As part of our program to develop the description of three-body effects in
quantum vacuum phenomena, we study the three-body interaction of two
anisotropically polarizable atoms with a perfect electrically conducting plate,
a generalization of earlier work. Three- and four-scattering effects are
important, and lead to nonmonotonic behavior.Comment: 10 pages, 5 figures, for the proceedings of the conference
  Mathematical Structures in Quantum Systems, Benasque, Spain, July 2012, to be
  published in Nuovo Ciment

Abalo, Elom

Milton, Kimball A.

Parashar, Prachi

Shajesh, K. V.

English

arXiv

As part of our program to develop the description of three-body effects in quantum vacuum phenomena, we study the three-body interaction of two anisotropically polarizable atoms with a perfect electrically conducting plate, a generalization of earlier work. Three- and four-scattering effects are important, and lead to nonmonotonic behavior

Abalo, E. K.

Scientific Open-access Literature Archive and Repository

DOI 10.1393/ncc/i2013-11531-5Colloquia: MSQS 2012IL NUOVO CIMENTO Vol. 36 C, N. 3 Maggio-Giugno 2013Three-body Casimir-Polder interactionsKimball A. Milton(1), E. K. Abalo(1), Prachi Parashar(1)and K. V. Shajesh(2)(1) Homer L. Dodge Department of Physics and Astronomy, University of OklahomaNorman, OK 73019, USA(2) Department of Materials Science and Engineering, Iowa State University of Scienceand Technology, Ames, IA 50011, USAricevuto l’11 Gennaio 2013Summary. — As part of our program to develop the description of three-bodyeffects in quantum vacuum phenomena, we study the three-body interaction of twoanisotropically polarizable atoms with a perfect electrically conducting plate, a gen-eralization of earlier work. Three- and four-scattering effects are important, andlead to nonmonotonic behavior.PACS 03.70.+k – Theory of quantized fields.PACS 11.80.La – Multiple scattering.PACS 34.35.+a – Interactions of atoms and molecules with surfaces.PACS 42.50.Lc – Quantum fluctuations, quantum noise, and quantum jumps.1. – IntroductionEven before the discovery of the Casimir effect [1], Casimir and Polder studied theretarded dispersion force between two atoms, and between one atom and a perfectly con-ducting surface [2]. The Casimir-Polder force between an atom and a conducting platewas, until relatively recently, only observed in one experiment [3], although the temper-ature dependence of this force has now been confirmed [4]. In contrast, the experimentalobservation of the Casimir-Polder force between atoms has remained beyond reach. Foran elementary review of aspects of Casimir-Polder forces see ref. [5].In this paper we wish to consider three-body Casimir-Polder energies, involving twopolarizable atoms and a perfectly conducting plate. This was considered recently, butin a somewhat restricted geometry [6]. A scalar analog was also recently examined byShajesh and Schaden [7]. (In the nonretarded regime, this was also considered by deMelo e Souza et al., following earlier work described in ref. [8].) Here we wish to examinethe situation a bit more generally. Our interest is not so much in finding observableeffects, which are probably difficult to detect, but to understand the general features ofthree-body interactions; this is a small part of our continuing efforts in this direction.c© Societa` Italiana di Fisica 183184 KIMBALL A. MILTON, E. K. ABALO, PRACHI PARASHAR and K. V. SHAJESHaZ 1Z 2α 1α 2Fig. 1. – Two anisotropically polarizable atoms interacting with a perfectly conducting plate.The atom a, a = 1, 2, has polarizability αa, and is a distance Za above the plate. The distancebetween the atoms in the direction parallel to the plate is a. The difference in heights abovethe plate is ΔZ = Z1 − Z2.2. – Two-body energyFor simplicity, we assume we are in the fully retarded regime, that is, the atoms aresufficiently far from each other or the plate so that we can describe them by their staticelectric polarizabilities,αa = αa(ω = 0), a = 1, 2.(1)We also neglect magnetic polarizability, and quadrupole and higher multipole momentsof the atoms. We will, however, assume that the atoms are not isotropically polarizable.In the isotropic case, the two-body interaction energy between the atoms, a distance rapart, is given by the famous formula (h¯ = c = 1)E12 = − 234πα1α2r7,(2)but it is considerably more complicated when the atoms are not isotropically polarizable,E12 = −13Tr(α1 ·α2)− 56 (rˆ ·α1 ·α2 · rˆ) + 63 (rˆ ·α1 · rˆ)(rˆ ·α2 · rˆ)8πr7 ,(3)where r = rrˆ is the relative position vector of the two atoms. This formula was apparentlyfirst given by Craig and Power [9]. In contrast, the interaction energy of a polarizableatom with a perfectly conducting plate a distance Z away is very simple,ECP = − Trα8πZ4 .(4)(One might recall that the dimension of α is (length)3, and that α is necessarily asymmetric matrix.)We now wish to consider what happens when all three bodies are present, the twoatoms and the perfectly conducting plate.3. – Three-body energyWe wish to consider two polarizable atoms in the neighborhood of a perfectly con-ducting plate. The geometry is sketched in fig. 1. The two atoms have polarizabilities αa,THREE-BODY CASIMIR-POLDER INTERACTIONS 185and are located at distances from the plate Za, a = 1, 2. The difference in heights fromthe plate is ΔZ = Z1 − Z2, and the horizontal distance (parallel to the plate) betweenthe two atoms is called a.To compute the interaction energy, we use the multiple-scattering formalism. Forthe case at hand, where only a single scattering with each atom need be considered, thetwo-body terms are (this is just the famous TGTG formula in weak coupling [10,11])Eab =i2∫dω2πTr T˜aT˜b,(5)where a, b = 1, 2, 3, and we denote the two atoms as bodies 1 and 2, while the perfectlyconducting plate is body 3. Here the modified scattering operator isT˜ = TΓ0 = V(1− Γ0V)−1Γ0.(6)The free propagator Γ0 is [12]Γ0(r, r′) = −e−|ζ|R4πR3[1u(|ζ|R)− RˆRˆv(|ζ|R)], R = r− r′,(7)where ζ = −iω is the imaginary frequency, andu(x) = 1 + x + x2, v(x) = 3 + 3x + x2.(8)The scattering matrix for the atom is just the potential,T1,2 = V1,2 = 4πα1,2δ(r− r1,2),(9)where ra are the positions of the atoms. The structure in which the scattering matrixfor the plate appears isΓ0T3Γ0 = Γ3 − Γ0;(10)Γ3 is the Green’s dyadic for the plate. In fact, with nˆ denoting the normal to the plate,(Γ3 − Γ0)(r, r′) = −Γ0(r, r′ − 2nˆnˆ · r′) · (1− 2nˆnˆ) ≡ −Γ(r, r′).(11)This is just the image construction for a perfect electric mirror [13].Using these constructions, the results in the previous section are easily derived. Ourconcern here is with the three-body terms. These are of three, related, types [14], wherethe subscripts on the right-hand side refer to the ordering of the scattering between thebodies,ΔE3 = E123 + E213 + E1323.(12)The three-scattering energy isE123 =i2∫dω2πTr T˜1T˜2T˜3 =12∫ ∞−∞dζ2πTrV1Γ0V2Γ,(13)186 KIMBALL A. MILTON, E. K. ABALO, PRACHI PARASHAR and K. V. SHAJESHwhile E213 is obtained by interchanging 1 and 2. The four-scattering contribution isE1323 =i2∫dω2πTr T˜1T˜3T˜2T˜3 = −12∫ ∞−∞dζ2πTrV1ΓV2Γ.(14)It is very straightforward to work out these contributions in the static approxima-tion (1), where we ignore the frequency dependence of the polarizability. Let us nowchoose the plate to lie in the x-y plane. Because of the operator appearing in Γ, it isconvenient to define βa = (1⊥ − zˆzˆ) · αa, 1⊥ = xˆxˆ + yˆyˆ. We can write all the energycontributions, including the two-atom ones, as follows:E12(r12) = −F (r12, r21;α1,α2),(15a)E123(r12, r21¯) = F (r12, r21¯;α2,β1),(15b)E213(r21, r12¯) = F (r21, r12¯;α1,β2),(15c)E1323(r21¯, r12¯) = −F (r21¯, r12¯;β1,β2).(15d)Here rab = ra − rb, and the positions of the images of the atoms are denoted by ra¯.These relative position vectors satisfyr21 + r12¯ + r2¯1¯ + r1¯2 = 0,(16a)(r21 + r12¯) · (r12¯ + r1¯2) = 0.(16b)The function appearing in the energies isF (x,y;α,β) =14πx3y3(x + y)[A(x, y)Tr(α · β)−B(x, y)(yˆ · β ·α · yˆ)(17)−B(y, x)(xˆ ·α · β · xˆ) + C(x, y)(xˆ ·α · yˆ)(yˆ · β · xˆ)],whereA(x, y) =8(x + y)4[x4 + 5x3y + 14x2y2 + 5xy3 + y4] x=y−→ 13,(18a)B(x, y) =8(x + y)4[3x4 + 15x3y + 26x2y2 + 10xy3 + 2y4] x=y−→ 28,(18b)C(x, y) =48(x + y)4[x4 + 5x3y + 9x2y2 + 5xy3 + y4] x=y−→ 63.(18c)4. – Special cases4.1. Isotropically polarizable atoms, equidistant from the plate. – Let us first considerisotropic atoms equidistant from the plate, so ΔZ = 0 and Z1 = Z2 = Z. Definingγ =√1 +4Z2a2,(19)we express the three-body interaction in terms of the two-atom Casimir-Polder energy,ΔE3 = E123 + E213 + E1323 = g(γ)E12,(20)THREE-BODY CASIMIR-POLDER INTERACTIONS 18700.51.0− 0.5− 1.00.2 0.4 0.6 0.8 1.0Zag(γ )three-scattering contributionfour-scattering contributiontotal three-body correctionaZα 1 α 2Fig. 2. – The three-body correction to the Casimir-Polder force between two atoms in thepresence of a perfectly conducting plate. Here it is assumed that the atoms are isotropicallypolarizable, and are equidistant from the plate. What is plotted is the ratio of the three-bodycorrection relative to the two-atom energy, plotted as a function of Z/a. The variable γ isdefined in eq. (19).where in this case E12 is given by eq. (2). Hereg(γ) = − 64(1 + 4γ)23γ3(1 + γ)4+1γ7,(21)where the last term is the four-scattering contribution. This function vanishes as γ →∞,that is, at large distances from the plate, and has a maximum negative deviation of about12%, and has the value on the plate, at Z = 0, of g(1) = 3/23 ≈ 0.13, as shown in fig. 2.(In the scalar analog considered in ref. [7], the correction on the plate is g(1) = −1, so thethree-body energy cancels the two-atom energy when the atoms touch the plate(1).) Notethat the three-scattering terms, which are always negative, dominate for Z/a > 0.16, butvery close to the plate the four-scattering term, which is always positive, causes the signof the correction to reverse. However, close to the plate, these corrections are negligiblecompared to the two-body atom-wall energy (4), becauseE12ECP=α2r3463(Zr)4 1,(22)since typical values of α ∼ (10−8 cm)3, while the separation consistent with our macro-scopic approximation is not going to be smaller than r ∼ 10−6 cm. We might note thatif the atoms and the plate are all equidistant, Z = a, the three-body effect is small, andthe four-scattering contribution is totally negligible.4.2. Anisotropically polarizable atoms, equidistant from the plate. – As a second ex-ample, we show the result considered in ref. [6], when the atoms are at equal heights,(1) g(γ) for the scalar analog reported in ref. [7] has an error of a factor of 2 in its first term.188 KIMBALL A. MILTON, E. K. ABALO, PRACHI PARASHAR and K. V. SHAJESH0123− 10.2 0.4 0.6 0.8 1.0Zag(γ )three-scattering contributionfour-scattering contributiontotal three-body correctionaZα 1z α 2zFig. 3. – The three-body correction to the Casimir-Polder force between two atoms in thepresence of a perfectly conducting plate. Here it is assumed that the atoms are completelyanisotropic, being polarizable only in the direction perpendicular to the plate, and are the samedistance from the plate. What is plotted is the ratio of the three-body correction relative to thetwo-atom energy, plotted as a function Z/a. The variable γ is again defined in eq. (19).ΔZ = 0, with a simple anisotropic polarizability,αa = diag(αa⊥, αa⊥, αaz), a = 1, 2.(23)ThenE123 =2πa71γ51(1 + γ)5[α1zα2z(−3− 15γ − 24γ2 + 10γ4 + 5γ5 + γ6)(24)+α1⊥α2⊥(3 + 15γ + 28γ2 + 20γ3 + 6γ4 − 5γ5 − γ6)] .This agrees with the result given in ref. [6], when the identical contribution of E213 isincluded.The four-scattering term may be given in a similar formE1323 = − 18πa7γ11 [α1zα2z(63− 70γ2 + 20γ4)(25)+α1⊥α2⊥(63− 56γ2 + 26γ4) + (α1⊥α2z + α1zα2⊥)63(γ2 − 1)].This is identical to the corresponding term in ref. [6]. These energies, of course, agreewith the result (21) in the isotropic case αa⊥ = αaz .We plot the contributions for the α1zα2z terms in fig. 3. That is, suppose the atomsare only polarizable in the z direction. Then the two-body Casimir-Polder interactionbetween the atoms isE12 = − 138πa7α1zα2z,(26)THREE-BODY CASIMIR-POLDER INTERACTIONS 189and we normalize the three- and four-scattering contributions relative to this value2E123E12= g3(γ),E1323E12= g4(γ), g(γ) = g3(γ) + g4(γ).(27)Again it is seen that the four-scattering contribution is significant only very close to theplate. Now the sign of the correction is negative only for Z/a > 0.485.4.3. Anisotropically polarizable atoms, unequal distances from the plate. – As a finalexample, let us assume that the atoms are only polarizable in the z direction, αaz = 0,αa⊥ = 0, but that the atoms are at different distances from the plate, ΔZ = 0. Then,with the distances between the atoms being r, and the distance between one atom andthe image of the other being R ≡ Γr, that isR2 = a2 + (Z1 + Z2)2, r2 = a2 + (Z1 − Z2)2,(28)we easily findE12 = −α1zα2z8πr7(20− 70a2r2+ 63a4r4),(29a)E123 = E213 = − 2α1zα2zπr7Γ5(1 + Γ)5[2Γ2(1 + Γ)2(1 + 3Γ + Γ2)(29b)−a2r2(1 + Γ)2(3 + 9Γ + 11Γ2 + 9Γ3 + 3Γ4)+6a4r4(1 + 5Γ + 9Γ2 + 5Γ3 + Γ4)],E1323 = − α1zα2z8πr7Γ7(20− 70 a2Γ2r2+ 63a4Γ4r4).(29c)We plot these functions versus Γ in fig. 4.The figure shows that although the four-scattering term is generally negligible com-pared to the three-scattering term, it is significant near the plate. In fact, for Γ = 1,which corresponds to one atom touching the plate, the following relation holds:E123 = E1323 = E12, r = R,(30)so the total energy is (again, we exclude the infinite two-body atom-wall energy)E = E12 + 2E123 + E1323 = 4E12(31)at that unphysical point. Nonmonotonic effects can be seen when a/r is sufficientlynear 1, or ΔZ/a sufficiently small.190 KIMBALL A. MILTON, E. K. ABALO, PRACHI PARASHAR and K. V. SHAJESHΓE-0.06-0.04-0.020.021.5 2.0 2.5 3.0three-scattering contributionfour-scattering contributiontotal three-body energytwo-body energyaZ 1 Z 2∆ Zα 1zα 2zFig. 4. – The three-body correction to the Casimir-Polder energy (in units of α1zα2z/r7) betweentwo atoms in the presence of a perfectly conducting plate. Here, it is assumed that the atomsare polarizable only in the direction perpendicular to the plate, and are at different distancesfrom the plate. Note that although for large distances, that is, for large values of Γ, the three-scattering contribution dominates, for values close to Γ = 1 the four-scattering contribution issignificant, and leads to nonmonotonicity in the slope. The energies plotted are for a/r = 0.75,or ΔZ/a ≈ 0.88, where the effect of the four-scattering term is most significant.To close this section, we consider the same geometry, but atoms that are only polar-izable transversely, αaz = 0, αa⊥ = 0. Then we immediately findE12 = −α1⊥α2⊥8πr7(26− 56a2r2+ 63a4r4),(32a)E123 = E213 =2πr7Γ5(1 + Γ)5[2Γ2(1 + 5Γ + 14Γ2 + 5Γ3 + Γ4)(32b)−a2r2(3 + 15Γ + 28Γ2 + 20Γ3 + 28Γ4 + 15Γ5 + 3Γ6)+6a4r4(1 + 5Γ + 9Γ2 + 5Γ3 + Γ4)],E1323 = − α1⊥α2⊥8πr7Γ7[26− 56 a2Γ2r2+ 63a4Γ4r4].(32c)Now the effects of the four-scattering terms are always much less dramatic. Forexample, for a/r = 0.5, in fig. 5 we plot the contributions to the correction to theCasimir-Polder energy between the atoms. The three-scattering contribution alwaysdominates, although the four-scattering term can be important near the plate. Now, atthe unphysical point Γ = 1 we haveE123 = −E1323 = −E12,(33)so the total energy is zero there. Thus, g(1) = −1 for this case, similar to the scalaranalog considered in ref. [7], where the three-body energy cancels the two-body energywhen the atoms touch the plate. Of course, this result does not include the infiniteCasimir-Polder energy (4) corresponding to the atom touching the plate.THREE-BODY CASIMIR-POLDER INTERACTIONS 191ΓE-1.0-0.50.51.01.5 2.0 2.5 3.0three-scattering contributionfour-scattering contributiontotal three-body energytwo-body energyaZ 1 Z 2∆ Zα1α 2⊥⊥Fig. 5. – Here we plot the three-body correction (in units of α1⊥α2⊥/r7) to the Casimir-Polderenergy E12 for a/r = 0.5, for atoms polarizable only in directions parallel to the plate. Nowthere are no nonmonotonic effects, and the three-scattering contribution is dominant, althoughthe four-scattering term can be significant near the plate.5. – ConclusionsWe have illustrated, in the simple context of two polarizable atoms interacting witheach other and a nearby perfectly conducting plate, the three-body interactions beyondthe two-body Casimir-Polder forces. These three-body energies break up into three-and four-scattering terms in this context, where the atom interactions are regarded asweak. This work generalizes that of ref. [6], which considered atoms equidistant fromthe plate, and that of ref. [7], which considered a scalar analog of the Casimir-Polderinteraction. We observe regions of nonmonotonicity in the energy, which is reminescentof results found earlier, numerically, in the context of rectangular objects near conductingsurfaces [15,16].These effects are of conceptual interest only, because they generally represent smallcorrections to the Casimir-Polder interaction between atoms, which itself has never beendirectly observed, due to lack of sufficiently refined experimental technique. Where thethree-body interactions are comparable to the two-body atom-atom energy, they aredwarfed by the two-body atom-wall interaction (4). The importance of this work liesin its contribution to our developing understanding of three-body effects in Casimiror quantum vacuum energy calculations. Further examples of these effects, in morenontrivial contexts, will appear elsewhere.∗ ∗ ∗This work was supported in part by grants from the US National Science Founda-tion, the US Department of Energy, and the Julian Schwinger Foundation. We thankFardin Kheirandish and Reinaldo de Melo e Souza for discussions. KVS wouldlike to thank Martin Schaden for contributions in the early stages of this work, andTom Sizmur for discussions.192 KIMBALL A. MILTON, E. K. ABALO, PRACHI PARASHAR and K. V. SHAJESHREFERENCES[1] Casimir H. B. G., Proc. Kon. Ned. Akad. Wetensch., 51 (1948) 793.[2] Casimir H. B. G. and Polder D., Phys. Rev., 73 (1948) 360.[3] Sukenik C. I., Boshier M. G., Cho D., Sundoghar V. and Hinds E. A., Phys. Rev.Lett., 70 (1993) 560.[4] Obrecht J. M., Wild R. J., Antezza M., Pitaevskii L. P., Stringari S. andCornell E. A., Phys. Rev. Lett., 98 (2007) 063201.[5] Milton K. A., Am. J. Phys., 79 (2011) 697.[6] Rodriguez-Lopez P., Rahi S. J. and Emig T., Phys. Rev. A, 80 (2009) 022519.[7] Shajesh K. V. and Schaden M., Int. J. Mod. Phys. Conf. Ser., 14 (2012) 521.[8] de Melo e Souza R., Kort-Kamp W. J. M., Sigaud C. and Farina C., Int. J.Mod. Phys. Conf. Series, 14 (2012) 281; arXiv:1204.2858; and poster presented at thePASI Frontiers of Casimir Physics Workshop, Ushuaia, Argentina, October 2012, http://physics.iupui.edu/sites/default/files/physics/deMeloeSouzaz-poster2.pdf.[9] Craig D. P. and Power E. A., Chem. Phys. Lett., 3 (1969) 195; Int. J. Quantum Chem.,3 (1969) 903.[10] Milton K. A., Parashar P. and Wagner J., Phys. Rev. Lett., 101 (2000) 160402.[11] Milton K., Parashar P., Wagner J. and Cavero-Pelaez I., J. Vac. Sci. Technol. B,28 (2010) C4A8.[12] Milton K. A., Parashar P. and Wagner J., in The Casimir Effect and Cosmology,edited byOdintsov S. D., Elizalde E. andGorbunova O. G. (Tomsk State PedagogicalUniversity Press) 2008, pp. 107–116.[13] Levine H. and Schwinger J., Comm. Pure Appl. Math. III, 4 (1950) 355, reprinted inMilton K. A. and Schwinger J., Electromagnetic Radiation (Springer, Berlin) 2006,pp. 543–579.[14] Shajesh K. V. and Schaden M., Phys. Rev. D, 83 (2011) 125032.[15] Zaheer S., Rodriguez A. W., Johnson S. G. and Jaffe R. L., Phys. Rev. A, 76 (2007)063816.[16] Rodriguez A., Ibanescu M., Iannuzzi D., Capasso F., Joannopoulos J. D. andJohnson S. G., Phys. Rev. Lett., 99 (2007) 080401.

Three-body Casimir-Polder interactions

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Three-body Casimir-Polder interactions

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