Casimir force in presence of multi layer magnetodielectric slabs

Annals of Physics 326 (2011) 657–667 Contents lists available at ScienceDirect Annals of Physics journal homepage: www.elsevier.com/locate/aop Casimir force in presence of multi layer magnetodielectric slabs Fardin Kheirandish ⇑, Morteza Soltani, Jalal Sarabadani Department of Physics, University of Isfahan, Isfahan 81746, Iran a r t i c l e i n f o Article history: Received 4 July 2010 Accepted 27 December 2010 Available online 31 December 2010 Keywords: Field quantization Path-integral Susceptibility Casimir force Magnetodielectric medium a b s t r a c t By using the path-integral formalism, electromagnetic field in the presence of some linear, isotropic magnetodielectric slabs is quantized and related correlation functions are found. In the framework of path-integral techniques, Casimir force between two infinitely large, parallel and ideal conductors, with a different number of magnetodielectric slabs in between, is obtained by calculating the Green’s function corresponding to each geometry. Ó 2011 Elsevier Inc. All rights reserved. 1. Introduction One of the most direct manifestations of the zero-point vacuum oscillations is the Casimir effect. This effect in its simplest form, is the attraction force between two neutral, infinitely large, parallel and ideal conductors in vacuum. The effect is completely quantum mechanical and is a result of electromagnetic field quantization in the presence of some boundary conditions. The presence or absence of boundary conditions cause a finite change of vacuum-energy which its variation with respect to the distance between the conductors gives the Casimir force [1]. The Casimir force is not necessarily attractive and there are some situations where the Casimir force is repulsive [2,3]. There are usually some main approaches to calculate the Casimir effect. In the first approach, which is the most appropriate when we deal with geometrically regular metallic boundaries, the energy difference in the presence and absence of some boundary conditions is found and the Casimir force is determined from variation of the energy difference with respect to a relevant parameter [4–7]. The second approach, is based on the pressure of quantum-vacuum [9] which is determined from ⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (F. Kheirandish), [email protected] (M. Soltani), [email protected] (J. Sarabadani). 0003-4916/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.aop.2010.12.007 658 F. Kheirandish et al. / Annals of Physics 326 (2011) 657–667 the energy–momentum tensor of the electromagnetic field in the presence of some geometrically regular dielectrics which also can be considered as a Green’s function approach [8–15]. For a review of these methods see [11,12] and for recent works [16–22]. For a very recent review on the long range interactions in the nanoscale see [23] and for a review on Casimir effect [24]. This approach is used to calculate the Casimir force between some dielectric or magnetodielectric slabs [8–22]. Also the Lifshitz interaction at finite temperature for a multilayered system has been studied [25] which in the limiting case of zero temperature, i.e. T ? 0 tends to our results as expected. The third approach is based on the powerful path-integral techniques which is the most effective technique for dealing with general geometries, see for example [26–31]. To calculate the vacuum-energy one can use either the canonical quantization scheme [32–35] or the path-integral method [26–31]. There are some situations where the path-integral method are more effective than the other methods. In these situations, the constraints imposed by the boundary conditions on the fields can be easily included into the process of quantization through a modified action. An interested reader is referred to the calculations of the normal and lateral Casimir force between two rough plates [26–29] and for the dynamic Casimir effect to [31]. Fig. 1. One magnetodielectric slab. Fig. 2. Two magnetodielectric slab. F. Kheirandish et al. / Annals of Physics 326 (2011) 657–667 659 Fig. 3. Three magnetodielectric slab. Here we have two conductors immersed in a magnetodielectric medium. Therefore first of all we should quantize the electromagnetic field in the presence of a magnetodielectric medium. This means that the medium should be included in the process of quantization, so it must be modeled by a Lagrangian density and added to the lagrangian density of the electromagnetic field, for details the interested reader is referred to [32–36]. Here we follow the path-integral approach and extend our previous work [36] to find the Casimir force in the presence of a multi-layer magnetodielectric slab (Figs. 1–3). The present approach is simple and systematic which is based on a Lagrangian and can be generalized to dynamical Casimir effects in the presence of a magnetodielectric medium and also some more complicated geometries with corrugated surfaces [37] which is under consideration. 2. Field quantization The Lagrangian of electromagnetic field in a magnetodielectric medium can be written as [35] L ¼ LEM þ L1mat þ L2mat þ Lint ; ð1Þ where LEM is LEM ¼ e0 E2 2  B2 : 2l0 ð2Þ In terms of scaler and vector potentials we have E ¼ A_  rU and B = r  A. Here we use a gauge in which U = 0. The magnetodielectric medium can be modeled in terms of two independent vector fields Xi (i = 1, 2) describing the electric and magnetic properties of the medium. Therefore the Lagrangian densities describing the medium can be written as 1 Z Limat ¼ dx 0   1 _2 1 ði ¼ 1; 2Þ: Xix  x2 X2ix 2 2 ð3Þ Now we define the polarization and magnetization fields as P¼ Z M¼ 1 dxm1 ðr; xÞX1x ; Z0 ð4Þ 1 dxm2 ðr; xÞX2x 0 and by using these definitions the interaction part of Lagrangian can be written as Lint ¼ A  P_ þ r  A  M: ð5Þ 660 F. Kheirandish et al. / Annals of Physics 326 (2011) 657–667 Generally a generating function is defined by Ryder [38] Z½J ¼ Z  Z  3þ1 D½u exp ı d x½LðuðxÞÞ þ JðxÞuðxÞ ð6Þ and by taking the repeated functional derivatives with respect to the source field J(x) we find the different correlation functions. To obtain the generating functional for the interacting fields, we first calculate the generating functional for the free fields Z Z 0 ½J0 ; fJ1x g;fJ2x g ¼ Z  Z 3þ1 D½AD½X1x D½X2x   exp ı d x½LEM þ L1x þ L2x þ J0  A þ dxðJ1x  X1x þ J2x  X2x Þ : ð7Þ Using the 3 + 1-dimensional version of Gauss’s theorem we find Z d 3þ1 xLem ¼  Z d 3þ1   x A  ð5  5  AÞ  A  @ 2t A : ð8Þ Also using integration by parts we find Z d 3þ1 xX_ ix  X_ ix ¼  Z 3þ1 d xXix  @2 X ix : @t2 ð9Þ From Eqs. (8) and (9), the free generating functional (7) can be written as Z 0 ½J0 ; J1x ; J2x  ¼  Z ı 3þ1  D½AD½X1x D½X2x  exp  d x A  5  5  A  A  @ 2t A 2 ! ! Z @2 @2 2 2 þ dxX1x ðxÞ þ x X1x þ X2x ðxÞ þ qx X2x ðxÞ þ J0 ðxÞ  AðxÞ @t 2 @t2  Z þ dxðJ1x  X1x ðxÞ þ J2x  X2x ðxÞÞ : Z ð10Þ The integral in Eq. (10) can be easily calculated from the field version of the quadratic integral formula and the result is   $ Z Z ı 3þ1 3þ1 d x d x0 J0  G0 ðx  x0 Þ  J0 Z 0 ½J0 ; J1x ; J2x  ¼ exp  2   Z $ $ þ dx J1x ðxÞ  G1x ðx  x0 Þ  J1x ðx0 Þ þ J2x ðxÞ  G2x ðx  x0 Þ  J2x ðx0 Þ ; $ ð11Þ $ where Gix ðx  x0 Þ and G0 ðx  x0 Þ are the propagators for free fields and satisfy the following equations and  $ $ 5  5  @ 2t G0 ðx  x0 Þ ¼ d ðx  x0 Þ ( ) $ @2 2 þ x Gixa;b ðx  x0 Þ ¼ dðx  x0 Þda;b : @t 2 ð12Þ ð13Þ Eq. (13) can be solved using the Fourier transformation and the solution is $ Gixab ðx  x0 Þ ¼ 1 2p Z 0 dx0 0 eıx ðtt Þ dðx  x0 Þdab : x2  x02 ð14Þ Here the space component of the point x 2 R3þ1 is indicated in bold by x 2 R3 and the time component by t or x0 2 R. For further use we define F. Kheirandish et al. / Annals of Physics 326 (2011) 657–667 661 JP ¼ Z dxmðx; xÞJ1x ; ð15Þ JM ¼ Z dxmðx; xÞJ2x : ð16Þ and The generating functional of the interacting fields can be written in terms of the free generating functional as [38] ı 1 R d3þ1 zLint d ; d ; d dJu ðzÞ dJP ðzÞ dJM ðzÞ Z ½J0 ; JP ; JM  ¼ Z ½0e Z 0 ½J0 ; J1x ; J2x   Z n 1 X 1 d @ d d d 3þ1 1 Z 0 ½J 0 ; J x ; ð17Þ ı d z  þ5  ¼ Z ½0 n! dJEM ðzÞ @z0 dJ1x ðzÞ dJEM ðzÞ dJ1x ðzÞ n¼0 from which the Green’s function of the electromagnetic field can be obtained as  d2 Z½J  Gab ðx  yÞ ¼ i  dJ a ðxÞdJ b ðyÞ $ ð18Þ : J¼0 To proceed with the calculations we take the time-Fourier-transform of the Green’s function (18). After some manipulations we find $ Z $ 0 dx1 dx2 G0 ðx  x1 ; xÞ dx0 m21 ðx1 ; x0 Þx2 Gx ðx1  x2 Þ   Z $ $  GEM ðx2  x0 ; xÞ þ dx1 dx2 5  G0 ðx  x1 ; xÞ  Z $ $ 0  dx0 m22 ðx1 ; x0 ÞGx ðx1  x2 Þ 5 GEM ðx2  x0 ; xÞ ð19Þ $ $ GEM ðx  x0 ; xÞ ¼ G0 ðx  x0 ; xÞ þ Z and one can show that this Green’s function satisfies the following equation $ $ $ r  ½1  vM ðr; xÞr  G ðr; r0 ; xÞ  x2 1 þ vP ðr; xÞ G ðr; r0 ; xÞ ¼ d ðr; r0 Þ;  where þ1 vP ðr; xÞ  Z vM ðr; xÞ  Z ð20Þ dx0 m21 ðr; x0 Þ x  x0  ıe ð21Þ dx0 m22 ðr; x0 Þ ; x  x0  ıe ð22Þ 1 and þ1 1 are electric and magnetic susceptibilities respectively, and lðr;1xÞ ¼ 1  vM ðr; xÞ. In a similar way we can find the following correlation functions among different fields $ $ GEM;P ðr; r0 ; xÞ ¼ ixvP ðr; xÞGEM ðr; r0 ; xÞ; $ $ GEM;M ðr; r0 ; xÞ ¼ vM ðr; xÞr  GEM ðr; r0 ; xÞ; $ GP;P ðr; r0 ; xÞ ¼ $ GM;M ðr; r0 xÞ ¼ ð23Þ ð24Þ $ m21 ðr; xÞ  x2 v2P ðxÞGEM ðr; r0 ; xÞ; x ð25Þ $ m22 ðr; xÞ 2 ~ r  GEM ðr; r0 ; xÞ  rr0 : þ vM ðr; xÞr x ð26Þ 662 F. Kheirandish et al. / Annals of Physics 326 (2011) 657–667 Eqs. (23) and (24) show that vP and vM are electric and magnetic susceptibilities, respectively. Also Eqs. (25) and (26) show that the path integral formalism like other quantization schemes satisfy the fluctuation–dissipation theorem [13,32,35]. In the next section we use the path integral method to calculate the Casimir energy in presence of different layers of magnetodielectric slabs (as shown in Figs. 1–3). The calculation is based on the Green’s function introduced in Eq. (19). To simplify the problem we separate the transverse electric (TE) and magnetic (TM) of the Green’s function. The TM and TE modes should satisfy the Dirichlet and Neumann boundary conditions, respectively. The longitudinal part of the Green’s function does not lead to any force since this part of the Green’s function is local and as we have proven in our pervious work the local fields do not lead to any Casimir force, so we only consider the TE and TM parts. The TM and TE modes can be represented by a scaler field satisfying the following equation f½1  vM ðr; xÞr2  x2 ½1 þ vP ðr; xÞgu ¼ 0: ð27Þ In a medium and on the boundaries, using the boundary condition satisfied by electric and magnetic fields, it can be shown that e(x)u and @ zu are continues for a TM mode and u and lð1xÞ @ z u should be continues for a TE modes. Therefore in what follows instead of solving the complicated Eq. (19) we calculate the Green’s function of Eq. (27) with given boundary conditions and knowing the Green’s function we can obtain the partition function of the electromagnetic field. 3. Casimir force In this Section we calculate the Casimir energy and force for the system illustrated by Figs. 1–3. For this purpose we consider the electromagnetic field as TM and TE wave modes represented by scalar fields which satisfy the Dirichlet uðX a Þ ¼ 0; ð28Þ or Neumann @ n uðX a Þ ¼ 0; ð29Þ boundary conditions where Xa, (a = 1, 2) is an arbitrary point on the conducting plates. To obtain the partition function from the Lagrangian we use the Wick’s rotation, (t ! ıs) and change the signature of space–time from Minkowski to Euclidean. The Dirichlet or Neumann boundary conditions can be represented by auxiliary fields wa(Xa) in field version or extended definition of Fourier transform [26] dðuðX a ÞÞ ¼ Z D½wa ðX a Þeı and dð@ n uðX a ÞÞ ¼ Z R dX a wðX a ÞuðX a Þ D½wa ðX a Þeı R ð30Þ dX a @ n wðX a ÞuðX a Þ ð31Þ : Using Eqs. (30) and (31) and Wick’s rotation the partition function for Dirichlet and Neumann BBCs can be written as Z D ¼ Z 1 0 Z D½u Z D½u 2 Y D½wa ðX a ÞÞeSD ½u ð32Þ D½wa ðX a ÞÞeSN ½u ; ð33Þ a¼1 and Z N ¼ Z 1 0 2 Y a¼1 respectively where SD(u) and SN(u) are defined by SD ½u ¼ Z 3þ1 d ( x LðuðxÞÞ þ uðxÞ 2 Z X a¼1 3 d XdðX  X a Þwa ðxÞ ) ð34Þ F. Kheirandish et al. / Annals of Physics 326 (2011) 657–667 663 and SN ½u ¼ Z d 3þ1 ( x LðuðxÞÞ þ uðxÞ 2 Z X ) 3 d XdðX  X a Þ@ n wa ðxÞ : a¼1 ð35Þ By comparing Eqs. (34), (35) and (6), we can rewrite Eqs. (34) and (35) as ZD ¼ Z Y 2 D½wa ðxÞZ a¼1 2 Z X ! 3 d XdðX  X a Þwa ðXÞ a¼1 ð36Þ and ZN ¼ Z Y 2 D½wa ðxÞZ a¼1 where Z P2 a¼1 R 2 Z X ! 3 d XdðX  X a Þ@ n wa ðXÞ ; a¼1 3 d X a dðX  X a Þwa ðXÞ and Z P2 a¼1 R ð37Þ 3 d X a dðX  X a Þ@ n wa ðXÞ are the generating func- tionals of interacting fields that have been defined in Eq. (17) with imaginary time. From Eqs. (36), (37) and (18) the relevant partition functions can be written as ZD ¼ Z Y 2 D½wa ðX a ÞeıSD ðw1 ;w2 Þ ð38Þ Z Y 2 D½wa ðX a ÞeıSN ðw1 ;w2 Þ ; ð39Þ a¼1 and ZN ¼ a¼1 where ıSD ðw1 ; w2 Þ ¼ ıJD GJ D ; ð40Þ ıSN ðw1 ; w2 Þ ¼ ıJN GJ N ; ð41Þ JD(X) and JN(X) are respectively defined by J D ðXÞ ¼ Z d XdðX  X a Þwa ðXÞ J N ðXÞ ¼ Z d XdðX  X a Þ@ n wa ðXÞ: 3 ð42Þ 3 ð43Þ and Now we define briefly G. The partition functions which have been defined by Eq. (38) and (39) are calculated straightforwardly [39] 1 Z D ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; det CD ðx; y; HÞ where ð44Þ 1 Z N ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; det CN ðx; y; HÞ CD ðx; y; HÞ ¼ Gðx  y; z1 ; z1 Þ Gðx  y; z2 ; z1 Þ Gðx  y; z1 ; z2 Þ Gðx  y; z2 ; z2 Þ " CN ðx; y; HÞ ¼  ð45Þ ð46Þ ; @ 2z Gðx  y; z1 ; z1 Þ @ 2z Gðx  y; z2 ; z1 Þ @ 2z Gðx  y; z1 ; z2 Þ @ 2z Gðx  y; z2 ; z2 Þ # : ð47Þ 664 F. Kheirandish et al. / Annals of Physics 326 (2011) 657–667 Here the parameter H is the distance between boundaries and the spatial components of x and y live on the boundaries. In order to calculate the Casimir force we define the effective action as Seff ¼ ı ln Z D ½H; ð48Þ from which the Casimir force can be obtained easily as F¼ @Seff ðHÞ : @H ð49Þ As can be seen from Figs. 1–3, the geometry of the system under consideration the Green’s function in Fourier-space can be diagonalized thus we have 3 lnfdet CD ðx; y; HÞg ¼ Z    Gðq; z1 ; z1 Þ Gðq; z2 ; z1 Þ   ln  Gðq; z1 ; z2 Þ Gðq; z2 ; z2 Þ  ð2pÞ lnfdet CN ðx; y; HÞg ¼ Z    @ 2 Gðq; z ; z Þ @ 2 Gðq; z ; z Þ  1 1 1 2   z z ln  ; ð2pÞ3  @ 2z Gðq; z2 ; z1 Þ @ 2z Gðq; z2 ; z2 Þ  and d q 3 ð50Þ 3 d q ð51Þ where Gðq; z1 ; z1 Þ is the Fourier transformation of Gðx  y; z1 ; z2 Þ. Then to calculate the Casimir force we should obtain the Green’s function of Eq. (27) with the introduced boundary conditions. In what follows we calculate the Green’s function for three different geometries which are depicted in Figs. 1–3. 3.1. Homogeneous magnetodielectric medium To calculate the Casimir energy for the geometry of Fig. 1 we can calculate the Green’s function of Eq. (27). If we take the Fourier transformation of the same equation we find ! 2 d  Q Gðq; z; z0 Þ ¼ dðz  z0 Þ; dz2 ð52Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eðiq0 Þlðiq0 Þq20 þ q21 þ q22 : ð53Þ where Q¼ By using the following relation ! 0 2 d eiajzz j 2 þ a ¼ dðz  z0 Þ dz2 2ia ð54Þ and taking a ¼ ıQ, the function Gðq; z; z0 Þ is obtained as 0 Gðq; z; z0 Þ ¼ eQjzz j : 2Q ð55Þ Now by using Eqs. (50) and (51) the effective action can be read as S N ¼ SD ¼ Z 3 d q ð2pÞ3  ln 1  e2QH ð56Þ and consequently the Casimir force is F c ¼ 2 Z 3 d q ð2pÞ 3 Q ; e2QH  1 this result coincides with the result obtained in Ref. [36]. ð57Þ F. Kheirandish et al. / Annals of Physics 326 (2011) 657–667 665 3.2. Two joined magnetodielectric slabs For the geometry of the system in Fig. 2 we should find the Green’s function of Eq. (27) in regions I and II, see Fig. 2. Note that for TM and TE modes we should also consider the homogeneous solution of Eq. (27) in order to satisfy the boundary conditions. Then for the region z0 < 0 we have 0 Gðq; z; z Þ ¼ ( eQ1 jzz 2Q1 ( eQ2 jzz 2Q2 0j þ R1 eQ1 z Q2 z T1e z < 0; z>0 ð58Þ and for z0 > 0 0 Gðq; z; z Þ ¼ 0j þ R2 eQ2 z Q1 z T2e z > 0; z < 0: ð59Þ For TM wave modes if we apply the boundary conditions then the variables R1, T1, R2 and T2 can be determined as 0 eQ1 z TM D ; Q1 12 e2 ðiq0 ÞeQ2 z0 ; T1 ¼ e2 ðiq0 ÞQ1 þ e1 ðiq0 ÞQ2 R1 ¼ 0 eQ2 z TM D ; Q2 21 e1 ðiq0 ÞeQ1 z0 T2 ¼ : e1 ðiq0 ÞQ2 þ e2 ðiq0 ÞQ1 ð60Þ R2 ¼ Similarly for TE wave modes we have 0 eQ1 z TE D ; Q1 12 e2 ðiq0 ÞeQ2 z0 T1 ¼ ; l2 ðiq0 ÞQ1 þ l1 ðiq0 ÞQ2 R1 ¼ 0 eQ2 z TE D ; Q2 21 e1 ðiq0 ÞeQ1 z0 ; T2 ¼ l1 ðiq0 ÞQ2 þ l2 ðiq0 ÞQ1 ð61Þ R2 ¼ where ei ðiq0 ÞQj  ej ðiq0 ÞQi ; ei ðiq0 ÞQj þ ej ðiq0 ÞQi li ðiq0 ÞQj  lj ðiq0 ÞQi DTE ij ¼ li ðiq0 ÞQj þ lj ðiq0 ÞQi DTM ij ¼ ð62Þ and i, j = 1, 2. Using Eqs. (50) and (51) we can easily show that effective action (Casimir energy) for the geometry which is illustrated in Fig. 2 can be obtained as TEðTMÞ Seff ¼ Z 3 d q ð2pÞ 3 h i TEðTMÞ TEðTMÞ ln 1  D12 e2Q1 d1 þ D12 e2Q2 d2  e2ðQ1 d1 þQ2 d2 Þ : ð63Þ 3.3. Three joined magnetodielectric slabs For the system composed of three joined magnetodielectric slabs which is depicted in Fig. 3 the Green’s function is written as follows. If z0 < d 666 F. Kheirandish et al. / Annals of Physics 326 (2011) 657–667 8 0 eQ1 jzz j Q1z > > < 2Q þ R1 e 0 Gðq; z; z Þ ¼ AeQ2 z þ BeQ2 z > > : T 1 eQ3 z z < d; 8 0 eQ3 jzz j Q3 z > > < 2Q3 þ R2 e 0 0 0 Q z Q Gðq; z; z Þ ¼ A e 2 þ B e 2 z > > : T 2 eQ1 z z > d; ð64Þ jzj < d; z>d 0 and for region z > d ð65Þ jzj < d; z < d: For TE and TM wave modes the Green’s function should satisfy the corresponding boundary conditions on the interfaces of the slabs. For TM wave modes we have T2 ¼ e1 ðiq0 Þe2 ðiq0 ÞeQ3 z ¼ T1 e2 ðiq0 Þe3 ðiq0 ÞeQ1 z edðQ1 þ2Q2 þQ3 Þ TM 4dQ2 ðe ðiq ÞQ þ e ðiq ÞQ Þðe ðiq ÞQ þ e ðiq ÞQ Þ DTM 2 1 2 3 1 2 3 2 0 0 0 0 12 D23  e ; ð66Þ Q 1 R1 eQ1 ð2dzÞ ¼ TM 4Q2 d DTM 12  D23 e TM TM D12 D23 þ e4Q2 d ð67Þ Q 3 R2 eQ3 ð2dþzÞ ¼ TE 4Q2 d DTE 23  D12 e : TE TE D12 D23 þ e4Q2 d ð68Þ and TE To obtain the Green’s function for TE wave modes we should replace DTM 12 with D12 and e1(iq0) with l1(iq0). Using Eqs. (50) and (51) and after some calculations we can read the effective action for TM wave modes as STM eff ¼ 3 d q Z ln nh 2Q1 d1 1  DTM 12 e ih 2Q3 d2 1  DTM 32 e ð2pÞ3 h ih i o 4Q2 d  e2Q1 d1  DTM e2Q3 d2  DTM 12 32 e i ð69Þ TE and the effective action for TE wave modes can be obtained by replacing DTM 12 with D12 . This result is in agreement with the results of [7]. 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