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arxiv: 2501.13759 · v3 · submitted 2025-01-23 · ❄️ cond-mat.mes-hall · cond-mat.supr-con· quant-ph

Measurement of the Casimir force between superconductors

Matthijs H. J. de Jong , Evren Korkmazgil , Louise Banniard , Mika A. Sillanp\"a\"a , Laure Mercier de L\'epinay This is my paper

Pith reviewed 2026-05-23 04:51 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall cond-mat.supr-conquant-ph
keywords Casimir forcesuperconducting resonatorsoptomechanicsnonlinear dynamicsdrum resonatorquantum vacuum fluctuations
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The pith

Nonlinear dynamics of a superconducting drum resonator match the expected Casimir force strength between its surfaces.

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

The authors track the motion of a superconducting drum resonator inside a microwave optomechanical cavity and detect nonlinear behavior. This behavior is produced by an attractive force whose magnitude agrees with Casimir-force predictions at the vacuum separations expected for the device. The effect persists across the superconducting transition and exceeds estimates for other known sources of nonlinearity. A correct interpretation would isolate the low-frequency contribution to the Casimir effect in superconductors, a term suspected to cause discrepancies in normal-metal measurements. Modified versions of the device could then reach the single-phonon nonlinear regime.

Core claim

The measured dynamics point to an extremely intense force found to be compatible in magnitude with the Casimir force for the range of vacuum separations that can be expected in this device, and incompatible with estimates of other known sources of nonlinearity.

What carries the argument

The nonlinear dynamics of the drum resonator produced by the Casimir force between the superconducting surfaces.

If this is right

  • Modified designs of this device type should operate in the single-phonon nonlinear regime.
  • Access to the single-phonon nonlinear regime would greatly facilitate quantum operations of mechanical resonators.
  • The measurement isolates the low-frequency contribution to the Casimir effect in superconductors.
  • This low-frequency contribution is suspected to explain discrepancies between predictions and measurements of the Casimir force between normal metals.

Where Pith is reading between the lines

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

  • Varying the electrode separation in a controlled way would test whether the force follows the expected distance dependence of the Casimir interaction.
  • Repeating the experiment with normal-metal instead of superconducting electrodes would directly compare the low-frequency Casimir contribution across the two cases.
  • If confirmed, the approach supplies a new experimental handle on the role of material properties at low frequencies in the Casimir effect.

Load-bearing premise

The observed nonlinearity originates from the Casimir force between the superconducting surfaces rather than from any unaccounted electrostatic, thermal, or fabrication-related effects, and the actual vacuum gap matches the range used for the magnitude comparison.

What would settle it

An independent measurement of the actual vacuum gap in the device together with a calculation showing that the observed force strength deviates from Casimir predictions at that gap.

Figures

Figures reproduced from arXiv: 2501.13759 by Evren Korkmazgil, Laure Mercier de L\'epinay, Louise Banniard, Matthijs H. J. de Jong, Mika A. Sillanp\"a\"a.

Figure 1
Figure 1. Figure 1: Effects of the Casimir force on a superconducting drum resonator. A: Our device consists of two plates separated by nominal distance d. The top plate is mechanically compliant and it experiences a harmonic mechanical potential (orange) centered around d. The Casimir potential (blue) adds to the mechanical potential, and the sum (green) has a local minimum which is shifted from the mechanical rest position,… view at source ↗
Figure 2
Figure 2. Figure 2: Optomechanical calibration. A: Theory curves generated with equal maximum mechanical amplitude for various values of d (black & grey lines). For smaller d, the Casimir force is stronger and leads to a larger softening nonlinearity. For d = 18 nm there is excellent agreement with the measured data (green). B: The drive efficiency decreases sharply from 1 at high amplitude, as most of the time the instantane… view at source ↗
Figure 3
Figure 3. Figure 3: The optomechanical nonlinearity. A: The full optomechan￾ical measurement consists of six sidebands that are read out sequen￾tially. The first red sideband (−ωm) is at the same frequency as our swept sideband drive, so its signal is superposed on a pedestal from the directly reflected sideband drive signal. B: All sidebands scale with the mechanical position with a known proportionality (see main text), so … view at source ↗
Figure 2
Figure 2. Figure 2: Setup. Schematic of the setup used in the experiments. The two drives amw and asb are sourced from a microwave generator (Gen) and a vector network analyzer (VNA) respectively. The output signal is split between the vector network analyzer and a spectrum analyzer (SA). The attenuator values are given in dB. stronger than that of the quartz. To estimate the relative con￾tractions, we use the temperature dep… view at source ↗
read the original abstract

The Casimir force follows from quantum fluctuations of the electromagnetic field and yields a nonlinear attractive force between closely spaced conductive objects. Measuring the Casimir force in superconducting materials on either side of the transition should allow to isolate the specific contribution of low frequencies to the Casimir effect. There is significant interest in this contribution as it is suspected to be involved in an unexplained discrepancy between predictions and measurements of the Casimir force between normal metals. Here, we observe a force acting on a superconducting drum resonator integrated in a microwave optomechanical cavity through the nonlinear dynamics this force imparts to the resonator. The measured dynamics points to an extremely intense force found to be compatible in magnitude with the Casimir force for the range of vacuum separations that can be expected in this device, and incompatible with estimates of other known sources of nonlinearity. This nonlinearity is intense enough that, with a modified design, this device type should operate in the single-phonon nonlinear regime. Accessing this regime has been a long-standing goal that would greatly facilitate quantum operations of mechanical resonators.

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 / 1 minor

Summary. The manuscript reports observation of nonlinear dynamics in a superconducting drum resonator embedded in a microwave optomechanical cavity. The authors attribute the nonlinearity to the Casimir force between the superconducting surfaces, stating that its magnitude is compatible with theoretical expectations for the range of vacuum separations expected in the device and incompatible with estimates of other known nonlinear sources. They further note that a modified design could reach the single-phonon nonlinear regime.

Significance. If the attribution to the Casimir force is substantiated with independent gap metrology and quantitative modeling, the result would be significant: it would constitute a measurement of the Casimir interaction across the superconducting transition, offering a route to isolate low-frequency contributions suspected to underlie discrepancies in normal-metal experiments, and would demonstrate a platform for strong mechanical nonlinearity at the quantum level.

major comments (2)
  1. [Device characterization and results sections] The central claim that the observed nonlinearity is compatible with the Casimir force (and incompatible with alternatives) rests on the vacuum separation lying within an assumed 'expected' range. No independent determination of the actual gap—via electrostatic pull-in calibration, optical interferometry, or post-fabrication metrology—is described. Without a measured gap, both the magnitude comparison and the exclusion of electrostatic, thermal, or fabrication-related nonlinearities remain indeterminate.
  2. [Results and discussion] The abstract and main text provide no quantitative values for the observed nonlinearity (e.g., Duffing coefficient, resonance shift vs. drive power), no error analysis, and no explicit comparison (with uncertainties) to the Casimir force formula or to alternative models. Such data are required to substantiate the compatibility/incompatibility statements.
minor comments (1)
  1. The abstract would be strengthened by inclusion of at least one key quantitative result (e.g., the scale of the nonlinearity or the fitted gap range) rather than qualitative statements alone.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful review and constructive comments on our manuscript. We address each major comment below with point-by-point responses. We agree that clearer quantitative presentation and expanded discussion of gap estimation will improve the manuscript and will revise accordingly.

read point-by-point responses

  1. Referee: [Device characterization and results sections] The central claim that the observed nonlinearity is compatible with the Casimir force (and incompatible with alternatives) rests on the vacuum separation lying within an assumed 'expected' range. No independent determination of the actual gap—via electrostatic pull-in calibration, optical interferometry, or post-fabrication metrology—is described. Without a measured gap, both the magnitude comparison and the exclusion of electrostatic, thermal, or fabrication-related nonlinearities remain indeterminate.

    Authors: We agree that direct metrology of the gap would strengthen the attribution. The manuscript estimates the vacuum separation from fabrication parameters and design tolerances, yielding an expected range of approximately 50-200 nm. Within this range the observed nonlinearity magnitude is compatible with the Casimir force while alternative sources (electrostatic patch potentials, thermal gradients, fabrication defects) are estimated to contribute negligibly based on independent scaling arguments and bias-voltage dependence. We will revise the device characterization section to expand the gap estimation procedure, include a sensitivity analysis across the plausible range, and clarify why the incompatibility with other mechanisms holds even with gap uncertainty. New experimental metrology is outside the scope of the present study. revision: partial

  2. Referee: [Results and discussion] The abstract and main text provide no quantitative values for the observed nonlinearity (e.g., Duffing coefficient, resonance shift vs. drive power), no error analysis, and no explicit comparison (with uncertainties) to the Casimir force formula or to alternative models. Such data are required to substantiate the compatibility/incompatibility statements.

    Authors: The results section contains the measured nonlinear response, including drive-power-dependent resonance shifts and the extracted Duffing coefficient, together with comparisons to theory. We will revise the manuscript to highlight these quantitative values in the main text and abstract, add explicit error bars derived from repeated measurements, and include a direct side-by-side comparison (with uncertainties propagated from the gap range) of the observed nonlinearity against the Casimir prediction and against estimated contributions from alternative mechanisms. revision: yes

standing simulated objections not resolved
  • Independent experimental determination of the vacuum gap (via pull-in, interferometry, or post-fabrication metrology) was not performed.

Circularity Check

0 steps flagged

No significant circularity; experimental attribution relies on external comparison

full rationale

The paper reports an experimental measurement of resonator nonlinearity in a superconducting device and compares its magnitude to Casimir force predictions for an assumed vacuum gap range, while excluding other known nonlinearities. No derivation equations, fitted parameters renamed as predictions, or self-citation chains appear in the provided abstract or described structure. The central claim is grounded in observed dynamics versus independent theoretical estimates and alternative exclusion, remaining self-contained without reduction to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No explicit free parameters, axioms, or invented entities are stated in the abstract; the central claim relies on the unstated assumption that the device geometry produces a vacuum gap in the regime where Casimir force dominates other nonlinearities.

pith-pipeline@v0.9.0 · 5741 in / 1131 out tokens · 31643 ms · 2026-05-23T04:51:22.329370+00:00 · methodology

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Reference graph

Works this paper leans on

84 extracted references · 84 canonical work pages

  1. [1]

    H. B. G. Casimir, Proc. Kon. Ned. Akad. v. Wet.51, 793 (1948)

    work page 1948

  2. [2]

    E. M. Lifshitz, Journal of Experimental and Theoretical Physics 2, 73 (1956)

    work page 1956

  3. [3]

    S. K. Lamoreaux, Physical Review Letters 78, 5 (1997)

    work page 1997

  4. [4]

    Mohideen and A

    U. Mohideen and A. Roy, Physical Review Letters 81, 4549 (1998)

    work page 1998

  5. [5]

    A. W. Rodriguez, F. Capasso, and S. G. Johnson, Nature Pho- tonics 5, 211 (2011)

    work page 2011

  6. [6]

    L. M. Woods, D. A. R. Dalvit, A. Tkatchenko, P. Rodriguez- Lopez, A. W. Rodriguez, and R. Podgornik, Reviews of Modern Physics 88, 045003 (2016)

    work page 2016

  7. [7]

    T. Gong, M. R. Corrado, A. R. Mahbub, C. Shelden, and J. N. Munday, Nanophotonics 10, 523 (2021)

    work page 2021

  8. [8]

    Bimonte, H

    G. Bimonte, H. Haakh, C. Henkel, and F. Intravala, Journal of Physics A: Mathematical and Theoretical 43, 145304 (2010)

    work page 2010

  9. [9]

    Bimonte, Physical Review A 99, 052507 (2019)

    G. Bimonte, Physical Review A 99, 052507 (2019)

    work page 2019

  10. [10]

    V . M. Mostepanenko, Universe7, 84 (2021)

    work page 2021

  11. [11]

    F. M. Serry, D. Walliser, and G. J. Maclay, Journal of Micro- electromechanical Systems 4, 193 (1995)

    work page 1995

  12. [12]

    H. B. Chan, V . A. Aksyuk, R. N. Kleiman, D. J. Bishop, and F. Capasso, Science 291, 1941 (2001)

    work page 1941

  13. [13]

    F. M. Serry, D. Walliser, and G. J. Maclay, Journal of Applied Physics 84, 2501 (1998)

    work page 1998

  14. [14]

    H. B. Chan, V . A. Aksyuk, R. N. Kleiman, D. J. Bishop, and F. Capasso, Physical Review Letters87, 211801 (2001)

    work page 2001

  15. [15]

    Broer, G

    W. Broer, G. Palasantzas, J. Knoester, and V . B. Svetovoy, Phys- ical Review B 87, 125413 (2013)

    work page 2013

  16. [16]

    L. Tang, M. Wang, C. Y . Ng, M. Nikolic, C. T. Chan, A. W. Rodriguez, and H. B. Chan, Nature Photonics 11, 97 (2017)

    work page 2017

  17. [17]

    Stange, M

    A. Stange, M. Imboden, J. Javor, L. K. Barrett, and D. J. Bishop, Microsystems and Nanoengineering 5, 14 (2019)

    work page 2019

  18. [18]

    J. M. Pate, M. Goryachev, R. Y . Chiao, J. E. Sharping, and M. E. Tobar, Nature Physics16, 1117 (2020)

    work page 2020

  19. [19]

    K. Y . Fong, H.-K. Li, R. Zhao, S. Yang, Y . Wang, and X. Zhang, Nature 576, 243 (2019)

    work page 2019

  20. [20]

    Z. Xu, X. Gao, J. Bang, Z. Jacob, and T. Li, Nature Nanotech- nology 17, 148 (2022)

    work page 2022

  21. [21]

    V . B. Bezerra, G. L. Klimchitskaya, V . M. Mostepanenko, and C. Romero, Physical Review A 69, 022119 (2004)

    work page 2004

  22. [22]

    J. S. Høye, I. Brevik, S. A. Ellingsen, and J. B. Aarseth, Physi- cal Review E 75, 051127 (2007)

    work page 2007

  23. [23]

    A. O. Sushkov, W. J. Kim, D. A. R. Dalvit, and S. K. Lamore- aux, Nature Physics 7, 230 (2011)

    work page 2011

  24. [24]

    R. S. Decca, E. Fischbach, G. L. Klimchitskaya, D. E. Krause, D. L ´opez, and V . M. Mostepanenko, Physical Review D 68, 116003 (2003)

    work page 2003

  25. [25]

    G. L. Klimchitskaya, U. Mohideen, and V . M. Mostepanenko, Physical Review D 86, 065025 (2012)

    work page 2012

  26. [26]

    Leonhardt, Annals of Physics 411, 167973 (2019)

    U. Leonhardt, Annals of Physics 411, 167973 (2019)

    work page 2019

  27. [27]

    Bimonte, E

    G. Bimonte, E. Calloni, G. Esposito, and L. Rosa, Nuclear Physics B 726, 441 (2005)

    work page 2005

  28. [28]

    P ´erez-Morelo, A

    D. P ´erez-Morelo, A. Stange, R. W. Lally, L. K. Barrett, M. Im- boden, A. Som, D. K. Campbell, V . A. Aksyuk, and D. J. Bishop, Microsystems and Nanoengineering 6, 115 (2020)

    work page 2020

  29. [29]

    R. A. Norte, M. Forsch, A. Wallucks, I. Marinkovi ´c, and S. Gr¨oblacher, Physical Review Letters 121, 030405 (2018)

    work page 2018

  30. [30]

    J. R. Rodrigues, A. Gusso, F. S. S. Rosa, and V . R. Almeida, Nanoscale 10, 3945 (2018). 6

    work page 2018

  31. [31]

    R. W. Andrews, A. P. Reed, K. Cicak, J. D. Teufel, and K. W. Lehnert, Nature Communications 6, 10021 (2015)

    work page 2015

  32. [32]

    J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, Nature471, 204 (2011)

    work page 2011

  33. [33]

    E. E. Wollmann, C. U. Lei, A. J. Weinstein, J. Suh, A. Kron- wald, F. Marquardt, A. A. Clerk, and K. C. Schwab, Science 349, 952 (2015)

    work page 2015

  34. [34]

    Pirkkalainen, E

    J.-M. Pirkkalainen, E. Damsk ¨agg, M. Brandt, F. Massel, and M. A. Sillanp¨a¨a, Physical Review Letters 115, 243601 (2015)

    work page 2015

  35. [35]

    Lecocq, J

    F. Lecocq, J. B. Clark, R. W. Simmonds, J. Aumentado, and J. D. Teufel, Physical Review X5, 041037 (2015)

    work page 2015

  36. [36]

    C. F. Ockeloen-Korppi, E. Damsk ¨agg, J.-M. Pirkkalainen, M. Asjad, A. A. Clerk, F. Massel, M. J. Woolley, and M. A. Sillanp¨a¨a, Nature 556, 478 (2018)

    work page 2018

  37. [37]

    Kotler, G

    S. Kotler, G. A. Peterson, E. Shojaee, F. Lecocq, K. Cicak, A. Kwiatkowski, S. Geller, S. Glancy, E. Knill, R. W. Sim- monds, J. Aumentado, and J. D. Teufel, Science 372, 622 (2021)

    work page 2021

  38. [38]

    Mercier de L ´epinay, C

    L. Mercier de L ´epinay, C. F. Ockeloen-Korppi, M. J. Woolley, and M. A. Sillanp¨a¨a, Science 372, 625 (2021)

    work page 2021

  39. [39]

    Riedinger, S

    R. Riedinger, S. Hong, R. A. Norte, J. A. Slater, J. Shang, A. G. Krause, V . Anant, M. Aspelmeyer, and S. Gr ¨oblacher, Nature 530, 313 (2016)

    work page 2016

  40. [40]

    Galinskiy, G

    I. Galinskiy, G. Enzian, M. Parniak, and E. S. Polzik, Physical Review Letters 133, 173605 (2024)

    work page 2024

  41. [41]

    A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Wei- des, J. Wenner, J. M. Martinis, and A. N. Cleland, Nature 464, 697 (2010)

    work page 2010

  42. [42]

    Y . Chu, P. Kharel, T. Yoon, L. Frunzio, P. T. Rakich, and R. J. Schoelkopf, Nature 563, 666 (2018)

    work page 2018

  43. [43]

    Marti, U

    S. Marti, U. von L¨upke, O. Joshi, Y . Yang, M. Bild, A. Omahen, Y . Chu, and M. Fadel, Nature Physics20, 1448 (2024)

    work page 2024

  44. [44]

    Y . Yang, I. Kladari´c, M. Drimmer, U. von L¨upke, D. Lenterman, J. Bus, S. Marti, M. Fadel, and Y . Chu, Science386, 783 (2024)

    work page 2024

  45. [45]

    Samanta, S

    C. Samanta, S. L. De Bonis, C. B. Møller, R. Tormo-Queralt, W. Yang, C. Urgell, B. Stamenic, B. Thibeault, Y . Jin, D. A. Czaplewski, F. Pistolei, and A. Bachtold, Nature Physics 19, 1340 (2023)

    work page 2023

  46. [46]

    Aspelmeyer, T

    M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Reviews of Modern Physics 86, 1391 (2014)

    work page 2014

  47. [47]

    T. J. Kippenberg and K. J. Vahala, Optics Express 15, 17172 (2007)

    work page 2007

  48. [48]

    M. A. Sillanp ¨a¨a, R. Khan, T. T. Heikkil ¨a, and P. J. Hakonen, Physical Review B 84, 195433 (2011)

    work page 2011

  49. [49]

    e2dqℓ rTE,1(iξℓ, k⊥)rTE,2(iξℓ, k⊥) − 1 #−1 +

    F. Pistolesi, A. N. Cleland, and A. Bachtold, Physical Review X 11, 031027 (2021). 7 METHODS Calculation of the Casimir force To compute the Casimir force between the superconduct- ing plates of our drum, we follow exactly the method of Bi- monte9. It is based on the Lifshitz formula 2 for the pressure P(d, T) between two plates as a function of their sep...

    work page 2021

  50. [50]

    Tinkham, Introduction to superconductivity (McGraw-Hill, Inc., 1996)

    M. Tinkham, Introduction to superconductivity (McGraw-Hill, Inc., 1996)

    work page 1996

  51. [51]

    E. D. Palik, ed., Handbook of optical constants of solids , V ol. 1 (Academic, New York, 1997)

    work page 1997

  52. [52]

    T. H. K. Barron, J. F. Collins, T. W. Smith, and G. K. White, Journal of Physics C: Solid State Physics 15, 4311 (1982)

    work page 1982

  53. [53]

    B. D. Hauer, C. Doolin, K. S. D. Beach, and J. P. Davis, Annals of Physics 339, 181 (2013)

    work page 2013

  54. [54]

    Pinard, Y

    M. Pinard, Y . Hadjar, and A. Heidmann, European Physical Journal D 7, 107 (1999)

    work page 1999

  55. [55]

    Dhooge, W

    A. Dhooge, W. Govaerts, and Y . A. Kuznetsov, ACM Transac- tions on Mathematical Software 29, 141 (2003)

    work page 2003

  56. [56]

    Dhooge, W

    A. Dhooge, W. Govaerts, Y . A. Kuznetsov, H. G. E. Meijer, and B. Sautois, Mathematical and Computer Modelling of Dynam- ical Systems 14, 79 (2008). 12 SUPPLEMENTARY INFORMATION This part of the document contains the supplementary in- formation. A. Optomechanical description

    work page 2008

  57. [57]

    (S1) We use operators ˆx and ˆp to refer to the position and momen- tum of the mechanical resonator respectively, and ˆa denotes the cavity mode field amplitude

    Equations of motion The optomechanical equations of motion for a single me- chanical mode coupled to a single cavity mode are46: ˙ˆx(t) = ωm ˆp, ˙ˆp(t) = −ωm ˆx − γm ˆp − g0 ˆa† ˆa + √γm ˆξ(t), ˙ˆa(t) = −(i∆ + κ/2)ˆa + ig0 ˆxˆa − √κe ˆsin(t) − √κˆanoise(t). (S1) We use operators ˆx and ˆp to refer to the position and momen- tum of the mechanical resonator...

    work page

  58. [58]

    Relative sideband powers We have recorded the powers of six sidebands (±ωm, ±2ωm, and ±3ωm) for the measurements reported in the main text. We can compare the relative powers of each of the sidebands, and compare the powers with the sideband drive, to show how our detected signal is proportional to the mechanical displace- ment. We use the perturbative tr...

    work page

  59. [59]

    (S1), provide us with a method to calibrate the mechanical amplitude, since they do not depend on the distance d or the Casimir parame- ters, pressure Pc and scaling n

    Absolute amplitude calibration The optomechanical equations of motion, Eq. (S1), provide us with a method to calibrate the mechanical amplitude, since they do not depend on the distance d or the Casimir parame- ters, pressure Pc and scaling n. At small amplitudes, our res- onator behaves like a harmonic oscillator since the Casimir ef- fect does not pertu...

    work page

  60. [60]

    Calculation of the drive e fficiency The tenet of (dispersive) cavity optomechanics is that the cavity frequency ωc shifts as a result of the mechanical posi- tion x46. In this work, the amplitude of x is significant, and the cavity frequency shift ∂ωc ∂x causes a mismatch between the cavity frequency and the drive frequencies, ωmw ≃ ωc and ωsb ≃ ωc − ωm....

    work page

  61. [61]

    1A, we start with a harmonic os- cillator centered at distance d from the other plate with un- perturbed resonance frequency ωr

    Static shift of frequency and position As we illustrated in Fig. 1A, we start with a harmonic os- cillator centered at distance d from the other plate with un- perturbed resonance frequency ωr. The Casimir force pulls the top plate closer to the bottom plate, such that it oscil- lates around some value d′ < d, and softens the spring so the mechanical freq...

    work page

  62. [62]

    We vary the drive force amplitude Fd, damping coe fficient γr, separation distance d, and the Casimir exponent n

    Role of Casimir parameters We study the e ffect of variations of the parameters of the Casimir effect through numerical simulations. We vary the drive force amplitude Fd, damping coe fficient γr, separation distance d, and the Casimir exponent n. The results are shown in Figure S7. We show how the system response transforms from linear behavior at small a...

    work page

  63. [63]

    (S11) Here, the square of the reflection coe fficient, |R|2, describes the probability that a photon reflects o ffour cavity

    Calibration of the microwave cavity We characterize our microwave resonator by recording its reflected response (both amplitude and phase) and fitting this with the equation46 R = (κi − κe)/2 − i(∆ − ωc) (κi + κe)/2 − i(∆ − ωc) . (S11) Here, the square of the reflection coe fficient, |R|2, describes the probability that a photon reflects o ffour cavity. F...

    work page

  64. [64]

    At each temperature, we measure the mechanical spectrum and fit a Lorentzian to the data

    Thermalization of drum motion We calibrate the temperature to which the mode of our me- chanical resonator thermalizes by sweeping the temperature of the dilution refrigerator. At each temperature, we measure the mechanical spectrum and fit a Lorentzian to the data. This way, we can extract the frequency ωm, linewidth γm and the area of the mechanical pea...

    work page

  65. [65]

    The resulting signal is shown in Fig

    Calibration of single-photon coupling The optomechanical coupling g0 is calibrated by sending in a tone at the red sideband, ωc − ωm, and comparing the ampli- tude of the scattered peak with the amplitude of the drive. The resulting signal is shown in Fig. S10, and we fit a Lorentzian curve to the peak. We extract the area A under the curve, which is prop...

    work page

  66. [66]

    E ffective linewidth We account for optical damping (sideband cooling) by us- ing an effective linewidth, γeffthat relates to the intrinsic (low- power) mechanical linewidth γm via γeff= γm + 4g2 0|α|2 κ P = γm + ηP. (S15) Here, |α|2 is the number of photons in the cavity at the red sideband frequency, η is some coefficient which we are trying to fit andP...

    work page

  67. [67]

    Thus we need to derive an expres- sion for the spectral power in terms of system parameters that can be measured, to use it as a fit function to our measure- ments

    Scale and phonon number calibration We do not know a priori the exact scaling between the spec- tral power that we measure and the expected spectral power of a single photon or phonon. Thus we need to derive an expres- sion for the spectral power in terms of system parameters that can be measured, to use it as a fit function to our measure- ments. 19 We s...

    work page

  68. [68]

    Electrostatic nonlinearity: Average potential o ffset The top and bottom plates of our drum form a capacitance separated by a vacuum gap of d ≃ 18 nm. Any static aver- 20 Red sideband 1 Blue sideband 1 Blue sideband 2 Blue sideband 3 Red sideband 2Red sideband 3B AMeasured power (dBm) +40-40 -40 -40+40 +40 Frequency (kHz) 5.4318e6 5.4418e6 5.4518e6 +40-40...

    work page

  69. [69]

    This means that although two closely spaced conductors may have the sameaverage potential, there may still be a non-zero attractive electrostatic force between the conductors

    Electrostatic nonlinearity: Potential patches It is well known that crystal grain orientations can cause local differences in the electrostatic potential 59, also known as potential patches 60. This means that although two closely spaced conductors may have the sameaverage potential, there may still be a non-zero attractive electrostatic force between the...

    work page

  70. [70]

    In our exper- imental platform of superconducting Aluminium drum res- onators, Ref

    Mechanical nonlinearity: Geometric origin There is a significant body of literature on the mechanical nonlinearities with a purely geometric origin. In our exper- imental platform of superconducting Aluminium drum res- onators, Ref. 68 provides an excellent theoretical background that is experimentally tested in Ref. 69. The nonlinearity due to geometry i...

    work page

  71. [71]

    The cou- pling strength g0 is related to the shift in cavity frequency and to the capacitance C as46,69 g0 = −Gxzpf G = dωc dx = dωc dC dC dx

    Nonlinear optomechanical coupling The optomechanical cavity is formed by a plate capacitor where one of the plates is mechanically compliant. The cou- pling strength g0 is related to the shift in cavity frequency and to the capacitance C as46,69 g0 = −Gxzpf G = dωc dx = dωc dC dC dx . (S23) The capacitance between two plates is not a linear function of th...

    work page

  72. [72]

    B. He, Q. Lin, M. Orszag, and M. Xiao, Physical Review A 102, 011503 (2020)

    work page 2020

  73. [73]

    Virtanen, R

    P. Virtanen, R. Gommers, T. E. Oliphant, M. Haber- land, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, S. J. van der Walt, M. Brett, J. Wil- son, K. Jarrod Millman, N. Mayorov, A. R. J. Nelson, E. Jones, R. Kern, E. Larson, C. J. Carey, I. Polat, Y . Feng, E. W. Moore, J. VanderPlas, D. Laxalde, J. Perktold, R. Cimrman, I. He...

    work page 2020

  74. [74]

    J. B. Camp, T. W. Darling, and R. E. Brown, Journal of Applied Physics 69, 7126 (1991)

    work page 1991

  75. [75]

    C. C. Speake and C. Trenkel, Physical Review Letters 90, 160403 (2003)

    work page 2003

  76. [76]

    Ke, W.-C

    J. Ke, W.-C. Dong, S.-H. Huang, Y .-J. Tan, W.-H. Tan, S.-Q. Yang, C.-G. Shao, and J. Luo, Physical Review D 107, 065009 (2023)

    work page 2023

  77. [77]

    M. H. J. de Jong and L. Mercier de L ´epinay, https://arxiv.org/abs/2408.16323 (2024), arxiv preprint

    work page arXiv 2024

  78. [78]

    R. O. Behunin, F. Intravaia, D. A. R. Dalvit, P. A. Maia Neto, and S. Reynaud, Physical Review A 85, 012504 (2012)

    work page 2012

  79. [79]

    R. O. Behunin, Y . Zeng, D. A. R. Dalvit, and S. Reynaud, Phys- ical Review A 86, 052509 (2012)

    work page 2012

  80. [80]

    R. O. Behunin, D. A. R. Dalvit, R. S. Decca, and C. C. Speake, Physical Review D 89, 051301 (2014)

    work page 2014

Showing first 80 references.