# Casimir Effect Invalidates the Drude Model for Transverse Electric Evanescent Waves

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Formalisms of the Lifshitz Theory in Terms of Real or Pure Imaginary Frequencies

## 3. Calculation of the Casimir Pressure between Metalic Plates Using the Drude and Plasma Models

## 4. Comparison Studies of Contributions from the Propagating and Evanescent Waves

#### 4.1. Transverse Magnetic Polarization

#### 4.2. Transverse Electric Polarization

## 5. Discussion: Failure of the Drude Model for Transverse Electric Evanescent Waves, the Role of Dissipation, and Possibilities of Alternative Tests

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Casimir, H.B.G. On the attraction between two perfectly conducting plates. Proc. Kon. Ned. Akad. Wetensch. B
**1948**, 51, 793–795. Available online: https://dwc.knaw.nl/DL/publications/PU00018547.pdf (accessed on 26 August 2023). - Lifshitz, E.M. The theory of molecular attractive forces between solids. Zh. Eksp. Teor. Fiz.
**1955**, 29, 94–110. (In Russian); English translation: Sov. Phys. JETP**1956**, 2, 73–83. Available online: http://jetp.ras.ru/cgi-bin/e/index/e/2/1/p73?a=list (accessed on 26 August 2023). - Mahanty, J.; Ninham, B.W. Dispersion Forces; Academic Press: London, UK, 1976. [Google Scholar]
- Israelachvili, J.N. Intermolecular and Surface Forces; Academic Press: San Diego, CA, USA; Elsevier: San Diego, CA, USA, 2011. [Google Scholar] [CrossRef]
- Milonni, P.W. The Quantum Vacuum. An Introduction to Quantum Electrodynamics; Academic Press, Inc.: San Diego, CA, USA, 1994. [Google Scholar] [CrossRef]
- Mostepanenko, V.M.; Trunov, N.N. The Casimir Effect and Its Applications; Clarendon Press/Oxford University Press, Inc.: Oxford, UK; New York, NY, USA, 1997. [Google Scholar]
- Milton, K.A. The Casimir Effect: Physical Manifestations of Zero-Point Energy; World Scientific: Singapore, 2001. [Google Scholar] [CrossRef]
- Parsegian, V.A. Van der Waals Forces: A Handbook for Biologists, Chemists, Engineers, and Physicists; Cambridge University Press: Cambridge, UK; New York, NY, USA, 2005. [Google Scholar] [CrossRef]
- Buhmann, S.Y. Disperson Forces I: Macroscopic Quantum Electrodynamics and Ground-State Casimir, Casimir–Polder and van der Waals Forces; Springer: Berlin/Heidelberg, Germany, 2012. [Google Scholar] [CrossRef]
- Buhmann, S.Y. Disperson Forces II: Many-Body Effects, Excited Atoms, Finite Temperature and Quantum Friction; Springer: Berlin/Heidelberg, Germany, 2012. [Google Scholar] [CrossRef]
- Langbein, D. Theory of Van der Waals Attraction; Springer: Berlin/Heidelberg, Germany, 2013. [Google Scholar] [CrossRef]
- Bordag, M.; Klimchitskaya, G.L.; Mohideen, U.; Mostepanenko, V.M. Advances in the Casimir Effect; Oxford University Press Inc.: New York, NY, USA; Oxford, UK, 2015. [Google Scholar] [CrossRef]
- Sernelius, B.E. Fundamentals of van der Waals and Casimir Interactions; Springer Nature, Switzerland AG: Cham, Switzerland, 2018. [Google Scholar] [CrossRef]
- Lamoreaux, S.K. Demonstration of the Casimir Force in the 0.6 to 6 μm Range. Phys. Rev. Lett.
**1997**, 78, 5–8. [Google Scholar] [CrossRef] - Speake, C.C.; Trenkel, C. Forces between conducting surfaces due to spatial variations of surface potential. Phys. Rev. Lett.
**2003**, 90, 160403. [Google Scholar] [CrossRef] [PubMed] - Bezerra, V.B.; Klimchitskaya, G.L.; Mohideen, U.; Mostepanenko, V.M.; Romero, C. Impact of surface imperfections on the Casimir force for lenses of centimeter-size curvature radii. Phys. Rev. B
**2011**, 83, 075417. [Google Scholar] [CrossRef] - Mohideen, U.; Roy, A. Precision measurement of the Casimir force from 0.1 to 0.9 μm. Phys. Rev. Lett.
**1998**, 81, 4549–4552. [Google Scholar] [CrossRef] - Decca, R.S.; López, D.; Fischbach, E.; Krause, D.E. Measurement of the Casimir force between dissimilar metals. Phys. Rev. Lett.
**2003**, 91, 050402. [Google Scholar] [CrossRef] - Decca, R.S.; Fischbach, E.; Klimchitskaya, G.L.; Krause, D.E.; López, D.; Mostepanenko, V.M. Improved tests of extra-dimensional physics and thermal quantum field theory from new Casimir force measurements. Phys. Rev. D
**2003**, 68, 116003. [Google Scholar] [CrossRef] - Decca, R.S.; López, D.; Fischbach, E.; Klimchitskaya, G.L.; Krause, D.E.; Mostepanenko, V.M. Precise comparison of theory and new experiment for the Casimir force leads to stronger constraints on thermal quantum effects and long-range interactions. Ann. Phys.
**2005**, 318, 37–80. [Google Scholar] [CrossRef] - Decca, R.S.; López, D.; Fischbach, E.; Klimchitskaya, G.L.; Krause, D.E.; Mostepanenko, V.M. Tests of new physics from precise measurements of the Casimir pressure between two gold-coated plates. Phys. Rev. D
**2007**, 75, 077101. [Google Scholar] [CrossRef] - Decca, R.S.; López, D.; Fischbach, E.; Klimchitskaya, G.L.; Krause, D.E.; Mostepanenko, V.M. Novel constraints on light elementary particles and extra-dimensional physics from the Casimir effect. Eur. Phys. J. C
**2007**, 51, 963–975. [Google Scholar] [CrossRef] - Bimonte, G.; López, D.; Decca, R.S. Isoelectronic determination of the thermal Casimir force. Phys. Rev. B
**2016**, 93, 184434. [Google Scholar] [CrossRef] - Bimonte, G.; Spreng, B.; Maia Neto, P.A.; Ingold, G.-L.; Klimchitskaya, G.L.; Mostepanenko, V.M.; Decca, R.S. Measurement of the Casimir Force between 0.2 and 8 μm: Experimental Procedures and Comparison with Theory. Universe
**2021**, 7, 93. [Google Scholar] [CrossRef] - Chang, C.-C.; Banishev, A.A.; Castillo-Garza, R.; Klimchitskaya, G.L.; Mostepanenko, V.M.; Mohideen, U. Gradient of the Casimir force between Au surfaces of a sphere and a plate measured using an atomic force microscope in a frequency-shift technique. Phys. Rev. B
**2012**, 85, 165443. [Google Scholar] [CrossRef] - Banishev, A.A.; Chang, C.-C.; Klimchitskaya, G.L.; Mostepanenko, V.M.; Mohideen, U. Measurement of the gradient of the Casimir force between a nonmagnetic gold sphere and a magnetic nickel plate. Phys. Rev. B
**2012**, 85, 195422. [Google Scholar] [CrossRef] - Banishev, A.A.; Klimchitskaya, G.L.; Mostepanenko, V.M.; Mohideen, U. Demonstration of the Casimir force between ferromagnetic surfaces of a Ni-coated sphere and a Ni-coated plate. Phys. Rev. Lett.
**2013**, 110, 137401. [Google Scholar] [CrossRef] - Banishev, A.A.; Klimchitskaya, G.L.; Mostepanenko, V.M.; Mohideen, U. Casimir interaction between two magnetic metals in comparison with nonmagnetic test bodies. Phys. Rev. B
**2013**, 88, 155410. [Google Scholar] [CrossRef] - Xu, J.; Klimchitskaya, G.L.; Mostepanenko, V.M.; Mohideen, U. Reducing detrimental electrostatic effects in Casimir-force measurements and Casimir-force-based microdevices. Phys. Rev. A
**2018**, 97, 032501. [Google Scholar] [CrossRef] - Liu, M.; Xu, J.; Klimchitskaya, G.L.; Mostepanenko, V.M.; Mohideen, U. Examining the Casimir puzzle with an upgraded AFM-based technique and advanced surface cleaning. Phys. Rev. B
**2019**, 100, 081406. [Google Scholar] [CrossRef] - Liu, M.; Xu, J.; Klimchitskaya, G.L.; Mostepanenko, V.M.; Mohideen, U. Precision measurements of the gradient of the Casimir force between ultraclean metallic surfaces at larger separations. Phys. Rev. A
**2019**, 100, 052511. [Google Scholar] [CrossRef] - Sushkov, A.O.; Kim, W.J.; Dalvit, D.A.R.; Lamoreaux, S.K. Observation of the thermal Casimir force. Nat. Phys.
**2011**, 7, 230–233. [Google Scholar] [CrossRef] - Klimchitskaya, G.L.; Bordag, M.; Mostepanenko, V.M. Comparison between experiment and theory for the thermal Casimir force. Int. J. Mod. Phys. A
**2012**, 27, 1260012. [Google Scholar] [CrossRef] - Klimchitskaya, G.L.; Mostepanenko, V.M. Experiment and theory in the Casimir effect. Contemp. Phys.
**2006**, 47, 131–144. [Google Scholar] [CrossRef] - Bimonte, G.; Emig, T.; Kardar, M.; Krüger, M. Nonequilibrium fluctuational quantum electrodynamics: Heat radiation, heat transfer, and force. Ann. Rev. Condens. Matter Phys.
**2017**, 8, 119–143. [Google Scholar] [CrossRef] - Milton, K.A.; Li, Y.; Kalauni, P.; Parashar, P.; Guérout, P.; Ingold, G.-L.; Lambrecht, A.; Reynaud, S. Negative entropies in Casimir and Casimir-Polder interactions. Fortschr. Phys.
**2017**, 65, 1600047. [Google Scholar] [CrossRef] - Svetovoy, V.B.; van Zwol, P.J.; Palasantzas, G.; De Hosson, J.T.M. Optical properties of gold films and the Casimir force. Phys. Rev. B
**2008**, 77, 035439. [Google Scholar] [CrossRef] - Bimonte, G. Making precise predictions of the Casimir force between metallic plates via a weighted Kramers-Kronig transform. Phys. Rev. A
**2011**, 83, 042109. [Google Scholar] [CrossRef] - Behunin, R.O.; Intravaia, F.; Dalvit, D.A.R.; Maia Neto, P.A.; Reynaud, S. Modeling electrostatic patch effects in Casimir force measurements. Phys. Rev. A
**2012**, 85, 012504. [Google Scholar] [CrossRef] - van Zwol, P.J.; Palasantzas, G.; De Hosson, J.T.M. Influence of random roughness on the Casimir force at small separations. Phys. Rev. B
**2008**, 77, 075412. [Google Scholar] [CrossRef] - Broer, W.; Palasantzas, G.; Knoester, J.; Svetovoy, V.B. Roughness correction to the Casimir force at short separations: Contact distance and extreme value statistics. Phys. Rev. B
**2012**, 85, 155410. [Google Scholar] [CrossRef] - Maia Neto, P.A.; Lambrecht, A.; Reynaud, S. Casimir effect with rough metallic mirrors. Phys. Rev. A
**2005**, 72, 012115. [Google Scholar] [CrossRef] - Canaguier-Durand, A.; Maia Neto, P.A.; Cavero-Pelaez, I.; Lambrecht, A.; Reynaud, S. Casimir interaction between plane and spherical metallic surfaces. Phys. Rev. Lett.
**2009**, 102, 230404. [Google Scholar] [CrossRef] [PubMed] - Fosco, C.D.; Lombardo, F.C.; Mazzitelli, F.D. Proximity force approximation for the Casimir energy as a derivative expansion. Phys. Rev. D
**2011**, 84, 105031. [Google Scholar] [CrossRef] - Bimonte, G.; Emig, T.; Jaffe, R.L.; Kardar, M. Casimir forces beyond the proximity force approximation. Europhys. Lett. (EPL)
**2012**, 97, 50001. [Google Scholar] [CrossRef] - Teo, L.P. Material dependence of Casimir interaction between a sphere and a plate: First analytic correction beyond proximity force approximation. Phys. Rev. D
**2013**, 88, 045019. [Google Scholar] [CrossRef] - Bimonte, G. Going beyond PFA: A precise formula for the sphere-plate Casimir force. Europhys. Lett.
**2017**, 118, 20002. [Google Scholar] [CrossRef] - Hartmann, M.; Ingold, G.-L.; Maia Neto, P.A. Plasma versus Drude modeling of the Casimir force: Beyond the proximity force approximation. Phys. Rev. Lett.
**2017**, 119, 043901. [Google Scholar] [CrossRef] - Klimchitskaya, G.L.; Mohideen, U.; Mostepanenko, V.M. The Casimir force between real materials: Experiment and theory. Rev. Mod. Phys.
**2009**, 81, 1827–1885. [Google Scholar] [CrossRef] - Mostepanenko, V.M. Casimir puzzle and conundrum: Discovery and search for resolution. Universe
**2021**, 7, 84. [Google Scholar] [CrossRef] - Klimchitskaya, G.L.; Mostepanenko, V.M. Current status of the problem of thermal Casimir force. Int. J. Mod. Phys. A
**2022**, 37, 2241002. [Google Scholar] [CrossRef] - Esquivel, R.; Svetovoy, V.B. Correction to the Casimir force due to the anomalous skin effect. Phys. Rev. A
**2004**, 69, 062102. [Google Scholar] [CrossRef] - Svetovoy, V.B.; Esquivel, R. Nonlocal impedances and the Casimir entropy at low temperatures. Phys. Rev. E
**2005**, 72, 036113. [Google Scholar] [CrossRef] [PubMed] - Sernelius, B.E. Effects of spatial dispersion on electromagnetic surface modes and on modes associated with a gap between two half spaces. Phys. Rev. B
**2005**, 71, 235114. [Google Scholar] [CrossRef] - Torgerson, J.R.; Lamoreaux, S.K. Low-frequency character of the Casimir force between metallic films. Phys. Rev. E
**2004**, 70, 047102. [Google Scholar] [CrossRef] - Bimonte, G. Comment on “Low-frequency character of the Casimir force between metallic films”. Phys. Rev. E
**2006**, 73, 048101. [Google Scholar] [CrossRef] [PubMed] - Intravaia, F.; Henkel, C. Casimir interaction from magnetically coupled eddy currents. Phys. Rev. Lett.
**2009**, 103, 130405. [Google Scholar] [CrossRef] [PubMed] - Intravaia, F.; Ellingsen, S.A.; Henkel, C. Casimir-Foucault interaction: Free energy and entropy at low temperature. Phys. Rev. A
**2010**, 82, 032504. [Google Scholar] [CrossRef] - Svetovoy, V.B.; Esquivel, R. The Casimir free energy in high- and low-temperature limits. J. Phys. A Math. Gen.
**2006**, 39, 6777–6784. [Google Scholar] [CrossRef] - Klimchitskaya, G.L.; Mostepanenko, V.M.; Svetovoy, V.B. Experimentum crucis for electromagnetic response of metals to evanescent waves and the Casimir puzzle. Universe
**2022**, 8, 574. [Google Scholar] [CrossRef] - Bordag, M. The Casimir effect for thin plasma sheets and the role of the surface plasmons. J. Phys. A Math. Gen.
**2006**, 39, 6173–6185. [Google Scholar] [CrossRef] - Palik, E.D. (Ed.) Handbook of Optical Constants of Solids. Volume 1; Academic Press, Inc.: San Diego, CA, USA, 1985. [Google Scholar] [CrossRef]
- Boström, M.; Sernelius, B.E. Thermal effects on the Casimir force in the 0.1–5 μm range. Phys. Rev. Lett.
**2000**, 84, 4757–4760. [Google Scholar] [CrossRef] [PubMed] - Bordag, M.; Geyer, B.; Klimchitskaya, G.L.; Mostepanenko, V.M. Casimir force at both nonzero temperature and finite conductivity. Phys. Rev. Lett.
**2000**, 85, 503–506. [Google Scholar] [CrossRef] [PubMed] - Greffet, J.-J.; Carminati, R. Image formation in near-field optics. Prog. Surf. Sci.
**1997**, 56, 133–237. [Google Scholar] [CrossRef] - Törmä, P.; Barnes, W.L. Strong coupling between surface plasmon polaritons and emitters: A review. Rep. Progr. Phys.
**2015**, 78, 013901. [Google Scholar] [CrossRef] - Culshaw, W.; Jones, D.S. Effect of a metal plate on total reflection. Proc. Phys. Soc. B
**1953**, 66, 859–864. [Google Scholar] [CrossRef] - Brady, J.J.; Brick, R.O.; Pearson, V.D. Penetration of microwaves into the rarer medium in total reflection. J. Opt. Soc. Am.
**1960**, 50, 1080–1084. [Google Scholar] [CrossRef] - Zhu, S.; Yu, A.W.; Hawley, D.; Roy, R. Frustrated total internal reflection: A demonstration and review. Am. J. Phys.
**1986**, 54, 601–606. [Google Scholar] [CrossRef] - Hsu, J.W.P. Near-field scanning optical microscopy studies of electronic and photonic materials and devices. Mater. Sci. Engin R Rep.
**2001**, 33, 1–50. [Google Scholar] [CrossRef] - Aigouy, L.; Lahrech, A.; Grésillon, S.; Cory, H.; Boccara, A.C.; Rivoal, J.C. Polarization effects in apertureless scanning near-field optical microscopy: An experimental study. Opt. Lett.
**1999**, 24, 187–189. [Google Scholar] [CrossRef] - Klimchitskaya, G.L.; Mostepanenko, V.M.; Svetovoy, V.B. Probing the response of metals to low-frequency s-polarized evanescent waves. Europhys. Lett. (EPL)
**2022**, 139, 66001. [Google Scholar] [CrossRef] - Ulvr, M. Design of PCB search coils for AC magnetic flux density measurement. AIP Adv.
**2018**, 8, 047505. [Google Scholar] [CrossRef] - Ramadan, Q.; Samper, V.; Poenar, D.; Yu, C. On-chip micro-electromagnets for magnetic-based bio-molecules separation. J. Magn. Magnet. Mater.
**2004**, 281, 150–172. [Google Scholar] [CrossRef] - Wensink, H.; Benito-Lopez, F.; Hermes, D.C.; Verboom, W.; Gardeniers, H.J.G.E.; Reinhoudt, D.N.; van den Berg, A. Measuring reaction kinetics in a lab-on-a-chip by microcoil NMR. Lab Chip
**2005**, 5, 280–284. [Google Scholar] [CrossRef] [PubMed] - Liu, Z.-X.; Wang, B.; Kong, C.; Si, L.-G.; Xiong, H.; Wu, Y. A proposed method to measure weak magnetic field based on a hybrid optomechanical system. Sci. Rep.
**2017**, 7, 12521. [Google Scholar] [CrossRef] - Murzin, D.; Mapps, D.J.; Levada, K.; Belyaev, V.; Omelyanchik, A.; Panina, L.; Rodionova, V. Ultrasensitive magnetic field Sensors for biomedical applications. Sensors
**2020**, 20, 1569. [Google Scholar] [CrossRef] - Huang, J.-H.; Duan, X.-Y.; Wang, G.-J.; Hu, X.-Y. Enhancing the precision of detecting weak magnetic fields based on weak-value amplification. J. Opt. Soc. Amer. B
**2022**, 39, 1289. [Google Scholar] [CrossRef] - Hannemann, M.; Wegner, G.; Henkel, C. No-slip boundary conditions for electron hydrodynamics and the thermal Casimir pressure. Universe
**2021**, 7, 108. [Google Scholar] [CrossRef] - Brevik, I.; Shapiro, B. A critical discussion of different methods and models in Casimir effect. J. Phys. Commun.
**2022**, 6, 015005. [Google Scholar] [CrossRef] - Klimchitskaya, G.L.; Mostepanenko, V.M. An alternative response to the off-shell quantum fluctuations: A step forward in resolution of the Casimir puzzle. Eur. Phys. J. C
**2020**, 80, 900. [Google Scholar] [CrossRef] - Klimchitskaya, G.L.; Mostepanenko, V.M. Casimir effect for magnetic media: Spatially non-local response to the off-shell quantum fluctuations. Phys. Rev. D
**2021**, 104, 085001. [Google Scholar] [CrossRef] - Klimchitskaya, G.L.; Mostepanenko, V.M. Theory-experiment comparison for the Casimir force between metallic test bodies: A spatially non-local dielectric response. Phys. Rev. A
**2022**, 105, 012805. [Google Scholar] [CrossRef]

**Figure 1.**The ratio of the Casimir pressures for Au plates computed at $T=300\phantom{\rule{0.166667em}{0ex}}$K using the Drude (D) or the plasma (p) model to the classical limit of the Casimir pressure ${P}_{\mathrm{D}}^{\phantom{\rule{0.166667em}{0ex}}0}$ (13) found using the Drude model, is shown as a function of separation.

**Figure 2.**The relative deviation between the Casimir pressures for Au plates computed at $T=300\phantom{\rule{0.166667em}{0ex}}$K using the simple Drude (D) or plasma (p) model and the optical data for Au extrapolated to zero frequency by the same models is shown as a function of separation. The inset: the region of large separations is shown on an enlarged scale.

**Figure 3.**(

**a**) The transverse magnetic contributions to the Casimir pressure for Au plates normalized to ${P}_{\mathrm{D}}^{\phantom{\rule{0.166667em}{0ex}}0}$ computed at $T=300$ K using the simple Drude or plasma model are shown as a function of separation by the solid and dashed lines, respectively. (

**b**) The relative deviation between these contributions is shown by the solid line.

**Figure 4.**The transverse magnetic contributions to the Casimir pressure for Au plates due to propagating and evanescent waves normalized to ${P}_{\mathrm{D}}^{\phantom{\rule{0.166667em}{0ex}}0}$ computed at $T=300\phantom{\rule{0.166667em}{0ex}}$K using the simple Drude model are shown as a function of separation by the top short-dashed and bottom long-dashed blue lines, respectively. The solid blue and long-dashed red lines for the normalized total transverse magnetic contributions to the Casimir pressure computed using the Drude and plasma models are reproduced from Figure 3a.

**Figure 5.**The transverse electric contributions to the Casimir pressure for Au plates due to propagating and evanescent waves normalized to ${P}_{\mathrm{D}}^{\phantom{\rule{0.166667em}{0ex}}0}$ computed at $T=300\phantom{\rule{0.166667em}{0ex}}$K using the simple Drude model and the total transverse electric contribution are shown as a function of separation by the top and bottom short-dashed, long-dashed lines, and the lower solid line, respectively. The upper solid line shows similar results for the transverse electric contribution computed using the simple plasma model.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Klimchitskaya, G.L.; Mostepanenko, V.M.
Casimir Effect Invalidates the Drude Model for Transverse Electric Evanescent Waves. *Physics* **2023**, *5*, 952-967.
https://doi.org/10.3390/physics5040062

**AMA Style**

Klimchitskaya GL, Mostepanenko VM.
Casimir Effect Invalidates the Drude Model for Transverse Electric Evanescent Waves. *Physics*. 2023; 5(4):952-967.
https://doi.org/10.3390/physics5040062

**Chicago/Turabian Style**

Klimchitskaya, Galina L., and Vladimir M. Mostepanenko.
2023. "Casimir Effect Invalidates the Drude Model for Transverse Electric Evanescent Waves" *Physics* 5, no. 4: 952-967.
https://doi.org/10.3390/physics5040062