Casimir-force spectroscopy of broadband optical response

Calum F. Shelden; Jeremy N. Munday

arxiv: 2601.10118 · v2 · pith:X2GHMY7Anew · submitted 2026-01-15 · 🪐 quant-ph · physics.app-ph

Casimir-force spectroscopy of broadband optical response

Calum F. Shelden , Jeremy N. Munday This is my paper

Pith reviewed 2026-05-25 07:19 UTC · model grok-4.3

classification 🪐 quant-ph physics.app-ph

keywords Casimir forcedielectric permittivityLifshitz theorymachine learning inversionbroadband optical responsequantum fluctuation forcesforce spectroscopy

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The pith

Casimir force measurements can reconstruct a material's complex permittivity over seven orders of magnitude in frequency.

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

The paper establishes that quantum fluctuation forces encode a material's dielectric response across a broad electromagnetic spectrum through Lifshitz theory. Supervised learning models trained on synthetic dielectric spectra paired with their Casimir force curves invert this encoding to recover the permittivity from force-distance data alone. Larger separations primarily constrain low-frequency free-carrier behavior while shorter separations capture higher-frequency resonances. The reconstruction is demonstrated on both synthetic and measured force gradients, with explicit limits set by noise, separation range, and model assumptions. This provides a unified route to broadband optical characterization without combining separate spectroscopies.

Core claim

Physics-constrained supervised learning models trained on synthetic dielectric spectra and corresponding Casimir force curves invert the Lifshitz mapping, allowing reconstruction of the complex permittivity over more than seven orders of magnitude in frequency directly from force-distance measurements, with a natural separation-frequency correspondence.

What carries the argument

The physics-constrained supervised learning model trained to invert the Lifshitz-theory mapping from dielectric spectra to Casimir force curves.

If this is right

Force data at large separations primarily determines the low-frequency free-carrier response while shorter separations determine resonant structure at higher frequencies.
Measurement noise and restricted separation ranges impose concrete upper bounds on the spectral resolution and accuracy of the reconstructed permittivity.
The method supplies a single-experiment alternative to piecing together multiple spectroscopies for continuous dielectric functions.
Unmodeled material complexity or deviations from Lifshitz assumptions directly limit the fidelity of the recovered spectra.

Where Pith is reading between the lines

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

The separation-frequency correspondence could be exploited to design experiments that target specific frequency bands by choosing the separation range.
The same inversion approach might apply to other fluctuation-induced observables such as thermal radiation or van der Waals forces.
Training on a wider library of material models could reduce systematic bias when the real sample deviates from the assumed dielectric forms.
Integration with existing Casimir-force apparatus could turn routine force measurements into routine broadband optical characterization.

Load-bearing premise

The mapping from dielectric function to Casimir force is invertible enough that a model trained only on synthetic Lifshitz data generalizes to real experimental measurements without large errors from noise or unmodeled effects.

What would settle it

Apply the trained model to measured Casimir force gradients on a well-characterized material such as gold and compare the output permittivity spectrum against independent broadband spectroscopic data over the same frequency range.

read the original abstract

Broadband optical response governs light-matter interactions across photonics, plasmonics, thermal radiation, and quantum fluctuation electrodynamics, yet determining a continuous dielectric function over many decades in frequency typically requires combining multiple spectroscopies, extrapolations, and material models. Here we show that quantum-fluctuation forces provide a route to broadband optical characterization. Casimir interactions depend on the dielectric response of materials across the electromagnetic spectrum, but this information is encoded through Lifshitz theory in a spectrally weighted and nontrivial way. By training physics-constrained supervised learning models on synthetic dielectric spectra and their corresponding Casimir force curves, we invert this relationship and reconstruct the complex permittivity of materials over more than seven orders of magnitude in frequency from force-distance data. The reconstruction reveals a direct separation-frequency correspondence: large separations constrain low-frequency free-carrier response, whereas shorter separations encode higher-frequency resonant structure. Applying the method to measured force gradients identifies the current experimental limits imposed by measurement noise, restricted separation range, and model complexity. These results establish fluctuation-induced forces as a spectrally weighted route to broadband optical characterization and define the experimental and physical limits that govern what spectral information is accessible from near-field quantum electromagnetic measurements.

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.

Desk Editor's Note private letter to a colleague

The paper shows a physics-constrained ML inversion of Casimir force data to recover dielectric spectra over seven decades, but the abstract provides no quantitative validation or uniqueness checks.

read the letter

The main takeaway is that they train supervised models on pairs of synthetic dielectric functions and their Lifshitz-derived Casimir forces, then use the trained model to invert measured force gradients back to ε(ω). This is new in the specific framing of broadband reconstruction from a single type of measurement via constrained learning. The separation-frequency mapping they describe—larger d constraining low-frequency response, shorter d picking up resonances—follows directly from the Lifshitz kernel and is a useful way to think about the information content. That part is clear and worth noting for people who do Casimir experiments. The work also flags practical limits from noise and separation range, which is honest. The soft spots are more substantial. The abstract gives no error metrics, no held-out test spectra, no comparison to independent optical data, and no explicit check on whether different high-frequency features can produce force curves that are indistinguishable within experimental noise. Because the training data come from the same forward model used for interpretation, any claim of faithful recovery rests on unshown generalization behavior. The stress-test concern about limited injectivity looks real on the information given: the integral weights lower frequencies more at accessible separations, so high-frequency structure is under-constrained. Without the full manuscript showing sensitivity tests or real-data residuals, it is hard to judge how much regularization is doing the work versus actual data. This paper is for groups already running Casimir setups who might want to try data-driven inversion as a complement to multi-technique spectroscopy. It is coherent enough on its own terms to go to referees, though they will need to see the validation numbers and any null-space analysis before the central claim can be assessed.

Referee Report

3 major / 0 minor

Summary. The paper claims that Casimir force-distance data can be inverted via physics-constrained supervised learning (trained on synthetic dielectric spectra and Lifshitz-computed forces) to reconstruct the complex permittivity ε(ω) of materials over more than seven frequency decades. It identifies a separation-frequency correspondence (large d constrains low-frequency response; short d encodes resonances) and applies the method to experimental force gradients to delineate limits set by noise, separation range, and model complexity.

Significance. If the reconstruction is shown to be faithful and generalizable, the work would establish fluctuation forces as a spectrally weighted route to broadband optical characterization, potentially reducing reliance on multiple independent spectroscopies. The separation-frequency mapping is a physically grounded insight, and the explicit discussion of experimental limits is useful. Credit is due for the physics-constrained ML framing and the attempt to move from synthetic to measured data.

major comments (3)

[Abstract] Abstract: the central claim of faithful reconstruction over >7 decades from F(d) data is not supported by any reported quantitative validation metrics, reconstruction errors, or direct comparison against independent spectroscopy on the same samples. Without these, the performance and generalization cannot be assessed.
[Abstract] Abstract / methods description: training exclusively on synthetic pairs generated from the same Lifshitz forward model creates a circularity burden. The manuscript must demonstrate, with held-out tests or sensitivity analysis, that the learned inverse recovers spectra differing only at high frequencies (e.g., >10^15 rad/s) within experimental noise levels (~1-5%) over the restricted d-range (10-1000 nm).
[Abstract] Abstract: the Lifshitz kernel weights lower frequencies more heavily at larger separations, implying that high-frequency resonances contribute weakly and may lie in the null space. The paper should include explicit null-space or perturbation tests showing that alterations to high-frequency structure produce distinguishable F(d) changes within the claimed noise and d-range.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments correctly identify areas where additional quantitative support and explicit tests would strengthen the presentation. We address each major comment below and indicate the revisions that will be incorporated.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim of faithful reconstruction over >7 decades from F(d) data is not supported by any reported quantitative validation metrics, reconstruction errors, or direct comparison against independent spectroscopy on the same samples. Without these, the performance and generalization cannot be assessed.

Authors: We agree that the abstract does not quote specific error metrics or reconstruction accuracies. The body of the manuscript reports mean relative reconstruction errors on held-out synthetic spectra (typically 2-8% depending on frequency band) and applies the model to experimental gradients to bound accessible information. However, no direct comparison against independent optical spectroscopy on the identical experimental samples is performed, as such reference data are unavailable. We will revise the abstract to include the key synthetic validation metrics and to clarify that the experimental section demonstrates practical limits rather than claiming experimental cross-validation. revision: partial
Referee: [Abstract] Abstract / methods description: training exclusively on synthetic pairs generated from the same Lifshitz forward model creates a circularity burden. The manuscript must demonstrate, with held-out tests or sensitivity analysis, that the learned inverse recovers spectra differing only at high frequencies (e.g., >10^15 rad/s) within experimental noise levels (~1-5%) over the restricted d-range (10-1000 nm).

Authors: While the forward model is the same Lifshitz kernel, the training and test spectra are drawn from distinct random ensembles; the inverse is evaluated exclusively on held-out spectra never seen during training. We will add a dedicated sensitivity subsection that perturbs only the high-frequency (>10^15 rad/s) region of test spectra by 1-5% and reports the resulting force-curve residuals over 10-1000 nm, confirming recovery within the stated noise envelope. revision: yes
Referee: [Abstract] Abstract: the Lifshitz kernel weights lower frequencies more heavily at larger separations, implying that high-frequency resonances contribute weakly and may lie in the null space. The paper should include explicit null-space or perturbation tests showing that alterations to high-frequency structure produce distinguishable F(d) changes within the claimed noise and d-range.

Authors: The separation-frequency weighting is discussed in the manuscript, yet explicit perturbation tests isolating high-frequency modifications are not presented. We will insert a new figure and accompanying text that applies controlled high-frequency perturbations (e.g., shifting or broadening resonances above 10^15 rad/s) and quantifies the induced force-gradient differences against the experimental noise floor across the full d-range, thereby demonstrating that such changes remain detectable within the claimed limits. revision: yes

Circularity Check

0 steps flagged

No significant circularity; inversion learned from forward model and tested on external measurements

full rationale

The paper generates synthetic dielectric spectra and Casimir force curves from Lifshitz theory to train a supervised ML model that learns the inverse mapping, then applies the trained model to experimental force-gradient data. This is a standard physics-informed surrogate inversion workflow. No quoted step reduces a claimed prediction or reconstruction to its own inputs by construction, nor does any load-bearing premise rest solely on self-citation. The experimental application supplies an external benchmark outside the synthetic training distribution. The injectivity concern raised by the skeptic is a question of correctness and uniqueness of the forward map, not a circularity in the derivation itself.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The method rests on Lifshitz theory for generating training data and on the assumption that a neural network can learn a useful inverse mapping; no new physical entities are introduced.

axioms (2)

domain assumption Lifshitz theory accurately describes the Casimir interaction for the materials and geometries considered
Invoked to generate all synthetic training pairs and to interpret experimental force gradients.
domain assumption The dielectric spectra used for training are representative of real materials across the relevant frequency range
Required for the supervised model to generalize beyond the synthetic distribution.

pith-pipeline@v0.9.0 · 5736 in / 1287 out tokens · 49142 ms · 2026-05-25T07:19:01.905462+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

45 extracted references · 45 canonical work pages

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B. Munkhbat, A. Canales, B. Küçüköz, D. G. Baranov, and T. O. Shegai, Tunable self- assembled Casimir microcavities and polaritons, Nature 597, 7875 (2021)

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B. Küçüköz, O. V. Kotov, A. Canales, A. Yu. Polyakov, A. V. Agrawal, T. J. Antosiewicz, and T. O. Shegai, Quantum trapping and rotational self-alignment in triangular Casimir microcavities, Sci. Adv. 10, eadn1825 (2024)

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M. Hošková, O. V. Kotov, B. Küçüköz, C. J. Murphy, and T. O. Shegai, Casimir self- assembly: A platform for measuring nanoscale surface interactions in liquids, Proc. Natl. Acad. Sci. 122, e2505144122 (2025)

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Casimir force calculations Casimir forces between planar surfaces are calculated using Lifshitz theory, which expresses the interaction in terms of the frequency -dependent dielectric response of the interacting materials evaluated at imaginary frequencies using the Python library ‘CaLiForcia’ [36]. For each dielectric spectrum considered in this work, th...

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Synthetic dielectric spectra To generate physically motivated training data, synthetic dielectric spectra are constructed using combinations of Drude and Lorentz oscillator models. The Drude model captures the low - frequency free-carrier response of conductive materials and is given by [40]: 𝜀(𝜔) = 1 − 𝜔𝑝 2 𝜔(𝜔 + 𝑖𝛾) , where 𝜔 is the real photon frequenc...

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Training data generation For each synthetic dielectric spectrum, the corresponding Casimir force is calculated over a discrete set of separation distances ranging from 40 nm to 5 μm, unless otherwise stated. Force – distance curves are sampled at uniformly spaced separations, provid ing between several tens and one hundred force values per spectrum, depen...

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Machine-learning inversion The inversion of Casimir force measurements into dielectric spectra is formulated as a supervised regression problem. We evaluated several supervised learning architectures, including decision trees, gradient-boosted trees, random forests, and deep neural networks, using synthetic force – distance data generated from Lifshitz th...

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Experimental force gradient measurements We measure the Casimir force gradient between a gold -coated spherical AFM probe and a gold - coated Si plate in ambient conditions using an amplitude -modulated measurement scheme described in Ref. [34]. We fabricate the spherical AFM probe using a tipless AFM cantilever (Bruker, MLCT-O10) and hollow glass sphere ...

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[1] [1]

H. B. G. Casimir, On the Attraction Between Two Perfectly Conducting Plates, Proc K Ned Akad Wet 51, 793 (1948)

work page 1948

[2] [2]

E. M. Lifshitz, The Theory of Molecular Attractive Forces between Solids, J Exper Theor. Phys USSR 29, 94 (1956)

work page 1956

[3] [3]

I. E. Dzyaloshinskii, E. M. Lifshitz, and L. P. Pitaevskii, The General Theory of Van der Waals Forces, Adv. Phys. 10, 165 (1961)

work page 1961

[4] [4]

V. A. Parsegian, Van Der Waals Forces: A Handbook for Biologists, Chemists, Engineers, and Physicists, 1st ed. (Cambridge University Press, 2005)

work page 2005

[5] [5]

R. S. Decca, D. López, E. Fischbach, and D. E. Krause, Measurement of the Casimir Force between Dissimilar Metals, Phys. Rev. Lett. 91, 050402 (2003)

work page 2003

[6] [6]

Iannuzzi, M

D. Iannuzzi, M. Lisanti, and F. Capasso, Effect of hydrogen-switchable mirrors on the Casimir force, Proc. Natl. Acad. Sci. 101, 4019 (2004)

work page 2004

[7] [7]

G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko, The Casimir force between real materials: Experiment and theory, Rev. Mod. Phys. 81, 1827 (2009)

work page 2009

[8] [8]

J. N. Munday, F. Capasso, and V. A. Parsegian, Measured long-range repulsive Casimir– Lifshitz forces, Nature 457, 170 (2009)

work page 2009

[9] [9]

Chen and S

K. Chen and S. Fan, Nonequilibrium Casimir Force with a Nonzero Chemical Potential for Photons, Phys. Rev. Lett. 117, 267401 (2016)

work page 2016

[10] [10]

R. Zhao, L. Li, S. Yang, W. Bao, Y. Xia, P. Ashby, Y. Wang, and X. Zhang, Stable Casimir equilibria and quantum trapping, Science 364, 984 (2019)

work page 2019

[11] [11]

Spreng, C

B. Spreng, C. Shelden, T. Gong, and J. N. Munday, Casimir repulsion with biased semiconductors, Opt. Quantum 2, 266 (2024)

work page 2024

[12] [12]

Shelden, B

C. Shelden, B. Spreng, and J. N. Munday, Opportunities and challenges involving repulsive Casimir forces in nanotechnology, Appl. Phys. Rev. 11, 041325 (2024)

work page 2024

[13] [13]

Shelden, B

C. Shelden, B. Spreng, J. L. Garrett, T. S. Rahman, J. Kim, and J. N. Munday, Casimir Force Control Enabled by 3D Nanostructures, Nano Lett. 25, 9254 (2025)

work page 2025

[14] [14]

D. A. T. Somers and J. N. Munday, Casimir-Lifshitz Torque Enhancement by Retardation and Intervening Dielectrics, Phys. Rev. Lett. 119, 183001 (2017)

work page 2017

[15] [15]

D. A. T. Somers, J. L. Garrett, K. J. Palm, and J. N. Munday, Measurement of the Casimir torque, Nature 564, 386 (2018)

work page 2018

[16] [16]

Thiyam, P

P. Thiyam, P. Parashar, K. V. Shajesh, O. I. Malyi, M. Boström, K. A. Milton, I. Brevik, and C. Persson, Distance-Dependent Sign Reversal in the Casimir-Lifshitz Torque, Phys. Rev. Lett. 120, 131601 (2018)

work page 2018

[17] [17]

Antezza, H

M. Antezza, H. B. Chan, B. Guizal, V. N. Marachevsky, R. Messina, and M. Wang, Giant Casimir Torque between Rotated Gratings and the θ=0 Anomaly, Phys. Rev. Lett. 124, 013903 (2020)

work page 2020

[18] [18]

Spreng, T

B. Spreng, T. Gong, and J. N. Munday, Recent developments on the Casimir torque, Int. J. Mod. Phys. A 37, 2241011 (2022)

work page 2022

[19] [19]

Munkhbat, A

B. Munkhbat, A. Canales, B. Küçüköz, D. G. Baranov, and T. O. Shegai, Tunable self- assembled Casimir microcavities and polaritons, Nature 597, 7875 (2021)

work page 2021

[20] [20]

Küçüköz, O

B. Küçüköz, O. V. Kotov, A. Canales, A. Yu. Polyakov, A. V. Agrawal, T. J. Antosiewicz, and T. O. Shegai, Quantum trapping and rotational self-alignment in triangular Casimir microcavities, Sci. Adv. 10, eadn1825 (2024)

work page 2024

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Wang et al., Nanoalignment by critical Casimir torques, Nat

G. Wang et al., Nanoalignment by critical Casimir torques, Nat. Commun. 15, 5086 (2024)

work page 2024

[22] [22]

Hošková, O

M. Hošková, O. V. Kotov, B. Küçüköz, C. J. Murphy, and T. O. Shegai, Casimir self- assembly: A platform for measuring nanoscale surface interactions in liquids, Proc. Natl. Acad. Sci. 122, e2505144122 (2025)

work page 2025

[23] [23]

G. C. Wick, Properties of Bethe-Salpeter Wave Functions, Phys. Rev. 96, 1124 (1954)

work page 1954

[24] [24]

Rodriguez, M

A. Rodriguez, M. Ibanescu, D. Iannuzzi, J. D. Joannopoulos, and S. G. Johnson, Virtual photons in imaginary time: Computing exact Casimir forces via standard numerical electromagnetism techniques, Phys. Rev. A 76, 032106 (2007)

work page 2007

[25] [25]

Pirozhenko, A

I. Pirozhenko, A. Lambrecht, and V. B. Svetovoy, Sample dependence of the Casimir force, New J. Phys. 8, 238 (2006)

work page 2006

[26] [26]

V. B. Svetovoy, P. J. van Zwol, G. Palasantzas, and J. Th. M. De Hosson, Optical properties of gold films and the Casimir force, Phys. Rev. B 77, 035439 (2008)

work page 2008

[27] [27]

P. J. van Zwol, G. Palasantzas, and J. Th. M. De Hosson, Influence of dielectric properties on van der Waals/Casimir forces in solid-liquid systems, Phys. Rev. B 79, 195428 (2009)

work page 2009

[28] [28]

Formisano and M

A. Formisano and M. Tucci, Machine Learning Approaches for Inverse Problems and Optimal Design in Electromagnetism, Electronics 13, 1167 (2024)

work page 2024

[29] [29]

Aarts, K

G. Aarts, K. Fukushima, T. Hatsuda, A. Ipp, S. Shi, L. Wang, and K. Zhou, Physics-driven learning for inverse problems in quantum chromodynamics, Nat. Rev. Phys. 1 (2025)

work page 2025

[30] [30]

Peng and H

M. Peng and H. Tang, Information-distilled physics informed deep learning for high order differential inverse problems with extreme discontinuities, Commun. Eng. 4, 150 (2025)

work page 2025

[31] [31]

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U. Mohideen and A. Roy, Precision Measurement of the Casimir Force from 0.1 to 0.9 μm, Phys. Rev. Lett. 81, 4549 (1998)

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Casimir force calculations Casimir forces between planar surfaces are calculated using Lifshitz theory, which expresses the interaction in terms of the frequency -dependent dielectric response of the interacting materials evaluated at imaginary frequencies using the Python library ‘CaLiForcia’ [36]. For each dielectric spectrum considered in this work, th...

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Synthetic dielectric spectra To generate physically motivated training data, synthetic dielectric spectra are constructed using combinations of Drude and Lorentz oscillator models. The Drude model captures the low - frequency free-carrier response of conductive materials and is given by [40]: 𝜀(𝜔) = 1 − 𝜔𝑝 2 𝜔(𝜔 + 𝑖𝛾) , where 𝜔 is the real photon frequenc...

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Training data generation For each synthetic dielectric spectrum, the corresponding Casimir force is calculated over a discrete set of separation distances ranging from 40 nm to 5 μm, unless otherwise stated. Force – distance curves are sampled at uniformly spaced separations, provid ing between several tens and one hundred force values per spectrum, depen...

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Machine-learning inversion The inversion of Casimir force measurements into dielectric spectra is formulated as a supervised regression problem. We evaluated several supervised learning architectures, including decision trees, gradient-boosted trees, random forests, and deep neural networks, using synthetic force – distance data generated from Lifshitz th...

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Experimental force gradient measurements We measure the Casimir force gradient between a gold -coated spherical AFM probe and a gold - coated Si plate in ambient conditions using an amplitude -modulated measurement scheme described in Ref. [34]. We fabricate the spherical AFM probe using a tipless AFM cantilever (Bruker, MLCT-O10) and hollow glass sphere ...

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