We study the Casimir interaction in the plane-sphere geometry in the
classical limit of high temperatures. In this limit, the finite conductivity of
the metallic plates needs to be taken into account. For the Drude model, the
classical Casimir interaction is nevertheless found to be independent of the
conductivity so that it can be described by a single universal function
depending only on the aspect ratio $x=L/R$ where $L$ is the interplate distance
and $R$ the sphere radius. This universal function differs from the one found
for perfect reflectors and is in principle amenable to experimental tests. The
asymptotic approach of the exact result to the Proximity Force Approximation
appears to be well fitted by polynomial expansions in $\ln x$.Comment: Updated version with minor modifications and addition of a referenc

A. Lambrecht

Antoine Canaguier-Durand

Astrid Lambrecht

E. M. Lifshitz

Gert-Ludwig Ingold

H. B. G. Casimir

Marc-Thierry Jaekel

Paulo A. Maia Neto

S. J. Rahi

Serge Reynaud

English

arXiv

We study the Casimir interaction in the plane-sphere geometry in the classical limit of high temperatures. In this limit, the finite conductivity of the metallic plates needs to be taken into account. For the Drude model, the classical Casimir interaction is nevertheless found to be independent of the conductivity so that it can be described by a single universal function depending only on the aspect ratio x=L/R, where L is the interplate distance and R is the sphere radius. This universal function differs from the one found for perfect reflectors and is in principle amenable to experimental tests. The asymptotic approach of the exact result to the proximity force approximation appears to be well fitted by polynomial expansions in ln x

Canaguier-Durand, Antoine

Ingold, Gert-Ludwig

Jaekel, Marc-Thierry

Lambrecht, Astrid

Maia Neto, Paulo A.

Reynaud, Serge

OPUS Augsburg

PHYSICAL REVIEW A 85, 052501 (2012)Classical Casimir interaction in the plane-sphere geometryAntoine Canaguier-Durand,1,* Gert-Ludwig Ingold,2 Marc-Thierry Jaekel,3 Astrid Lambrecht,1Paulo A. Maia Neto,4 and Serge Reynaud11Laboratoire Kastler Brossel, ENS, UPMC, Centre National de la Recherche Scientifique, F-75252 Paris, France2Institut für Physik, Universität Augsburg, D-86135 Augsburg, Germany3Laboratoire de Physique Théorique, ENS, UPMC, Centre National de la Recherche Scientifique, F-75231 Paris, France4Instituto de Fı́sica, UFRJ, CP 68528, Rio de Janeiro RJ 21941-909, Brazil(Received 6 March 2012; published 1 May 2012)We study the Casimir interaction in the plane-sphere geometry in the classical limit of high temperatures. Inthis limit, the finite conductivity of the metallic plates needs to be taken into account. For the Drude model,the classical Casimir interaction is nevertheless found to be independent of the conductivity so that it can bedescribed by a single universal function depending only on the aspect ratio x = L/R, where L is the interplatedistance and R is the sphere radius. This universal function differs from the one found for perfect reflectors andis in principle amenable to experimental tests. The asymptotic approach of the exact result to the proximity forceapproximation appears to be well fitted by polynomial expansions in ln x.DOI: 10.1103/PhysRevA.85.052501 PACS number(s): 31.30.jh, 03.70.+k, 05.70.−a, 78.20.CiI. INTRODUCTIONThe Casimir effect arises due to the confinement of thequantum fluctuations of electromagnetic fields between re-flecting bodies. Its discussion is often focused on the ideal caseintroduced by H. Casimir [1] of a pair of perfectly reflectingparallel plates at zero temperature. In this case, the energyis given by a universal expression EP0 = −h̄cπ2A/720L3depending only on geometrical parameters, the mirrors’separation L and surface A, and fundamental constants h̄ andc. However, this idealization does not hold for any experimentand reliable descriptions of the Casimir interaction have toaccount for the optical response of matter [2,3].In the present article, we want to study another casewhere the Casimir interaction becomes universal, namely,the high-temperature limit indicated in the following by thesubscript HT. This limit has been thoroughly studied for twoperfectly reflecting parallel plates [4–6] where the free energyFPHT = −ζ (3)kBT A/8πL2 depends only on temperature Tand geometrical parameters L and A. ζ is the Riemannfunction and ζ (3)  1.202 and the superscript P refers toperfect mirrors. When the Drude model is used for describingthe effect of conduction electrons in metallic mirrors, the freeenergy FDHT = FPHT/2 is one half of that obtained for perfectmirrors [7–9] (the superscript D refers to the Drude model).This reduction by a factor of 2 at large temperatures,independent of the parameters of the Drude model (see below),is confirmed by microscopic descriptions of the interactionbetween two metallic bulks [10–12]. This difference can inprinciple be tested at room temperature by measuring the forceat large distances. Experiments are difficult in this domain be-cause the force decreases with distance. However, experimentshave recently been performed at distances up to 7 μm wherethe thermal effect is large. The results have been interpreted bythe authors [13] as being in agreement with the Drude modelprediction once an electrostatic patch contribution is subtracted[14]. This conclusion stands in contradiction with the results of*Present address: ISIS—Université de Strasbourg.other Casimir force measurements, performed at smaller dis-tances, up to 0.75 μm, which were interpreted by their authorsas excluding the dissipative Drude model and agreeing withthe lossless plasma model [15]. This contradiction remains amatter of debate (see discussions and references in [16,17]).It is also worth stressing that most precise experiments areperformed in the plane-sphere geometry. The evaluation of theforce is often done through the so-called proximity force ap-proximation (PFA) [18], which amounts to averaging the forcebetween parallel plates over the distribution of local interplatedistances. This trivial treatment of geometry cannot reproducethe rich relation expected between Casimir effect and geometry[17,19–21]. Theoretical treatments going beyond the PFA haverecently been proposed, in particular through a multipolarexpansion well adapted to the plane-sphere geometry. In thefollowing, we will use results known for perfect or metallicmirrors coupled to electromagnetic fields [22,23].The aim of the present article is to investigate the classicalCasimir interaction for the geometry of a plane and a spherewith an arbitrary aspect ratio. We will focus our attention onthe models of perfect mirrors and dissipative mirrors and usescattering methods discussed in [23]. The results will turnout to be given by universal functions of the aspect ratio, fordissipative as well as perfect mirrors, with the two functionsdiffering from each other. The limit of high temperatures willallow us to reach a much better numerical accuracy than inprevious works [23]. This improvement in the accuracy willbe used to sharpen the discussion of the asymptotic approachof the exact result to the PFA. This asymptotic behavior willappear to be well fitted by polynomial expansions of thelogarithm of the aspect ratio.II. THE SCATTERING FORMULA ATTHE CLASSICAL LIMITWe start from the Matsubara formula giving the Casimirfree energy F between a plane and a sphere [23]:F = kBT′∑nln detD(iξn),(1)D = I − R2e−KLR1e−KL.052501-11050-2947/2012/85(5)/052501(4) ©2012 American Physical SocietyANTOINE CANAGUIER-DURAND et al. PHYSICAL REVIEW A 85, 052501 (2012)Here ξn = 2πnkBT/h̄ are the Matsubara frequencies and theprime indicates a sum over integers n = 0 . . . ∞ with a factor12 for n = 0; R1 and R2 are the reflection operators on theplane and the sphere, respectively, discussed below in terms ofFresnel reflection amplitudes and Mie scattering amplitudes;e−KL describes the one-way propagation on the distance L =L + R between the center of the sphere and the plane, withR being the radius of the sphere and L being the minimaldistance between the plane and the sphere; K =√k2 + ξ 2/c2is the longitudinal wave vector after a Wick rotation from realto imaginary frequencies, with k being the modulus of thetransverse wave vector.The nonzero Matsubara frequencies ξn=0  2πkBT/h̄ in-crease with temperature and their contributions to the sumEq. (1) become exponentially small because of the propaga-tion factor e−KL with K  ξ/c  2πkBT/h̄c. At the high-temperature limit kBT  h̄c/L, the free energy is dominatedby the first term n = 0 in the sum Eq. (1) and, therefore, it isproportional to temperature:FHT = −kBT ,  = − 12 ln detD(0), (2)D(0) = I − RS(0)e−kLRP (0)e−kL. is a dimensionless function of the parameters which does notdepend on temperature. Hence the entropy S is independentof temperature and the energy E vanishes:SHT = −∂T F = kB,(3)EHT = FHT + T SHT = 0.This implies that the classical limit of the Casimir interactionhas a purely entropic origin [24,25].The Drude model is the simplest description of a dissipativemirror. It corresponds to a dielectric function ε[iξ ] = 1 + σ/ξat imaginary frequencies ω = iξ , without a magnetic response(μ = 1). σ [iξ ] = ω2p/(γ + ξ ) measures the conductivity asso-ciated with conduction electrons (the conductivity in SI unitsis ε0σ ). The Drude model is characterized by two dimensionalparameters, the squared plasma frequency ω2p, proportional tothe density of the conduction electrons, and their relaxationrate γ . It meets the important property of ordinary metalsto have a finite static conductivity σ0 = ω2p/γ . The losslessplasma model γ → 0 corresponds to an infinite value for σ0,in contradiction with tabulated optical data at low frequencies[26,27].The Fresnel reflection amplitudes rTE and rTM for planewaves with polarization TE or TM are well known from studiesof the plane-plane geometry [7–9]. They are rPTE → −1 andrPTM → 1 for perfect mirrors whereas they go to rDTE → 0 andrDTM → 1 for the Drude model at low frequencies. For thelossless plasma model, they are found to interpolate betweenperfect and Drude models, a property found also below forthe Mie amplitudes. The disappearance of the TE reflectionamplitude is responsible for the ratio 2 between the classicallimits of the free energies obtained from perfect and Drudemodels in the plane-plane geometry.A similar discrepancy arises also for the Mie amplitudes aand b which describe the scattering on the sphere [24]. For per-fect mirrors and low frequencies, we get aP  (−1)(+1)ξ̃ (2+1)(2+1)!!(2−1)!!and bP  − aP +1 . The electric and magnetic multipoles a andb progressively decrease with increasing multipole index ,but they have the same power law in the reduced frequencyξ̃ = ξR/c for a given . The situation is different for the Drudemodel where aD  aP and bD  bP σ̃0 ξ̃(2+3)(2+1) (σ̃0 = σ0R/c isthe reduced frequency associated with the static conductivity).Incidentally, the results for the lossless plasma model arefound to interpolate between perfect and Drude models: bhas the same form as bP (bD ) for large spheres (small spheres).The crossover between these two regimes is determined bythe parameter ω̃p being, respectively, larger or smaller than 1(ω̃p = ωpR/c is the reduced plasma frequency).The function P is a universal expression of the aspect ratiofor perfect mirrors, as there exist no other parameters. Thediscussion of the preceding paragraph entails that it is also thecase for the function D calculated for Drude mirrors [28],although the latter is associated with dimensional parameterslike σ0. As a matter of fact, b is negligible with respect toa for any given value of , and it follows that the classicalhigh-temperature limit is the same as if bD were simply zero.In spite of the fact that D and P are universal functionsof the aspect ratio, these two functions differ. As anotherimportant consequence of this discussion is that the losslessplasma model result does not obey the universality propertymet by perfect as well as Drude mirrors. For this reason, wewill not discuss this model any longer in this article.III. NUMERICAL EVALUATIONS AND RESULTSIn the remainder of this article, we present the results of thenumerical evaluation of the functions D,P which determinethe classical Casimir free energy between a plane and a sphere,for Drude and perfect mirrors, respectively. We write thesefunctions through a comparison with the often used PFAapproximation (the superscript D,P means D or P):D,P(x) = CD,PxD,P(x), x ≡ LR,(4)CD ≡ ζ (3)8, CP ≡ ζ (3)4.FIG. 1. (Color online) Classical (high-temperature) Casimir in-teraction between plane and spherical mirrors, represented by the ratioD,P of the exact result to the PFA result vs the aspect ratio x = L/R;the full red line corresponds to Drude mirrors and the dashed blueline corresponds to perfect mirrors.052501-2CLASSICAL CASIMIR INTERACTION IN THE PLANE- . . . PHYSICAL REVIEW A 85, 052501 (2012)The first factors in the expressions of D,P are the PFA resultswhile the second ones, D,P, represent deviations from PFA.The factors D,P go to one for large spheres x → 0, whichconfirms the validity of PFA as an asymptotic approximation.The fact that D,P and D,P depend on a single geometricalparameter, the aspect ratio x, corresponds to the universalitydiscussed above.The explicit numerical evaluation proceeds as in [23].Due to the relatively simple form of the matrices in thelow-frequency limit, the numerics can be pushed farther thanfor the general case. Precisely, the computation can be pushedto a larger maximum value max for the multipole index , andit follows that the numerical results are now more accurate forsmall values of the aspect ratio x. The results of the numericalevaluations shown on Fig. 1 have been calculated with maxup to 5000 at our minimal value for the aspect ratio 10−3. Thenumerical accuracy δ is estimated to be of the order of 10−4at x = 10−3 and of 10−8 at x = 2 × 10−3.Though both of them are universal, the functions D(x) andP(x) drawn in Fig. 1 do not coincide. In other words, the ratioP(x)/D(x) = 2P(x)/D(x) between Casimir free energiesFIG. 2. (Color online) Deviation from PFA of the classicalCasimir interaction between plane and spherical mirrors, describedby the additive correction βD,P(x) vs ln x (upper plot) and thelogarithmic derivatives − dβD,Pd ln x = −x dβD,Pdx (lower plot); the fullred lines correspond to Drude mirrors and the dashed blue linescorrespond to perfect mirrors.calculated for perfect and Drude mirrors is itself a function ofx. This ratio reaches the value 2 at the PFA limit (x → 0), asknown from the studies on plane-plane geometry [7], and thevalue 3/2 at large distances (x → ∞) [23].IV. ASYMPTOTIC APPROACH TO PFAWe finally discuss the asymptotic approach to PFA for smallvalues of x. To this aim, we introduce another representationof the deviation from PFA:D,P(x) ≡ 1 + xβD,P(x). (5)βD,P(x) is the slope of the line which joins the points (0,1) to(x,D,P(x)) on the plots of Fig. 1. It may also be thought of asan additive correction to PFA in Eq. (4):D,P(x) = CD,P(1x+ βD,P(x)). (6)The numerical accuracy δ discussed above is now translatedinto a numerical accuracy δβ = δ/x for β. This accuracy δβis estimated to be of the order of 10−1 at x = 10−3 and of5 × 10−6 at x = 2 × 10−3. The following discussions of theasymptotic approach to PFA are possible only thanks to thisgood accuracy.The quantities βD,P are shown as the upper plot in Fig. 2for Drude and perfect mirrors, as functions of ln x. The lowerFIG. 3. (Color online) Best fits of the numerical results with thetrial functions Eq. (8) for the Drude (red lines, upper plot) and perfectmirrors (blue lines, lower plot); the full, dotted, and dashed linescorrespond, respectively, to βl , βx , and βm; the full circles correspondto numerical points not used for the fits; the insets zoom on thesmallest values of x.052501-3ANTOINE CANAGUIER-DURAND et al. PHYSICAL REVIEW A 85, 052501 (2012)plot in Fig. 2 represents the quantities − dβD,Pd ln x which appear inthe Casimir force:F D,P(x) = −CD,PkBTL(1x− dβD,P(x)d ln x). (7)The plots in Fig. 2 suggest that the asymptotic approach toPFA is well described for small values of x by a polynomialexpansion in the variable ln x. In order to check this idea,we have performed best fits of the numerical results with thefollowing trial functions, defined with the same number ofparameters:βl(x) = a0 + a1 ln x + a2 ln2 x + a3 ln3 x,βx(x) = b0 + b1x + b2x2 + b3x3, (8)βm(x) = c0 + c1 ln x + c2x + c3x2.The first one, βl , is a polynomial form in the variable ln x assuggested by Fig. 2 (see also [29]). The second one, βx , isa polynomial form in the variable x, which was sufficient torepresent functions ρ obtained from older calculations with alesser accuracy [22,30]. It corresponds to the free energy beinga Laurent series in the variable x. The third one, βm, is a mixedform which results, in the classical limit (high temperatures),from the assumption [31] that the force (not the free energy)may be written as a Laurent series in x.In each case, the best-fit functions are defined from thenumerical results obtained in the window ln x ∈ [−5, − 3].Their comparison in Fig. 3 with the numerical values obtainedin a broader window shows that βl is a better representation ofthe numerical results than βm, itself better than βx . In particular,the insets in Fig. 3 show the deviations of the best fits of βxand βm from the exact results at the smallest values of theaspect ratio ln x ∈ [−6.2, − 5]. The assumptions that the freeenergy or force is a Laurent series in x cannot be consideredas generally valid.The difference between the cases of Drude and per-fect mirrors is in principle amenable to experimental tests.Direct comparison with exact calculations would requiremeasurements at large distances (high-temperature limit). Thischallenge could be met with experimental parameters not sofar from those in the experiment [13].ACKNOWLEDGMENTSThe authors thank the European Science FoundationResearch Networking Program CASIMIR (www.casimir-network.com) for providing excellent possibilities for dis-cussions and exchange. Financial support from internationalprograms Capes-Cofecub and Procope is gratefully acknowl-edged.[1] H. B. G. Casimir, Proc. K. Ned. Akad. Wet. 51, 793 (1948).[2] E. M. Lifshitz, Sov. Phys. JETP 2, 73 (1956).[3] M.-T. Jaekel and S. Reynaud, J. Physique I 1, 1395 (1991).[4] J. Mehra, Physica 37, 145 (1967).[5] L. S. Brown and G. J. Maclay, Phys. Rev. 184, 1272 (1969).[6] J. Schwinger, L. L. de Raad, and K. A. Milton, Ann. Phys. (NY)115, 1 (1978).[7] M. Boström and B. E. Sernelius, Phys. Rev. Lett. 84, 4757(2000).[8] I. Brevik, S. E. Ellingsen, and K. A. Milton, New J. Phys. 8, 236(2006).[9] G.-L. Ingold, A. Lambrecht, and S. Reynaud, Phys. Rev. E 80,041113 (2009).[10] B. Jancovici and L. Šamaj, Europhys. Lett. 72, 35 (2005).[11] P. R. Buenzli and P. A. Martin, Europhys. Lett. 72, 42 (2005).[12] G. Bimonte, Phys. Rev. A 79, 042107 (2009).[13] A. O. Sushkov, W. J. Kim, D. A. R. Dalvit, and S. K. Lamoreaux,Nat. Phys. 7, 230 (2011).[14] R. O. Behunin, F. Intravaia, D. A. R. Dalvit, P. A. Maia Neto,and S. Reynaud, Phys. Rev. A 85, 012504 (2012).[15] R. S. Decca, D. López, E. Fischbach, G. L. Klimchitskaya,D. E. Krause, and V. M. Mostepanenko, Ann. Phys. (NY) 318,37 (2005); Phys. Rev. D 75, 077101 (2007).[16] G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko,Rev. Mod. Phys. 81, 1827 (2009).[17] A. Lambrecht, A. Canaguier-Durand, R. Guérout, and S.Reynaud, in Casimir Physics, edited by D. A. R. Dalvit et al.,Lecture Notes in Physics, Vol. 834 (Springer-Verlag, New York,2011), p. 97.[18] B. V. Derjaguin, I. I. Abrikosova, and E. M. Lifshitz, Q. Rev.Chem. Soc. 10, 295 (1968).[19] R. Balian and B. Duplantier, Ann. Phys. (NY) 104, 300 (1977);112, 165 (1978).[20] A. Weber and H. Gies, Phys. Rev. D 82, 125019 (2010).[21] S. J. Rahi, T. Emig, and R. L. Jaffe, in Casimir Physics, editedby D. A. R. Dalvit et al., Lecture Notes in Physics, Vol. 834(Springer-Verlag, New York, 2011), p. 129.[22] P. A. Maia Neto, A. Lambrecht, and S. Reynaud, Phys. Rev.A 78, 012115 (2008); A. Canaguier-Durand, P. A. Maia Neto,I. Cavero-Pelaez, A. Lambrecht, and S. Reynaud, Phys. Rev.Lett. 102, 230404 (2009).[23] A. Canaguier-Durand, P. A. Maia Neto, A. Lambrecht, andS. Reynaud, Phys. Rev. Lett. 104, 040403 (2010); Phys. Rev.A 82, 012511 (2010).[24] J. Feinberg, A. Mann, and M. Revzen, Ann. Phys. (NY) 288,103 (2001).[25] L. Spruch, Phys. Rev. A 66, 022103 (2002).[26] A. Lambrecht and S. Reynaud, Eur. Phys. J. D 8, 309(2000).[27] V. B. Svetovoy, P. J. van Zwol, G. Palasantzas, and J. T. M. DeHosson, Phys. Rev. B 77, 035439 (2008).[28] R. Zandi, T. Emig, and U. Mohideen, Phys. Rev. B 81, 195423(2010).[29] M. Bordag and V. Nikolaev, Phys. Rev. D 81, 065011(2010).[30] T. Emig, J. Stat. Mech.: Theory Exp. (2008) P04007.[31] G. Bimonte, T. Emig, R. L. Jaffe, and M. Kardar, Europhys.Lett. 97, 50001 (2012).052501-4

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Classical Casimir interaction in the plane-sphere geometry

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