Casimir-Polder interaction in the presence of parallel walls

Transcription

1 Csimir-Polder interction in the presence of prllel wlls rxiv:qunt-ph/2v 6 Nov 2 F C Sntos, J. J. Pssos Sobrinho nd A. C. Tort Instituto de Físic Universidde Federl do Rio de Jneiro Cidde Universitári - Ilh do Fundão - Cix Postl Rio de Jneiro RJ, Brsil. October 29, 28 Abstrct Mking use of the quntum correltors ssocited with the Mxwell field vcuum distorted by the presence of plne prllel mteril surfces we derive the Csimir-Polder in the presence of plne prllel conducting wlls nd in the presence of conducting wll nd mgneticlly permeble one. PACS:.. -z; m e-mil: fildelf@if.ufrj.br e-mil:tort@if.ufrj.br

2 In 948, Csimir nd Polder [ tking into ccount suggestion mde by experimentlists evluted the interction potentil between two eletricl polrizble molecules seprted by distnce r including the effects due to the finiteness of the speed of propgtion of the electromgnetic interction, i.e.: of the retrdment. Csimir nd Polder showed tht the retrdment cuses the interction potentil to chnge from r 6 power lw to r 7 power lw. In the sme pper, Csimir nd Polder lso nlyzed the retrded interction between n tom nd conducting wll nd showed tht the interction potentil in this cse vries ccording to r 4 power lw, where now r is the distnce between the tom nd the wll. For n introduction on these subjects see [2. Here we wish to show how it is possible with the help of the so clled renormlized electromgnetic field correltors, in our cse the ones tht tke into ccount the presence of the boundry conditions imposed on the fields, to reobtin the piece of Csimir nd Polder s result for the tom-wll interction tht depends on the distortion of the vcuum oscilltions of the electromgnetic field cused by the presence of prllel wlls. The electromgnetic field correltors for the cse of two prllel perfectly conducting surfces seprted by distnce were evluted in [ nd in [4. For the cse of perfectly conducting plne wll nd perfectly permeble plne wll, setup first introduced by Boyer [5, they were clculted in [4. These mthemticl objects, closely relted to the pertinent electromgnetic Green s functions, were lso employed to obtin n lterntive view of the Csimir effect [6 through the quntum version of the Lorentz force between the wlls [7. Let us first recll some spects concerning electriclly nd mgneticlly polrizble bodies [8. From clssicl point of view the induced eletricl polriztion density P cn be thought of s function of the electric nd mgnetic fields E nd B. In mny cses only the dependence on the eletric field is relevnt. It cn be shown tht under conditions for which the effects of the retrdment (i.e., of the finiteness of the speed of light must be tken into ccount it suffices to consider the sttic eletricl polrizbility α( only, see for instnce [2 nd references therein. If the electric field chnges by δe, the interction between the polrizble body nd the electric field will chnge ccording to δv = P[E δe = α(e δe. Therefore, if the field chnges from zero to finite vlue E, the interction energy is V E = α(e 2 /2. In the quntum version of this interction potentil we must replce E 2 by its vcuum expecttion vlue, Ê2. The sme rguments hold when we consider the mgnetiztion M. The quntum interction potentil between mgneticlly polrizble tom nd the mgnetic field is given by V M = β( B 2 /2, where β( is the sttic mgnetic polrizbility. In order to proceed we must know the vcuum expecttion vlues of the quntum field opertors E 2 nd B 2. This mens to evlute explicitly the vcuum expecttion vlues of the so clled electromgnetic field correltors E i (r,te j (r,t, B i (r,tb j (r,t, nd E i (r,tb j (r,t, in the presence of externl conditions, i.e., boundry conditions. A regulriztion recipe will lso be necessry. Fortuntely these objects were clculted before nd we cn limit ourselves to mke use of the results. For the cse of two prllel conducting wlls seprted by fixed distnce we hve [, 4 E i (r,te j (r,t = π [ ( δ +δ ij 2 +δ ijf(ξ, ( where δ ij := δ ixδ jx +δ iy δ jy nd δ ij := δ izδ jz. The function F (ξ with ξ := πz/ is defined by F (ξ := 8 d cot(ξ, (2 dξ 2 2

3 nd its expnsion bout ξ = is given by F (ξ 8 ξ O( ξ 2. ( Ner ξ = π (which corresponds to z = we mke the replcement ξ ξ π. Notice tht due to the behvior of F (ξ ner ξ =,π, divergences control the behvior of the correltors ner the pltes. The mgnetic field correltors re [, 4 B i (r,tb j (r,t = π [ ( δ +δ ij 2 δ ijf(ξ. (4 A direct evlution shows tht the correltors < E i (r,tb j (r,t re zero. For the cse of perfectly conducting plne wll nd mgneticlly permeble one results re [4 nd Êi (r,têj(r,t ˆBi (r,t ˆB j (r,t = = π π Observe tht ner ξ = the function G(ξ behves s G(ξ = 8 ξ ner ξ = π, however, its behvior is slightly different [ ( 7 ( δ +δ ij 8 2 [ ( 7 ( δ +δ ij 8 2 +δ ij G(ξ δ ij G(ξ, (5. (6 2 +O( ξ 2, (7 G(ξ = 8 (ξ π O[ (ξ π 2. (8 Agin, direct clcultion shows tht Êi (r,t ˆB j (r,t = for this cse lso. As before the divergent behvior of the correltors ner the pltes we re interested in is n effect of the distortions of the electromgnetic oscilltions with respect to sitution where the pltes re not present. The correltors given by (, (4, (5, nd (6 llow us to obtin in strightforwrd wy expressions for the interction potentil energy between n electriclly or mgneticlly polrizble tom plced between the wlls nd the wlls. Let us consider first the cse of n electriclly polrizble tom or molecule plced between two perfectly conducting prllel wlls. Suppose tht the tom is plced t distnce z from the conducting wll plced t z =. The interction potentil between the tom nd the wlls is given by V E (z = Ê2 2 α( (z, (9 where α( is the sttic polrizbility of the molecule. Mking use of ( we cn evlute Ê2 (z nd using the bove eqution we obtin V E (z = α(π 4 [ F z. ( 2

4 Mking use of ( nd tking the limit we obtin the single wll limit of the interction potentil between n electriclly polrizble tom nd conducting wll, V E (z = α( 8πz 4, ( in greement with[9, ; see lso[2. Consider now mgneticlly polrizble tom or molecule plced between the two conducting wlls. The interction potentil in this cse will be given by V M (z = + β(π 4 [ F z +, (2 2 where we mde use of (4. If the tom or molecule is simultneously electriclly nd mgneticlly polrizble the interction potentil will be simply V (z = V E (z + V M (z, tht is ( V (z = (α( β( π πz π 4F +(α(+β( 64. ( The single conducting wll limit ( of ( is esily obtined with the help of (. The result is: V (z (α( β(, (4 8πz4 which is in greement with [9,. The polrizble tom or molecule cn be lso plced between conducting plte t z = nd permeble one t z =. In this cse, mking use of (5 e (6 strightforwrd clcultion leds to the following result ( V (z = (α( β( π πz ( 4G +(α(+β( 7 π (5 There re now two single wlls limits to be considered. Ner the conducting plte t z = the potentil is given by (4, but ner the perfectly permeble plte t z =, the potentil is repulsive nd given by V (z + 4 (α( β(, (6 8π(z where we mde use of (8. These lst results re to our knowledge new. It is importnt to keep in mind tht, s mentioned before, we hve delt with prt of the interction between n tom nd two or one wlls. The contribution of the interction between the electric/mgnetic dipole moment nd its imges ws utterly neglected. Therefore, the results refer only to the contribution of the quntum vcuum distorted by one or two wlls to the totl interction potentil. With this proviso we cn stte tht the Csimir-Polder interction shows certin spects of the quntum structure of the vcuum confined between the plne surfces in question. The tom cts s probe of the confined quntum vcuum, prticulrly ner the wlls. 4

5 References [ H.B.G. Csimir nd D. Polder, Phys. Rev 7, 6 (948. [2 P.W. Milonni, The Quntum Vccum: An introduction to Quntum Electrodynmics,(Acdemic Press, New York, 994. [ C.A. Lütken nd F. Rvndl, Phys. Rev. A, 282 (985; see lso: G. Brton, Phys. Lett. B 27, 559 (99; M. Bordg, D. Robschik nd E. Wieczorek, Ann. Phys. (NY 65, 92 (985. [4 M. V. Cougo-Pinto, C. Frin, F. C. Sntos nd A. C. Tort, J. of Phys. A 2 (999, 446. [5 T.H. Boyer Phys. Rev A 9, 278 (974. [6 H. B. G. Csimir, Proc. K. Ned. Akd. Wet. 5, 79 (948; Philips Res.Rep (95. [7 C. Frin, F. C. Sntos nd A.C. Tort, Eur. J. Phys. 24, N-N5 (2. [8 J. D. Jckson, Clssicl Electrodynmics, rd. ed., (John Wiley, New York 999. [9 H.B.G. Csimir, J. Chim. Phys. 46, 47 (948. [ T.H. Boyer, Phys. Rev 8, 9 (969. 5