We study the classical electromagnetic radiation due to the presence of a
dipole layer on a plane that performs a bounded motion along its normal
direction, to the first non-trivial order in the amplitude of that motion. We
show that the total emitted power may be written in terms of the dipole layer
autocorrelation function. We then apply the general expression for the emitted
power to cases where the dipole layer models the presence of patch potentials,
comparing the magnitude of the emitted radiation with that coming from the
quantum vacuum in the presence of a moving perfect conductor (dynamical Casimir
effect).Comment: 5 pages, no figure

Fosco, Cesar D.

Mazzitelli, Francisco D.

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arXiv

César D. Fosco

Francisco D. Mazzitelli

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Physical Review A

Radiation from a moving planar dipole layer: Patch potentials versus dynamical Casimir effect

We study the classical electromagnetic radiation due to the presence of a dipole layer on a plane that performs a bounded motion along its normal direction, to the first nontrivial order in the amplitude of that motion. We show that the total emitted power may be written in terms of the dipole layer autocorrelation function. We then apply the general expression for the emitted power to cases where the dipole layer models the presence of patch potentials, comparing the magnitude of the emitted radiation with that coming from the quantum vacuum in the presence of a moving perfect conductor (dynamical Casimir effect).Fil: Fosco, Cesar Daniel. Comisión Nacional de Energía Atómica. Centro Atómico Bariloche; Argentina. Comisión Nacional de Energía Atómica. Gerencia del Área de Energía Nuclear. Instituto Balseiro; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Mazzitelli, Francisco Diego. Comisión Nacional de Energía Atómica. Centro Atómico Bariloche; Argentina. Comisión Nacional de Energía Atómica. Centro Atómico Bariloche; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

Fosco, Cesar Daniel

Mazzitelli, Francisco Diego

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PHYSICAL REVIEW A 89, 062513 (2014)Radiation from a moving planar dipole layer: Patch potentials versus dynamical Casimir effectCésar D. Fosco and Francisco D. MazzitelliCentro Atómico Bariloche, Comisión Nacional de Energı́a Atómica, R8402AGP Bariloche, Argentinaand Instituto Balseiro, Universidad Nacional de Cuyo, R8402AGP Bariloche, Argentina(Received 8 March 2014; published 30 June 2014)We study the classical electromagnetic radiation due to the presence of a dipole layer on a plane that performsa bounded motion along its normal direction, to the first nontrivial order in the amplitude of that motion. Weshow that the total emitted power may be written in terms of the dipole layer autocorrelation function. We thenapply the general expression for the emitted power to cases where the dipole layer models the presence of patchpotentials, comparing the magnitude of the emitted radiation with that coming from the quantum vacuum in thepresence of a moving perfect conductor (dynamical Casimir effect).DOI: 10.1103/PhysRevA.89.062513 PACS number(s): 31.30.jh, 41.60.−m, 42.50.−pI. INTRODUCTIONDue to the unavoidable presence of imperfections, impuri-ties, and even spatial variations in its chemical composition,the surface of a real metal is not a perfect equipotential. In otherwords, those effects manifest themselves in the existence of anelectrostatic potential, with a nontrivial space dependence, onthe metallic surface [1]. These residual “patch potentials” mayeven produce forces between different metallic surfaces, whichbecome relevant for high sensitivity experiments, in manyareas of physics [2]. In particular, they can be crucial whendetermining the vacuum forces of quantum electromagnetic(EM) origin between neutral objects (static Casimir effect) andin experiments looking for modifications to the gravitationalinverse-square law [2–5]. In those experimental setups, theexistence of patch potentials results in the presence of aforce, electrostatic in origin, and the theoretical goal is thento have a proper, and hopefully simple, way to quantify it.Ideally one should be able to disentangle its contributionin any experimental attempt to determine forces which areof a different nature. For static situations, one is usuallyable to derive general expressions of the effects due tothe patch potentials in terms of just their autocorrelationfunctions.In this paper we quantify an effect due to the presenceof patch potentials in a rather different, complementarysituation: namely, we consider a metallic object that undergoesaccelerated motion and study the resulting EM radiation. Thephysical reason to expect such radiation becomes clear whenone recalls that patch potentials can be thought of as due to theexistence of a (space-dependent) dipole layer on the surface ofan otherwise neutral body [3]. Therefore, when accelerated, themoving dipole layer shall emit radiation. Our main goal hereis to evaluate the power emitted by a flat surface containingpatch potentials, in terms of the acceleration of the plane andthe characteristics of the patch potentials.One of the motivations that lead us to this calculation is tomake a quantitative comparison between the magnitudes of thisclassical effect with the power emitted by an accelerated ideal,perfectly conducting mirror. This “motion-induced radiation”or “dynamical Casimir effect” (DCE) [6] is a purely quantumeffect, that, as we will see, could also be interpreted as comingfrom a moving dipole layer with an ad hoc autocorrelationfunction.The radiation field of a single time-dependent dipole atrest is a classical problem, described in almost all texts onclassical electrodynamics. The radiation field of a movingdipole, however, is not so widely known, although it hasalready been investigated in the sixties [7]. To our knowledge,the spectrum of classical EM radiation due to the presenceof a dipole layer on a plane that moves rigidly has not beencomputed before. Therefore, since it is a crucial ingredient forour study, we present this calculation in Sec. II. As we shallshow, there is a rather simple formula for the spectral densityassociated to the radiated energy, in terms of a two-pointfunction that describes the correlation of the dipoles at differentpoints of the surface. Using the model of Ref. [3], this leadsimmediately to an expression for the emitted power by movingpatch potentials. In this section, we also comment on the caseof time dependent dipole layers. In Sec. III we compute thespectral density for the particular autocorrelations previouslyused in the literature to describe patch potentials, and make acomparison between the classical emitted power and the DCE.Section IV contains the conclusions of our work.II. RADIATIONIn this section we evaluate the classical EM radiation dueto the presence of a dipole layer on a plane that movesrigidly along the direction defined by its normal (we useCGS-Gaussian units throughout). The instantaneous positionof the plane may be defined in terms of a single functionq(t) such that x3 = q(t). Here x3 is one of the three Cartesiancoordinates (x1,x2,x3), for which we shall also use the notationx‖ ≡ (x1,x2). The dipole layer density D shall then be afunction of the two coordinates parallel to the plane, namely,with the notation just introduced, D = D(x‖).To proceed to the calculation of the emitted radiation, weneed the charge and current densities ρ and j, which for thesystem we are considering are given byρ(x,t) = −D(x‖) δ′(x3 − q(t)),(1)j(x,t) = −D(x‖) δ′(x3 − q(t)) q̇(t) ê3,where ê3 is the unit vector along the direction of motion.Finally, we shall assume that the motion is bounded,namely, that there is a length l such that |q(t)|  l, ∀t .1050-2947/2014/89(6)/062513(4) 062513-1 ©2014 American Physical SocietyCÉSAR D. FOSCO AND FRANCISCO D. MAZZITELLI PHYSICAL REVIEW A 89, 062513 (2014)To determine the radiated power, we introduce the retardedpotentials, and use the Lorentz gauge fixing condition, obtain-ing for the potentials the inhomogeneous wave equationsφ(x,t) =∫d3x ′ dt ′ G(x,t ; x′,t ′) ρ(x′,t ′),(2)A(x,t) = 1c∫d3x ′ dt ′G(x,t ; x′,t ′)j(x′,t ′),where G denotes the retarded Green’s function for the waveequation, which satisfies(c−2∂2t − ∇2x)G(x,t ; x′,t ′) = 4π δ(x − x′) δ(t − t ′). (3)A more explicit expression may be obtained by introducingthe Fourier transformation:G(x,t ; x′,t ′) =∫dω2πd3k(2π )3e−iω(t−t′)+ik·(x−x′)G̃(k‖,k3,ω),(4)withG̃(k‖,k3,ω) = 4πk2‖ + k23 −(ωc+ iη)2 . (5)The next step in the derivation of the emitted power is theintroduction of the Poynting vector, S = c4π E × B, whereE = −∇φ − 1c∂∂tA,(6)B = ∇ × A.Of course, for an arbitrary dipole layer, the radiation flux mayhave a rather cumbersome spatial dependence on the details ofthe layer. On the other hand, one is presumably more interestedin the global effect, namely, the average flux of energy, sincethe spatial dependence of that flux is hardly detectable. Thegeometry of the system suggests to evaluate the total flux ofradiated energy due to the moving plane. It is then convenientto evaluate the third component of S, on a constant-x3 plane,far from the region where the plane moves (|x3| > l).We see that the third component of S may be written asfollows:S3 = c4πεij EiBj , (7)where the indices i, j shall be assumed, from now on, to runfrom 1 to 2. Since A points in the ê3 direction, we see that thecomponents of the electric and magnetic field relevant to thecalculation of (7) are given byEi = −∂iφ, Bi = εij ∂jA3. (8)Thus S3 becomesS3 = c4π∂jφ ∂jA3. (9)The total flux of energy through one such plane shall be, ingeneral, a divergent quantity, something that may be dealt withby dividing it by the total area. Besides, it is also convenient tocalculate the total radiated energy since, when written in termsof the Fourier transforms of the time-dependent functions, itwill allow us to extract the spectral density of radiation.Thus, the (average) radiated energy per unit area through aconstant-x3 plane becomesUrad(x3) = 1L2∫dt∫d2x‖ S3(x‖,x3,t), (10)where L2 is the area of the plane, assumed temporarily largeand finite, but an L → ∞ limit at the end is assumed. Weexpect it to be independent of x3 far from the planes.After some algebra, we see that, to second order in q(t),Urad(x3) may be written as follows:Urad(x3)= −4π∫d2k‖4π2dω2πdk32πdp32π×{|k‖|2k23p3ω[k2‖ + k23 −(ωc+ iη)2][k2‖ + p23 − (ωc − iη)2]× ̃(k‖) |q̃(ω)|2eix3(k3+p3)}, (11)where we have introduced the Fourier transform of the dipolelayer autocorrelation function(x‖) = 1L2∫d2y‖D(y‖)D(x‖ + y‖). (12)It is worth noting that, in natural ( = 1 and c = 1) units,̃ is a dimensionless quantity. We mention at this pointthat a similar expression to the one above could have beenobtained if one had a random patch potential distribution, witha translation invariant stochastic correlation. Namely, evenwithout evaluating the average over a constant-x3 plane, thetranslation invariance of the system does produce an entirelyanalogous expression to the one above, now interpreting  asthe result of an average with a statistical weight.We then evaluate the integrals over k3 and p3, which canbe performed, for example, by using Cauchy’s theorem in arather straightforward way, obtaining a result that contains bothconvection and radiation terms. The latter are, for x3 > 0, inde-pendent of x3. On the other hand, an evaluation of Urad(−x3),the average energy flux through a plane symmetrically locatedwith respect to x3 = 0, yields the same result as Urad(x3).Next we introduce Urad, the total radiated energy per unitarea:Urad = Urad(x3) + Urad(−x3) = 2Urad(x3), (13)which, moreover, may be conveniently written as follows:Urad =∫ ∞0dω2πP(ω), (14)where the spectral density P(ω) isP(ω) = |ω||q̃(ω)|2∫ ω/c0dk‖ k3‖ ̃(k‖)√(ωc)2− k2‖, (15)where we have assumed the autocorrelation function to beisotropic.Equations (14) and (15) constitute the main result of thissection, namely, a rather general and compact expression forthe spectral density of emitted energy in terms of the two mainingredients that characterize the system: its motion q̃(ω) andthe autocorrelation function of the dipole layer density.062513-2RADIATION FROM A MOVING PLANAR DIPOLE LAYER: . . . PHYSICAL REVIEW A 89, 062513 (2014)Assuming that the correlation length is much smallerthan c/ω, we can approximate ̃(k‖) 	 ̃(0). With thisapproximationUrad 	 215̃(0)∫dt ȧ2, (16)where a is the acceleration. From this equation, one can showthat the radiation reaction force per unit area on the dipolelayer isfrad 	 215 ̃(0)...a . (17)Notably, in this approximation the results coincide with thoseof a single moving dipole d with d2 ≡ ̃(0)c5 [7,8].Until now we considered a dipole layer. In order todescribe an imperfect conductor with patch potentials, fol-lowing Ref. [3] we can consider a dipole layer close to agrounded perfect conductor. For this configuration, and fornonrelativistic motion of the mirror, the radiated power can becomputed using the method of images: in the presence of thegrounded conductor, the dipole density is increased by a factor2, and therefore the spectral density by a factor 4.Finally, we consider the case of time dependent dipolelayers, namely, D = D(x‖,t). This time dependence maybe produced by an external agent or may describe intrinsicfluctuations of the material. The generalization of (1) to thissituation isρ(x,t) = −D(x‖,t) δ′(x3 − q(t))j(x,t) =[−D(x‖,t) δ′(x3 − q(t)) q̇(t) (18)+ ∂D(x‖,t)∂tδ(x3 − q(t))]ê3,where the presence of the last term is required by currentconservation.A lengthier but otherwise entirely analogous calculationallows one to obtain the spectral density for this case:P(ω) = 1L2|ω|∫ ω/c0dk‖ k3‖√(ωc)2− k2‖∫dν2πdν ′2π× q̃(ν) D̃(k‖,ω − ν)D̃(−k‖, − ω − ν ′) q̃(ν ′), (19)in terms of the space and time Fourier transforms on the patchpotentials [9].The derivation was performed without any assumptionabout the origin of the time dependence. Let us now focus onthe case in which the time dependences are correlated. In theabsence of external agents producing that time dependence,it is reasonable to assume that they depend only on the timedifference between the two potentials. In Fourier space, thatamounts to1L2D̃(k‖,ω)D̃(−k‖,ω′) → ̃(k‖,ω) (2π )δ(ω + ω′), (20)which inserted into (19) yieldsP(ω) = 1L2|ω|∫ ω/c0dk‖ k3‖√(ωc)2− k2‖×∫dν2πq̃(−ν) ̃(k‖,ω − ν) q̃(ν). (21)The last equation reduces to the static one for instantaneouscorrelation, namely, wheñ(k‖,ω) = ̃(k‖)(2π )δ(ω). (22)The calculation for time dependent dipole layers could be auseful starting point to develop a microscopic approach of theDCE. One should consider both electric and magnetic dipolesas sources, with a particular time-dependent correlation func-tion to describe the quantum fluctuations.III. EXAMPLES AND COMPARISON WITH THE DCEAs a first example of a patch potential autocorrelation, wefirst consider the Gaussian approximation to the quasilocalcorrelation function proposed in Ref. [10]:̃(k‖) = π8V 2rms2 exp[− 116|k‖|22], (23)where Vrms is the variance of the potential and  a characteristiclength. For this particular correlation, the spectral density readsP(ω) = V 2rms|q̃(ω)|2ω4c3f (ω/4c) , (24)withf (x) = 2πx3[3x − (3 + 2x2)D+(x)], (25)where D+ denotes the Dawson function (note that in theequations above we have included the factor 4 coming fromthe image dipole layer). For a fixed frequency, the spectraldensity is a nonmonotonous function of the correlation length. Indeed, P vanishes for  → 0 (no patch potentials) andalso vanishes in the opposite limit  → ∞, since by a simpleapplication of symmetry arguments and Gauss’s law one seesthat a uniform density of charges or dipoles on a planecannot radiate. Therefore, it must have a maximum at anintermediate value. A plot of the function f (x) shows thatthe maximum is located at x ∼ 1.5. As a consequence, ifthe plane moves with a definite frequency ω0, the radiationemitted is maximum when the characteristic size of the patchesis  ∼ 6c/ω0.As a second example, we will consider the sharp-cutoffmodel proposed in Ref. [3]:̃(k‖) = 4πV2rmsk2max − k2minθ (|k‖| − kmin)θ (kmax − |k‖|), (26)which yields for the spectral densityP(ω) = 64πV2rms15c5(k2max − k2min)ω6|q̃(ω)|2×[1 −(kmincω)2]3/2 [2 + 3(kmincω)2], (27)where we assumed that kmincω< 1 and kmaxcω> 1. Note thatP(ω)vanishes for kmincω> 1. As a consequence, for the particularcase in which the plane moves with a definite frequency ω0,there is a threshold to have a nonvanishing emitted radiationkmin < ω0/c.062513-3CÉSAR D. FOSCO AND FRANCISCO D. MAZZITELLI PHYSICAL REVIEW A 89, 062513 (2014)In both examples one can consider the limiting case ofsmall correlation length ω/c  1, which is physically themore relevant limit. For the quasilocal correlations we obtainP(ω) 	 π60V 2rms2 ω6c5|q̃(ω)|2 . (28)A similar expression can be obtained for the sharp-cutoffmodel.We will now compare the last result with that comingform the DCE. A single accelerated perfect mirror producesphotons due to the interaction with the quantum fluctuationsof the electromagnetic field. On dimensional grounds, inthe nonrelativistic limit we expect the dissipative force perunit length on the mirror (fDCE) to be proportional to...a /c4. This corresponds to a spectral density proportionalto ω6|q̃(ω)|2/c4. An explicit calculation yields [6]PDCE(ω) = 30π2ω6c4|q̃(ω)|2 . (29)Defining ξ = P/PDCE = frad/fDCE, we obtainξ = π3V 2rms22c	 V2rms(40 mV)22(100 nm)2, (30)where we have written the result in terms of typical values thatcharacterize the patch potentials. This shows that the reactionforce due to the classical radiation could be comparable oreven larger than the one in the DCE.IV. CONCLUSIONSWe have shown that a real, accelerated metallic surfaceproduces classical EM radiation due to the unavoidablepresence of patch potentials. The calculation of the totalemission spectrum for a flat surface undergoing boundedmotion is a rather straightforward exercise in classical elec-trodynamics, and the result depends only on the most relevantphysical quantity characterizing the patch potentials: theirautocorrelation function.Remarkably, when the correlation length of the patchpotentials is sufficiently small, the emitted radiation coincideswith that of a single moving dipole and has the same frequencydependence than the radiation induced by the motion of aperfect conductor through the quantum vacuum. Althoughthese facts could have been anticipated by dimensionalanalysis, the explicit calculations in this paper allowed us tocompare the classical radiation of the moving patches withthe quantum radiation associated to the DCE. Dependingon the characteristics of the patches, the dissipative effectsassociated to the classical radiation could be comparablewith those coming from the quantum vacuum for perfectconductors.We have considered the simplest situation, correspondingto a single accelerated mirror. It is well known that thedynamical Casimir effect for this configuration is far frombeing optimal regarding the possibility of its experimentaldetection. It would be interesting to assess whether classicalradiation from patch potential masks the dynamical Casimireffect or not in more realistic experimental settings, such as aclosed cavity with variable length, in a regime of parametricresonance [6].ACKNOWLEDGMENTSWe would like to thank D. Dalvit for useful comments. Thiswork was supported by ANPCyT, CONICET, and UNCuyo.[1] R. Smoluchowski, Phys. Rev. 60, 661 (1941); N. D. Lang andW. Kohn, Phys. Rev. B 3, 1215 (1971); N. Gaillard et al., Appl.Phys. Lett. 89, 154101 (2006).[2] See, for instance, S. E. Pollack, S. Schlamminger, and J. H.Gundlach, Phys. Rev. Lett. 101, 071101 (2008); L. Deslauriers,S. Olmschenk, D. Stick, W. K. Hensinger, J. Sterk, andC. Monroe, ibid. 97, 103007 (2006); J. D. Carter and J. D.D. Martin, Phys. Rev. A 83, 032902 (2011).[3] C. C. Speake and C. Trenkel, Phys. Rev. Lett. 90, 160403 (2003).[4] C. D. Fosco, F. C. Lombardo, and F. D. Mazzitelli, Phys. Rev.A 88, 062501 (2013).[5] R. O. Behunin, D. A. R. Dalvit, R. S. Decca, and C. C. Speake,Phys. Rev. D 89, 051301(R) (2014).[6] For recent reviews see G. Barton, V. V. Dodonov, andV. I. Man’ko, J. Opt. B: Quantum Semiclass. Opt. 7, S1 (2005);V. V. Dodonov, Phys. Scripta 82, 038105 (2010); D. A. R. Dalvit,P. A. Maia Neto, and F. D. Mazzitelli, Lect. Notes Phys. 834,419 (2011).[7] J. R. Ellis, Proc. Cambridge Philos. Soc. 59, 759 (1963); G. N.Ward, Proc. R. Soc. London A 279, 562 (1964); ,Proc. CambridgePhilos. Soc. 61, 547 (1965); J. J. Monaghan, J. Phys. A 1, 112(1968).[8] J. A. Heras, Am. J. Phys. 62, 1109 (1994).[9] Since we are mainly interested in the radiation produced bythe motion of the dipole layer, in this formula we omitted thecontribution coming from a static layer, that is present when thedipole density D is time dependent.[10] R. O. Behunin, F. Intravaia, D. A. R. Dalvit, P. A.Maia Neto, and S. Reynaud, Phys. Rev. A 85, 012504(2012).062513-4

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Radiation from a moving planar dipole layer: patch potentials vs dynamical Casimir effect

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