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REVIEW 1 major objections 5 minor 82 references

Optically accessible high-finesse millimeter-wave resonator for cavity quantum electrodynamics with atom arrays

T0 review · 1 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper reports a superconducting millimeter-wave Fabry–Pérot cavity that achieves finesse $5.8(1)\times10^7$ while leaving a numerical aperture of 0.56 open for optical access, and shows that modeling post-paraxial mode mixing is what…

desk verdict Solid device paper with a direct finesse measurement that fills a real gap, but the model-guided-tuning story is partly post-hoc because the mode-coupling V is fitted. read the letter →

arxiv 2506.05804 v1 pith:SHYTVDEZ submitted 2025-06-06 quant-ph physics.atom-ph

Tony Zhang , Michelle Wu , Sam R. Cohen , Lin Xin , Debadri Das , Kevin K.S. Multani , Nolan Peard , Anne-Marie Valente-Feliciano
show 4 more authors
Paul B. Welander Amir H. Safavi-Naeini Emilio A. Nanni Monika Schleier-Smith
This is my paper · ORCID
classification quant-phphysics.atom-ph
keywords millimeter-wavecavityquantumelectrodynamicsRydbergatomsFabry-Pérotresonatorpost-paraxialopticsmodehybridizationsuperconductingmirrorscooperativity
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper reports a superconducting millimeter-wave Fabry–Pérot cavity that achieves a finesse of $5.8(1)\times10^7$ while leaving a numerical aperture of 0.56 open for optical access. Its aim is to show that the near-confocal geometry needed for both strong coupling and imaging can be made to work despite post-paraxial mode hybridization that otherwise couples the $\mathrm{TEM}_{00}$ mode to lossy higher-order modes. Modeling these corrections and lengthening the cavity away from confocality suppresses the loss, yielding a cooperativity of $\eta=2.7\times10^6$ for a 92 GHz circular-Rydberg transition. If right, this removes a key obstacle to combining high-cooperativity cavity QED with trapped atom arrays.

What carries the argument

The load-bearing object is the post-paraxial round-trip phase operator $\hat{H}$ (Eq. S6): a Hermitian operator on transverse mode profiles whose eigenvectors are the cavity resonances, built from the paraxial harmonic-oscillator Hamiltonian plus fourth-order propagation, wavefront, asphericity, vector spin–orbit, and astigmatism terms. With four fitted parameters—cavity length $L$, harmonic-mean curvature radius $R$, astigmatism $\eta_{\mathrm{astig}}$, and aspheric coefficient $\tilde{p}$—it predicts mode frequencies and mixtures across the spectrum. It identifies the errant fourth-order mode responsible for the avoided-crossing loss near confocality and quantifies how much the cavity must be lengthened to move that mode away from $\mathrm{TEM}_{00}$, which is the tuning step that produces the high finesse.

What would settle it

Scan the cavity length continuously between $g=-0.029$ and $g=-0.109$ while recording $S_{21}$ spectra and ringdown finesse; the model predicts the avoided crossing moves monotonically away from $\mathrm{TEM}_{00}$ and the finesse rises correspondingly. Failure to see the finesse dip move and vanish, or disagreement between the predicted higher-order-mode frequencies and the measured spectrum across 67–115 GHz, would settle the claim.

Watch

Extended reading notes

Core claim

The central discovery is a cavity whose finesse reaches $5.8(1)\times10^7$ in a geometry with wide transverse access, obtained by identifying and avoiding a loss channel. In the geometry closest to confocality ($g_0=-0.029$), spectroscopy reveals an avoided crossing between the $\mathrm{TEM}_{00}$ mode series and a fourth-order transverse mode series; post-paraxial frequency shifts bring these modes into near-degeneracy, and the admixture of the larger higher-order mode increases clipping loss. After fitting the post-paraxial operator model to the spectra, the authors lengthen the cavity to $g_1=-0.068$ and $g_2=-0.109$, where the higher-order modes move away, probe-coupling measurements confirm improved mode localization, and ringdown spectroscopy gives a plateau finesse of $5.8(1)\times10^7$ limited mainly by trapped magnetic flux. The corresponding cooperativity for circular Rydberg states is $\eta=2.7\times10^6$, with $(g,\kappa,\Gamma)=2\pi\times(22\times10^3,55,13)$ Hz.

Load-bearing premise

The tuning remedy rests on the fitted post-paraxial model: if the four-parameter operator Hamiltonian misdescribes mode frequencies and mode mixing near confocality, the claimed cause of the excess loss is unsupported, although the high finesse measured in the longer $g_1$ and $g_2$ geometries is direct.

Editorial extensions

If this is right

  • A cooperativity of $\eta=2.7\times10^6$ on the 92 GHz $|41C\rangle\leftrightarrow|42C\rangle$ transition puts the cavity deep in the strong-coupling regime, with $(g,\kappa,\Gamma)=2\pi\times(22\times10^3,55,13)$ Hz.
  • The measured loss budget implies deterministic cavity-mediated entangling gates between circular Rydberg atoms with fidelity $\gtrsim98\%$ are feasible.
  • The cavity can operate up to about 1.3 K without degrading finesse, relaxing heat-load constraints from scattered trapping light in cryogenic atom arrays.
  • If the ambient magnetic field normal to the mirrors is suppressed to $\lesssim10$ mG, the remaining roughness limit would place the finesse near $3\times10^8$, leaving substantial headroom.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper leaves implicit that numerical optimization of the mirror profile, using the fast finite-element mode-loss estimates it demonstrates, could suppress mode mixing directly rather than only by detuning the cavity length.
  • Extrapolating from the paper's loss budget, improved magnetic shielding should allow an open-access cavity to approach finesse $\sim3\times10^8$, within about a factor of three of the finesse previously reported for closed superconducting millimeter-wave cavities.
  • The frequency-dependent finesse data imply that Rydberg transitions below about 90 GHz would need even stronger detuning from confocality, or optimized mirrors, to reach the same finesse plateau; this follows from the model but is not demonstrated in the paper.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 5 minor

Summary. This paper reports the design, fabrication, and characterization of a superconducting millimeter-wave Fabry–Pérot resonator intended for cavity QED with arrays of optically trapped Rydberg atoms. The main experimental result is a finesse of F=5.8(1)×10^7 at low temperature (plateau up to ~1.3 K) for fundamental TEM00 modes in two near-confocal cavity geometries, with an open transverse numerical aperture of 0.56. The finesse is obtained from ensemble-averaged ringdown decays (Fig. 4(c)) and reproduced across multiple modes and both geometries. The authors further demonstrate that a more confocal geometry (g0=-0.029) suffers excess loss due to hybridization of the TEM00 mode with a higher-order transverse mode, identify the responsible near-degeneracy using a post-paraxial operator Hamiltonian (Eq. S6), and argue that lengthening the cavity to g1 and g2 evades the loss. The paper closes with a loss budget (surface roughness and trapped flux) and a projected cooperativity η=2.7×10^6 for the 92 GHz circular Rydberg transition, along with a simulated iSWAP gate at ~98% fidelity.

Significance. The result, if correct, is a significant advance: it combines, for the first time in a millimeter-wave cavity, a finesse in excess of 10^7 with the transverse optical access needed for individual atom trapping and addressing. The finesse measurement itself is strong, being based on ringdown spectroscopy that is insensitive to the cryostat vibrations which otherwise broaden frequency-domain lines (Sec. S3C). The consistency of the measured finesse across modes and geometries, the agreement between measured probe couplings and finite-element simulations, and the detailed loss-budget analysis are all commendable. The paper is also unusually transparent about its limitations, e.g., the cylindrical-symmetry approximation in Sec. S4 and the empirical nature of the coupling V in Sec. S2C. If the projected Rydberg parameters are realized, the cavity would enable strong-coupling cavity QED with atom arrays, with plausible applications to nonlocal gates and many-body physics.

major comments (1)
  1. [Sec. S2A, Eq. S6; Sec. S2C, Eq. S8; Abstract] The post-paraxial Hamiltonian in Eq. S6 is block-diagonal in transverse order N, as stated in Sec. S2A, and therefore cannot by itself generate the avoided crossing shown in Fig. 2(a). The coupling V in Eq. S8 is introduced as a free parameter and fitted to the same spectra, so the abstract's claim that 'modeling these corrections allows for tuning the cavity geometry to evade this loss' overstates what the model actually provides: the model predicts the near-degeneracy of the fourth-order mode with TEM00, but the hybridization strength is not derived and is attributed to an unmodeled mechanism (mirror truncation, roughness, or misalignment). In addition, the justification in Sec. S2A that dropping excitation-nonconserving terms is valid 'as long as the cavity is not near-planar (g̸≈1), ensuring that modes of equal longitudinal order q but differing transverse order N are far detuned' is incorrect: in the near-planar limit g→1 the transverse mode spacing vanishes, making modes degenerate rather than far detuned. I recommend that the authors correct this justification, explicitly state that V is an empirical parameter, and temper the causal claim in the abstract (for example, 'identifies a near-degeneracy that motivates tuning the geometry to evade the resulting loss'). The measured finesse in the g1 and g2 geometries is a direct result and is not affected by this issue, but the design narrative as written is only partially supported.
minor comments (5)
  1. [Sec. S2B] The four geometric parameters (L, R, η_astig, p̃) are fitted to the same spectra that are subsequently labeled using the fitted Hamiltonian; please report the number of modes included in the fit and the rms residual of the mode frequencies, and, if practical, provide a cross-check in which a subset of modes is held out to demonstrate that the four-parameter model is not overfitting the mode labels. The agreement across many FSRs in Fig. S4 is encouraging, but a quantitative statement would strengthen the identification of the TEM00 modes used for the finesse measurements.
  2. [Main text, Fig. 4 and Abstract] The headline finesse is measured at 0.4 K (Fig. 4(c)), while the abstract states the value 'at a temperature of 1 K.' The main text notes that performance is unchanged up to 1.3 K, but it should state this explicitly at the point of the plateau claim so that the reader can connect the abstract statement to the ringdown measurement.
  3. [Fig. 4(c) caption] The caption describes 'κ=2π×55.1(1.5) Hz decay of the fitted exponential curve' without specifying whether this is the amplitude or power decay rate; since Eq. S22 defines κ through the decay of the squared signal, please define the convention at the first use of κ in a measurement context.
  4. [Sec. S4 and Fig. 3(b)] The authors appropriately note that the cylindrical-symmetry approximation cannot capture the dominant g0 hybridization; however, because Fig. 3(b) compares g0 and g1 simulations, the main text should alert the reader that the g0 simulation underestimates the mode mixing that motivates the geometry change, so that the comparison is not misread as a quantitative reproduction of the effect.
  5. [References] Reference [33] appears as a placeholder URL; please provide a stable DOI or journal identifier for the Supplemental Material, or state that it is available in the ancillary files.

Circularity Check

2 steps flagged · score 4.0 of 10

Fitted post-paraxial model is presented as predictive for the mode-hybridization explanation, but the central finesse result is a direct ringdown measurement and unaffected.

  1. fitted input called prediction [Main text, paragraph applying Ref. [35]; Fig. 2(e)-(f)]
    "The theory predicts mode frequencies as a function of four geometric parameters—cavity length L, mirror curvature radii Rx,y, and a coefficient p̃ describing the fourth-order curvature of the mirrors—which we fit to observed spectra [33]. Figures 2(e)–(f) plot the S21 cavity spectrum around TEM26,0,0 with model predictions of mode frequencies and mode field patterns, implicating an errant fourth-order mode in the finesse dip observed in Fig. 2(b)."

    The four parameters were fit to the same observed spectra against which the 'model predictions' are compared, so the agreement in Figs. 2(e)-(f) is a fit residual, not an independent prediction. The identification of the 'errant fourth-order mode' and the claim that post-paraxial effects explain the degeneracy therefore rest on the fitted model rather than on an out-of-sample test. This does not affect the direct ringdown finesse measurement, but it weakens the causal claim that modeling allowed tuning to evade loss.

  2. fitted input called prediction [Supplemental Material, Sec. S2 C, Eq. S8]
    "We reconstruct the mode mixture fractions in Figs. 2(a)–(c) by fitting a cavity geometry to the observed mode frequencies of the modes involved in the avoided crossing feature using the theory described in Sec. S2 A together with the following simple model of the hybridization."

    The avoided-crossing data are used twice: first to fit the geometry (hence the detuning δ) and then to fit an ad hoc off-diagonal coupling V. The resulting 'mode mixture fractions' are therefore outputs of a fit to the same data they are used to explain. The post-paraxial Hamiltonian of Eq. S6 is explicitly block-diagonal in transverse order N and does not predict V; the hybridization strength that causes clipping loss is not derived from first principles. Thus the loss mechanism is partially circular: it is fit, not predicted.

full rationale

The central performance claim—finesse 5.8(1)×10^7 and NA 0.56—comes from direct cavity ringdown measurements and geometric characterization that do not depend on the post-paraxial model. The model (Ref. [35], an external prior work, not a self-citation) is used only in the interpretive layer: identifying the fourth-order mode responsible for the g0-geometry finesse dip and motivating the detuning to g1/g2. In that layer the paper uses 'predict' for outputs of a model whose four parameters were fit to the same spectra, and the hybridization coupling V in Sec. S2 C is an additional fitted parameter, so the explanation of the loss and the model-guided-tuning claim are weaker than the abstract implies. There is no load-bearing self-citation chain or definitional equivalence; the measured improvement in finesse is empirical and independent of the model. Score 4 reflects partial circularity in the explanatory narrative, not in the headline device performance.

Assumptions & free parameters 7 free parameters · 6 assumptions · 0 invented entities

The central finesse claim rests on measured ringdown decay, but the quoted Fmax=5.8(1)×10^7 is the variance-weighted average of two fitted plateaus (Flim free parameters in Fig. 4a), and the mode-identification and explanation of the avoided crossing relies on the post-paraxial Hamiltonian of Ref. [35] with four fitted geometric parameters. No new particles or forces are introduced.

free parameters (7)
  • harmonic mean mirror curvature R = R = 42.53 mm
    Determined from cavity spectra in the three geometries (g0,1,2 = -0.029, -0.068, -0.109); used in the post-paraxial Hamiltonian to predict mode frequencies and the avoided-crossing location.
  • astigmatism parameter η_astig = not stated in main text
    One of the four fitted Hamiltonian parameters (Eq. S6); needed to reproduce the polarization-dependent mode frequencies and the deliberate polarization splitting.
  • aspheric coefficient p̃ = not stated in main text
    Fourth-order curvature coefficient of the mirrors; fitted to observed spectra and controlling the post-paraxial frequency shifts that produce the accidental degeneracy.
  • limiting finesse Flim = 6.16(21)×10^7 and 5.74(10)×10^7 (g1, g2)
    Free parameter in the fits of Fig. 4(a) used to extract the plateau finesse; the quoted Fmax = 5.8(1)×10^7 is the inverse-variance-weighted average of the two fitted plateaus.
  • mode expansion factor b = not stated
    Free parameter in the modified clipping-loss fit of Fig. 4(a), introduced to account for mode mixing in an empirical way.
  • residual surface resistance R0 = implied by Fig. 4(b) fits
    Free parameter in the temperature-dependence fit (Eq. S38), representing temperature-independent loss.
  • pure-material penetration depth λ0 = λ0 = 37 nm
    Free parameter in the BCS-resistance fit; the extracted value is close to but not equal to the literature value.
assumptions (6)
  • domain assumption The post-paraxial Hamiltonian (Eq. S6) of Ref. [35] correctly describes mode frequencies and mode mixing in this toroidal near-confocal cavity.
    Invoked in Sec. S2 A-B to compute mode frequencies, identify TEM00 modes, and explain the avoided crossing. The four parameters are fitted, so the functional form is assumed.
  • domain assumption The cavity mode is a Gaussian TEM00 mode whose clipping can be described by the analytic F_clip formula Eq. S28.
    Used in Fig. 1(c) to size mirrors and choose the geometry; used in modified form in Fig. 4(a).
  • domain assumption BCS surface resistance plus Gurevich trapped-vortex loss (Ref. [67]) account for the residual losses.
    Sec. S5 C-D; used to infer flux-trapping limitation and the roughness limit.
  • domain assumption The local London limit applies for the BCS-resistance model, even though ξ≈λ in this film.
    Sec. S5 C: the authors explicitly state this assumption is not entirely self-consistent; they use it anyway and find a reasonable model.
  • domain assumption The toroidal cavity can be approximated as cylindrically symmetric for the FEM mode-loss simulations.
    Sec. S4: 'our simulations approximate the cavity as cylindrically symmetric, replacing the toroidal mirrors with spherical mirrors matching the harmonic mean radius of curvature.'
  • standard math Standard cooperativity formulas (Eq. 1, Eq. S42-S49) and the Rydberg dipole matrix element from Ref. [74] give the projected g, κ, Γ.
    Sec. S6; used to project η=2.7×10^6 and gate fidelity. The dipole matrix element is from an external numerical code.

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Pith. "Pith review of Optically accessible high-finesse millimeter-wave resonator for cavity quantum electrodynamics with atom arrays." pith.science (2026). https://pith.science/paper/SHYTVDEZ

@misc{pith2026250605804,
  author       = {Pith},
  title        = {Pith review of: Optically accessible high-finesse millimeter-wave resonator for cavity quantum electrodynamics with atom arrays},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SHYTVDEZ}},
  note         = {Machine review of arXiv:2506.05804}
}
abstract

Cavity quantum electrodynamics (QED) is a powerful tool in quantum science, enabling preparation of non-classical states of light and scalable entanglement of many atoms coupled to a single field mode. While the most coherent atom-photon interactions have been achieved using superconducting millimeter-wave cavities coupled to Rydberg atoms, these platforms so far lack the optical access required for trapping and addressing individual atomic qubits. We present a millimeter-wave Fabry-P\'erot cavity with finesse $5.8(1) \times 10^7$ at a temperature of 1 K providing generous transverse optical access (numerical aperture 0.56). Conflicting goals of strong atom-photon coupling and optical access motivate a near-confocal geometry. Close to confocality, however, post-paraxial corrections to the cavity spectrum introduce unexpected degeneracies between transverse modes, leading to excess cavity loss. Modeling these corrections allows for tuning the cavity geometry to evade this loss, producing a high finesse that will enable cavity QED experiments with trapped atoms deep in the strong coupling regime.

Figures

Figures reproduced from arXiv: 2506.05804 by the authors.

Figure 1
Figure 1. FIG. 1. Cavity design for combining strong coupling with optical [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Cavity spectroscopy of mode hybridization in the [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. (a) summarizes the improved finesse as a function of mode frequency in the g1 -geometry (red markers) and the longer g2 -geometry (blue markers). For both cavity lengths, the finesse of the TEM00 modes increases with increasing frequency ν before plateauing to a common value Fmax = 5.8(1)×107 for ν ≳ 90GHz. We attribute the lower finesses at low frequencies to residual mode mixing, which is more significant for the … view at source ↗

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