Casimir force between partially transmitting mirrors

12 Pages·0.18 MB·English

by  Marc-Thierry Jaekel

Casimir force between partially transmitting mirrors
Marc Thierry Jaekel (a) and Serge Reynaud (b)
(a) Laboratoire de Physique Th´eorique de l’ENS ∗, 24 rue Lhomond, F75231 Paris Cedex 05 France
(b) Laboratoire de Spectroscopie Hertzienne †, 4 place Jussieu, case 74, F75252 Paris Cedex 05 France
(Journal de Physique I 1(1991) 1395-1409)
The Casimir force can be understood as resulting from the radiation pressure exerted by the
vacuum fluctuations reflected by boundaries. We extend this local formulation to the case of par-
tially transmitting boundaries by introducing reflectivity and transmittivity coefficients obeying
conditions of unitarity, causality and high frequency transparency. We show that the divergences
associated with the infiniteness of the vacuum energy do not appear in this approach. We give
explicit expressions for the Casimir force which hold for any frequency dependent scattering and
anytemperature. Thecorrespondingexpressions fortheCasimir energy areinterpretedin termsof
1
phase shifts. The known results are recovered at the limit of a perfect reflection.
0
PACS: 03.65 - 42.50 - 12.20
0
2

n
INTRODUCTION
a
J
6 As a consequence of Heisenberg inequalities, the electromagnetic field exhibits quantum fluctuations even in the
1 vacuumstate. These“vacuumfluctuations”havebeenextensivelystudiedinthedomainofquantumopticstheselast
years [1]. Casimir [2] observed that the resulting vacuum energy (1¯hω per field mode) depends upon the boundary
2
1 conditionsandparticularlyuponthepositionsofreflectingbodies. Asaconsequence,thevacuumfluctuationsmanifest
v
themselves through macroscopic forces.
7
These Casimir forces are usually computed by comparing the mode density in the absence and in the presence of
6
perfectly reflectingboundaries [3]. Ina localformulation,they canalsobe understoodasresulting fromthe radiation
0
1 pressure exerted by the vacuum fluctuations reflected by the boundaries [4,5]. In the simplest case of a scalar field in
0 a two-dimensional (2D) spacetime, one gets the following force between two pointlike mirrors at a distance q
1
0 ¯hcπ
F = (1)
h/ 24q2
p
- Fortheelectromagneticfieldinafour-dimensional(4D)spacetime,onegetstheCasimirpressure(forcemeasuredper
nt unit area) between two parallel plane and infinite mirrors [6]
a
u ¯hcπ2
F = (2)
q 240q4
:
v
i From now on, we use natural units where c=ε0 =kB =1; however, we keep ¯h as a scale for vacuum fluctuations.
X
Aprobleminanycalculationofvacuuminducedeffectsis todisposeofthe divergencesassociatedwiththe infinite-
r ness of the total vacuum energy. Usually, this is done by arbitrarily cutting off the mode density. The divergences of
a
the energyinthe absenceandinthe presenceofthe boundariescanceleachother,whichleadsto afinite netresultat
the limit of infinite cutoff frequency. In the local formulation, the field correlation function diverges when the fields
are evaluated at the same point. A regularized stress tensor is obtained by splitting the two points and ignoring the
divergent part [4,7].
Clearly, a more natural regularization should be provided by studying partially transmitting mirrors. Any real
mirrors are certainly transparent at high frequencies so that the expression of the force should be regular. Using the
physicalproperties of a dielectric constant, Lifshitz [8] has obtained regular expressionsfor the Casimir-Polderforces
betweenmacroscopicdielectricbodies[9,10]. Thisapproachisnotfreefromproblems,suchastheeffectofdissipation
and the expression of energy density in a dispersive dielectric medium, which have been discussed in detail [11,12].
In this paper, we use a scattering approach to avoid these problems. We use the local formulation and intro-
duce frequency dependent reflectivity and transmittivity coefficients for the two mirrors, which are assumed to obey
∗Unit´epropreduCentreNationaldelaRechercheScientifique,associ´ee`al’EcoleNormaleSup´erieureet`al’Universit´eParis-Sud
†Unit´e de l’Ecole Normale Sup´erieure et de l’Universit´e Pierre et Marie Curie, associ´ee au Centre National de la Recherche
Scientifique
1
conditions of reality, unitarity, causality and high frequency transparency. The scattering coefficients determine the
vacuum stress tensor and therefore the Casimir force on the mirrors. Explicit expressions, which come out as finite
integrals, are obtained for any frequency dependent scattering. Eventually, the reflectivity function appears as a
physicalregulatorandthe knownresults (1)and(2) arerecoveredwhenthe mirrorsare totally reflectingovera large
enough frequency interval.
These results do not rely upon a detailed microscopic analysis of the scattering process and are not limited to
dielectric mirrors. Dissipative processes are disregarded (unitary scattering) and all field fluctuations originate from
theinputvacuum. Wealsogivetheresultsforanonzerotemperatureinputstate. TheCasimirenergycanbededuced
byintegratingtheforceandisdifferentfromtheintegratedfieldenergywhenthereflectivitiesarefrequencydependent.
This expression, interpreted in terms of phase shifts, makes the connection with the mode density approach.
A first part details the simple case where two pointlike mirrors are placed in the vacuum state, or in a thermal
state, of a scalar field in a 2D spacetime. We come then to the problem of parallel plane and infinite mirrors which
scatter the electromagneticfield in a 4D spacetime. FollowingLifshitz [8], the effect ofevanescent wavesis accounted
for. Expressions of the forces as integrals over imaginary frequencies [8] are given in an appendix.
Thefollowingconventionisusedthroughoutthepaper: anyfunctionf(t)definedinthetimedomainanditsFourier
transform f[ω] are related through 1
dω
f(t) = f[ω]e−iωt
2π
Z
ONE MIRROR IN 2D VACUUM
Ina2Dspacetime(onetimecoordinatet,onespacecoordinatex), afreefieldisthesumoftwocounterpropagating
fields
Φ = ϕ(t−x)+ψ(t+x)
It can also be described by a twofold column matrix in the frequency domain
ϕ[ω]
Φ[ω] =
ψ[ω]
(cid:18) (cid:19)
We consider now that the field is scattered by a mirror located at point q
Φ = θ(q−x)(ϕ (t−x)+ψ (t+x))+θ(x−q)(ψ (t+x)+ϕ (t−x))
in out in out
where ϕ and ψ are the input fields, ϕ and ψ the output fields (see Figure 1).
in in out out
FIG.1. Onemirror scatters the two counterpropagating fields.
With the hypothesis of a perfect reflection, the field vanishes on the mirror
ϕ (t) = −ψ (t+2q) ψ (t) = −ϕ (t−2q)
out in out in
Equivalently
ϕ [ω] = −ψ [ω]e−2iωq ψ [ω] = −ϕ [ω]e2iωq
out in out in
1
The notation used in the original paper for Fourier transforms has been changed to a more convenient one.
2
Inthe generalcaseofapartiallytransmittingmirror,the scatteringofthe fieldisdescribedbyafrequencydependent
S matrix
s[ω] r[ω]e−2iωq
Φ [ω]=S[ω]Φ [ω] S[ω]= (3)
out in r[ω]e2iωq s[ω]
(cid:18) (cid:19)
We assume that the S matrix obeys the following conditions. The values of S are real in the temporal domain
s[−ω]=s[ω]∗ r[−ω]=r[ω]∗ (4)
The scattering is causal [13]
s(t)=r(t)=0 for t<0 (5)
s[ω] and r[ω] are analytic and regular for Imω >0
The scattering matrix is unitary (dissipation inside the mirror is neglected)
S[ω]S[ω]† =1
|s[ω]|2+|r[ω]|2 =1 s[ω]r[ω]∗+r[ω]s[ω]∗ =0 (6)
The mirror is transparent at high frequencies and the reflectivity decreases sufficiently rapidly
s[ω]→1 ω|r[ω]|→0 for ω →∞ (7)
Because of the transparency condition (7), the reflectivity will provide a natural regulator to the ultraviolet diver-
gence of the vacuum energy. The perfectly reflecting mirror corresponds to r = −1 and s = 0 at all frequencies and
does not obey this condition. So, it will be preferable to consider the perfect mirror as the limit of a model obeying
the conditions (4-7). This amounts to consider the effect of non zero reflection delays and to let them go to zero at
the end of the calculations.
Quantum field theory provides the field correlation functions corresponding to the vacuum state. We will use the
symmetric correlation functions which are the mean values of the anticommutators
c(t)
h{ϕ′(t+τ),ϕ′(t)}i =h{ψ′(t+τ),ψ′(t)}i=
2
h{ϕ′(t+τ),ψ′(t)}i =0
ϕ′(t)=∂ ϕ(t) (8)
t
c[ω]
iωiω′h{ϕ[ω],ϕ(ω′)}i =iωiω′h{ψ[ω],ψ(ω′)}i=2πδ(ω+ω′)
2
iωiω′h{ϕ[ω],ψ(ω′)}i =0
with
¯h 1 1
c(τ)= lim +
ǫ→0+ 2π "(ǫ+iτ)2 (ǫ−iτ)2#
c[ω]=h¯|ω| (9)
As usual in the local formulation, one evaluates the force as the difference between the radiation pressures exerted
upon the left and right sides of the mirror. In a 2D spacetime, the component T of the stress tensor is equal to the
xx
energy density and one gets
F =e −e
L R
e = ϕ′ 2(t−q)+ψ′ 2(t+q) e = ψ′ 2(t+q)+ϕ′ 2(t−q) (10)
L in out R in out
Substituting the output fie(cid:10)lds in terms of the input(cid:11)fields and u(cid:10)sing the unitarity conditi(cid:11)on, one obtains
c(0)
ϕ′ 2(t−q) = ψ′ 2(t+q) = ϕ′ 2(t−q) = ψ′ 2(t+q) =
out out in in 4
Hence (cid:10) (cid:11) (cid:10) (cid:11) (cid:10) (cid:11) (cid:10) (cid:11)
∞ dω
e =e = c[ω] (11)
L R 2π
Z0
Theradiationpressuresexerteduponthe twosidesofthemirrorareinfinite. However,they canceleachotherexactly
and the net force is zero. This is connected to the property that the impulsion density is zero in the vacuum state
of a free field while the energy density is infinite. Note that the cancellation between infinite quantities is obtained
inside a single physical situation and not by comparing two different situations.
3
TWO MIRRORS IN 2D VACUUM
We study now the situation where two pointlike mirrors are placed in the 2D vacuum at positions q and q (see
1 2
Fig.2).
FIG.2. Two mirrors scatter thetwo counterpropagating fields.
As is usual in the theory of a Fabry Perot cavity, the global scattering may be deduced from the elementary
scattering matrices given by (3) and associated with the two mirrors
ϕcav[ω] = s1[ω] r1[ω]e−2iωq1 ϕin[ω]
ψout[ω] r1[ω]e2iωq1 s1[ω] ψcav[ω]
(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19)
ϕout[ω] = s2[ω] r2[ω]e−2iωq2 ϕcav[ω]
ψcav[ω] r2[ω]e2iωq2 s2[ω] ψin[ω]
(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19)
One can solve these equations to express the outcoming and the intracavity fields in terms of the input ones
Φ [ω]=S[ω]Φ [ω]
out in
Φ [ω]=R[ω]Φ [ω]
cav in
The global scattering matrix S[ω] and the resonance matrix R[ω] are given by
s [ω]s [ω]
S [ω]=S [ω]= 1 2
11 22 d[ω]
s [ω]2r [ω]eiωq
S [ω]=r [ω]e−iωq+ 2 1
12 2 d[ω]
s [ω]2r [ω]eiωq
S [ω]=r [ω]e−iωq+ 1 2 (12)
21 1 d[ω]
R [ω]= s1[ω] R [ω]= s2[ω]r1[ω]e−2iωq1
11 d[ω] 12 d[ω]
R [ω]= s1[ω]r2[ω]e2iωq2 R [ω]= s2[ω] (13)
21 d[ω] 22 d[ω]
where
d[ω] = 1−r[ω]e2iωq r[ω] = r [ω]r [ω] q = q −q
1 2 2 1
The matrices S and R are real in the time domain, they are causal functions and the matrix S is unitary.
Substituting the output fields in terms of the input fields and using the unitarity of the matrix S[ω], one deduces
that the energy densities e and e , again defined by (10), are still givenby (11). But the intracavity energy density
L R
e is different
cav
∞ dω
e = ϕ′ 2 +ψ′ 2 = c[ω]g[ω]
cav cav cav 2π
Z0
(cid:10)|R |2+|R (cid:11)|2+|R |2+|R |2 1−|r[ω]|2
g[ω]= 11 12 21 22 =
2 |1−r[ω]e2iωq|2
Thefunctiong[ω](calculatedfromequation(13))describesanenhancement(resp. suppression)ofvacuumfluctuations
seen inside the cavity at frequencies inside (resp. outside) the Airy peaks [14,15]. One deduces the Casimir force as
the difference between the outer and inner radiation pressures
4
F =e −e =F F =e −e =−F (14)
1 L cav 2 cav R
∞ dω
F = c[ω](1−g[ω]) (15)
2π
Z0
InthestandardderivationoftheCasimirforce,oneconsidersmirrorswhichareperfectlyreflectingatallfrequencies.
In this case, the Airy peaks reduce to Dirac distributions
π π
1−g[ω] = 1− δ(ω−n )
q q
n
X
The integral (15) is then regularized by introducing a cutoff in the mode density and computed by applying the
Euler-Mac Laurin summation rule [6]
B π2 B π4 ¯hω
F = 2 ∂ + 4 ∂3 +...
2! q2 ω 4! q4 ω 2π
(cid:26)(cid:18) (cid:19) (cid:27)ω=0
where the B are the Bernouilli numbers B = 1; B = − 1 ; .... This leads to the expression (1) of the Casimir
2k 2 6 4 30
force.
However, the introduction of a cutoff in the mode density is not necessary since the integral (15) is finite as soon
as the reflectivity functions obey the conditions (4-7). One can write
∞ dω −r[ω]e2iωq
F = c[ω] +c.c. (16)
2π 1−r[ω]e2iωq
Z0
As c[ω] is an even function of ω, one deduces from the reality condition (4)
∞ dω −r[ω]
F = c[ω] (17)
2π e−2iωq−r[ω]
Z−∞
One obtains the force as a sum
∞ dω
F = F(ℓ) F(ℓ) =− e2ℓiωqrℓ[ω]c[ω] (18)
2π
ℓ>0 Z−∞
X
where each term F(ℓ) corresponds to a number ℓ of roundtrips inside the cavity. This term can be transformed into
a convolution product in the time domain, more precisely into an integral over ℓ reflection delays t , ... t
1 ℓ
∞ ∞
F(ℓ) =− dt r(t )... dt r(t )c(2ℓq+t +...+t ) (19)
1 1 ℓ ℓ 1 ℓ
Z0 Z0
The reflection delays are positive and the two point correlation function c is evaluated at distances between the
points greater than 2q. The expression of the force comes out free from the divergences usually associated with the
infiniteness of the vacuum energy and the regulator ǫ may be omitted in the expression (9) of c.
In the limiting case where the reflection delays (t , ... t ) are much shorter than the distance q between the two
1 ℓ
mirrors(largedistance approximation),one canneglect them inthe function c(2ℓq+t +...+t ). Itfollows that the
1 ℓ
force depends only upon the reflectivities evaluated at zero frequency
F = − rℓc(2ℓq) r = r[0] = r [0]r [0]
0 0 1 2
ℓ>0
X
that is
¯h
F = ζ (2) (20)
4πq2 r0
∞ xℓ
ζ (p)= (21)
x ℓp
ℓ=1
X
5
If the two mirrors are perfectly reflecting at zero frequency, one finds the expression (1) for the Casimir force
ζ (2)= π2 .
1 6
(cid:16) The equat(cid:17)ion (18) allows one to evaluate the correction to the limit of ideal mirrors. It can be written as a formal
series
¯h
F(ℓ) = −i(2ℓq−i∂ )−1 ωr[ω]ℓ +c.c.
ω
2π ω=0
= ¯h n(2ℓq−i∂ )−2r[ω(cid:0)]ℓ (cid:1)+oc.c. (22)
ω
2π ω=0
n o
When r[ω] is a very flat function, one recovers the large distance approximation where the force depends upon the
value r evaluated at zero frequency. The terms of the formal series which depend upon the derivatives of r[ω] may
0
be considered as corrections to this approximation.
These discussions show that the reflectivity function r[ω] plays the same role as a regulator for the mode density.
However, it is a regulator obeying the conditions (4-7) which cannot be real for real frequencies. We show in the
appendix that it is possible to write the force as an integral over imaginary frequencies and that r[iω] appears as a
real regulator [8].
CASIMIR ENERGY
The local formulationled us to the expressionof the force. One can deduce a Casimir energy U by integrating this
force (see equation (14))
dU =−F dq −F dq =Fdq (23)
1 1 2 2
We want now to compute this Casimir energy and to compare it with the integrated field energy [4].
One deduces from equation (16)
∞ dω−¯h∆[ω]
F =∂ U U = (24)
q
2π 2
Z0
1−r[ω]e2iωq
∆[ω]=iLog (25)
1−r[ω]∗e−2iωq
One deduces respectively from the reality condition (4) and from the transparency condition (7)
∆[0] = 0 ω∆[ω] → 0 for ω → ∞
so that the energy can be written
∞ dω¯hω
U = ∂ ∆[ω] (26)
ω
2π 2
Z0
This is the standard form for the phase shift representation of the Casimir energy [3]. Actually, one checks from the
expression (12) of the S matrix that
detS[ω] = detS [ω]detS [ω]ei∆[ω]
1 2
where S [ω] and S [ω] are the elementary S matrices associated with the two mirrors; ∆[ω] is the q−dependent part
1 2
of the sum of the two phase shifts at frequency ω.
Now, one gets from equation (25)
(2qr[ω]−i∂ r[ω])e2iωq
∂ ∆[ω]= ω +c.c.
ω 1−r[ω]e2iωq
=−(1−g[ω])(2q+∂ δ)−g[ω]sin(2ωq+δ)∂ Log 1−ρ2 (27)
ω ω
where δ and ρ are the frequency dependent phase and modulus of the reflectivit(cid:0)y funct(cid:1)ion r. Usually, the Casimir
effect is computed for frequency independent reflectivities. In this case, one deduces that the Casimir energy U is
identical to the integrated field energy [4]
U = q(e −e ) = −qF
cav L
Thisresultiscompatiblewiththedefinition(23)ofU sincetheforcescalesasq−2 inthiscase[7]. Butthereflectivities
have to be frequency dependent in order to fulfill the transparencycondition (7). The resulting terms in the Casimir
energy can be considered as corrections associated with the reflection delays upon the mirrors.
6
CASIMIR FORCE FOR NON ZERO TEMPERATURES
Now,wederivetheexpressionoftheCasimirforcefornonzerotemperatures[3,4]. Wesupposethatthereflectivities
are independent of the temperature. Therefore, we have only to change the correlation function of the input fields.
The Casimir force is given by the integrals (15) or (17) with a modified function c
1
c [ω]=c [ω]n [ω] n [ω]= (28)
T 0 T T
tanh h¯|ω|
2T
(cid:16) (cid:17)
where c [ω] corresponds to the zero temperature and n [ω] is the mean number of thermal photons per mode. The
0 T
equations (19) are still valid with a function c [ω] which is the Fourier transform of
T
¯h α2 πT
c (τ) = − α =
T πsinh2(ατ) ¯h
It is known that the correlation function c is the same as for an accelerated vacuum [16–18], which also suggests
T
that there are relations between the Casimir effect for non zero temperatures and the physics of black holes [19].
At the large distance approximation, one gets
F = − rℓc (2ℓq)
0 T
ℓ>0
X
At the low temperature limit, one has the same results as previously. At the high temperature limit, one obtains an
exponentially small force since
4π 2πTτ
c (τ) ≈ − T2exp − for Tτ ≫ ¯h
T
¯h ¯h
(cid:18) (cid:19)
The thermal contribution exactly cancels the vacuum contribution for temperatures T higher than h¯.
q
In the general case (no approximation on the distance), we obtain the Casimir free energy F by integrating the
force [4]
∞ dω−¯h∆[ω]
F =∂ F(q,T) F = n [ω] (29)
q T
2π 2
Z0
where ∆[ω] is still given by (25). Noting that (S is the entropy)
U =F +TS =F −T∂ F
T
ω∂ n [ω]=−T∂ n [ω]
ω T T T
one also obtains
∞ dω−¯h∆[ω]
U = ∂ (ωn [ω])
ω T
2π 2
Z0
An integration by parts then leads to the modified phase shift representation
∞ dω¯hω
U = n [ω]∂ ∆[ω] (30)
T ω
2π 2
Z0
Usingequation(27),onereachesthesameconclusionsaspreviouslyaboutthecomparisonbetweentheCasimirenergy
and the integrated field energy.
TWO MIRRORS IN THE ELECTROMAGNETIC VACUUM
We considernowthattwoparallel,planeandinfinite mirrorsareplacedinthe vacuumstate ofthe electromagnetic
field (plate separation q along the x direction). The scattering on each mirror is still described by a S matrix which
obeys conditions of reality, unitarity, causality and high frequency transparency. Now, the reflection coefficients
7
associated with the two mirrors depend not only upon the frequency, but also upon the propagation direction and
the polarization. We will denote
γ = cosϕ κ = ωcosϕ = ωγ κ′ = ωsinϕ
where ϕis the incidence angle,κ the wavevectoralongthe x directionandκ′ the lengthofthe transversewavevector.
The calculations of the Casimir force (measured per unit area) are mostly the same as in the 2D case, but one
obtains it as a sum over the two polarizations p and p
1 2
F = F +F
p1 p2
Each contribution F can be written as an integral over the modes
p
dω dγ
F = ω2 θ(ω)c[ω](1−g [ω,κ])γ2 (31)
p p
2π 2π
Z Z
The function c is the same as in the 2D case. The function g is computed as previously, but its resonances are now
p
determined by the wavevector measured along x
1−|r [ω,κ]|2
g [ω,κ]= p
p |1−r [ω,κ]e2iκq|2
p
r [ω,κ]=r [ω,κ]r [ω,κ]
p 1p 2p
where r and r are the reflectivities associated with the two mirrors. In comparison with the 2D computation,
1p 2p
there is in equation (31) an extra geometrical factor γ2 which is the ratio between the component T of the stress
xx
tensor (which gives the radiation pressure upon the mirror) and the energy density [20].
If the mirrors are supposed perfectly reflecting at all frequencies, Dirac distributions appear in the formula (31)
which can be computed by a regularizationand the application of the Euler-Mac Laurin summation rule [6]
B π2 B π4 ¯hκ2(K−κ)
F = 2 ∂ + 4 ∂3+... (32)
p 2! q2 κ 4! q4 κ 4π2
(cid:26)(cid:18) (cid:19) (cid:27)κ=0
The constant K goes to infinity with the cutoff frequency. But the even derivatives do not appear in the Euler-Mac
Laurin sum and this constant is eliminated, which leads to expression (2) of the force.
In the general case of a frequency dependent reflectivity, one writes
dωdγ −r [ω,ωγ]
F = θ(ω)c[ω]ω2γ2 p +c.c. (33)
p 2π 2π e−2iωγq−r [ω,ωγ]
p
Z Z
Now, one must specify more precisely the integrationpath for γ. Without further investigation,it could be admitted
that it coincides with the segment [0,1[. Using the same methods as for the 2D computation, one shows however
that this integration path leads to a Casimir force which diverges at the limit of a perfectly reflecting mirror: there
remains a contribution to the force analogous to the constant K appearing in (32).
Actually, the integration path for γ must also include the upper part of the imaginary axis ]+i∞,0]. Imaginary
values of γ not too far from 0 correspond to values of κ′ greater than, but not too much greater than ω. They are
associated with vacuum fluctuations which enter the system through the side boundaries of the mirror and which
contribute to the field outside the mirrors as evanescent waves. We will suppose that r [ω,κ] may be analytically
p
continued [21] for values of κ in the quadrant [Reκ>0,Imκ>0] and also that the contribution of the evanescent
waves to the force is given by the analytical continuation of the ordinary expression [22]. Imaginary values of γ
farther from0 correspondto largervalues of κ′. They areassociatedto “closed”input ports and haveactually a zero
contribution to the force. Assuming that |r [ω,κ]| decreases sufficiently fast for large values of κ and that [23]
p
|r [ω,κ]| < 1 for ω real and [Reκ>0,Imκ>0]
p
one moves the integration path for γ to ]1,∞[ so that
∞ dω ∞ dγ r [ω,ωγ]
F = c[ω]ω2γ2 p +c.c. (34)
p 2π 2π e−2iωγq−r [ω,ωγ]
Z0 Z1 p
This result is regular at the limit of perfect reflection. As for the 2D computation, one writes the force (34) as a
sum over a number ℓ of roundtrips
8
∞ dκ
F = F(ℓ) F(ℓ) = e2ℓiκq¯hκ2R(ℓ)[κ]
p p p 2π p
ℓ>0 Z−∞
X
κ dω
R(ℓ)[κ]= r [ω,κ]ℓ (35)
p 2π p
Z0
(we have used a reality condition r [ω,κ]∗ =r [−ω,−κ]). This expression can be written as a formal series
p p
¯h
F(ℓ) = i(2ℓq−i∂ )−1 κ2R(ℓ)[κ] +c.c.
p 2π κ p κ=0
n (cid:16) (cid:17)o
At the limit of a large separation, we retain the terms which depend upon the reflectivities evaluated around zero
frequency but not those which depend upon the derivatives of the reflectivities. As
κ κ dω
R(ℓ)[κ] = r [κ,κ]ℓ− ω∂ r [ω,κ]ℓ
p 2π p 2π ω p
Z0
we write
κ
R(ℓ)[κ] ≈ rℓ r = {r [κ,κ]}
p 2π 0 0 p κ→0
The value r is evaluated at zero frequency and at normalincidence. This value is the same for the two polarizations
0
and one obtains finally the total Casimir force as
6h¯
F = rℓC(2ℓq) C(τ)= (36)
0 π2τ4
ℓ>0
X
that is
3h¯
F = ζ (4) (37)
8π2q4 r0
where ζ (p) is defined by (21). At the limit of perfectly reflecting mirrors, one recovers the expression (2) for the
x
Casimir force ζ (4)= π4 .
1 90
As in the 2D(cid:16)case, the e(cid:17)xpressions of the force can be written in terms of the reflectivities evaluated for imaginary
frequencies (see the appendix).
A phase shift representationcan be obtained for any distance by integrating the force. One writes the energy U as
a sum over the ordinary and evanescent waves
∞ dω 1 dγ¯hγω2
U = (−∆[ω,ωγ])
2π 2π 2
Z0 Zi∞
where ∆[ω,κ] is obtained by summing all the phase shifts at frequency ω and wavevectorκ
∆[ω,κ]=∆ [ω,κ]+∆ [ω,κ]
p1 p2
1−r [ω,κ]e2iκq
∆ [ω,κ]=iLog p
p 1−r [ω,κ]∗e−2iκq
p
Moving the integration path leads to expressions in terms of the unphysical waves
∞ dω ∞ dγ¯hγω2
U = ∆[ω,ωγ]
2π 2π 2
Z0 Z1
These expressions can be written in terms of ∂ ∆[ω,ωγ] by an integration by parts
ω
∞ dω 1 dγ¯hγω3
U = ∂ ∆[ω,ωγ]
ω
2π 2π 6
Z0 Zi∞
∞ dω ∞ dγ¯hγω3
= (−∂ ∆[ω,ωγ])
ω
2π 2π 6
Z0 Z1
9
Writing ∂ ∆[ω,ωγ] as in equation (27), one obtains the Casimir energy as the sum of the integratedfield energy and
ω
of corrections associated with the reflection delays upon the mirrors. The integrated field energy is here
qF
U = −
3
In the limit of a perfect reflection, this relationis compatible with the definition (23) of U [7] since the force scales as
q−4.
Theforcefornonzerotemperaturesisgivenbyequations(33-34)withthefunctioncmodifiedaccordingtoequation
(28). It is also given by equation (35) with
κ dω
R(ℓ)[κ] = r [ω,κ]ℓn [ω]
p 2π p T
Z0
At the large distance approximation, one obtains
¯h α πT
F = rℓC(2ℓq) C(τ) = ∂2 α =
0 π2 τtanh(ατ) ¯h
ℓ>0
X
The low temperature limit gives the known result. For high temperatures, one obtains by omitting exponentially
small terms
2T
C(τ)≈ for Tτ ≫¯h
πτ3
T
F = ζ (3) (38)
4πq3 r0
where ζ (p) is defined by (21). High temperatures correspond to a classical limit so that the expression does not
x
contain h¯. For a perfect mirror, one recovers the known result [4].
In the general case, one also obtains a phase shift representation of the free energy
∞ dω 1 dγ¯hγω2
F = n [ω](−∆[ω,ωγ])
T
2π 2π 2
Z0 Zi∞
∞ dω ∞ dγ¯hγω2
= n [ω]∆[ω,ωγ]
T
2π 2π 2
Z0 Z1
Acknowlegdements
We thank C. Cohen-Tannoudji, C. Fabre, E. Giacobino, S. Haroche, A. Heidmann and H.M. Nussenzveig for
stimulating comments.
APPENDIX A: EXPRESSIONS OF THE FORCES AS INTEGRALS OVER IMAGINARY FREQUENCIES
Following the computation by Lifshitz of the Casimir-Polder forces [8], one can write the forces as integrals over
imaginary frequencies.
In the 2D case, one first shows that |r[ω]| remains smaller than 1 for any ω in the upper half plane Imω > 0.
The modulus|r[ω]| is less than 1 on the real axis (from the unitarity and reality conditions); it goes to 0 for large
values of ω (from the transparency condition); as r[ω] is analytic for Imω >0, the conclusion is a consequence of the
Phragm´en-Lindel¨oftheorem [13].
Therefore,the integrandappearing in (17) has no poles in the upper half plane. Using the transparencycondition,
one can then move the integration path [0,∞[ to the upper part of the imaginary axis [0,i∞[
∞ ¯hω r[iω]
F = dω +c.c.
2π e2ωq−r[iω]
Z0
Furthermore, r[iω] is real since it is the Laplace transform of the causal and real function r
∞
r[iω] = dt r(t)e−ωt
Z0
10