Casimir-Lifshitz force out of thermal equilibrium

Casimir-Lifshitz force out of thermal equilibrium

Mauro Antezza,1,

*

Lev P. Pitaevskii,1,2Sandro Stringari,1and Vitaly B. Svetovoy3 1_{Dipartimento di Fisica, Università di Trento and CNR-INFM R&D Center on Bose-Einstein Condensation,}

Via Sommarive 14, I-38050 Povo, Trento, Italy 2

Kapitza Institute for Physical Problems, ul. Kosygina 2, 119334 Moscow, Russia 3

MESA⫹ Research Institute, University of Twente, PO 217, 7500 AE Enschede, The Netherlands

共Received 13 June 2007; revised manuscript received 8 November 2007; published 5 February 2008兲

We study the Casimir-Lifshitz interaction out of thermal equilibrium, when the interacting objects are at different temperatures. The analysis is focused on the surface-surface, surface-rarefied body, and surface-atom configurations. A systematic investigation of the contributions to the force coming from the propagating and evanescent components of the electromagnetic radiation is performed. The large distance behaviors of such interactions is discussed, and both analytical and numerical results are compared with the equilibrium ones. A detailed analysis of the crossing between the surface-surface and the surface-rarefied body, and finally the surface-atom force is shown, and a complete derivation and discussion of the recently predicted nonadditivity effects and asymptotic behaviors is presented.

DOI:10.1103/PhysRevA.77.022901 PACS number共s兲: 34.35.⫹a, 12.20.⫺m, 37.10.Vz, 42.50.Nn

I. INTRODUCTION

The Casimir-Lifshitz force is a dispersion interaction of electromagnetic origin acting between neutral dispersive bodies without permanent polarizations. The original Ca-simir intuition about the presence of such a force between two parallel ideal mirrors 关1兴 共or between an atom and a mirror, i.e., the so-called Casimir-Polder force 关2兴兲 was readily extended to real materials by Lifshitz关3–5兴. He used the theory of electromagnetic fluctuations developed by Ry-tov关6兴 to formulate the most general theory of the dispersion interaction in the framework of the statistical physics and macroscopic electrodynamics 共see also 关7兴兲. The Lifshitz theory is still the most advanced one; today it is extensively accepted providing a common tool to deal with dispersive forces in different fields of science共physics, biology, chem-istry兲 and technology.

It is useful to stress here that the geometry of the system is relevant for the explicit calculation of the force, but does not affect the nature of the interaction that preserves all its peculiar characteristics and relevant length scales. For this reason we refer to the Casimir-Lifshitz force for all geo-metrical configurations. In particular, in this paper we are interested in the force between flat and parallel surfaces of two macroscopic bodies, and between a surface and an indi-vidual atom.

The Lifshitz theory is formulated for systems at thermal equilibrium. In this theory the pure quantum effect at T = 0 is clearly separated from the finite temperature effect. The former gives a dominant contribution at small separation 共⬍1␮m at room temperature兲 between the bodies and was readily confirmed experimentally with good accuracy (see 关8兴 共surface-atom兲, 关9–12兴 共surface-sphere兲, and 关13兴 共surface-surface兲).

The thermal component prevails at larger distances and was measured only recently at JILA in experiments with cold atoms 关16兴. These experiments are based on the

measure-ment of the shift of the collective oscillations of a Bose-Einstein condensate共BEC兲 of trapped atoms close to a sur-face关14,15兴. The JILA group measured the Casimir-Lifshitz force at very large distances共⬃10␮m兲 and showed the ther-mal effects of the Casimir-Lifshitz interaction共and indeed of any dispersion interaction兲, in agreement with the theoretical predictions关17兴. This measurement was done out of thermal equilibrium关18兴, where thermal effects are stronger.

There was an interest in configurations out of thermal equilibrium since the work by Rosenkrans et al.关19兴 共atom-atom兲. Surface-atom interaction was analyzed by Henkel et al. 关20兴 and by Antezza et al. 关17,21–24兴. Surface-surface force was investigated by Dorofeyev et al.关25,26兴 and An-tezza et al.关23,24兴. For a review of nonequilibrium effects, see also关27兴.

Further nonequilibrium effects were explored by Polder and Van Hove 关28兴, who calculated the heat-flux between two parallel plates, and Bimonte 关29兴, who expressed fluc-tuations of fields for the metal-metal configuration in terms of surface impedance.

The principal interest in the study of systems out of ther-mal equilibrium is connected to the possibility of tuning the interaction in both strength and sign 关17,23兴. Such systems also give a way to explore the role of thermal fluctuations, usually masked at thermal equilibrium by the T = 0 compo-nent which dominates the interaction up to very large dis-tances, where the actual total force results to be very small. A crucial role in explaining the peculiarity of the nonequi-librium surface-atom force is played by cancellation effects between the fluctuations of the different components of the radiations, as the incident to and emitted by the surface关17兴. In this paper we present a detailed study of the Casimir-Lifshitz force out of thermal equilibrium, with particular at-tention devoted to the surface-surface and surface-atom in-teractions. We perform a systematic investigation of the contributions to the force coming from the propagating and evanescent components of the electromagnetic radiation. The large distance behaviors of these interactions are extensively discussed, both analytically and numerically, and

compari-*antezza@science.unitn.it

sons with the equilibrium results are done. We perform a detailed analysis of the relation between the surface-surface interaction when one body is rarefied共surface-rarefied body force兲 and the surface-atom force. We also present a com-plete derivation and discussion of the recently predicted non-additivity effects and asymptotic behaviors noted in关23兴.

We are interested in the force occurring between two pla-nar bodies, which are kept at different temperatures and separated by a distance l. We consider that the bodies are thick enough, in order to exclude possible effects of the pres-ence of the vacuum gap on the radiation outside the two bodies. We also assume that each body is in local thermal equilibrium, the whole system being in a stationary state. In our configuration the left-side body, 1, has a complex dielec-tric function␧1共␻兲=␧1

⬘

共␻兲+i␧1

⬙

共␻兲, occupies the volume V1

and is held at temperature T1. The right-side body, 2, has a

complex dielectric function ␧2共␻兲=␧2

⬘

共␻兲+i␧2

⬙

共␻兲, occupies

the volume V2and is held at temperature T2. First we assume

that each body fills an infinite half-space, in particular V1and

V2 coincide with the left and right half-spaces, respectively.

Later we consider a more general situation of two parallel thick slabs with the external regions shined by the thermal radiations at arbitrary temperatures. In this case additional distance-independent contributions to the pressure are present. Finally, we will consider the case in which one of the two bodies is rarefied. In this case the interplay between the finite thickness of the body and the nonequilibrium con-figuration leads to different interesting behaviors of the pres-sure.

The general problem can be set in the following way, for two bodies occupying the two half-spaces. Let us choose the origin of the coordinate system at the boundary of the space 1 and let us set the z axis in the direction of the half-space 2 共see Fig. 1兲. The electromagnetic pressure between the two bodies along z can be calculated as关30,31兴

Pneq共T1,T2,l兲 = 具Tzz共r,t兲典, 共1兲

that should be regularized by subtracting the same expres-sion at separation l→⬁. In Eq. 共1兲, r is a generic point between the two bodies, and

Tzz共r,t兲 = −

⌳␣␤

8␲关E␣共r,t兲E␤共r,t兲 + B␣共r,t兲B␤共r,t兲兴, 共2兲 is the zz component of the Maxwell stress tensor in the vacuum gap. Here⌳_␣␤ is a diagonal matrix with⌳11=⌳22

= 1 and⌳33= −1.

To calculate the pressure共1兲 one must average over the state of the electromagnetic field the squares of the spatial components of the electric and magnetic field E共r,t兲 and

B共r,t兲, which appear in Eq. 共2兲.

Before starting with the analysis of the problem we men-tion the structure of this work in the following outline. In Sec. II we develop the formalism, introduce the role and the description of the fluctuations of the electromagnetic field, and specify the approach we adopt to deal with the surface optics. In Sec. III we recall the main results of the surface-surface Casimir-Lifshitz interaction at thermal equilibrium, and in particular specify the distinction between the T = 0 共purely quantum兲 and the thermal contribution to the force, generated by the radiation pressure of the thermal radiation. In Sec. IV we present a detailed derivation of the surface-surface pressure out of thermal equilibrium Pneq_共T

1, T2, l兲. In

Sec. V we show an alternative and useful expression for Pneq共T1, T2, l兲, together with numerical results relative to

par-ticular couples of dielectric materials 共i.e., fused silica-silicon and sapphire-fused silica兲. In Sec. VI we deal with the distance-independent terms in the pressure due to the finite thickness of the two bodies, and the eventual effect of exter-nal radiation at different temperature impinging the exterexter-nal surfaces. In Sec. VII we derive the large distance behavior of the surface-surface pressure out of thermal equilibrium, and discuss the role of the propagating waves共PW兲 and evanes-cent waves共EW兲 contributions. We also make a comparison with the corresponding terms of the pressure at thermal equi-librium. In Sec. VIII we consider the interaction between a surface and a rarefied body and derive the large distance behaviors of the PW and EW components. In the same sec-tion we stress the presence of nonadditivity in the interacsec-tion out of equilibrium共in contrast with the equilibrium case兲 and show the analysis of the crossing between different asymptotic behaviors. In Sec. IX we show the transition from the surface-rarefied body to surface-atom interactions out of thermal equilibrium, and demonstrate the essential role of finite thickness of the rarefied body. Finally, in Sec. X, we provide our conclusions.

In Appendix A we give some details on the expression of the Green functions we used in our calculation and in Ap-pendix B we discuss in detail the force acting between a surface and a rarefied body of finite thickness.

II. FORMALISM

Our approach is based on the theory of the fluctuating electromagnetic共EM兲 field developed by Rytov 关6兴. In this approach it is assumed that the field is driven by randomly fluctuating current density or, alternatively, by randomly fluctuating polarization field. In this respect the Maxwell equations become of Langevin-type. For a monochromatic field in a nonhomogeneous, linear, and nonmagnetic medium

1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1 00 0 0 00 0 0 00 0 0 00 0 0 00 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 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1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 l z 1 2 T1 T2

FIG. 1. Schematic figure of the surface-surface system out of thermal equilibrium. Here the two bodies occupy infinite half-spaces.

with the dielectric function ␧共␻, r兲 the Maxwell equations become

⵱ ∧ E关␻;r兴 − ikB关␻;r兴 = 0, 共3兲 ⵱ ∧ B关␻;r兴 + ik␧共␻;r兲E关␻;r兴 = − 4␲ikP关␻;r兴, 共4兲 where k =␻/c is the vacuum wave number and ∧ is the vec-tor product symbol. The source of the electromagnetic fluc-tuations is described by the electric polarization P关␻; r兴, re-lated to the electric current density as J关␻; r兴=−i␻P关␻; r兴. We use the following notations for the frequency Fourier transforms A关␻; r兴 of the quantity A共t,r兲:

A共t,r兲 =

冕

−⬁ +⬁_d␻

2␲e

−i␻t_{A关␻;r兴. 共5兲}

To find the solution of the Maxwell equations we use the Green functions formalism. A Green function is a solution of the wave equation for a point source in presence of surround-ing matter. When this solution is known one can construct the solution due to a general source using the principle of linear superposition. This method takes into account the ef-fects of nonadditivity, which originates from the fact that the interaction between two fluctuating dipoles is influenced by the presence of a third dipole. Employing this formalism we can express the electric field at the observation point r as the convolution

E关␻;r兴 =

冕

G¯ 关␻;r,r

⬘

兴 · P关␻;r

⬘

兴dr

⬘

. 共6兲 Here P关␻; r

⬘

兴 is the random polarization at the source point

r

⬘

and G¯ 关␻; r , r

⬘

兴 is the dyadic Green function of the sys-tem. Then it is clear that the Green function plays the role of the response function in a linear-response theory. The Green function is the solution of the following equation关32兴

兵⵱ ∧ ⵱ ∧ − k2_{␧共␻,r兲其G¯ 关␻;r,r}

_⬘

_{兴 = 4␲k}2¯_{I␦共r − r}

_⬘

_{兲, 共7兲} where I¯ is the identity dyad. This equation, resulting from the Maxwell equations共3兲 and 共4兲 and convolution 共6兲, has to be solved with proper boundary conditions characterizing the fields components at the interfaces, as well as the condi-tion required by a retarded Green’s funccondi-tion 关35,36兴, i.e.,

G¯ 关␻; r , r

⬘

兴→0 as 兩r−r

⬘

兩→⬁.

Finally, it is useful to recall the relations G_␣␤关␻; r , r

⬘

兴 = G_␤␣关␻; r

⬘

, r兴 and G_␣␤* 关␻; r , r

⬘

兴=G_␣␤关−␻; r , r

⬘

兴 that are the consequence of the microscopic reversibility in the linear-response theory and the reality of the time dependent fields, respectively.

A. Field correlation functions

From Eq. 共1兲 it is evident that we are interested in the time correlations between different components of the elec-tric 共magnetic兲 field at equal times. In the quantum theory such correlations are described by the averages of symme-trized products of the field components:

具E␣共r,t兲E␤共r

⬘

,t兲典sym⬅

1

2具E␣共r,t兲E␤共r

⬘

,t兲 + E␤共r

⬘

,t兲E␣共r,t兲典. 共8兲 Notice that, although in this paper we are using symmetrized correlations, other possible forms of the correlation functions could be more appropriate in other situations关33兴. The cor-relations共8兲 in terms of their Fourier transforms can be pre-sented as

具E␣共r,t兲E␤共r

⬘

,t兲典sym

=

冕冕

d␻ 2␲ d␻

⬘

2␲e −i共␻−␻⬘兲t_具E ␣关␻;r兴E␤†关␻

⬘

;r

⬘

兴典sym. 共9兲

Using Eq. 共6兲 these correlations can be expressed via the correlations of the polarization field P, which obeys the fluctuation-dissipation theorem关30兴 具P␣关␻;r兴P␤†关␻

⬘

;r

⬘

兴典sym= ប␧

⬙

共␻,r兲 2 coth

冉

ប␻ 2kBT

冊

␦共␻ −␻

⬘

兲␦共r − r

⬘

兲␦_␣␤, 共10兲 expressed via the Fourier transformed P关␻; r兴. Due to the presence of the ␦共r−r

⬘

兲 factor these fluctuations are local. Fluctuations of the sources in different points of the material are non-coherent. This permits to assume that in the nonequi-librium situation, when temperature T is different in different points, the sources correlations are given by the same equa-tions. We must emphasize that this assumption, even being quite reasonable, is still a hypothesis, which is worth both of further theoretical investigation and experimental verifica-tion. The problem was discussed previously共see particularly 关41兴兲, but in our opinion the conditions of applicability of the theory has not been still established. The same assumption was used by Polder and Van Hove 关28兴 to calculate the ra-diative heat transfer between two bodies with different tem-peratures.

The assumption共10兲 共local source hypothesis兲 represents the starting point of our analysis allowing for an explicit calculation of the electromagnetic field also if the system is not in global thermal equilibrium.

It is now evident that EM field in the vacuum gap is given by the sum of the fields produced by the fluctuating polar-izations in the materials filling respectively the half-space 1, with the dielectric function ␧1共␻兲 and temperature T1, and

the half-space 2 with the dielectric function␧2共␻兲 and

tem-perature T2. Then the Fourier transform of the electric field

correlations can be presented as 具E␣关␻;r兴E␤†关␻

⬘

;r

⬘

兴典sym=

冋

ប␧1

⬙

共␻兲 2 coth

冉

ប␻ 2kBT1

冊

S_␣␤共1兲关␻;r,r

⬘

兴 +ប␧2

⬙

共␻兲 2 coth

冉

ប␻ 2kBT2

冊

⫻S_␣␤共2兲关␻;r,r

⬘

兴

册

␦共␻−␻

⬘

兲, 共11兲 where S_␣␤共i兲 共i=1,2兲 is defined as convolution of two Green functions

S_␣␤共i兲关␻;r,r

⬘

兴 =

冕

V_i

dr

⬙

G_␣␥关␻;r,r

⬙

兴G_␥␤* 关␻;r

⬙

,r

⬘

兴. 共12兲 Here V1 and V2 are the volumes occupied by the left and

right body, respectively, and the two terms in Eq.共11兲 corre-spond to the parts of the pressure generated by the sources in each body separately.

It is interesting to see how the global equilibrium is re-stored when T1→T2= T in Eq.共11兲. In this case Eq. 共11兲 can

be written as

具E␣关␻;r兴E␤†关␻

⬘

;r

⬘

兴典sym=

ប 2 coth

冉

ប␻ 2kBT

冊

␦共␻−␻

⬘

兲 ⫻

冕

V₁+V2 dr

⬙

␧

⬙

共␻,r

⬙

兲G_␣␥关␻;r,r

⬙

兴 ⫻G_␥␤* _关␻;r

_⬙

_,r

_⬘

_{兴. 共13兲}

The integral over the product of two Green functions is con-nected with the imaginary part of the single Green function by the important关41,42兴 relation

冕

⍀

dr

⬙

␧

⬙

共␻,r

⬙

兲G_␣␥关␻;r,r

⬙

兴G_␥␤* 关␻;r

⬙

,r

⬘

兴

= 4␲Im G_␣␤关␻;r,r

⬘

兴, 共14兲 where⍀ is a volume restricted by a surface where the Green function vanishes. Keeping in mind that in the vacuum gap ␧

⬙

= 0, one can extend the integration in Eq.共13兲 over the the whole space and using Eq.共14兲 one recovers the well-known form of the electric fields fluctuation-dissipation theorem 关32兴 valid at a global thermal equilibrium:

具E␣关␻;r兴E␤†关␻

⬘

;r

⬘

兴典sym

= 2␲ប coth

冉

ប␻ 2kBT

冊

Im G_␣␤关␻;r,r

⬘

兴␦共␻−␻

⬘

兲. 共15兲 Notice that all fluctuations presented in this section include both the vacuum共T=0兲 and the thermal fluctuations. These can be identified with the first and second terms, respec-tively, of the right-hand side共RHS兲 of the identity

coth

冉

ប␻ 2kBT

冊

= sgn共␻兲

冉

1 + 2

eប兩␻兩/kBT_{− 1}

冊

, ␻⫽ 0.

共16兲

B. Pressure in terms of fluctuations

The pressure共1兲 can be presented in terms of the Fourier transformed fields correlations:

Pneq共T1,T2,l兲 = − 1 8␲

冕冕

d␻ 2␲ d␻

⬘

2␲e −i共␻−␻⬘兲t ⫻ ⌳␣␤兩关具E␣关␻;r兴E␤†关␻

⬘

;r

⬘

兴典 +具B_␣关␻;r兴B_␤†关␻

⬘

;r

⬘

兴典兴兩r=r⬘. 共17兲

Here the electric and magnetic contributions to the total pres-sure are explicit, and r is a point inside of the vacuum gap. The stress tensor is in fact constant in the vacuum gap due to the momentum conservation required by a stationary con-figuration共see discussion in Sec. IV A兲. In this equation we omitted the symmetrization index since the average is taken at the same point r = r

⬘

. Using Eq.共3兲 it is useful to rewrite expression共17兲 in terms of the electric fields only 关34兴 as

Pneq共T1,T2,l兲 = − 1 8␲

冕冕

d␻ 2␲ d␻

⬘

2␲e −i共␻−␻⬘兲t ⫻ ⌰␦␯兩关具E␦关␻;r兴E†␯关␻

⬘

;r

⬘

兴典兴兩r=r⬘. 共18兲 Here the pressure is expressed in terms of the correlations 共11兲, and the operator

⌰_␦␯=⌳␣␤

冉

␦␣␦␦␤␯+ 1

k2⑀␣␥␦⑀␤␩␯⳵␥⳵␩

⬘

冊

共19兲 selects the electric and magnetic contributions, given by the first and the second term in Eq.共19兲, respectively.

From Eq.共16兲 it is possible to express the total pressure as the sum

Pneq共T1,T2,l兲 = P0共l兲 + Pthneq共T1,T2,l兲, 共20兲

where the contribution of the zero-point共T=0兲 fluctuations, P₀共l兲, is separated from that produced by the thermal fluc-tuations, P_thneq共T₁, T₂, l兲. Furthermore, thanks to Eq. 共11兲 it is possible to express the thermal component of the pressure acting between the bodies as the sum of two terms

P_thneq共T1,T2,l兲 = Pneqth 共T1,0,l兲 + Pthneq共0,T2,l兲. 共21兲

The pressure at thermal equilibrium Peq_{共T,l兲, being a}

par-ticular case of Eq.共20兲, can be written as Peq共T,l兲 = P0共l兲 + Pth

eq_{共T,l兲. 共22兲}

The pressures P0共l兲 and Pth

eq_{共T,l兲 are given by Eq. 共18兲,}

where the field fluctuations are provided by Eq.共15兲 after the substitution, respectively, of coth

冉

ប␻ 2kBT

冊

→ sgn共␻兲, 共23兲 coth

冉

ប␻ 2kBT

冊

→ 2 sgn共␻兲 eប兩␻兩/kBT_{− 1}. 共24兲

If one simply performs such substitutions, it is well known that Eq. 共18兲 diverges at T=0, and contains constant 共l-independent兲 terms in the thermal part. The divergence has the same origin as the usual divergence of the zero-point fields energy in quantum electrodynamics, while the constant terms are related to the fact that we consider infinite bodies, and hence we neglect the pressure of the radiation exerted on the remote, external surfaces of the two bodies. To recover the exact finite value for the pressures P0共l兲, and exclude the

constant terms in P_theq共T,l兲, one should regularize the Green function in the RHS of Eq.共15兲 by subtracting the bulk part Gij

bu

, corresponding to a field produced by a pointlike dipole in an homogeneous and infinite dielectric关7,39,40兴. In fact,

the Green function with both the observation point r and the source point r

⬘

in the vacuum gap 共see Appendix A 1兲 is given by the sum

Gij关␻;r,r

⬘

兴 = Gij sc_关␻

;r,r

⬘

兴 + Gij bu_关␻

;r,r

⬘

兴 共25兲 of a scattered and a bulk term. The subtraction of the bulk term corresponds to the subtraction of the pressure at l→⬁, as prescribed after Eq.共1兲. The expressions for the pressure at thermal equilibrium are given explicitly in Sec. III.

Concerning the thermal pressure out of thermal equilib-rium P_thneq共T₁, T₂, l兲 of Eq. 共20兲, it can be obtained from Eq. 共18兲 by using Eq. 共11兲 and the substitution 共24兲. Also in this case the thermal pressure P_thneq共T1, T2, l兲 contains an

l-dependent and a constant term, as it happens for P_theq共T,l兲 before being regularized. Differently from the equilibrium case, here the origin of the constant terms is not only due to the absence of the pressure acting on the remote surfaces, but is also related to the fact that out of thermal equilibrium there is a net momentum transfer between the bodies. In this case the constant terms can remain also after considering bodies of finite thickness, and can even be different for the two bodies, depending on the external radiations. In Secs. IV and V we will calculate P_thneq共T1, T2, l兲 for two bodies filling two

infinite half-spaces, and we will mainly discuss the pure l-dependent component. The constant terms will be dis-cussed in Sec. VI for the general case of bodies of finite thickness, with impinging the external radiations at different temperatures.

C. Electromagnetic waves in surface optics

In this work we formulate the electromagnetic problem in terms of s- and p-polarized vector waves and in terms of the Fresnel coefficients for the interfaces 关37兴. Such notations are very useful in surface optics. We will also employ the angular spectrum representation for the description of the EM and polarization vectors.

If xˆ, yˆ, and zˆ are the coordinate unit vectors 共with real norm equal to 1兲, one can write the position vector as r=R + zzˆ, where the capital letter refers to vectors parallel to the interface 关R⬅共Rx, Ry, 0兲兴. Let us write the electromagnetic

共complex兲 wave vector in the medium m with the complex dielectric function␧m共␻兲=␧m

⬘

共␻兲+i␧m

⬙

共␻兲 as

q共m兲共⫾兲 = Q ⫾ qz共m兲zˆ. 共26兲

Here the sign共⫹兲 corresponds to an upward-propagating 共or evanescent兲 wave, and the sign 共⫺兲 corresponds to a downward-propagating共or evanescent兲 wave. The vector Q ⬅共Qx, Qy, 0兲 is the projection 共always real兲 of q共m兲共⫾兲 on the

interface and the z component of the wave vector, and qz共m兲=

冑

␧mk2− Q2, 共27兲

is a complex number with a positive imaginary part, with positive real part in case Im q_z共m兲= 0. Real and imaginary parts of q_z共m兲are expressed by the following relations:

Re qz共m兲=

冑

1 2兵兩␧m共␻兲k 2_{− Q}2_{兩 + 关␧} m

⬘

共␻兲k2_{− Q}2_{兴其, 共28兲} Im qz共m兲=

冑

1 2兵兩␧m共␻兲k 2_{− Q}2_{兩 − 关␧} m

⬘

共␻兲k2_{− Q}2_{兴其. 共29兲}

Then, if the medium m is nonabsorbing 共␧m

⬙

= 0兲, for Q

艋

冑

␧m

⬘

k the wave vector qz

共m兲 _{is real and corresponds to a}

wave propagating in the medium m, while for Q⬎

冑

␧m

⬘

k the

wave vector q_z共m兲is imaginary and corresponds to evanescent wave in the medium m. The following identities will be use-ful: 2 Im qz共m兲Re qz共m兲= k2␧m

⬙

共␻兲, 共30兲 共Q2_+兩q z 共m兲_兩2_{兲Re q} z 共m兲_{= k}2_Re关␧ m *_共␻兲q z 共m兲_{兴, 共31兲} 共Q2_−兩q z 共m兲_兩2_{兲Im q} z 共m兲_{= k}2_Im关␧ m *_共␻兲q z 共m兲_{兴. 共32兲}

It is worth noticing that the wave vectors q共m兲共⫾兲 lie in the plane of incidence spanned by Qˆ and zˆ. Then one can intro-duce the s- and p-unit complex polarization vectors

es共m兲共⫾兲 = Qˆ ∧ zˆ, 共33兲 e共m兲p 共⫾兲 = es共m兲共⫾兲 ∧ qˆ共m兲共⫾兲 =

Qzˆ⫿ qz共m兲Qˆ

冑

␧m共␻兲k

, 共34兲 that are vectors transversal and longitudinal to that plane, respectively. Usually the polarization vector e_s共m兲共⫾兲

关ep

共m兲_{共⫾兲兴 is called transverse electric 共TE兲 关transverse}

mag-netic 共TM兲兴 since it corresponds to the electric 共magnetic兲 field transverse to the plane of incidence.

Our geometry consists of two half-spaces labeled with m = 1 , 2 separated by a vacuum gap. Inside of the gap the wave vector q and the polarization vectors e_␮共⫾兲 are not labeled and are obtained, respectively, from the definitions 共26兲, 共27兲, 共33兲, and 共34兲 by omitting the apices 共m兲_{, and}

setting␧m= 1.

Finally we can introduce the well known reflection and transmission Fresnel coefficients for the vacuum gap-dielectric interfaces, which for the s- and p-wave compo-nents are rm s =qz− qz 共m兲 qz+ qz共m兲 , rm p =qz␧m− qz 共m兲 qz␧m+ qz共m兲 , 共35兲 tm s = 2qz 共m兲 qz共m兲+ qz , tm p = 2

冑

␧m共␻兲qz 共m兲 qz共m兲+ qz␧m共␻兲 . 共36兲

In particular, the coefficients rm relate the radiation in the

vacuum gap impinging the interface m and its part reflected back into the vacuum gap. The coefficients tmrelate the

ra-diation impinging the interface m from the interior of the dielectric m and its part transmitted into the vacuum gap共see Appendix A兲.

III. PRESSURE AT THERMAL EQUILIBRIUM

In this section, we briefly recall the main results of the pressure in a system at thermal equilibrium. We present the

thermal component of the pressure as the sum of PW and EW components, and in terms of real frequencies, which will prove useful for the rest of the discussion. The results we show for the pressure at equilibrium are regularized关see dis-cussion after Eq.共24兲兴.

The Lifshitz surface-surface pressure at thermal equilib-rium can be expressed in terms of real frequencies as

Peq共T,l兲 = − ប 2␲2

冕

0 ⬁ d␻coth

冉

ប␻ 2kBT

冊

⫻ Re

冋

冕

0 ⬁ dQ Qqzg共Q,␻兲

册

, 共37兲 where g共Q,␻兲 =

兺

␮=s,p r₁␮r₂␮e2iqzl D_␮ =_␮=s,p

兺

关共r1 ␮_r 2 ␮_兲−1_e−2iqzl_{− 1}兴−1_. 共38兲 In the previous equation the multiple reflections are de-scribed by the factor

D_␮= 1 − r₁␮r₂␮e2iqzl_, 共39兲

and the reflection Fresnel coefficients rm␮ for the

vacuum-dielectric interfaces are defined in Eq.共35兲.

By performing the Lifshitz rotation on the complex plane it is possible to write Eq.共37兲 in terms of imaginary frequen-cies: Peq_{共T,l兲 =} kBT 16␲l3

冕

₀ ⬁ dx x2

冋

共␧10+ 1兲共␧20+ 1兲 共␧10− 1兲共␧20− 1兲 ex_{− 1}

册

−1 + kBT ␲c3

兺

_n=1 ⬁ ␰n3

冕

1 ⬁ dp p2g共p,i␰n兲, 共40兲 where p =

冑

1 + c2_Q2_/␰ n

2_{. The dielectric functions that enter to}

g共p,i␰n兲 must be evaluated at imaginary frequencies ␧1,2

=␧_1,2共i␰n兲, where␰n= 2␲kBTn/ប. In the first term of Eq. 共40兲

we have also introduced the static values of the dielectric functions␧10=␧1共0兲 and ␧20=␧2共0兲.

The pressure at thermal equilibrium includes contribu-tions from zero-point fluctuacontribu-tions P0共l兲 and from thermal

fluctuations P_theq共T,l兲 as Eq. 共22兲 shows. P₀共l兲 can be ex-tracted from Eq.共37兲 with the substitutes Eq. 共23兲 or from Eq.共40兲 as the limit of continuous imaginary frequency. The final result for the T = 0 pressure is

P0共l兲 = ប 2␲2c3

冕

0 ⬁ d␰

冕

1 ⬁ dp p2␰3g共p,i␰兲. 共41兲 The pressure P₀共l兲 admits two important limits, i.e., the van der Waals–London and the Casimir-Polder behaviors, valid at small and large distances, respectively, in respect to the characteristic length scale␭optfixed by the absorption

spec-trum of the bodies 共typically is of the order of fraction of microns兲.

The behavior of the thermal component P_theq共T,l兲 is related to a second length scale, i.e., the thermal wavelength

␭T⬅

បc kBT

, 共42兲

which at room temperature is⬇7.6␮m.

Then, the zero-point fluctuations dominate over the ther-mal contribution at sther-mall distances lⰆ␭T. In this limit

be-havior of the pressure is determined by the characteristic length scale␭_optⰆ␭T. In the interval␭optⰆlⰆ␭Tone enters

the Casimir-Polder regime where the pressure decays like 1/l4_{. For distances lⰆ␭}

optthe force instead exhibits the 1/l3

van der Waals–London dependence. The possibility of iden-tifying the Casimir-Polder regime depends crucially on the value of the temperature. The temperature should be in fact sufficiently low in order to guarantee the condition ␭T

Ⰷ␭opt.

The last part of this section focuses on the thermal com-ponent of the pressure that will often be used along the rest of the paper. The pressure P_theq共T,l兲 can be obtained from Eq. 共37兲 by using 共24兲. Since such a component of the pressure will be compared with that out of thermal equilibrium, we show here explicitly its expression for PW and EW contri-butions: P_theq,PW共T,l兲 = − ប ␲2

冕

0 ⬁ d␻ 1 eប␻/kBT_{− 1}

冕

0 k dQ Qqz ⫻

兺

␮=s,p Re共r1␮r2␮e2iqz l_{兲 − 兩r} 1 ␮_r 2 ␮_兩2 兩D␮兩2 , 共43兲 P_theq,EW共T,l兲 = ប ␲2

冕

0 ⬁ d␻ 1 eប␻/kBT_{− 1}

冕

k ⬁ dQ Q Im qze−2l Im qz ⫻

兺

␮=s,p Im共r1␮r2␮兲 兩D␮兩2 . 共44兲

In particular at high temperatures, or equivalently at large distances defined by the condition

lⰇ ␭T, 共45兲

the leading contribution to the pressure is given by the ex-pression for the total force关7兴

P_theq共T,l兲 = kBT 16␲l3

冕

0 ⬁ dx x2

冋

␧10+ 1 ␧10− 1 ␧20+ 1 ␧20− 1 ex− 1

册

−1 . 共46兲 It corresponds to the first term in Eq.共40兲 and is entirely due to the thermal fluctuations of the EM field. In Ref.关7兴 the asymptotic behavior 共46兲 has been found after the contour rotation in the complex␻plane of the EW term共44兲, that is partially canceled by the PW term共43兲.

One can note that in this regime only the static value of the dielectric functions is relevant. The pressure共46兲 is pro-portional to the temperature and is independent from the Planck constant as well as from the velocity of light. We will call this equation the Lifshitz limit. The pressure共46兲 can be obtained from the thermal free energyF=E−TS of the elec-tromagnetic field共per unit area兲 according to the thermody-namic identity P = −共⳵F/⳵l兲T, whereE and S are the thermal

energy and entropy, respectively. It is interesting to note that, differently from the free energy, the thermal energy E de-creases exponentially with l, which means that the pressure 共46兲 has pure entropic origin 关38兴.

It is important that at large separations only the p polar-ization contributes to the force共see, for example, in 关24兴, the detailed derivation of the PW and EW components兲. The reason is that for low frequencies the s-polarized field is nearly pure magnetic, but the magnetic field penetrates freely into a nonmagnetic material关43兴.

In the limit ␧₁₀,␧₂₀→⬁ we find the force between two metals

P_theq,met共T,l兲 = kBT

8␲l3␨共3兲. 共47兲 Let us empathize that this result was obtained for interaction between real metals关44兴. For “ideal mirrors” considered by Casimir, both polarizations of electromagnetic fields are re-flected. In this case there will be an additional factor 2 in Eq. 共46兲 due to the contribution of the s polarization. This ideal case can be realized using superconducting mirrors.

It is useful to note that the surface-surface pressure Peq共T,l兲 given by the Lifshitz result 共40兲 hides a nontrivial cancellation between the components of the pressure related to real and imaginary values of the EM wave vectors, lead-ing, respectively, to the propagating 共PW兲 and evanescent 共EW兲 wave contributions 关17,45兴. This study deserves care-ful investigation since for a configuration out of thermal equilibrium such cancellations are no longer present, and the PW and EW contribution will provide different asymptotic behaviors. The new effect, as we will see, is particularly important if one of the two bodies is a rarefied gas.

IV. PRESSURE OUT OF THERMAL EQUILIBRIUM BETWEEN TWO INFINITE DIELECTRIC HALF-SPACES

As was discussed above 关see Eqs. 共11兲 and 共21兲兴 each body contributes separately to the thermal pressure. In par-ticular, the pressure resulting from the thermal fluctuations in the body 1 is P_thneq共T,0,l兲 = − ប 16␲3

冕

₀ ⬁ d␻ ␧1

⬙

共␻兲 eប␻/kBT_{− 1} ⫻Re兩关⌰_␦␯S␦共1兲␯关␻;r1,r2兴兴兩r₁=r₂, 共48兲

where r1= r2is a point in the vacuum gap and the function S

is defined in Eq.共12兲. In Eq. 共48兲 we used the parity prop-erties ␧

⬙

共␻兲=−␧

⬙

共−␻兲 and S_␦␯关␻; r₁, r₂兴=S_␦*_␯关−␻; r₁, r₂兴 to restrict the range of integration to the positive frequencies. It is evident that P_thneq共0,T,l兲 can be expressed similarly to Eq. 共48兲, but with ␧₁

⬙

共␻兲→␧₂

⬙

共␻兲 and S_␦共1兲_␯→S_␦共2兲_␯.

Below we specify the expressions of the tensors S_␦共1兲_␯ and S_␦共2兲_␯ 共Sec. IV A兲, calculate the electric and magnetic contri-butions to the pressure 共Sec. IV B兲, and finally provide the result for the total pressure in terms of PW and EW compo-nents共Sec. IV C兲. The total pressure will be rewritten in a different form in Sec. V by using a powerful expansion in multiple reflections. In the present and in the next Sec. V the

pressure is calculated for two infinite bodies共see discussion at the end of Sec. II B兲.

A. S functions

In this subsection we show the result for the tensors S_␦共1兲_␯ and S_␦共2兲_␯ defined by Eq. 共12兲. In terms of the lateral Fourier transforms s_␦共1兲_␯关␻; Q , z1, z2兴 and s_␦共2兲_␯关␻; Q , z1, z2兴 one has

S_␦␯关␻;r1,r2兴 =

冕

d2Q

共2␲兲2e

iQ·共R1−R2兲_s

␦␯关␻;Q,z1,z2兴. 共49兲

By choosing the x axis parallel to the vector D = R1− R2and

defining␾ as the angle between Q and D one gets that Qx

= Q cos␾, Qy= Q sin␾and the polarization vectors become es共m兲共⫾兲 = 共兩sin␾兩,− cos␾sin␾/兩sin␾兩,0兲, 共50兲 ep共m兲共⫾兲 =

1

冑

␧mk

共⫿qz共m兲cos␾,⫿ qz共m兲sin␾,Q兲. 共51兲

Here it is evident that兩e_s共m兲共⫾兲兩2_{= 1 and}

兩ep共m兲共⫾兲兩2=

Q2+兩qz共m兲兩2

兩␧m兩k2

. 共52兲

After explicit calculation using the Green function given in Appendix A we find for the s_␦共1兲_␯ and s_␦共2兲_␯ functions the explicit expressions s_␦共1兲_␯关␻;Q,z₁,z₂兴 =4␲ 2_k2 ␧1

⬙

共␻兲 Re qz共1兲 兩qz共1兲兩2 ␮=s,p

兺

兩t1␮兩2 兩D␮兩2兩e␮ 共1兲_{共+ 兲兩}2

⫻关e␮,␦共+ 兲e␮,␯* 共+ 兲ei共qzz1−qz*z2兲 + e_␮,␦共+ 兲e_␮,␯* 共− 兲ei共q_zz₁+q_z*z₂兲_e−2iq_z*l_r

2 ␮*

+ e_␮,␦共− 兲e_␮,␯* 共+ 兲e−i共qzz1+qz

*_z

2兲_e2iqzl_r

2 ␮

+ e_␮,␦共− 兲e_␮,␯* 共− 兲e−i共qzz1−qz

*_z 2兲 ⫻e−4 Im qzl兩r 2 ␮_兩2_{兴, 共53兲} s_␦共2兲_␯关␻;Q,z1,z2兴 = 4␲2k2 ␧2

⬙

共␻兲 Re qz共2兲 兩qz共2兲兩2 e−2l Im qz

兺

␮=s,p 兩t2␮兩2 兩D␮兩2兩e␮ 共2兲_{共− 兲兩}2

⫻ 关e␮,␦共− 兲e␮,␯* 共− 兲e−i共qzz1−qz

*_z

2兲

+ e_␮,␦共− 兲e_␮,␯* 共+ 兲e−i共qzz1+qz

*_z 2兲_r 1 ␮* + e_␮,␦共+ 兲e_␮,␯* 共− 兲ei共qzz1+qz *_z 2兲_r 1 ␮ + e_␮,␦共+ 兲e_␮,␯* 共+ 兲ei共qzz1−qz *_z 2兲兩r 1 ␮_兩2_{兴, 共54兲}

where D_␮ is defined in Eq.共39兲.

It is worth noticing that in the nonequilibrium but station-ary regime the fields correlation functions s共1,2兲 are not uni-form in the vacuum cavity, while on the contrary the Max-well stress tensor Tzz 共which is related to the momentum

flux兲 has the same value in each point of the vacuum gap. This is valid also at equilibrium, and is a direct consequence of the momentum conservation required by a stationary

con-figuration. To show this property one can set z1= z2= z in Eq.

共53兲, where the dependence on z appears only in the expo-nential factors关the same would happen for Eq. 共54兲兴.

Let us note that now the first and the last terms in such an expression are proportional to e2z Im qz _{and e}−2z Im qz_,

respec-tively, while the second and the third terms are proportional to e−2iz Reqz_{and e}2iz Reqz_{, respectively. As it will be clear in the}

next Sec. IV B the first and the last terms will be responsible for the PW contribution to the pressure共for which Im qz= 0兲,

while the second and the third terms will be responsible for the EW contribution共for which Re qz= 0兲. It is then evident

that the position z disappears in the Maxwell stress tensor.

B. Electric and magnetic contributions to the pressure

The pure electric contribution to the pressure P_thneq共T,0,l兲 is due to the first term in Eq.共19兲

⌳␣␤␦␣␦␦␤␯S␦共1兲␯兩关␻;r1,r2兴兩r₁=r₂,z₁=0 =兩关S₁₁共1兲+ S₂₂共1兲− S₃₃共1兲兴兩r₁=r₂,z₁=0 =2␲k 2 ␧1

⬙

冕

0 ⬁ dQ QRe qz 共1兲 兩qz共1兲兩2 ⫻

冋

兩t1 s_兩2 兩Ds兩2 兩es共1兲共+ 兲兩2共1 + r₂ s_* e−2iqz *_l + r₂se2iqzl +兩r₂s兩2e−4l Im qz兲 + 兩t1 p_兩2 兩Dp兩2 兩ep共1兲共+ 兲兩2

冉

兩qz兩2− Q2 k2 −兩qz兩 2_{+ Q}2 k2 r2 p_* e−2iqz *_l −兩qz兩 2_{+ Q}2 k2 r2 p e2iqzl +兩qz兩 2_{− Q}2 k2 兩r2 p_兩2_e−4l Im qz

冊

册

_, 共55兲

while the magnetic contribution is related by the second term in Eq.共19兲, and is given by

1 k2⌳␣␤⑀␣␥␦⑀␤␩␯⳵␥⳵␩

⬘

S␦␯ 共1兲_兩关␻;r 1,r2兴兩r₁=r₂,z₁=0 = 1 k2

冕

₀ ⬁_dQ 2␲Q

冕

₀ 2␲_d␾ 2␲e iQD cos␾_兩兵⳵ 3⳵3

⬘

共s11+ s22兲 +关Q2共s33− s22− s11兲 + Qx 2 s11+ Qy 2 s22+ QxQy共s12+ s21兲兴 +关i⳵3共Qys23+ Qxs13兲 − i⳵

⬘

3共Qys32+ Qxs31兲兴其兩D=0,z₁=z₂=0, 共56兲 where s = s共1兲. One can show that, as it happens for the equi-librium case, the magnetic contribution共56兲 coincides with the electric one共55兲, after the interchange of the polarization indexes s↔p.

C. Final expression for the pressure

Taking the sum of Eqs.共55兲 and 共56兲 one finds that the pressure P_thneq共T,0,l兲 in Eq. 共48兲 is

P_thneq共T,0,l兲 = − ប 8␲2

冕

₀ ⬁ d␻ 1 eប␻/kBT_{− 1}

冕

0 ⬁ dQ QRe qz 共1兲 兩qz共1兲兩2 ⫻

再

兩t1 s_兩2 兩Ds兩2 关共qz2+兩qz兩2兲共1 + 兩r2 s_兩2_e−4 Im qzl兲 + 2共q z 2 −兩qz兩2兲Re共r2 s e2iqzl兲兴 + 兩t1 p_兩2 兩Dp兩2 Q2+兩qz共1兲兩2 兩␧1共␻兲兩k2 关共qz2 +兩qz兩2兲共1 + 兩r2 p_兩2_e−4 Im qzl兲 + 2共q z 2 −兩qz兩2兲Re共r2 p e2iqzl兲兴

冎

_. 共57兲

From this general expression one can extract the contribution of the propagating waves共PW兲 in the empty gap, for which qz is real and hence qz2=兩qz兩2, and the contribution of the

evanescent waves共EW兲, for which qzis pure imaginary and

hence qz2= −兩qz兩2: P_thneq,PW共T,0,l兲 = − ប 4␲2

冕

0 ⬁ d␻ 1 eប␻/kBT_{− 1}

冕

0 k dQ QRe qz 共1兲 兩qz共1兲兩2 qz 2 ⫻

冋

兩t1 s_兩2 兩Ds兩2 共1 + 兩r2 s_兩2_{兲 +} 兩t1 p_兩2 兩Dp兩2 Q2_+兩q z 共1兲_兩2 兩␧1共␻兲兩k2 共1 + 兩r2 p_兩2_兲

册

_, 共58兲 P_thneq,EW共T,0,l兲 = − ប 2␲2

冕

0 ⬁ d␻ 1 eប␻/kBT_{− 1}

冕

k ⬁ dQ QRe qz 共1兲 兩qz共1兲兩2 qz 2 e−2l Im qz ⫻

冋

兩t1 s_兩2 兩Ds兩2 Re共r2 s_{兲 +} 兩t1 p_兩2 兩Dp兩2 Q2_+兩q z 共1兲_兩2 兩␧1共␻兲兩k2 Re共r2 p_兲

册

. 共59兲 Now, using helpful identities关46兴

Re qz共1兲兩t1 s_兩2 兩qz共1兲兩2 =Re qz共1 − 兩r1 s_兩2_{兲 + 2 Im q} zIm r1 s 兩qz兩2 , 共60兲 Re关␧₁*共␻兲qz共1兲兴兩t1 p_兩2 兩␧1共␻兲兩兩qz共1兲兩2 =Re qz共1 − 兩r1 p_兩2_{兲 + 2 Im q} zIm r1 p 兩qz兩2 , 共61兲 and similar ones for 1↔2, it is possible to express P_thneq,PW共T,0,l兲 and P_thneq,EW共T,0,l兲 as

P_thneq,PW共T,0,l兲 = − ប 4␲2

冕

₀ ⬁ d␻ 1 eប␻/kBT_{− 1}

冕

0 k dQ Qqz ⫻

兺

␮=s,p 共1 − 兩r1␮兩2兲共1 + 兩r2␮兩2兲 兩D␮兩2 , 共62兲

P_thneq,EW共T,0,l兲 = ប ␲2

冕

0 ⬁ d␻ 1 eប␻/kBT_{− 1}

冕

k ⬁ dQ Q ⫻Im qze−2l Im qz

兺

␮=s,p Im共r1␮兲Re共r2␮兲 兩D␮兩2 . 共63兲 Note that the PW term共62兲 contains a distance independent contribution that will be discussed in the next section.

The pressure P_thneq共0,T,l兲 can be obtained following the same procedure but using the function s_ij共2兲given by Eq.共54兲. The result can be obtained without calculation simply by the interchange r₁␮↔r₂␮in Eqs.共62兲 and 共63兲.

V. ALTERNATIVE EXPRESSION FOR THE PRESSURE

The thermal pressure between two bodies in a configura-tion out of thermal equilibrium was derived in the previous section, and expressed in terms of Eqs.共62兲 and 共63兲. In this section we present an alternative expression for such a pres-sure, explicitly in terms of the pressure at thermal equilib-rium. In Sec. V A we discuss the case of bodies made of identical materials␧₁=␧₂, in Sec. V B we discuss the general case of bodies made of different materials, and finally in Sec. V C we show numerical results for the pressure between dif-ferent bodies held at difdif-ferent temperatures.

A. Pressure between identical bodies

In the case of two identical materials the pressure between bodies can be found without any calculations using the fol-lowing simple consideration. Let the body 1 be at tempera-ture T and the body 2 be at T = 0, then the thermal pressure will be P_thneq共T,0,l兲. Because of the material identity the pres-sure will be the same if we interchange the temperatures of the bodies: P_thneq共T,0,l兲= P_thneq共0,T,l兲. In general, we know from Eq.共21兲 that the thermal part of the pressure is given by the sum of two terms each of them corresponding to a con-figuration where only one of the bodies is at nonzero tem-perature, i.e., P_thneq共T1, T2, l兲= Pneqth 共T1, 0 , l兲+ Pthneq共0,T2, l兲. It is

now evident that at equilibrium, where T1= T2= T, the latter

equation gives P_thneq共T,0,l兲= P_theq共T,l兲/2 and we find for the total pressure P_thneq共T1,T2,l兲 = P_theq共T₁,l兲 2 + P_theq共T₂,l兲 2 . 共64兲

Therefore, the pressure between identical materials is ex-pressed only via the equilibrium pressures at T1and T2. The

same result was obtained by Dorofeyev 关25兴 by an explicit calculation of the pressure. It is interesting to note that Eq. 共64兲 is valid not only for the plane-parallel geometry, but for any couple of identical bodies of any shape displaced in a symmetric configuration with respect to a plane.

B. Pressure between different bodies

It is convenient to present the general expression of the pressure in a form which reduces to Eq.共64兲 in the case of

identical bodies. It can be done using Eq. 共21兲 where P_thneq共T,0,l兲 is given by Eqs. 共62兲 and 共63兲, and P_thneq共0,T,l兲 is obtained from P_thneq共T,0,l兲 after the interchange r₁␮↔r₂␮.

In P_thneq共T,0,l兲 we can separate symmetric and antisym-metric parts in respect to permutations of the bodies 1↔2. The factors sensitive to such a permutations in Eqs.共62兲 and 共63兲 are, respectively,

共1 − 兩r1兩2兲共1 + 兩r2兩2兲 = 共1 − 兩r1r2兩2兲 + 共兩r2兩2−兩r1兩2兲, 共65兲

Im共r₁兲Re共r₂兲 =1

2 Im共r1r2兲 + 1

2关Im共r1兲Re共r2兲 − Re共r1兲Im共r2兲兴, 共66兲 where we omitted the index ␮. The symmetric parts, 共1 −兩r1r2兩2兲 for PW and Im共r1r2兲/2 for EW, are responsible for

the equilibrium term P_theq共T,l兲/2 in the nonequilibrium pres-sure as Eq. 共64兲 shows. Concerning the EW terms, if one takes the symmetric part of Eqs. 共63兲, one obtains exactly P_theq,EW共T,l兲/2, where P_theq,EW共T,l兲 coincides with the equilib-rium EW component 共44兲. The analysis of the PW term is more delicate; in fact, if one takes the symmetric part 共1 −兩r1r2兩2兲 of Eq. 共62兲, one obtains P¯theq,PW共T,l兲/2, where

P ¯ th eq,PW_{共T,l兲 = −} ប 2␲2

冕

0 ⬁ _d␻ eប␻/kBT_{− 1} ⫻

冕

0 k dQ Qqz

兺

␮=s,p 1 −兩r₁␮r₂␮兩2 兩D␮兩2 . 共67兲

The above equation is different from P_theq,PW共T,l兲 given by Eq.共43兲.

The difference has a clear origin. In fact the pressure out of equilibrium, from which Eq.共67兲 is derived, is calculated for bodies occupying two infinite half-spaces. On the con-trary the equilibrium pressure P_theq,PW共T,l兲 was obtained after proper regularization, and hence taking into account the pres-sure exerted on the external surfaces of bodies of finite thick-ness关see discussion after Eq. 共24兲兴. Then the difference be-tween Eqs.共43兲 and 共67兲 is just a constant:

P ¯ th eq,PW_{共T,l兲 = P} th eq,PW_{共T,l兲 −}4␴T4 3c , 共68兲 where ␴=␲2_k B

4_/60c2_ប3 _{is the Stefan-Boltzmann constant.}

Using the following multiple-reflection expansion of the fac-tor兩D_␮兩−2_: 1 兩1 − Re2iqzl兩2= 1 1 −兩R兩2

冉

1 + 2 Re

兺

_n=1 ⬁ Rne2inqzl

冊

_, 共69兲

where R = r₁␮r₂␮, it is not difficult to show explicitly that Eqs. 共43兲 and 共67兲 are related by Eq. 共68兲. The constant term in Eq.共68兲 comes from the first term of this expansion.

Collecting together the symmetric and antisymmetric parts we can finally present the nonequilibrium pressure in the following useful form:

P_thneq,PW共T1,T2,l兲 = P_theq,PW共T1,l兲 2 + P_theq,PW共T2,l兲 2 − B共T1,T2兲 +⌬P_thPW共T₁,l兲 − ⌬P_thPW共T₂,l兲, 共70兲 P_thneq,EW共T1,T2,l兲 = P_theq,EW共T1,l兲 2 + P_theq,EW共T2,l兲 2 +⌬Pth EW_共T 1,l兲 −⌬P_thEW共T2,l兲. 共71兲

This is one of the main results of this paper. Here B共T1, T2兲=2␴共T1

4

+ T₂4兲/3c is a l-independent term, discussed in Eq. 共68兲. The equilibrium pressures P_theq,PW共T,l兲 and P_theq,EW共T,l兲 are defined by Eqs. 共43兲 and 共44兲 and do not contain l-independent terms. The expressions⌬P_thPW共T,l兲 and ⌬Pth

EW_{共T,l兲 are antisymmetric with respect to the interchange}

of the bodies 1↔2 and are defined as ⌬Pth PW_{共T,l兲 = −} ប 4␲2

冕

0 ⬁ d␻ 1 eប␻/kBT_{− 1}

冕

0 k dQ Qqz ⫻

兺

␮=s,p 兩r2␮兩2−兩r1␮兩2 兩D␮兩2 , 共72兲 ⌬PthEW共T,l兲 = ប 2␲2

冕

0 ⬁ d␻ 1 eប␻/kBT_{− 1}

冕

k ⬁ dQ Q Im qze−2l Im qz ⫻

兺

␮=s,p

Im共r1␮兲Re共r2␮兲 − Im共r2␮兲Re共r1␮兲

兩D␮兩2 . 共73兲

Let us note that the EW term共73兲 goes to 0 for l→⬁ because evanescent fields decay at large distances. However, the PW term 共72兲 contains a l-independent component since in the nonequilibrium situation there is momentum transfer be-tween bodies. This l-independent component can be directly extracted from Eq. 共72兲 using the expansion 共69兲. This ex-pansion shows explicitly the contributions from multiple re-flections. The distance independent term corresponds to the first term in the expansion 共69兲, and it is related with the radiation that pass the cavity only once, i.e., without being reflected. Finally it is possible to write⌬P_thPW共T,l兲 as the sum ⌬Pth

PW_{共T,l兲=⌬P} th,a

PW_共T兲+⌬P th,b

PW_{共T,l兲, where the constant and}

the pure l-dependent terms are respectively ⌬Pth,a PW_{共T兲 = −} ប 4␲2

冕

₀ ⬁ d␻ 1 eប␻/kBT_{− 1}

冕

0 k dQ Qqz ⫻

兺

␮=s,p 兩r2␮兩2−兩r1␮兩2 1 −兩r₁␮r₂␮兩2, 共74兲 ⌬Pth,b PW_{共T,l兲 = −} ប 2␲2

兺

n=1 ⬁ Re

再

冕

0 ⬁ d␻ 1 eប␻/kBT_{− 1}

冕

0 k dQ Qqz ⫻

兺

␮=s,p 兩r2␮兩2−兩r1␮兩2 1 −兩r₁␮r₂␮兩2共r1 ␮_r 2 ␮_兲n e2inqzl

冎

_. 共75兲

At thermal equilibrium T1= T2= T the sum of Eqs.共70兲 and

共71兲 provides the Lifshitz formula except for the term −4␴T4_{/3c, which is canceled due to the pressure exerted on}

the remote external surfaces of the bodies, as explicitly shown in the next section. Out of thermal equilibrium, but for identical bodies, r₁␮= r₂␮, the antisymmetric terms disap-pear: ⌬P_thPW共T,l兲=⌬P_thEW共T,l兲=0. In this case, Eq. 共64兲 is reproduced.

It is now clear that, due to the antisymmetric terms, Eq. 共64兲 is not valid if the two bodies are different. The problem of the interaction between two bodies with different tempera-tures was previously considered by Dorofeyev 关25兴 and Dorofeyev, Fuchs, and Jersch 关26兴. The authors used a dif-ferent method, based on the generalized Kirchhoff’s law关6兴. The general formalism of关25兴 agrees with our Eqs. 共74兲 and 共75兲. However, our results are in disagreement with the re-sults of关26兴, where Eq. 共64兲 was found to be valid also for bodies of different materials, so that we argue that the results of the last paper were based on some inconsistent derivation.

C. Numerical results for the pressure between two different bodies out of thermal equilibrium

In this section we show the results of the calculation of the pressure between two different bodies, for configurations both in and out of thermal equilibrium. In Figs.2and3we show the numerical results of the pressure for a system made of fused silica共SiO₂兲 for the left-side body 1 and low con-ductivity silicon共Si兲 for the right-side body 2. In both cases the experimental values of the dielectric functions in a wide range of frequencies were taken from the handbook关47兴. In particular in Fig. 2 we show the thermal pressure P_thneq共T1, T2, l兲, sum of Eqs. 共70兲 and 共71兲, as a function of the

separation l between 0.5␮m and 5 ␮m. Here we omit the l-independent terms. The pressure is presented for the con-figuration共T1= 300 K, T2= 0 K兲 关solid line兴 and for the

con-figuration 共T₁= 0 K, T₂= 300 K兲 关dashed兴. We plot also the thermal part of the force at thermal equilibrium, which is the sum of Eqs.共43兲 and 共44兲, at the temperature T=300 K 共dot-ted兲. The sum of the two configurations out of thermal equi-librium provides the force at thermal equiequi-librium. In Fig.3 we show the relative contribution P_th/ P₀of the thermal

com-1 −4 1x10 2 3 4 5 separation [µm] Pressure [dine/cm 2 ] T 1= T2= 300K (dotted) T₁= 300K , T₂= 0K (solid) T₁= 0K , T₂= 300K (dashed) 1x10−6 −5 1x10 −3 1x10

FIG. 2. Thermal component共only l-dependent part兲 of the

pres-sure out of equilibrium for fused silica-silicon system in the

con-figuration 共T₁= 300 K, T₂= 0 K兲 共solid兲 and in the configuration

共T1= 0 K, T2= 300 K兲 共dashed兲. We plot also the thermal part of the