authors would provide a calculation of the time dependent optical potential\nthrough the splitting process, probably in the supplementary information.\nWe understand the difficulties. We have modified the Fig. 1. Now the arrow of time appears\nonly linked to the manipulation of the lasers’ intensities on the left. The space arrow is detached\nfrom the time arrow so to the “integrated profiles of the BEC density” and “trapping potential”\nappear disconnected from the time axis.\nAll details about the effective potential are reported in the supplementary material, including\nV , V , V , and V . In order to clarify the differences between the real potential and the effective\n1 2 3 eff\none, we have added, in the supplementary material, the tunneling energies and the atomic\nwavefunctions as a function of time. We confirm that the fast modulation of the wavefunction of\nthe atoms in the two modes is rather strong. However the analogy between the two configurations\nis valid due to the similarity of the first two energy levels spacing.\nThe description „An external homogeneous force provides a linear potential energy that affects\neach interferometer with an equal energy mismatch ϵ between the two wells.„ Is very cryptic,\nreader needs to go to supplementary information to find out that there is an additional off-center\nbeam applied.\nWe have clarified this in the main text adding \"A spatial-dependent force realized in our\nexperiment with an additional vertical dipole trap laser, ....\"\nThe Fig. 4 caption refers to the methods section [43] to explain, but the histogram announced\nin the caption of figure 4b is actually missing – there is just one panel\nWe have revised Fig. 4(b) in the supplementary and we have added the histogram, which was\nmistakenly not included.\nThe inset of Fig. 4 is not clear: is this a simulated (artificially broadened) wave function,\nor has any tomography been performed?\nIt is an artificial plot used to illustrate the broadening.\neffect is studied. However, the interactions merely turn the predominantly technical fluctuations\ninto phase diffusion – which can be (partially) reversed using echo sequences? Should this really\nbe called decoherence?\nWe agree that our use of \"decoherence\" and \"dephasing\" was misleading (see discussion above).\nAccordingly, we have replace \"decoherence\" with \"dephasing\" in the title of this figure.\nThe sequences for the “splitter” and the “recombiner” seem to be different. What is the reason?\nThe Referee is correct: the two sequences are different. Contrary to the \"recombiner\" (called by\nus \"Tunneling atom beam-splitter\" and described in Fig. 2), the “splitter” at the beginning of the\ninterferometer is implemented in a shorter time, thus minimizing the effect of imperfections. We\nare planning to realize a Mach-Zehnder interferometer with two \"tunneling atom beam-splitters\"\nbut we leave this as future investigations.\nReviewer 2\nI co-reviewed this manuscript with one of the reviewers who provided the listed reports. This\nis part of the Nature Communications initiative to facilitate training in peer review and to provide\nappropriate recognition for Early Career Researchers who co-review manuscripts.\nReviewer 3\nIn this manuscript the authors report on the realization of several Mach-Zehnder atom inter-\nferometers based on Bose-Einstein condensates of 39K confined in an array of optical double-well\npotentials. The potentials are formed by a beat-note superlattice generated with three laser\nfrequencies, which allow for tuning the barrier height between two individual wells. Additionally,\ninterparticle interactions are suppressed by tuning the interaction strength with a broad magnetic\nFeshbach resonance, allowing the realization of a linear matter-wave beam splitter based on\ncoherent tunneling between adjacent wells. Ultimately, the operation of multiple interferometers\nenables differential phase measurements and the demonstration of a trapped atom-gradiometer for\na harmonic potential, created with an additional vertical laser beam. Furthermore, a spin-echo\n10\nprotocol is employed to mitigate technical dephasing and extend the coherence time of the\ninterferometers to nearly one second. Phase shift and noise are analyzed using a multiparameter\nmaximum likehood method and the experimental results are supported by a theoretical model\nincluding interaction-induced decoherence, beamsplitter imperfections and technical dephasing.\nThe supplemental material provides a detailed derivation of the used noise model, the double-well\npotentials and a technical explanation of the magnetic field calibration, all of which are relevant to\nsupporting the main results. The manuscript presents a compelling advance for quantum sensing\nwith trapped atom interferometers. By combining coherent tunneling with interaction tuning and\na spin-echo protocol the authors address a long-standing issue and achieve significant improvement\nin coherence time. The manuscript thereby presents a clear conceptual advancement and offers\nexperimental validation of a previously unrealized interferometric architecture with potential\napplications in fundamental physics and applied sensing, which will be of interest to a broader\ncommunity in those fields. The manuscript is globally well written, the methods are clearly\ndescribed, and the results are supported by appropriate modeling and data analysis. However, I\nhave the following remarks and questions, which I think can strengthen the manuscript:\nWe thank the Referees for carefully reading our manuscript and for the positive general\nassessment (\"compelling advance for quantum sensing with trapped atom interferometers\").\nPlease, find below an answer to all comments raised by the Referees.\nThe authors state a 30-fold improvement in coherence time compared to other realizations of\ntrapped atom interferometers. I think a quantitative comparison to these experiments, e.g. as a\ntable, would strengthen the significance of the achieved results while also highlighting potential\ndifferences between the experiments conducted.\nAs briefly reviewed in the introduction, there are different realizations of trapped interfer-\nometers with BEC. The reported 30-fold improvement refers to a comparison with the only\nexperiment in the literature where the full Mach-Zehnder sequence was completed in a double-well\ninterferometer using tunneling between trapped BECs [Berrada et al., Nat. Commun., 4, 2077,\n(2013)]. In that work, the condensates were coupled via tunneling through the separating barrier,\nand the interferometric phase was mapped onto the final population imbalance—all while the\natoms remained trapped. Listing the various characteristics and differences between trapped\ninterferometric platforms would be certainly useful but goes beyond the scope of our work as well\n11\nas the journal’s style.\nThe first beam splitter is performed by classically raising a potential barrier in the center of the\nBose-Einstein condensate. How stable is this method over time? Are there significant shot-to-shot\nfluctuations in the splitting ratio or long term drifts which have to be taken into account?\nThe control and the characterization of the noise affecting the first beam splitter is a crucial\npoint of our experiment. In the Supplementary Information Fig. 4 (a) we report the measurements\nof the atomic imbalance z after the first beam splitter over the time scale needed to collect one\nfull ellipse, i.e. approximately 20 minutes. The standard deviation of z (see Fig. 4 b) in the\nSupplementaryMaterial)isσ ∼0.07and, asexplainedinthemaintext, itiscrucialtodetermine\nBS\nthe amount of uncorrelated phase noise of the gradiometer signal (using Eq. 2 in the main text).\nIn addition we have discovered that larger amplitude drifts occur over longer timescales. For this\nreason at the beginning of every set of measurements, for each ellipse, we always performed a cycle\nof experiments to balance again the double-well. This has been done by finely tuning (few MHz)\nthe frequency of the laser at 1120 nm. The origin of short term fluctuations and long term drifts\nare probably due to air turbulence and temperature, pressure and humidity changes that cause\nboth fast common and slow relative phase shifts of the three lattices, propagating in air for 25 cm\nto reach the retroreflecting mirror. We have recently performed measurements of these phase shifts\nimplementing an optical Michelson interferometer simultaneously for the three wavelengths. The\nresults of the measurements and the simulated effect of these shifts on the atomic imbalance after\nthe first beam splitter are a good indication of the validity of our assumptions/hypotheses. Further\ninvestigations are needed to estimate the effect of other possible noise sources, like magnetic field\nfluctuations. Vibrations of the retroreflecting mirror are currently negligible as estimated by\nsimulations based on measurements with an accelerometer.\nCould the authors please comment on the scalability of their approach? While here two\ndouble-wells were used in the gradiometer, is it possible to operate a larger number of wells and\nthereby increase the spacing between interferometers? What are the potential limitations?\nThe use of BNSLs allows to operate in principle several double-well interferometers simultane-\nously. The residual harmonic confinement along the lattice direction is of the order of few Hz,\nmainly caused by the finite Rayleigh range of the radial beam (∼ 300µm), collinear with the lat-\n12\ntices and responsible of the confinement of the BECs along the directions orthogonal to the lattice\nbeams. So far, a maximum of 8 double wells have been loaded symmetrically in our experiment.\nThe limitation is set by the harmonic confinement (∼ 1 Hz) that would prevent balanced tunneling\nsplitting.\nRegarding the possibility of increasing the separation between the modes, larger spacing BNSLs\ncould be used, but this would increase the tunneling time. After the splitting we could try to tune\nthe wavelengths of the lattices, but increasing the separation between the modes, for example from\n5 microns to 50 microns, would require single mode operation of the lasers, while tuning their\nwavelengths over tens of nm. This is doable, but not easy.\nIt is stated that the central double-wells are loaded with N = 3000 atoms for the interfer-\nometry scheme. Since interaction induced dephasing can effectively be controlled by magnetic\nFeshbach tuning, can the wells also be loaded with more atoms to enhance the sensitivity of the\ninterferometer?\nA non-interacting condensate confined in a harmonic trap exhibits an in-trap Gaussian\nwavefunction, with a width determined by the harmonic oscillator length, independently from\nthe atom number. For potassium atoms in our double-well potentials, this length is on the\norder of 1µm. Increasing the atom number would enhance the density and thus three-body\nrecombination losses. To give an order-of-magnitude estimate, using the three-body loss coefficient\nK = 10−29 cm6s−1 as reported in previous studies, we estimate that under our current experi-\n3\nmental conditions, approximately 5% of the atoms are lost within 1 second. Since the three-body\nlossratescalesquadraticallywiththeatomnumber, increasingN quicklyleadstosignificantlosses.\nThe authors convincingly demonstrate differential phase measurements between two inter-\nferometers, consistent with a gradiometric setup. However, a sensitivity figure is not provided.\nIncluding one would greatly strengthen the manuscript as it would elevate the scheme from a\nproof-of-principle experiment to a functional sensor and allows to contextualize the performance\nrelative to other interferometric schemes.\nAlthough we would like to emphasize the broader significance of our achievement—which sets a\nlong-sought benchmark in the field—we fully agree that the potential applications of our sensor in\nreal-world measurements are of equal importance.\nmeasurementofforceswithhighspatialresolution. Thebestwaytoevaluatetheperspectivesoffered\nby our system is a direct comparison with the results recently reported in [67], that represents the\nstate of the art in the high precision measurement of forces with high spatial resolution using\nultracold atoms. In [67] the most precise measurement of Casimir-Polder forces between an atom\nandasurfacewasachievedusingaWannier-Starkatominterferometeroperatedwiththermalatoms.\nThat experiment reached a spatial resolution of 3 µm.\nFirst of all the gradiometer demonstrated in our work could provide improved spatial resolution,\nset by the size of the BEC itself—potentially below 1 µm.\nAs reported in [67] the main limitation on the sensitivity of the sensor was the vibrations of the\nmirror used to realize the optical lattice potential. Using our gradiometric approach, the operation\nof two double well interferometers, one close and one far away from the mirror surface, could allow\nfor common-mode rejection of trap-induced vibrations.\nThe second limitation of the work [67] is the quantum projection noise derived by the use of 500\natoms. Such low number is due to the need to reduce the overall atomic density and suppress the\ninteraction induced dephasing, even if the cloud has a size of 3 microns along the lattice direction\nand45micronsintheradialdirections. PleasenotethatourBECwithtunableinteractionhas3000\natoms and a rms size of 1 microns in all the directions and still not affected by interaction induced\ndephasing. Not only our atomic probe size is smaller but also its shot noise. Finally, the use of\na BEC rather than a thermal atomic ensemble opens the possibility for future quantum-enhanced\nsensingviaentanglement,potentiallysurpassingthestandardquantumlimit. Suchadvantagecould\nofferalmostanorderofmagnitudeincreaseinsensitivityincomparisontotheaforementionedsensor\nimplementingthermalatoms. Notethatoursystemhastheuniquepossibilitytoexploitinterparticle\ninteractions to create quantum entangled states and later to tune the collisional scattering length\nto zero to suppress dephasing.\nDespite these promising perspectives, our current gradiometric measurements exhibit a large\nphasenoise(seeforexampleFig. 5)of0.2radalreadyat70msinterrogationtime. Thiscorresponds\ntoacurrentdifferentialaccelerationsensitivityof10−4g/shot,tobecomparedwiththeonereported\nin [67] of approximately 6×10−5g/shot.\nTheoriginofthislargephasenoisehasnotbeenidentifiedyet,butitcouldbeduetoinstabilities\nof our external potential that is not exactly harmonic and not stable in amplitude. A significant\ncontribution is at very low frequency and we could measure it operating the gradiometer for almost\ntwo hours and recording the temporal evolution of the differential phase. Results are reported in\ninterferometers operating in parallel and applying the vertical dipole trap, during the interferometric time\nT=70ms,wemonitortheevolutionof∆ϕfortwohoursleavingunchangedalltheexperimentalparameters.\nFig. 1 of this reply.\nNote that ∆ϕ, mod 2π, changes from 0.1 to 0.3 rad over the time interval needed to collect a full\nellipse, i.e. approximately 15-20 minutes. Although we haven’t been able to suppress this noise yet,\nthe spin echo measurements allow to investigate what sensitivity could be achieved once suppressed\nthis noise source. An rms noise of the differential phase of 0.6 rad after 800 ms interrogation time\n(see Fig. 5 in the main paper) would correspond to a sensitivity of 2×10−5g/shot, already better\nthan what has been reported in Ref. [67].\nBesides these promises, we are aware that much has to be done before exploiting our sensor for\nhigh precision measurements, including the increase of the repetition rate of the measurements\n(important step forward has been recently reported in A. Herbst, et al. Phys. Rev. Research 6,\n013139(2024))andcontroltheeffectofexternalmagneticfields(seethereplytothenextquestion).\nIt is stated that the interferometer operates in the absolute ground state |1,1>, which allows to\npresicly tune the scattering length around the Feshbach resonance at approximately 350 G. This\n15\nimplies a substantial sensitivity to magnetic fields. Since 39K features broad Feshbach resonance\nfor all mF-states in F = 1, have the authors considered using a different mF-state with a reduced\nsensitivity to stray magnetic fields? Could environmental magnetic noise limit the application of\nthe scheme to unknown gradients?\nWe agree with the Referee that the accuracy of our gradiometer could be affected by spurious\nmagnetic fields. Although the influence of a uniform magnetic field gradient could be canceled by\na differential analysis—such as in high-spatial-resolution force measurements close to a dielectric\nsurface, with one interferometer placed near the surface and one far away—schemes to eliminate\nmagnetic forces are still needed.\nFirst of all, the atoms themselves could be employed as sensitive probes of residual magnetic\nfield gradients, enabling cancellation of this noise source.\nA second possible route involves the use of radio-frequency (rf)–coupled mixtures, properly\nchosen to reduce the magnetic moment and the interaction energy of the mixture (see for example\nJ. Sanz, PRL 128, 013201 (2022)).\nAnother interesting possibility is offered by the use of a different internal atomic state, such as\nthe F = 1,m = −1 state, for which the magnetic field can be tuned to a so-called “magic” value\nF\nof approx 100 Gauss, where the magnetic moment vanishes (see Fig. 5 of C. D’Errico et al. New\nJ. Phys. 9, 223 (2007)). At this point, the residual scattering length is only 7 a , which is smaller\n0\nthan in rubidium and thus favorable for interferometry. Most importantly for a few hundred atoms\nit could be possible to explore the occurrence of the revival of the coherence (see M. Greiner,\net al. Nature 419, 51 (2002)) after a complete dephasing of the interferometric phase. According to\nour simulations, we expect that after approximately one second of evolution, our system should\nhave a window of several ms during which the interferometric phase can be measured with shot\nnoiseresolution. Thismeasurementisinourprioritylistoffutureexperiments, becauseifsuccessful\ncouldallowtooperatetheinterferometerwithnonmagneticBECsandnotbelimitedbyinteraction\ninduced dephasing.\nIt is not excluded that in the future BECs of fully non-magnetic atoms could be used in our\nexperimental platform, like strontium BEC characterized by a zero magnetic moment and a rather\nsmall value of the scattering lengths, i.e. ≈ −2 a .\neffect of external magnetic fields can be canceled with a differential analysis, is by far the most\nversatile system where many proposals could be tested. Just to name a few we remember the\n16\nrealization of a gradiometer with sub shot noise sensitivity or interaction induced manipulation of\nquantum entanglement with both repulsive or attractive interactions.\nMinor spelling remarks: Page 5: “In conclusion, we report the realization of Mach-Zehnder atom\ninterferometry with trapped BEC, where the linear operations are enabled by the the precise can-\ncelingofthescatteringlength.“ Page5: “coopropagating“ ->co-propagatingPage5: “wavelenghts”\n-> wavelengths\nThanks for the remarks. We have corrected the text as suggested.\nReviewer 4\nThe manuscript describes an experimental realization of an atom interferometer using a Rb\nBEC. The beamsplitter operations are achieved through coherent tunneling through a double-well\npotential formed by optical lattices. The fidelity of the beamsplitter is quite good, but not perfect\n( 70%).\nWejustremindtheRefereethatweareusingPotassiumatoms(withtunableinteraction)rather\nthan Rubidium 87 (with constant scattering length of ∼ 100 Bohr radii).\nFigure 2c, where the Referee has extrapolated the visibility of the beam splitter, was not\noptimized to maximize the visibility and it is shown only to prove that neighboring double\nwells have the same tunneling frequency. The amplitude of oscillations is lower than the inter-\nferometric one (see below) because data in Figure 2c are collected starting from a condensate\nfully loaded in the left well and lowering the barrier in 10 ms, while the energy mismatch\nbetween the two wells is balanced. During this time interval a small fraction of atoms starts\nto tunnel to the right well reducing the initial atomic imbalance z ∼ 1 to z ∼ 0.75. This\nfinal value set the amplitude of the following Rabi oscillation. An estimation of the visibility of\nourinterferometercanbeextractedfromtheellipsereportedinFig.3(a),thatisslightlybelow90%.\nTheauthorsthendescribeagradiometerthatusesconjugateinterferometerstosuppresscommon\nmode noise and measure differences in phase accumulated between the interferometers. They use\nthis interferometer to measure the harmonic potential of an additional laser beam. The results\nagree with expectation, with sensitivity at the part per 100 level.\nThe authors then describe a spin-echo sequence, which is used to extend measurement time and\npossibly increase sensitivity.\nSignificance of scientific results.\nfor a while, so this is a nice update for these type of methods. The experimental scheme used\nis clever and gets around some of the limitations of the method. The data and analysis look\nconvincingandconsistent. Overall,thismanuscriptdescribesasignificantstepforwardforthistype\nof atom interferometers, although I have some major reservations with regards to the possibility\nof this method becoming competitive with other state-of-the-art light-pulse atom interferometry\ntechniques.\nWe thank the Referees for carefully reading our manuscript and for the positive general\nassessment (\"a significant step forward for this type of atom interferometers\"). Please, find below\nan answer to all comments raised by the Referees. We apologize if some of our replies refer to\nexplanations already included in our responses to the other Referees.\n1. The measurement shown here has precision at the part per 100 level. It is unclear how could\nthis be improved by more than 9 orders of magnitude to reach part per 1011 precision of the best\natom gravimeters (see fine structure constant and equivalence principle measurements). I suggest\nthe authors clarify the application(s) where this method is superior, otherwise there is strong\ncontradiction with the last sentence in the abstract: “Our interferometer will find applications\nin precision measurements of forces with a high spatial resolution and in linear manipulation of\nquantum entangled states for sensing with sub shot-noise sensitivity.”\nThe primary sensing application of our interferometer is the measurement of forces with high\nspatialresolution. Forthiskindofmeasurements,freefallingatominterferometersareofnousecon-\nsideringthattheymeasuretheaverageforceoverlargeinterrogationareas. Forexample10−11g/shot\nprecision has been achieved in a 10 m drop tower set-up. On the contrary, trapped atomic samples,\ncan resolve atom surface interactions with micrometric spatial resolution. We hope the answer to\nReferees 2 and 3 (page 13-14), where we describe the possible advantages offered by our system, is\nsatisfactory also for Referee 4.\nIn addition we would like to comment on “the one part per 100 level precision” stressed by the\nReferee. After reading such comment we have investigated the origin of such uncertainty and we\nhave found that our apparatus is currently affected by a drift of the differential phase (see above\nFig. 1 of this report). Over a timescale of 15-20 minutes, that is the time needed to record a full\nellipse, ∆ϕ drifts an amount of 0.1-0.3 rad. This explains why in Fig.3 the single measurements of\n∆ϕ have uncertainties of 0.04-0.06 rad, but they are often well above or below the optimal linear fit\n18\nby0.1-0.3rad, i.e. theestimatedvalueofthedriftduringanellipsemeasurementtime. Considering\nthat the value of ∆ϕ for T ≈ 85 ms, is 20 rad it is expected that the overall relative uncertainty\nof the trapping frequency measurement is 0.1-0.3 rad/20 rad ∼ 1/100. We then realized that the\nchoice of a vertical optical dipole trap to estimate the gradiometer sensitivity was not the optimal\nchoice. Considering the finite waist of the beam, i.e. 50 µm, a shift of the beam with respect to the\ndouble well potential center of only 3 µm is responsible for a 1/100 relative change of the harmonic\nconfinement. Such choice was driven by its simplicity and the focus of just demonstration the\noperations of the gradiometer, and not its ultimate performances. However we think that the spin\necho measurement is a good estimation for the sensitivity of the gradiometer in case no drift of the\nexternal harmonic confinement occurs. We should already have a competitive sensitivity in com-\nparison with the state of the art for the measurement of forces with micrometric spatial resolution\n[67] (see details in the reply to Referee 2 at page 13-14). We are planning future measurements\nwithaweakharmonicmagneticpotentialthatcanbemorestableandlesssensitivetomisalignment.\n2. Expandingonquestion1,therearesignificantunknownsinthepaper: themainresidualnoise\n(“extra stochastic noise”) is not understood so path to improvements are unclear. Entanglement is\nmentioned as a possibility in the system, but entanglement only becomes useful as a resource one\nthe experiment is at shot-noise, which is not the case here.\nAs explained in the answer to question one, the first step will be to realize a more stable\nand uniform harmonic confinement. In this way we should be less sensitive to misalignment and\ntemporal variations of the external harmonic confinement. An analysis extended to three or more\ndouble well interferometers could allow to take trace of common trapping frequency variations.\nRegarding the comment on the entanglement we agree that we haven’t achieved the shot noise\nlimit yet. Hopefully, we will get closer to such limit once our gradiometer will be operated on a\nmore stable and controlled external potential with a reduced number of atoms. Please note that\nmeasurements with high spatial resolution using ultra-cold atoms must implement small atomic\nsamples with low atom number to prevent three-body losses. So entanglement is a useful resource\nwhen reducing the projection noise by increasing the atom number is not possible. Likely the\ntunability of interactions can in principle easily allow to create and later exploit entanglement in\nour gradiometer.\n3. How well will the magnetic field offset have to be stabilized and gradients controlled to be\n19\nable to achieve competitive precision?\nRegarding the problem of the sensitivity of our interferometer to external spurious magnetic\nfieldgradientswehavealreadyansweredtoReferee2atpage15. Wehopetheanswerissatisfactory\nalso for Referee 4. Regarding the level of control of the magnetic field needed to make negligile the\ninteraction induced dephasing, we have to refer to the calculations reported in the Supplementary\nInformations. If we assume no technical noise during the first beam splitter, i.e. σ = 0 from\nBS\nSI.14 we get a quantum uncertainty of the gradiometric differential phase σ2 = 2 +Nτ2. If we\n∆ϕ N\nwant that the interaction induced dephasing term, i.e. Nτ2 is smaller than the shot noise limit,\ni.e. 2, we need N2τ2 < 2. From Eq. (3) in the main text we have that τ = (0.072a /a )Hz ·T.\nN s 0\nWith straightforward calculations we get that for an interferometric time T= 1 sec and N=3000\natoms, this condition is fulfilled if a is tuned to zero with a precision of 0.005 a . Thanks to the\ns 0\nhigh degree of control of the scattering length on the zero crossing of the Feshbach resonance of\n0.6 a /G, a stability of the magnetic field of 10 mG is needed. Such level is within reach of our\n0\ncurrent apparatus (see Fig. 3 of the Supplementary Materials), but this requirement could be also\nrelaxed working with a smaller number of atoms. Please note that this level of phase noise, i.e.\nshot noise at 1 sec of interferometric time T, is much smaller than what reported in state of the\nart measurements of forces with micrometric spatial resolution [67] that have achieved maximum\ncoherence time of 150 ms.\nTo conclude, the level of control of interactions for competitive measurements is not a major\nissue in our setup.\n4. For the spin-echo sequence, it is as if the two interferometer arms are switched. Why\ndoes this not cancel the measured gradient along with the noise? Related, I ssugest showing the\nmeasured phase for the spin-echo experiment not only noise, same as in Figure 3b. Please compare\nto the expected harmonic potential values.\nThe Referee’s intuition is correct. The spin-echo sequence cancels the external noise as well the\nsignal from the external harmonic confinement. This is confirmed from the inset of Fig. 5 where a\none to one correlation of the outputs of the two interferometers z1 vs z2 is reported. Considering\nthat the differential phase is still sensitive to interaction-induced dephasing, the spin-echo protocol\nprovides an estimate of the sensitivity of the gradiometer in absence of slow drifts of the external\n20\nharmonic confinement.\nIn conclusion, I enjoyed reading the article and the experimental realizations, but I have some\nquestionsonthetechnicalsideandthepresentationofthismethodasaprecisionsensingtechnique.\nSpecific comments and suggestions\nThe paper is quite well written in general, but there are a few points that could use some\nimprovement.\nWe thank the Referee for the positive comments. We hope to have addressed all the questions\nthey raised.\n“Moreover, the quantum-limited size of a BEC provides Heisenberg-limited spatial resolution\nfor sensing applications.” - It is unclear what “Heisenberg-limited” spatial resolution means. Could\nyou give a number? Earlier sentences claim that a BEC has very narrow momentum spread,\ndoesn’t that naively necessarily mean large position uncertainty per the uncertainty principle?\nConsidering the unclear definition of spatial resolution at the Heisenberg limit, which has\nalso raised concerns from Referee 2, we have decided to modify the text accordingly. To\nclarify, the BEC has an rms width equal to the harmonic oscillator length for a quantum\nparticle in a harmonic potential, i.e. ≈ 1µm, provided a trapping harmonic confinament of\n≈ 100 Hz. Instead, as correctly guessed by the Referee, in free falling atom interferometers,\nBECswithlargesizesoftheorderoffewtensofµmallowverysmallvelocityspreadsof20-50µm/s.\n“Each MZI (see the scheme in Fig. 1) consists of a beam splitter, an interrogation time, and\na final beam splitter that maps the accumulated relative phase into a population imbalance.” -\nI\nwould move this into a new paragraph. The way that paragraph is written is confusing, I suggest\nto finish the introduction and only then go into experiment description\nThe sentence pointed out by the Referee describes one of the main novelty of our work: i.e. that\nwe are able to perform a full Mach-Zehnder interferometer sequence with a trapped BEC, where\nthe phase information is mapped on the population signal. This sentence is not the beginning of\nthe experiment description. We prefer to maintain the current sentence in the introduction.\n21\n“The energy difference δ can be then estimated by a differential analysis, as illustrated below.”\n– This is out of sequence, please move to where this is explained\nWe agree with the Referee and we have removed the sentence as it was indeed out of sequence.\n“The second beam-splitter exploits the linear evolution of non-interacting atoms tunneling”–\nit’s unclear to me what the authors mean by linear evolution?\nFormally speaking it means that the evolution of the system is described only by rotations\nof the Bloch vector. This is possible for a single spin or for a system composed by multiple\nnon-interacting spins. In our case this occurs thanks to the cancellation of the collisional scattering\nlength. We have clarified this in the introduction.\n“the interferometric phase acquired by the the j-th interferometer.” – interferometer arm\nThe index j refers to the jth interferometer, not to the arm. The interferometric phase is the\nrelative phase acquired between the two arms of the interferometer.\n“As shown by the previous analysis, tech is the major source of noise that limits the operation\nof our gradiometer close to Bmin, thanks to our high control of the scattering length.”- Strange\nwording, perhaps: “Near Bmin, interactions are minimized and the primary source of noise is\nσ .” “by the the precise canceling of the scattering length.”\ntech\nThanks. we have modified the text accordingly.\n“Our work demonstrates a long-sought achievement in the ultracold atom community – pursued\nforovertwodecades.” -ThisisaverysubjectivestatementthatIreallydoubt,Isuggestremovingit.\nFirst of all we would like to stress that the other Referees seem to agree on this statement\nconsidering their comments. Referee 1 recognizes that \"This combination and level of experimental\ncontrol has been a long-standing goal for over a decade\". The Referee 3 states that \"By combining\ncoherent tunneling with interaction tuning and a spin-echo protocol the authors address a long-\nstanding issue and achieve significant improvement in coherence time.\" However we agree with\ndecided to revise it to make it more broadly acceptable, while preserving the significance of our\nresults.\nThe authors explain nicely in the review the type of application of this interferometer: measuring\npotentials with um scale resolution. This, however, still doesn’t come through in the paper. The\nwords \"high spatial resolution\" appear in the abstract and introductory paragraph. Could a scale\nbe given next to that? In addition, can this be compared with similarly tuned mechanical systems,\nlike membranes?\nWe have specified in the abstract that high spatial resolution means sub-micron spatial\nresolution. We fully agree that membrane-based sensors provide outstanding spatial locality\nfor short-range force measurements. Our trapped-atom sensor is not intended to compete on\nspatial resolution, but to offer a complementary measurement platform characterized by different\nsystematic effects. Owing to the well-defined and highly controllable nature of atomic probes, this\napproach may enable more accurate modelling and a more controlled comparison between theory\nand experiment at micron-scale distances.\nThe authors also explain some of the limitations of noise and drift in page 13 of the review.\nWhy is this only made available to the authors and not included in the supplementary materials\nof the manuscript? Figure 1 seems important to add....\nWe have revised the supplementary info by adding a section on the Gradiometer Sensitivity\nwith the figure required by the Referee and the corresponding description.\nThe spin-echo phase is still not included in a plot, I would still be very interested to see it, not\nonly noise.\nWe would like to clarify that after a spin-echo sequence the expected phase is zero. This is\nconfirmed by the good correlation between the output atomic imbalances of the two interferometers\nshown in the inset of Fig. 5. We attribute the non perfect correlation to residual phase noise.\nWe have now explained in text that the spin-echo phase is assumed equal to zero in order to\nestimate the phase noise from the broadening of the correlated plot in the inset of Fig.5"}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors have carefully addresses all our queries as well as the ones of the other referees. They have found a good balance between providing sufficient \"depth\" and maintaining accessibility to a broad readership. I believe this reviewing iteration has improved the quality of the manuscript and now recommend it for publication without reservation. Congratulations on the fine work!", "reviewer_2": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.", "reviewer_3": "The authors have responded to all my comments and remarks satisfactorily. They have revised the manuscript where necessary and clarified ambiguous statements. Considering the comments of the other reviewers and the corresponding responses as well, I recommend publication of the revised manuscript.", "reviewer_4": "The authors accept some of the suggestions, particularly related to statements about putting the work into context and explaining the significance. However, there are a couple of points that I feel were not addressed sufficiently. 1. The authors explain nicely in the review the type of application of this interferometer: measuring potentials with um scale resolution. This, however, still doesn't come through in the paper. The words \"high spatial resolution\" appear in the abstract and introductory paragraph. Could a scale be given next to that? In addition, can this be compared with similarly tuned mechanical systems, like membranes? 2. The authors also explain some of the limitations of noise and drift in page 13 of the review. Why is this only made available to the authors and not included in the supplementary materials of the manuscript? Figure 1 seems important to add... 3. The spin-echo phase is still not included in a plot, I would still be very interested to see it, not only noise. Otherwise, the paper is a nice result and can be published."}, "author_response": ""}]

Decision letter