Classical Casimir interaction of perfectly conducting sphere and plate

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by  Giuseppe Bimonte

Classical Casimir interaction of perfectly conducting sphere and plate.
Giuseppe Bimonte
Dipartimento di Fisica E. Pancini, Universita` di Napoli Federico II,
Complesso Universitario di Monte S. Angelo, Via Cintia, I-80126 Napoli, Italy and
INFN Sezione di Napoli, I-80126 Napoli, Italy∗
We study the Casimir interaction between perfectly conducting sphere and plate in the classical
limitofhightemperatures. Bytakingthesmall-distanceexpansionoftheexactscatteringformula,
wecomputetheleadingcorrectiontotheCasimirenergybeyondthecommonlyemployedproximity
forceapproximation. WefindthatforasphereofradiusRatdistancedfromtheplatethecorrection
is of the form ln2(d/R), in agreement with indications from recent large-scale numerical computa-
tions. Wedevelopafast-convergingnumericalschemeforcomputingtheCasimirinteractiontohigh
precision,basedonbisphericalpartialwaves,andweverifythattheshort-distanceformulaprovides
7 precise values of the Casimir energy also for fairly large distances.
1
0 PACSnumbers: 12.20.-m,03.70.+k,42.25.Fx
2

b
I. INTRODUCTION Abreakthroughoccurredintheearly2000’swhen,gen-
e
eralizing early findings by Balian and Duplantier [8] and
F
Over the last two decades a new wave of precision ex- Langbein[9],anexactscatteringformulafortheCasimir
3
interaction between two (or more) dielectric objects of
2 periments spurred a strong resurgence of interest in the
any shape was finally worked out [10–12]. Shortly later,
Casimireffect[1],thetinylong-rangeforcebetween(neu-
scattering formulae have been derived for the Casimir
] tral)macroscopicpolarizablebodies,duetoquantumand
h force and the power of heat transfer between two bod-
thermal charge fluctuations within the bodies. For re-
p ies at different temperatures [13–15]. At thermal equi-
views we address the reader to Refs.[2–5].
-
t A distinctive feature of the Casimir force is its non- librium, the scattering formula has the form of a sum
n
overso-calledMatsubara(imaginary)frequenciesoffunc-
a additivity, a consequence of the inherent many-body
tional determinants involving the T-operators of the in-
u characteroffluctuationforces. Thispropertyenormously
q complicates the task of computing the Casimir force in volved bodies. In principle, the formula allows to ex-
[ actly compute the Casimir interaction between two bod-
arbitrary geometries. Indeed, Casimir himself was able

ies whose T-operators are either known (as it is the case
2 tocomputetheforceonlyforthehighlyidealizedgeome-
for planes, spheres or cylinders [16]) or can be computed
v try of two perfectly conducting plane-parallel surfaces at
numerically (for example for periodic rectangular grat-
1 zerotemperature. Amajorstepforwardwasmadeafew
6 years later by Lifshitz [6], who derived an exact formula ings [17]). Numerical schemes inspired by the scattering
4 formalismhavebeendevelopedoverthelastfewyears,by
for the Casimir interaction between two plane-parallel
6 whichitisnowpossibletoestimatenumericallywithrea-
dielectric slabs at finite temperature. After these early
0 sonableprecisiontheCasimirforcebetweenobjectswith
successes in the planar geometry, the task of computing
.
1 complicatedshapes(see[18]andRefs. therein). Veryre-
the force in non-planar geometries remained untractable
0 cently,thescatteringformulahasbeenevaluatedtoyield
for decades. This limitation has represented a serious
7 the exact classical Casimir interaction between a sphere
practical problem because, in order to avoid the insur-
1
and a plate subjected to Drude (Dr) boundary condi-
: mountable parallelism issues posed by two plane-parallel
v tions [19], and between two perfectly conducting spheres
surfaces separated by a sub-micron separation, practi-
i in four Euclidean dimensions [20].
X cally all Casimir experiments adopt non planar geome-
tries,likethesphere-plate. Foralongtime,theonlytool The scattering formula undoubtedly constitutes a ma-
r
a to estimate the Casimir force between two non-planar jor advancement, however its practical utility has been
gently curved surfaces has been the old-fashioned Prox- limitedsofarbecausethereisnoknownmethodtocom-
imity Force Approximation (PFA) [7], which expresses pute efficiently the T-operator for material surfaces of
theCasimirforceastheaverageoftheplane-parallelforce arbitraryshapes. Eveninthefewcases(spheresorcylin-
over the local separation between the surfaces. Being an ders) for which the scattering operator is known exactly,
uncontrollable approximation, the problem remained of useofthescatteringformulaishamperedbyitsslowcon-
addressing the systematic theoretical error engendered vergence rate. Consider as an example the experimen-
by the PFA, an issue of great importance for a proper tally important geometry of a metallic sphere of radius
interpretation of modern high precision Casimir experi- R at a minimum distance d from a plane. It has been
ments. found that to accurately compute the Casimir interac-
tion, it is necessary to include in the scattering formula
all multipoles up to an order l (cid:38) R/d or so. At the
max
moment of this writing, the largest numerical computa-
∗Electronicaddress: [email protected] tion [21] has lmax = 100, which allows to estimate the
2
Casimir interaction only for aspects ratios d/R smaller correction ∼ d/R, a larger logarithmic ∼ d/R ln(d/R)
than 0.02 or so. For comparison, it should be consid- correction had been found in [24]. A successive recalcu-
eredthatinordertoincreasethemagnitudeoftheforce, lationbysomeoftheauthorsof[24]detectedasignmis-
Casimirexperimentsusealargesphereatsmalldistances take in their original computation, and finally led to full
from the plate, with typical aspects ratios of the order agreementwiththeDEexpansionalsoinemandNcases.
of one thousandth. For so small aspect ratios, a precise The DE for a D and N scalar at zero and finite tempera-
computation of the Casimir interaction requires multi- tureinanynumberofspace-timedimensionswasworked
pole orders of several thousands, which are out of reach outin[28],whiletheexperimentallyimportantcaseofdi-
for now. electriccurvedsurfacesatfinitetemperatureispresented
in [29]. It is worth stressing that the NTLO correction
Fromapracticalstandpoint,themainroleoftheexact
predicted by the DE is also in full agreement with the
scattering formula has perhaps been to serve as a guide
short distance expansion of the exact sphere-plate and
towards systematically deriving approximation schemes
sphere-sphere classical Casimir energies both for Dr bc
in various regimes, going beyond the old PFA. For sur-
[19]aswellasforPbcinfourEuclideandimensions[20].
facescarryingcorrugationsofsmallamplitude,asystem-
The DE has been also used to study curvature effects in
atic perturbative expansion of the Casimir interaction
the Casimir-Polder interaction of a particle with a gen-
in powers of the small corrugation amplitude has been
tly curved surface [30, 31]. The same method has been
workedout[22]. Severalresearchershaveinsteadendeav-
usedveryrecentlytoestimatetheshiftsoftherotational
oredtocomputecurvaturecorrectionstotheCasimirin-
levels of a diatomic molecule due to its van der Waals
teraction, in the experimentally important limit of small
interaction with a curved dielectric surface [32].
surface separations. This is clearly a problem of out-
most practical importance, for the purpose of interpret- In this paper we study the sphere-plate Casimir inter-
ing current small-distance precision experiments. There action for P bc, in the high-temperature (HT) or classi-
are presently two approaches to compute curvature cor- cal limit. In this limit, the Casimir interaction reduces
rections to the PFA. The first one consists in working to the zero-frequency Matsubara-term of the full finite-
outtheasymptoticsmall-distanceexpansionoftheexact temperaturescatteringformula. Thezero-frequency(i.e.
scattering formula. The method is rigorous, but it has the classical) term becomes dominant for sphere-plate
thedrawbackthattheexpansionhastobeworkedoutab separations d that are larger than the thermal length
initio for each model, and for each surface geometry. By λ = (cid:126)c/(k T) (λ = 7.5 microns at room temper-
T B T
following this route, the next-to-leading-order (NTLO) ature). A perfect conductor constitutes the idealized
correction to the Casimir energy has been computed for limit of a superconductor, i.e. a conductor with per-
the cylinder/plate and the sphere/plate geometries, ini- fect Meissner effect [33]. Ohmic metals are better mod-
tially for a free scalar field obeying Dirichlet (D) bound- eled as Drude conductors, since normal metals do not
aryconditions(bc)[23],andthenfortheelectromagnetic impede static magnetic fields. Quite surprisingly, sev-
(em) field with perfect-conductor (P) bc [24]. Later the eral short-distance precision experiments (see [34] and
same approach was applied to a free scalar field obey- Refs. therein) with metallic plates at room temperature
ing D, Neumann (N) and mixed ND bc on two parallel areinbetteragreementwithasuperconductor-likemodel
cylinders [25]. (i.e. the plasma model) for the dielectric function of the
plates, while a single large distance experiment [35] fa-
An alternative route to compute the NTLO correction
vors the Drude model. For a thorough discussion of this
to PFA assumes that the Casimir energy functional ad-
delicateproblemweaddressthereadertothemonograph
mitsaderivativeexpansion(DE)inpowersofderivatives
[4].
ofthesurfacesheightprofiles. ThecoefficientsoftheDE
are computed by matching the DE with the perturba- The HT limit of the Casimir interaction for P bc has
tive expansion of the Casimir-energy functional in the beeninvestigatedin[36],wheretheasymptoticsmalldis-
common domain of validity (for details see [26, 27]). An tance expansion of the scattering formula was shown to
advantageofferedbytheDE,incomparisonwiththepre- reproduce in leading order the PFA. The authors of [36]
vious approach, is that once the DE is worked out for a did not study though corrections to PFA. Determining
specificmodel,itcanbestraightforwardlyappliedtosur- the form of the the NTLO correction is an interesting
facesofanyshape. In[26]theDEwasworkedoutforaD problem, for the following reason. In the HT limit, the
scalar field in the cylinder and sphere/plate geometries, Casimir interaction for P bc is mathematically equiva-
giving results in agreement with the asymptotic small- lent to the sum of the classical Casimir energies for a Dr
distanceexpansionofthescatteringformulain[23]. The or D scalar field (depending on whether the plates are
DE for the em field with P bc, as well as for a scalar grounded or not) plus a N scalar field. The HT limit
field obeying N and mixed DN bc was later worked out of the sphere-plate Casimir interaction for D and Dr bc
in [27], where the DE was also generalized to the case have been computed exactly not long ago [19]. How-
of two curved surfaces. Interestingly, the NTLO correc- ever, the N and P cases have been intractable so far.
tion for the sphere/plate geometry with P bc obtained Working out the NTLO correction to PFA for these two
in [27] by using the DE was in disagreement with the re- models is of great interest, because in the HT limit the
sult reported in [24]: while the DE predicted an analytic perturbativekernelsforNandPbcdisplayasingularbe-
3
havior for small in-plane momenta, invalidating the DE term is taken with weight 1/2. In Eq. (1), Tˆ(j) de-
[19, 28] [42]. The DE has been shown to fail also for the notes the T-operator of object j, evaluated for imagi-
plasma model in the HT limit in [37]. As a result, the nary frequency iξ , and Uˆ is the translation operator
n
analytic form of the NTLO for N and P bc is so far un- that translates the scattering solution from the coordi-
known. A large-scale numerical computation including nate of one object to the one of the other object. When
up to 5000 partial waves [38] suggests a ln2(d/R) form considered in a plane-wave basis |k,Q(cid:105), where k is the
for the NTLO term. However, the data of [38] appeared in-plane wave-vector and Q = E,M is the polarization
to support a ln2(d/R) also for the Dr model, and from (E and M denote, respectively, transverse magnetic and
the exact solution in [19] we now know that the correct transverse electric modes), the translation operator Uˆ is
NTLO correction for the Dr model is actually a ln(d/R) diagonal,withmatrixelementse−dqn wheredisthemin-
(cid:112)
term, in accordance with the DE. To resolve the matter, imum distance between the objects, q = k2+ξ2/c2
n n
itisclearlyofinteresttoseeiftheNTLOforNandPbc withk =|k|,andcthespeedoflight. Thisshowsthatin
canbeworkedoutanalytically. AswesaidabovetheDE the HT limit k T (cid:29)(cid:126)c/d, the free energy is dominated
B
cannot help now, and therefore we attacked the problem by the first term n=0 in the sum Eq. (1):
using the method based on the asymptotic expansion of
1
the scattering formula [23, 25]. We find that the NTLO F =−k T Φ, Φ=− Trln[1−Mˆ(0)]. (2)
is indeed of the ln2(d/R) form as it was argued in [38], HT B 2
and we determine its coefficient. We also develop an ef- Here, Φ is a dimensionless temperature-independent
ficient numerical implementation of the exact scattering function, depending on the static em response functions
formula, based on the use of bispherical multipoles [19]. of the two bodies. Since the free-energy is proportional
The fast convergence of our scheme allowed us to probe to the temperature, the HT (or classical) limit of the
extremely small aspect ratios down to d/R = 10−5. We Casimir interaction has a purely entropic character.
verify that the approximate formula obtained by taking We are interested in the classical Casimir interaction
the asymptotic short-distance expansion of the Casimir F of a sphere of radius R placed at a (minimum) dis-
HT
interactionisactuallyveryaccurateuptorelativelylarge tance d from a plate, bot subjected to P bc. We take
values of the aspect ratio. the surface of the plate to coincide with the (x,y) plane
The paper is organized as follows: in Sec II we discuss of a cartesian coordinate system, whose z axis passes
theHTlimitofthescatteringformula,andbrieflyreview through the sphere center C (see Fig.1). We define the
the exact HT sphere-plate solution for D and Dr bc of aspect ratio x of the system as x = d/R. According to
[19]. InSec. IIIwecomputetheshort-distanceexpansion Eq. (2) the computation of F involves scattering of
HT
of the HT scattering formula in the sphere-plate geom- staticemfieldsbythetwosurfaces. Staticemfieldswith
etry for P bc, and we obtain an approximate formula E and M polarizations represent, respectively, electro-
for the Casimir interaction valid for small separations. static and magnetostatic fields which do not mix under
By taking its short-distance limit, we compute explicitly scatteringbyadielectricsurfaceofanyshape. Therefore
the leading correction of the Casimir interaction beyond modes with E and M polarizations give separate contri-
PFA. In Sec. IV we present a fast-convergent numerical butionstotheCasimirenergyF . Moreover,itiseasily
HT
scheme to compute the Casimir energy based on the use seen that in the static limit the em scattering problem
of bispherical multipoles, and compare our numerical re- is mathematically equivalent to the scattering problem
sults with the approximate formula derived in Sec. III. for a free scalar field obeying the Laplace Equation. For
In Sec. V we present our conclusions. perfect conductors, the bc obeyed by the scalar field are
as follows. For E polarization, the scalar field is sub-
jected to either D or Dr bc on the surfaces of the two
II. THE CASIMIR ENERGY IN THE bodies,dependingonwhethertheplatesaregroundedor
CLASSICAL LIMIT not [19, 39, 40], while for M polarization the scalar field
obeys N bc. The dimensionless function Φ(P) providing
theclassicalCasimirinteractionforPbccanbethusde-
We start from the general scattering formula [10–12]
composed as the sum of two independent contributions
for the Casimir free energy of two objects (denoted as 1
Φ(D/Dr) and Φ(N), corresponding respectively to a D/Dr
and 2) in vacuum:
and a N scalar field:
F =k T (cid:88)(cid:48)Trln[1−Mˆ(iξ )], Φ(P) =Φ(D/Dr)+Φ(N) . (3)
B n
n≥0
In the limit of vanishing separations x, the Casimir en-
ergy approaches the PFA limit:
Mˆ =Tˆ(1)UˆTˆ(2)Uˆ . (1) Φ(P) ζ(3)
Φ(D) =Φ(Dr) =Φ(N) = PFA = . (4)
PFA PFA PFA 2 8x
Here k is Boltzmann’s constant, T is the temperature,
B
ξ =2πnk T/(cid:126) arethe(imaginary) Matsubarafrequen- The exact expression of the functions Φ(D/Dr) was deter-
n B
cies, and the prime in the sum indicates that the n = 0 mined in [19] by taking advantage of the separability of
4
z Theenergyforungroundedperfect-conductorsisaccord-
ingly represented as:
Φ(P) =Φ(Dr)+Φ(D)+δΦ, (9)
while for grounded conductors we write:
R Φ(P)| =2Φ(D)+δΦ. (10)
gr
In the next Section we shall work out an asymptotic for-
mulaforδΦ,validinthelimitofsmallseparations,while
C in Sec. IV δΦ shall be computed numerically using the
exact scattering formula Eq. (1).
F
III. A SHORT-DISTANCE FORMULA FOR δΦ
BeforewestartthecomputationofδΦ, itisimportant
tonoticethat,duetothepresenceofthetraceinthegen-
eral scattering formula Eq. (1), the Casimir interaction
dependsonlyontheequivalenceclass[[M]]formedbyall
d matrices M that represent the operator Mˆ, where two
elements M and M(cid:48) of [[M]] differ by a similarity trans-
formation by an invertible matrix A: M(cid:48) =AMA−1.
x
The matrix M(N) for N bc is easily computed in a
spherical multipole basis with origin at the sphere center
FIG. 1: Geometry of a sphere and plate. Shown are the C. In this basis the regular and outgoing eigenfunctions
centerC ofthesphereanditsfocusF. Thesystemischarac- of the Laplace Equation have the familiar form φ(reg) =
lm
terizedbyitsaspectratiox=d/R. Thethinsolidanddashed rlY (θ,φ), and φ(out) = r−(l+1)Y (θ,φ), with l ≥ 0,
linescorrespondtocurvesofconstantbisphericalcoordinates lm lm lm
and m = −l,··· ,l. By rotational symmetry around the
µ and η respectively.
azimuthal axis zˆ, the matrix M(N) commutes with J
z
and hence is block diagonal. We let M(N|m) the block
Laplace Equation in bisperical coordinates [41]: corresponding to the value m of Jz. One finds:
Φ(D) =−1(cid:88)∞ (2l+1)ln[1−Z2l+1], (5) Mˆ(N|m) =(cid:34)(cid:34) l (l+l(cid:48))! (cid:18) 1 (cid:19)l+l(cid:48)+1(cid:35)(cid:35) ,
2 l+1 (l+m)!(l(cid:48)−m)! 2(1+x)
l=0
(11)
with l,l(cid:48) ≥ |m|. Apart from the factor l/(l + 1), the
1(cid:40)(cid:88)∞ matrix Mˆ(N|m) coincides with the corresponding matrix
Φ(Dr) =− (2l+1)ln[1−Z2l+1]
2 Mˆ(D|m) for D bc:
l=1
(cid:34)(cid:34) (l+l(cid:48))! (cid:18) 1 (cid:19)l+l(cid:48)+1(cid:35)(cid:35)
(cid:34) (cid:88)∞ 1−Z2l (cid:35)(cid:41) Mˆ(D|m) = (l+m)!(l(cid:48)−m)! 2(1+x) ,
+ln 1−(1−Z2) Z2l+1 , (6)
1−Z2l+1 (12)
l=1 EachblockMˆ(N|m) contributesseparatelytotheCasimir
where the parameter Z depends on the aspect ratio x: energy, and we denote by Φ(N) the corresponding con-
m
tribution to Φ(N). Of course, opposite values of m
(cid:112)
Z =[1+x+ x(2+x)]−1 . (7) give identical contributions to the Casimir energy, i.e.
Φ(N) =Φ(N). We can thus write Φ(N) as:
The parameter Z is less than one for all positive values m −m
of x, and as x increases from 0 to ∞, Z decreases mono- (cid:88)
Φ(N) =2 (cid:48)Φ(N) , (13)
tonically from 1 towards zero. m
Unfortunately, for N bc the Casimir interaction Φ(N) m≥0
cannot be computed exactly. We find convenient to in-
where the prime again denotes that the m = 0 term is
troduce the difference δΦ between the N and D energies
taken with a weight 1/2 and
Φ(N) and Φ(D):
m
1
δΦ=Φ(N)−Φ(D) . (8) Φ(mN) =−2Trln[1−Mˆ(N|m)]. (14)
5
A. Contribution of the modes with m=0. where l ≡ l . Next, for 0 < i ≤ s we perform on
s+1 0
the indices l the shift: l = l +l(cid:48), where we set l :=
i i i
Luckily enough the contribution Φ(N) of the m = 0 l0. For small separations x (cid:28) 1, the Casimir energy is
0
modes can be computed exactly. By a similarity trans- dominated [23–25] by multipoles such that:
formation with the diagonal matrix A =(l+1)δ the
ll(cid:48) ll(cid:48)
matrix M(N|0) in Eq. (11) is transformed to the matrix
√ √
M˜(N|0) l∼1/x, |l(cid:48)|∼1/ x, |m|∼1/ x. (20)
i
Forsmallxthediscretesumsoverlandl(cid:48) inEq. (19)can
M˜(N|0) = l (l+l(cid:48))!(cid:18) 1 (cid:19)l+l(cid:48)+1 , (15) be replaced by integrations (this corresiponds to taking
ll(cid:48) l(cid:48)+1 l!l(cid:48)! 2(1+x) theleadingtermintheAbel-Planasummationformula):
with l,l(cid:48) ≥0. The first row of the matrix M˜(N|0) is zero,
while its l-th row with l = 1,2,··· is identical to the 1(cid:90) ∞ (cid:104)
(l−1)-th row of the matrix M(D|0) in Eq. (12) with its Φ(mN/D) = 2 dl Ml(,Nl/D|m)
first column deleted: M˜(N|0) = M(D|0) , l = 1,2,···, 0
ll(cid:48) l−1,l(cid:48)+1
l(cid:48) = 0,1,2,···. By a second similarity transformation
with the upper diagonal matrix A˜(Z):
l(cid:48)!
A˜ll(cid:48)(Z)=Zl(cid:48)−l(l(cid:48)−l)!l! , (16) +(cid:88)∞ 1 (cid:32)(cid:89)s (cid:90) ∞dl(cid:48)(cid:33)(cid:89)s M(N/D|m) (cid:35) , (21)
withA˜−1(Z)=A˜(−Z),thematrixM˜(N|0)istransformed s=1 s+1 i=1 −∞ i i=0 l+li(cid:48),l+li(cid:48)+1
into a lower triangular matrix M¯(N|0), with diagonal el-
ements equal to M¯(N|0) = Z2l+3, l = 0,1,2,···. This
ll andwesetl(cid:48) =l(cid:48) ≡0. InwritingtheaboveEquation,
implies at once: 0 s+1
we considered that the integration over l extending from
Φ(N) =−1(cid:88)ln[1−Z2l+3]. (17) |m| to ∞ can be replaced by an integration from zero
0 2 to ∞ because, according to Eq. (20), in the limit of
l≥0 small separations m is negligibly small compared to l.
We similarly replaced the integration over l(cid:48) extending
Thisresultcanbecontrastedwiththeanalogousformula i
from |m| − l to ∞ by an integration from −∞ to ∞
for D bc [19]:
because, compared to l(cid:48), (|m|−l) can be identified with
i
Φ(0D) =−12(cid:88)ln[1−Z2l+1]. (18) −nu∞m.beNrsexlt+, wl(cid:48)e,lo±bsmervaertehaaltlblayrgveirtinuteegoefrsEqfo.r(s2m0)altlhxe
l≥0 and therefore the factorials in Eqs. (11) and (12) can be
computed using Stirling’s formula:
B. Contribution of modes with m(cid:54)=0.
(cid:18) (cid:19)
1 1 1
Unfortunately, the contributions Φ(N) of the modes lnn!= n+ lnn−n+ ln2π+ +··· (22)
m 2 2 12n
with m (cid:54)= 0 cannot be computed exactly. By using the
technique of Refs. [23–25] it is however possible to prove
a short-distance formula for Φ(N), or more precisely for
m At this point, we Taylor expand the difference
the difference δΦ =Φ(N)−Φ(D). We start by expand-
m m m δM(m) =M(N|m) −M(D|m) amongthema-
ing the logarithm in Eq. (14): l+l(cid:48),l+l(cid:48) l+l(cid:48),l+l(cid:48) l+l(cid:48),l+l(cid:48)
i i+1 i i+1 i √i+1 √
trices M(N) and M(D) in powers of x (powers of x
 
∞ s ∞ s are reckoned according to the estimates in Eqs. (20)).
Φ(mN/D) = 12(cid:88)s+1 1(cid:89) (cid:88) (cid:89)Ml(iN,l/i+D1|m) , (19) Up to terms of order x2 we find:
s=0 i=0li=|m| i=0
1 (cid:18) l (cid:19) (cid:20) (l(cid:48) −l(cid:48) )2 m2(cid:21)
δM(m) = √ −1 exp −2xl− i i+1 − +o(x2). (23)
l+li(cid:48),l+li(cid:48)+1 4πl l+1 4l l
On the other hand, by taking the Taylor expansion of Eq. (12) we find:
1 (cid:20) (l(cid:48) −l(cid:48) )2 m2(cid:21)
M(D|m) = √ exp −2xl− i i+1 − +o(x). (24)
l+li(cid:48),l+li(cid:48)+1 4πl 4l l
6
The two formulae above confirm correctness of the es- The problem with this substitution is that it leads to an
timates in Eq. (20). There is a tricky but important infra-red divergence in the integral over l. To avoid this
point to stress here: following the logic of the Taylor ex- problem, we keep the complete factor [l/(l+1)−1] in
pansion, onemightfindappropriatetoreplacethefactor Eq.(23).
[l/(l +1)−1] in the r.h.s. of Eq. (23) by its first or- Starting from Eq. (21), and making use of Eqs. (23)
der Taylor approximation [l/(l+1)−1]=−1/l+o(x2). and (24), we obtain the following expression for δΦ :
m
1(cid:90) ∞ dl (cid:40)(cid:18) l (cid:19) (cid:18) m2(cid:19) 1(cid:88)∞ 1 (cid:34)(cid:18) l (cid:19)s+1 (cid:35)
δΦ = √ −1 exp −2xl− + −1
m 2 4πl l+1 l 2 s+1 l+1
0 s=1
(cid:32)(cid:89)s (cid:90) ∞ dl(cid:48) (cid:33)(cid:89)s (cid:20) (l(cid:48) −l(cid:48) )2 m2(cid:21)(cid:41)
× √ i exp −2xl− i i+1 − +o(x), (25)
4πl 4l l
i=1 −∞ i=0
Performing the gaussian integrals over l(cid:48), we then obtain the following estimate for δΦ accurate to order x1/2:
i m
1(cid:88)∞ 1 (cid:90) ∞ dl (cid:34)(cid:18) l (cid:19)s+1 (cid:35) (cid:20) (cid:18) m2(cid:19)(cid:21)
δΦ(1/2) = √ −1 exp (s+1) −2xl− . (26)
m 2 (s+1)3/2 4πl l+1 l
s=0 0
The sum over s can be expressed in terms the polylogarithm function Li (z)=(cid:80)∞ zk/kn:
n k=1
1(cid:90) ∞ dl (cid:26) (cid:20) l (cid:18) m2(cid:19)(cid:21) (cid:20) (cid:18) m2(cid:19)(cid:21)(cid:27)
δΦ(1/2) = √ Li exp −2xl− −Li exp −2xl− . (27)
m 2 4πl 3/2 l+1 l 3/2 l
0
1 1 lnµ−γ
Combiningtheaboveformulawiththeexactexpressions Φ(Dr) =Φ(D)− ln(γ −lnµ)− 2µ2+o(µ4),
of Φ(N) (Eq. (17)) and Φ(D) (Eq. (18)) we obtain for δΦ 2 1 12lnµ−γ1
0 0 (30)
the approximate small distance formula:
with γ(cid:48) = 0.0874485, γ = 1.270362, γ = 1.35369. We
1(cid:88) (cid:18)1−Z2l+3(cid:19) (cid:88) used µ0as a variable for1the expansion,2for it provides a
δΦ(0) =− ln +2 δΦ(1/2) . (28)
2 1−Z2l+1 m very accurate result also at larger distances. Both the
l≥0 m>0 D and Dr energies depend only on lnµ and even powers
We expect that this formula for δΦ is accurate to order of µ. This implies that the energies depend only on lnx
x0. Weshalllaterseethat,despitetheassumptionx(cid:28)1 and integer powers of x. In particular, the force for the
madeinitsderivation,theaboveformulaprovidesavery D case, once expanded in x, is a Laurent series starting
precise value of δΦ also for relatively large separations from1/x2. However,fortheDrcasetherearelogarithmic
(see Fig. 2). termsintheforceaswell. TheleadingcorrectiontoPFA
isthesamelnµtermforbothmodels,anditscoefficientis
in agreement with the DE. Interestingly, for practically
C. Expressions at small distances relevant separations the subleading double logarithmic
term in Eq. (30) dominates over the leading logarithmic
With exact expressions for the Casimir energies in the term, andthereforetheDandDrenergiesdisplayrather
D and Dr models, one can compute explicitly the inter- different behaviors.
action in the limit of short distances x (cid:28) 1. This limit ToworkouttheleadingcorrectiontoPFAoftheNen-
correspondstoZclosetounity,andonecancomputethe ergy, we start from Eq. (28). It is convenient to use for
series in Eqs. (5) and (6) using the Abel-Plana formula. δΦ(1/2) theexpressioninEq. (26). Inthelimitofvanish-
m
WesetZ =exp(−µ),andthenexpandforsmallµ,where ing separation, the sum over the angular index m can be
(cid:112)
µ=ln[1+x+ x(2+x)]. The resulting analytical ex- replacedbyanintegrationovermextendingfrom−∞to
pressions for the Casimir interaction were worked out in ∞. Performing the straightforward gaussian integral we
[19], and we reproduce them here for the convenience of find:
the reader:
Φ(D) = ζ4(µ32)−214lnµ−116+γ0(cid:48)+57760µ2+o(µ4), (29) δΦas = 41(cid:88)s∞=0(s+11)2 (cid:90)0∞dl(cid:34)(cid:18)l+l 1(cid:19)s+1−1(cid:35)e−2(s+1)xl
7
= 1(cid:90) ∞dl(cid:20)Li (cid:18) l e−2xl(cid:19)−Li (cid:0)e−2xl(cid:1)(cid:21) . (31) separable in bispherical coordinates, and its regular and
4 2 l+1 2 outgoing eigenfunctions are:
0
We computed analytically the asymptotic expansion of φreg/out =(cid:112)coshµ−cosηY (η,φ)exp[±(l+1/2)µ],
lm lm
the above formula for x→0 and found its leading term: (35)
for l ≥ 0, m = −l,··· ,l. Relative to the sphere (plane)
1
δΦ =− ln2x+o(lnx). (32) outgoing and regular eigenfunctions correspond, respec-
as 16
tively to the upper (lower) and lower (upper) sign in
Since in the D model the leading correction to the PFA theexponential. Scatteringsolutionscanbeexpandedin
is a lnx term (see Eq. (29)), the leading correction to these eigenfunctions. It is a simple matter to verify that
the PFA for the N model coincides with the ln2x term inthebisphericalbasisofEq. (35)thetranslationmatrix
of δΦ: U is diagonal with elements Ulml(cid:48)m(cid:48) = Zl+1/2δll(cid:48)δmm(cid:48).
where Z = exp(−µ ). For D bc the T-matrix for both
1
ζ(3) 1
Φ(N) = − ln2x+o(lnx). (33) the plane and sphere are minus the identity operator.
8x 16 Therefore, in the bispherical basis the M(D) matrix for
Earlier we pointed out that the leading correction to the DbcisdiagonalwithelementsMl(mDl)(cid:48)m(cid:48) =Z2l+1δll(cid:48)δmm(cid:48),
PFA for the Dr and the D model is the same lnx term. and thus evaluation of the scattering formula Eq. (1)
It then follows from Eq. (9) that the ln2x term of Φ(N) is straightforward yielding the result quoted in Eq. (5).
represents also the leading correction to the PFA for P ThecaseofDrbcismoreelaborate,asonehastoremove
bc: thecontributionofmonopolesfromthem=0block. De-
tails can be found in [19]. For N bc the T-matrix of the
Φ(P) = ζ(3) − 1 ln2x+o(lnx). (34) µ = 0 plane is equal to the identity operator. However,
4x 16 theT-matrixofthesphereisunfortunatelynon-diagonal.
Of course, the T-matrix is still block diagonal with re-
Thus our analytical results provide a rigorous proof of
the ln2(x) form of the leading curvature correction to spect to the angular index m, and it is convenient to
decompose its blocks as T(2|m) = 1+δT(N|m). By an
PFA, in accordance with indications obtained from the
explicit computation in the bispherical basis, it is found
high-precision numerical data of [38].
that the matrix δT(N|m) satisfies the linear system
B(m) δT(N|m) =−2sinhµ 1, (36)
1
IV. NUMERICAL COMPUTATION OF δΦ
where B(m) is the matrix of elements
The HT limit of the (ungrounded) sphere-plate
B(m) =[(2l+1)coshµ +sinhµ ]δ
Casimir energy with P bc was computed in [38] by a ll(cid:48) 1 1 ll(cid:48)
large-scale numerical computation of the exact scatter-
ing formula Eq. (1) using the standard spherical basis −(l−m)δl,l(cid:48)+1−(l(cid:48)+m)δl+1,l(cid:48) . (37)
with origin at the sphere center C. The computation in
with l,l(cid:48) ≥ |m|. The linear system Eq. (36) cannot be
[38] included up to 5000 partial wave orders, which al-
solvedanalytically,butitcanbeeasilysolvednumerically
lowed the authors of Ref. [38] to accurately estimate the
after truncation in the multipole order l,l(cid:48) <l .
functions Φ(P) and Φ(Dr) for aspectsratios x≥2×10−3. max
At this point it would seem that nothing is really
Earlier we saw that the classical Casimir energy for
gained by using bispherical multipoles, because we still
ungrounded perfect conductors is the sum of the ener-
face the problem of computing determinants of infinite-
gies for a Dr scalar plus a N scalar (see Eq. (3)). The
dimensional matrices, as we had to do anyhow in the
(normalized) energy Φ(Dr) can be computed exactly in
standard base of spherical multipoles. Indeed the situa-
the sphere-plate geometry (see Eq. (6)), while for N bc
tion seems even worse now, because earlier at least the
an exact formula exists for m = 0 modes. In the previ-
matrix M(N|m) had a simple expression (see Eq. (11)),
ous Section we derived an asymptotic formula, Eq. (28),
while now the matrix δT(N|m) has to be itself computed
validforsmall-distances,forthedifferenceδΦamongthe
numericallybysolvinganinfinite-dimensionallinearsys-
HT Casimir energies for N and D bc. In this Section
tem. Thisshortcomingofbisphericalcoordinatesishow-
the energy-difference δΦ is computed numerically, using
ever rewarded by the crucial advantage of a much faster
the exact scattering formula Eq. (1). As we shall see,
rate of convergence with respect to the maximum value
δΦ can be computed very efficiently by using a basis of
l of the multipole index l. To see this, consider the
bispherical multipoles [19]. max
expression of the M matrix for N bc in bispherical coor-
Bispherical coordinates (µ,η,φ) [41] are defined
dinates:
by (x,y,z) = a(sinηcosφ,sinηsinφ,sinhµ)/(coshµ −
cosη), where a identifies the focus F of the sphere de- M(N|m) =Z2l+1(δ +δT(m)), (38)
fined by µ = µ > 0 (see Fig.1). The sphere has radius ll(cid:48) ll(cid:48) ll(cid:48)
1
R = a/sinhµ , and L = acothµ is the distance of its with l,l(cid:48) ≥|m|. When this expression is substituted into
1 1
center C from the µ=0 plane. The Laplace Equation is the scattering formula Eq. (1), it is easy to factor out
8
the D contribution, and one ends up with the following 0
exact representation for the energy-difference δΦ defined
in Eq. (2): -2
(cid:88)
δΦ=− (cid:48)Trlog[1+V(m)δT(N|m)], (39) -4
m≥0 xϕ()
δ
-6
where V(m) is the diagonal matrix of elements:
1 -8
V(m) = δ . (40)
ll(cid:48) 1−eµ1(2l+1) ll(cid:48)
The exponential in the denominator of V(m) shows -5 -4 -3 -2 -1
ll(cid:48) Log10(x)
that the multipoles contributing to δΦ are those with
l,l(cid:48) (cid:46) 1/µ . For small x, µ = −lnZ = ln[1 + x +
1 √ 1 FIG.2: DifferenceδΦ=Φ(N)−Φ(D)amongthesphere/plate
(cid:112)
x(2+x)]−1 (cid:39) 2x and then we see that the order l NandDnormalizedCasimirenergiesasafunctionoflog (x):
(cid:112) 10
of the relevant partial waves scales like R/d, which is numericaldata(dots)computedusingthescatteringformula
only the square root of the multipole order l ∼ R/d in a bispherical basis Eq. (39), small-distance formula Eq.
max
(see Eq. (20)) needed in the spherical basis. (28) (solid line), leading term Eq. (32) (dashed line).
Todemonstratethefastrateofconvergenceofthescat-
tering formula in bispherical coordinates, we take as an
example x = 2 × 10−3, which is the smallest aspect The error made by using Eq. (28) varies from 0.16 % for
ratio considered in [38]. By including in the scatter- x=10−5toamaximumof1.2%forx=0.1. Thedashed
ing formula 5000 partial (spherical) waves, the authors line of Fig.2 corresponds to the leading term Eq. (32).
of [38] computed Φ(P) = 146.812 and Φ(Dr) = 74.5962. By a fit procedure, we verified that a very good agree-
On the other hand, using the exact formula in Eq. (5) ment between the dashed curve and the numerical data
we find Φ(D) = 75.2936. From Eq. (9) we then get in Fig. 2 can be obtained by adding to the expansion in
δΦ = −3.07737. In Table I we quote the values of δΦ Eq. (34) a subleading logarithmic term proportional to
obtainedfromEq. (39)withinclusionofbisphericalmul- lnx.
tipoles of order l ≤ lmax, for lmax = 20,40,80,120. As A convenient representation of deviations from the
it can be seen, δΦ converges quickly, and already with PFA energy is provided by the function β(P)(x) defined
lmax =80theerroronδΦisassmallas1.2×10−5. With such that [38]:
l = 120 we reproduce the value computed in [38] us-
max
ing 5000 partial waves. It is interesting to compare the ζ(3)(cid:18)1 (cid:19)
numerical value of δΦ with the estimate provided by the Φ(P) =−kBT 4 x +β(P)(x) . (41)
asymptotic formula Eq. (28). Evaluation of Eq. (28)
gives δΦ(0) = −3.068, which differs from the numerical
We similarly set:
value of δΦ by less than 0.3 %.
(cid:18) (cid:19)
ζ(3) 1
l 20 40 80 120 Φ(D/Dr/N) =−k T +β(D/Dr/N)(x) . (42)
max B 8 x
δΦ -2.92435 -3.06243 -3.07725 -3.07737
The exact expressions for the functions β(D/Dr)(x) can
TABLE I: Numerical values of δΦ for aspect ratio x =
2×10−3 obtained from the scattering formula in bispheri- be easily worked out starting from the exact solutions
cal coordinates Eq.(39) with inclusion of multipoles of order for the energies Eq. (5) and (6). On the other hand,
l≤l . The value of δΦ for l =120 is in perfect agree- β(N)(x) can be expressed in terms of δΦ:
lmax max
mentwiththevalueobtainedin[38]usingsphericalmultipoles
up to lmax =5000. 8
β(N)(x)=β(D)(x)+ δΦ. (43)
ζ(3)
In Fig.2 we plot δΦ as a function log (x). The dots
10
were computed using the scattering formula for δΦ in
Recalling Eq. (9), β(P)(x) can be decomposed as:
a bispherical basis Eq. (39). The fast convergence of
Eq. (39) allowed us to accurately compute δΦ for as-
(cid:18) (cid:19)
1 8
pect ratios as small as x=10−5 by using less than 1000 β(P)(x)= β(Dr)(x)+β(D)(x)+ δΦ . (44)
partial waves. The solid line Fig.2 was computed using 2 ζ(3)
the asymptotic formula for δΦ, Eq. (28). It can be seen
that the asymptotic formula Eq. (28) provides a precise In Fig. 3 we display a plot of β(P)(x), where dots repre-
estimate of δΦ over the entire range of aspect ratios dis- sent our numerical data, while the solid line is computed
playedintheFigure, uptothefairlylargevaluex=0.1. using the small-distance formula Eq. (28) for δΦ.
9
0 tions. Taking the asymptotic expansion of the small dis-
tance formula we found a ln2(d/R) correction in the en-
-5
ergy,beyondthecommonlyusedproximityforceapprox-
-10 imation. Theln2(d/R)formofthecorrectionisinagree-
mentwithafitoflarge-scalenumericaldata[38]. Wede-
P)x() -15 veloped a fast-converging numerical scheme for comput-
(β -20 ing the Casimir energy, based on a system of bispherical
partial waves. In bispherical coordinates, convergence of
-25
the exact scattering formula is achieved at multipole or-
(cid:112)
-30 der l (cid:39) R/d, while in the standard approach based
max
on spherical multipoles convergence is achieved only at
-5 -4 -3 -2 -1
order l (cid:39) R/d. Using the bispherical basis, we could
max
log10(x) accurately compute the Casimir energy for very small
aspects ratio d/R = 10−5. Comparison with the high
FIG.3: Additivecorrectionβ(P)tothePFAclassicalCasimir
precisionnumericaldatashowsthattheanalyticalsmall-
energy for perfectly conducting sphere-plate as a function of
distance formula precisely estimates the energy also for
log (x): numericaldata(dots)computedusingthescattering
10 fairly large values of the aspect ratio.
formulainabisphericalbasisEq. (39),small-distanceformula
for δΦ Eq. (28) (solid line), leading term for δΦ Eq. (32)
(dashed line).
Acknowledgments
V. CONCLUSIONS
TheauthorthanksT.Emig,N.Graham,M.Kruger,R.
We studied the Casimir interaction between a sphere L.JaffeandM.Kardarforvaluablediscussionswhilethe
and a plate, both perfectly conducting, in the classical manuscript was in preparation. Warm thanks are due
limit of high temperatures. We worked out an analytical also to the authors of [38] for sharing with the author
formulafortheenergy,validforsufficientlysmallsepara- their numerical data.
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