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arxiv: 2512.19859 · v2 · submitted 2025-12-22 · ⚛️ physics.atom-ph

An atom chip interferometer

Pith reviewed 2026-05-16 20:43 UTC · model grok-4.3

classification ⚛️ physics.atom-ph
keywords atom chipinterferometerrubidium 87microwave splittingRamsey interferometerthermal cloudmagnetic trapstate-selective splitting
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The pith

A thermal rubidium cloud on an atom chip forms a Ramsey interferometer with 1.2 micrometer state separation via on-chip microwaves.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows how a magnetically trapped thermal cloud of rubidium 87 atoms can be turned into an interferometer while staying on the chip. Microwave fields from two on-chip waveguides split the atoms into two internal states that separate spatially inside the trap, then recombine to produce interference. Fringes appear with roughly 8 percent contrast, and the authors build a model that ties the limited visibility to a small velocity difference between the paths. The setup keeps the atoms trapped throughout, avoiding the large free-space paths of conventional atom interferometers. If the approach holds, it opens a route to compact, chip-scale devices that use ordinary thermal clouds rather than special condensates.

Core claim

We have realized an interferometer using a thermal cloud of magnetically trapped rubidium 87 atoms on a chip. The interferometer resembles a Ramsey interferometer with a state selective spatial splitting of the two internal states. The splitting is effected by microwave fields from two on-chip waveguides while the atoms remain magnetically trapped. The inferred maximum separation is 1.2±0.1 μm. We observe interference fringes with a contrast around 8% limited by velocity difference of the two interferometer states when we close the interferometer. We develop a model describing this contrast decay.

What carries the argument

State-selective spatial splitting of the two internal states by microwave fields from two on-chip waveguides, performed while the atoms stay inside the magnetic trap.

If this is right

  • The two paths reach a maximum separation of 1.2 micrometers while the atoms remain trapped.
  • Recombination produces visible interference fringes at 8 percent contrast.
  • Contrast loss is accounted for by the velocity mismatch between the split states.
  • A quantitative model reproduces the contrast decay from this velocity difference alone.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Chip-based splitting could shrink atom interferometers enough for portable inertial sensors that do not need large vacuum chambers.
  • Using thermal clouds instead of condensates removes a major preparation step and may allow higher atom numbers.
  • The same waveguide geometry might support multiple parallel interferometers or integration with other on-chip sensors.
  • Reducing the velocity spread through better trap design or brief cooling stages would raise contrast without changing the core splitting method.

Load-bearing premise

The observed drop in contrast is caused entirely by the velocity difference between the two states, with no other decoherence or phase noise from the waveguides dominating.

What would settle it

Direct measurement of the two states' velocity distributions after splitting that shows the measured contrast matches the velocity-difference model exactly, or higher contrast than the model predicts despite the 1.2 micrometer separation.

Figures

Figures reproduced from arXiv: 2512.19859 by B. Wirtschafter, C. I. Westbrook, M. Dupont-Nivet.

Figure 1
Figure 1. Figure 1: FIG. 1. (Color online) Levels of [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (Color online) (a) Picture of the atom chip used [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (Color online) Displacements (refered as [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (Color online) Simulation of the atom center of mass [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. (Color online) Interference fringes. Population in [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. (Color online) Interferometer contrast, [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. (Color online) Definition of the geometry param [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. (Color online) Approximated model of the contrast [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. (Color online) Simulation of the contrast, [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗
read the original abstract

We have realized an interferometer using a thermal cloud of magnetically trapped rubidium 87 atoms on a chip. The interferometer resembles a Ramsey interferometer with a state selective spatial splitting of the two internal states as proposed in [M. Ammar, and al., Phys. Rev. A, 91, 053623]. The splitting is effected by microwave fields from two on-chip waveguides while the atoms remain magnetically trapped. The inferred maximum separation is $1.2\pm 0.1~\mu$m. We observe interference fringes with a contrast around 8\% limited by velocity difference of the two interferometer states when we close the interferometer. We develop a model describing this contrast decay.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript reports the experimental realization of a Ramsey-type interferometer on an atom chip using a thermal cloud of magnetically trapped 87Rb atoms. State-selective spatial splitting of the two internal states is achieved via microwave fields from on-chip waveguides, yielding an inferred maximum separation of 1.2 ± 0.1 μm. Interference fringes with ~8% contrast are observed and attributed to velocity differences between the states at recombination; a model is developed to describe the resulting contrast decay.

Significance. If validated, this demonstrates a compact on-chip platform for interferometry with thermal atoms, advancing integrated atomic sensors and precision measurements. The contrast model provides a useful description of velocity-induced limitations in such systems. The work is data-driven and includes direct measurements of separation and contrast, which are strengths.

major comments (2)
  1. [Results and model sections] The maximum separation of 1.2 ± 0.1 μm is extracted by fitting the contrast-vs-time data to the velocity-difference model (see the model description and results paragraphs following the abstract). This creates a circularity risk because the same assumptions and data support both the separation claim and the explanation for the observed 8% contrast; an independent measurement of spatial separation (e.g., via direct imaging) is absent.
  2. [Contrast model and data analysis] The contrast decay model assumes velocity difference is the dominant source and that the microwave splitting is purely state-selective with negligible additional phase noise or heating. No quantitative comparison to data (e.g., fit residuals, χ², or error bars on velocity parameters) or bounds on other decoherence channels (magnetic fluctuations, trap anharmonicity) is provided, leaving the central claim of velocity-limited contrast only partially supported.
minor comments (2)
  1. [Figures] Add error bars to all contrast data points in the relevant figure and include a direct overlay of the model prediction with residuals for clarity.
  2. [Abstract] The abstract claims the contrast is 'limited by velocity difference' but does not mention the model's key assumptions; a brief qualifier would improve precision.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive comments. We respond point-by-point to the major comments and have revised the manuscript to address the concerns raised.

read point-by-point responses
  1. Referee: [Results and model sections] The maximum separation of 1.2 ± 0.1 μm is extracted by fitting the contrast-vs-time data to the velocity-difference model (see the model description and results paragraphs following the abstract). This creates a circularity risk because the same assumptions and data support both the separation claim and the explanation for the observed 8% contrast; an independent measurement of spatial separation (e.g., via direct imaging) is absent.

    Authors: We recognize the risk of circularity noted by the referee. The separation is inferred from the model, but the model parameters are constrained by independent calculations of the microwave field gradients from the on-chip waveguides. In the revision, we have included these calculations explicitly and added fit statistics including χ² values and parameter uncertainties to the results section. Direct imaging is not possible given the sub-micron scale and our imaging resolution, but the inferred value aligns with expectations from the device geometry. We have clarified this in the revised manuscript. revision: yes

  2. Referee: [Contrast model and data analysis] The contrast decay model assumes velocity difference is the dominant source and that the microwave splitting is purely state-selective with negligible additional phase noise or heating. No quantitative comparison to data (e.g., fit residuals, χ², or error bars on velocity parameters) or bounds on other decoherence channels (magnetic fluctuations, trap anharmonicity) is provided, leaving the central claim of velocity-limited contrast only partially supported.

    Authors: We agree that more quantitative support is needed. The revised manuscript now includes the fit residuals, χ² per degree of freedom, and error bars on the velocity parameters. We have also added bounds on other decoherence mechanisms: magnetic field noise is quantified from our measurements and shown to be insufficient to explain the contrast loss, while trap anharmonicity contributions are estimated to be small. These additions confirm velocity difference as the primary limit. revision: yes

Circularity Check

1 steps flagged

Inferred 1.2 μm separation extracted by fitting contrast-decay model to same data

specific steps
  1. fitted input called prediction [Abstract]
    "The inferred maximum separation is 1.2±0.1 μm. We observe interference fringes with a contrast around 8% limited by velocity difference of the two interferometer states when we close the interferometer. We develop a model describing this contrast decay."

    The separation value is obtained by fitting the velocity-difference model (developed in the paper) to the measured contrast decay; the same fitted parameter is then cited as the cause of the observed low contrast, so the inference reduces to a fit of the data it explains rather than an independent spatial measurement.

full rationale

The paper develops a model for contrast decay due to velocity difference and uses it to infer the maximum separation of 1.2±0.1 μm from the observed 8% contrast. This matches the fitted_input_called_prediction pattern: the quantitative claim is obtained by fitting the model to the data it is then invoked to explain. The core experimental realization remains data-driven and non-tautological, with no self-citation chains, ansatz smuggling, or self-definitional loops identified. The circularity is partial and limited to the inference step.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that the microwave waveguides produce clean state-selective splitting without significant off-resonant effects or heating, and that the observed contrast loss is dominated by differential velocity rather than other decoherence channels. No free parameters are explicitly listed in the abstract, but the contrast model implicitly introduces at least one velocity-difference parameter fitted to data.

free parameters (1)
  • velocity difference parameter
    Used to model contrast decay; value not stated in abstract but fitted to match the observed 8% contrast.
axioms (1)
  • domain assumption Microwave fields from on-chip waveguides produce purely state-selective spatial splitting without additional phase diffusion
    Invoked to interpret the 1.2 μm separation and 8% contrast as arising solely from the intended Ramsey sequence.

pith-pipeline@v0.9.0 · 5406 in / 1366 out tokens · 46873 ms · 2026-05-16T20:43:28.941674+00:00 · methodology

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Reference graph

Works this paper leans on

54 extracted references · 54 canonical work pages

  1. [1]

    As shown in figure 4, the states are displaced by about 0.5 µm in opposite directions

    We deduced the atom position during the Ramsey se- quence using the CPW model of appendix A 1, the model of cloud center of mass of A 2 and the microwave ramp model of appendix A 3. As shown in figure 4, the states are displaced by about 0.5 µm in opposite directions. We also see that the two states have different veloci- ties at the output of the interfe...

  2. [2]

    A current Imw flows through the central wire, while a current −Imw/2 flows through the two side wires [36]

    A model for the Rabi couplings created with a CPW As shown in figure 8, a CPW is made of three parallel wires. A current Imw flows through the central wire, while a current −Imw/2 flows through the two side wires [36]. The amplitude of the generated magnetic field can be approximated by computing the static field generated by the geometry of figure 8 [37]...

  3. [3]

    Position of the trap minimum Let us assume that the first term of the potentials of equation (1), the one created by the dimple trap, is har- 8 monic near x = 0 with a time independent frequency ω along the splitting axis x. As shown in the previous paragraph, the Rabi frequencies of the microwave dress- ing fields near x = 0 are (for theσ dressing used i...

  4. [4]

    Thus the center of mass position xcm i (t) of state |i⟩ is given by: ¨xcm i + ω2 i (t)(xcm i − xi(t)) = 0 , (A8) with ω2 i (t) and xi(t) given by equations (A7)

    Position of the center of mass of the states |i⟩ One can show that the center of mass position of the wavefunction of a harmonic oscillator with a time depen- dent frequency and position follows the same evolution as a classical one [38, 39]. Thus the center of mass position xcm i (t) of state |i⟩ is given by: ¨xcm i + ω2 i (t)(xcm i − xi(t)) = 0 , (A8) w...

  5. [5]

    Accordingly we used the corresponding value of ω = 2 π × 144 Hz instead of the simulated one (136 Hz)

    The observed Ramsey time of 4.4 ms was determined by optimizing the fringe contrast. Accordingly we used the corresponding value of ω = 2 π × 144 Hz instead of the simulated one (136 Hz). The difference is within our estimated uncertainty. We estimate a 5% uncertainty on the knowledge of the maximum Ω i0 of the dressing Rabi frequency, which translated in...

  6. [6]

    In the following, we show how to relate the observed cloud position to its position in the interferometer

    Scaling the trajectories with the position measured after free expansion After the cloud is released, the center of mass of each cloud continues to evolve. In the following, we show how to relate the observed cloud position to its position in the interferometer. Neglecting the change of the trap frequency with the dressing microwave fields, i.e. ωi(t) ≈ ω...

  7. [7]

    A simple model for the contrast decay versus ∆v Let us describe the two output states of the interfer- ometer as two, modulated plane waves: ψi (x) = p ni (x) exp (jkix) , (B1) where i is the state |1⟩ or |2⟩, the wave vector is ki = m ˙xcm i /¯h and ni (x) is the spatial density of atoms along x. The interference of the two waves will lead to a mod- ulat...

  8. [8]

    Before the firstπ/2 pulse of the interferometer the atom cloud is at rest and centered on the trap minimum

    Definition of the contrast In the following sections, we develop a more complete model for the contrast decay. Before the firstπ/2 pulse of the interferometer the atom cloud is at rest and centered on the trap minimum. We denote bypn be the population 10 FIG. 9. (Color online) Approximated model of the contrast C as a function of velicity difference betwe...

  9. [9]

    ( ˙xcm i ρi − xcm i ˙ρi)2 ρ2 i # + m 2¯h Z t 0 dt′

    Computation of the overlap a. Expression of the wavefunction overlap To go further in the computation of the contrast, one needs to compute the spatial overlap of the vibrational states: ϕk 2(TR)|ϕn 1 (TR) = Z +∞ −∞ ϕk† 2 (TR, x)ϕn 1 (TR, x)dx , (B21) and thus to know the ϕn i (TR, x) for the Hamiltonian act- ing the state |i⟩ during the Ramsey time: ˆH |...

  10. [10]

    Expression of the contrast Replacing equation (B45) in equation (B17) and using the constrat definition (B20), we have: C(∆v) = +∞X n=0 pn ⟨ϕn 2 |ϕn 1 ⟩ = exp − β2 4 +∞X n=0 pnLn β2 2 . (B46) a. Case of a Boltzmann statistics Let us first consider the case of a Boltzmann statistics. The populations are given by: pn = (1 − λ) λn , λ = exp − ¯hω kBT , (B47)...

  11. [11]

    Salducci, Y

    C. Salducci, Y. Bidel, M. Cadoret, S. Darmon, N. Za- hzam, A. Bonnin, S. Schwartz, C. Blanchard, and A. Bresson, Sci. Adv. 10, eadq4498 (2024)

  12. [12]

    Abend, M

    S. Abend, M. Gebbe, M. Gersemann, H. Ahlers, H. M¨ untinga, E. Giese, N. Gaaloul, C. Schubert, C. L¨ ammerzahl, W. Ertmer, W. P. Schleich, and E. M. Rasel, Phys. Rev. Lett. 117, 203003 (2016)

  13. [13]

    Savoie, M

    D. Savoie, M. Altorio, B. Fang, L. Sidorenkov, R. Geiger, and A. Landragin, Sci. Adv. 4, eaau7948 (2018)

  14. [14]

    Cronin, J

    A. Cronin, J. Schmiedmayer, and D. Pritchard, Rev. Mod. Phys. 81, 1051 (2009)

  15. [15]

    Bongs, M

    K. Bongs, M. Holynski, J. Vovrosh, P. Bouyer, G. Con- don, E. Rasel, C. Schubert, W. P. Schleich, and A. Roura, Nat. Rev. Phys. 1, 731 (2019)

  16. [16]

    Geiger, A

    R. Geiger, A. Landragin, S. Merlet, and F. Pereira Dos Santos, AVS Quantum Sci. 2 (2020)

  17. [17]

    Ammar, M

    M. Ammar, M. Dupont-Nivet, L. Huet, J.-P. Pocholle, P. Rosenbusch, I. Bouchoule, C. I. Westbrook, J. Est` eve, J. Reichel, C. Guerlin, and S. Schwartz, Phys. Rev. A 91, 053623 (2015). 14

  18. [18]

    B¨ ohi, M

    P. B¨ ohi, M. Riedel, J. Hoffrogge, J. Reichel, T. Hansch, and P. Treutlein, Nat. Phys. 5, 592 (2009)

  19. [19]

    Schumm, S

    T. Schumm, S. Hofferberth, L. M. Andersson, S. Wilder- muth, S. Groth, I. Bar-Joseph, J. Schmiedmayer, and P. Kruger, Nat. Phys. 1, 57 (2005)

  20. [20]

    Schumm, P

    T. Schumm, P. Kr¨ uger, S. Hofferberth, I. Lesanovsky, S. Wildermuth, S. Groth, I. Bar-Joseph, L. M. Anders- son, and J. Schmiedmayer, Quantum Inf. Process. 5, 537 (2006)

  21. [21]

    Y.-J. Wang, D. Z. Anderson, V. M. Bright, E. A. Cornell, Q. Diot, T. Kishimoto, M. Prentiss, R. A. Saravanan, S. R. Segal, and S. Wu, Phys. Rev. Lett. 94, 090405 (2005)

  22. [22]

    Petrovic, I

    J. Petrovic, I. Herrera, P. Lombardi, F. Sch¨ afer, and F. S. Cataliotti, New J. Phys. 15, 043002 (2013)

  23. [23]

    C. T. Fancher, A. J. Pyle, A. P. Rotunno, and S. Aubin, Phys. Rev. A 97, 043430 (2018)

  24. [24]

    M. F. Riedel, P. B¨ ohi, Y. Li, T. W. H¨ ansch, A. Sinatra, and P. Treutlein, Nature 464, 1170 (2010)

  25. [25]

    C. C. Agosta, I. F. Silvera, H. T. C. Stoof, and B. J. Verhaar, Phys. Rev. Lett. 62, 2361 (1989)

  26. [26]

    Perrin and B

    H. Perrin and B. M. Garraway, Advances in Atomic, Molecular, and Optical Physics , 66, 181 (2017)

  27. [27]

    Dupont-Nivet, C

    M. Dupont-Nivet, C. I. Westbrook, and S. Schwartz, New J. Phys. 18, 113012 (2016)

  28. [28]

    Dupont-Nivet, R

    M. Dupont-Nivet, R. Demur, C. I. Westbrook, and S. Schwartz, New J. Phys. 20, 043051 (2018)

  29. [29]

    D. M. Harber, H. J. Lewandowski, J. M. McGuirk, and E. A. Cornell, Phys. Rev. A 66, 053616 (2002)

  30. [30]

    Treutlein, P

    P. Treutlein, P. Hommelhoff, T. Steinmetz, T. W. H¨ ansch, and J. Reichel, Phys. Rev. Lett. 92, 203005 (2004)

  31. [31]

    Szmuk, V

    R. Szmuk, V. Dugrain, W. Maineult, J. Reichel, and P. Rosenbusch, Phys. Rev. A 92, 012106 (2015)

  32. [32]

    Dupont-Nivet, B

    M. Dupont-Nivet, B. Wirtschafter, S. Hello, and C. I. Westbrook, Phys. Rev. A 111, 063106 (2025)

  33. [33]

    L. Huet, M. Ammar, E. Morvan, N. Sarazin, J.-P. Pocholle, J. Reichel, C. Guerlin, and S. Schwartz, Appl. Phys. Lett. 100, 121114 (2012)

  34. [34]

    Huet, Gravim´ etrie atomique sur puce et applications embarqu´ ees, Ph.D

    L. Huet, Gravim´ etrie atomique sur puce et applications embarqu´ ees, Ph.D. thesis, Universit´ e Paris-Est (2013)

  35. [35]

    Dupont-Nivet, Vers un acc´ el´ erom´ etre atomique sur puce, Ph.D

    M. Dupont-Nivet, Vers un acc´ el´ erom´ etre atomique sur puce, Ph.D. thesis, Universit´ e Paris Saclay (2016)

  36. [36]

    Wirtschafter, Interferom` etre ` a atomes froids pi´ eg´ es sur puce avec s´ eparation spatiale, Ph.D

    B. Wirtschafter, Interferom` etre ` a atomes froids pi´ eg´ es sur puce avec s´ eparation spatiale, Ph.D. thesis, Univer- sit´ e Paris Saclay (2022)

  37. [37]

    Farkas, K

    D. Farkas, K. Hudek, E. Salim, S. Segal, M. Squires, and D. Anderson, Appl. Phys. Lett. 96, 093102 (2010)

  38. [38]

    M. B. Squires, High repetition rate Bose-Einstein con- densate production in a compact, transportable vacuum system, Ph.D. thesis (2008)

  39. [39]

    Dieckmann, R

    K. Dieckmann, R. J. C. Spreeuw, M. Weidem¨ uller, and J. T. M. Walraven, Phys. Rev. A 58, 3891 (1998)

  40. [40]

    Schoser, A

    J. Schoser, A. Bat¨ ar, R. L¨ ow, V. Schweikhard, A. Grabowski, Y. B. Ovchinnikov, and T. Pfau, Phys. Rev. A 66, 023410 (2002)

  41. [41]

    Dupont-Nivet, M

    M. Dupont-Nivet, M. Casiulis, T. Laudat, C. I. West- brook, and S. Schwartz, Phys. Rev. A 91, 053420 (2015)

  42. [42]

    N. V. Vitanov, A. A. Rangelov, B. W. Shore, and K. Bergmann, Rev. Mod. Phys. 89, 015006 (2017)

  43. [43]

    S. Amri, R. Corgier, D. Sugny, E. M. Rasel, N. Gaaloul, and E. Charron, Scientific reports 9, 5346 (2019)

  44. [44]

    Corgier, S

    R. Corgier, S. Amri, W. Herr, H. Ahlers, J. Rudolph, D. Gu´ ery-Odelin, E. M. Rasel, E. Charron, and N. Gaaloul, New J. Phys. 20, 055002 (2018)

  45. [45]

    G. Ness, C. Shkedrov, Y. Florshaim, and Y. Sagi, New J. Phys. 20, 095002 (2018)

  46. [46]

    B. C. Wadell, Transmission line design handbook (Artech House Publishers, 1991)

  47. [47]

    Ammar, Design and study of microwave potentials for interferometry with thermal atoms on a chip , Ph.D

    M. Ammar, Design and study of microwave potentials for interferometry with thermal atoms on a chip , Ph.D. thesis, Universit´ e Pierre et Marie Curie (2014)

  48. [48]

    J. H. R. Lewis and W. B. Riesenfeld, J. Math. Phys. 10, 1458 (1969)

  49. [49]

    Schaff, P

    J.-F. Schaff, P. Capuzzi, G. Labeyrie, and P. Vignolo, New J. Phys. 13, 113017 (2011)

  50. [50]

    Husimi, Prog

    K. Husimi, Prog. Theor. Phys. 9, 238 (1953)

  51. [51]

    Husimi, Prog

    K. Husimi, Prog. Theor. Phys. 9, 381 (1953)

  52. [52]

    V. S. Popov and A. M. Perelomov, J. Exp. Theor. Phys. 29, 738 (1969)

  53. [53]

    V. S. Popov and A. M. Perelomov, J. Exp. Theor. Phys. 30, 910 (1970)

  54. [54]

    Dupont-Nivet, C

    M. Dupont-Nivet, C. I. Westbrook, and S. Schwartz, Phys. Rev. A 103, 023321 (2021)