The Scattering Approach to the Casimir Force

0.16 MB·English

by  S. Reynaud

THE SCATTERING APPROACH TO THE CASIMIR FORCE
S. REYNAUD∗, A. CANAGUIER-DURAND, R. MESSINA, A. LAMBRECHT1 and and P.A. MAIA NETO2
1Laboratoire Kastler Brossel, ENS, UPMC, CNRS, Jussieu, 75252 Paris, France
∗E-mail: [email protected]
2Instituto de F´ısica, UFRJ, CP 68528, Rio de Janeiro, RJ, 21941-972, Brazil
We present the scattering approach which is nowadays the best tool for describing the Casimir
forcein realistic experimentalconfigurations. Afterremindersonthesimple geometries of1d space
and specular scatterers in 3d space, we discuss the case of stationary arbitrarily shaped mirrors in
electromagnetic vacuum. We then review specific calculations based on the scattering approach,
dealing for example with the forces or torques between nanostructured surfaces and with the force
between a plane and a sphere. In these various cases, we account for the material dependence
of the forces, and show that the geometry dependence goes beyond the trivial Proximity Force
Approximation often used for discussing experiments.
The many facets of the Casimir effect Fexp have to be assessed independently from each other
0
andshouldforbidanyoneto use theory-experimentcom-
1
parison for proving (or disproving) some specific experi-
0 The Casimir effect [1] is a jewel with many facets.
2 First, it is an observable effect of vacuum fluctuations in mental result or theoretical model.
n themesoscopicworld,whichdeservescarefulattentionas Casimircalculatedtheforcebetweenapairofperfectly
smooth, flat and parallel plates in the limit of zero tem-
a a crucial prediction of quantum field theory [2–7].
J Then, it is also a fascinating interface between quan- perature and perfect reflection. He found universal ex-
pressions for the force F and energy E
9 tum field theory and other important aspects of funda- Cas Cas
1 mental physics. It has connections with the puzzles of ~cπ2A ~cπ2A
] gravitationalphysicsthroughtheproblemofvacuumen- FCas = 240L4 , ECas =−720L3 (1)
h ergy[8,9]aswellaswiththeprincipleofrelativityofmo-
p tion through the dynamical Casimir-like effects [10–12]. with L the distance, A the area, c the speed of light and
- Effects beyond the Proximity Force Approximation also ~ the Planck constant. This universality is explained by
t
n make apparent the extremely rich interplay of vacuum the saturation of the optical response of perfect mirrors
a energy with geometry (references and more discussions which reflect 100% (no less, no more) of the incoming
u below). fields. Clearly, this idealization does not correspond to
q
Casimir physics also plays an important role in the any realmirror. In fact, the effect of imperfect reflection
[
tests of gravity at sub-millimeter ranges [13, 14]. Strong is large in most experiments, and a precise knowledge

1 constraintshavebeenobtainedinshortrangeCavendish- of its frequency dependence is essential for obtaining a
v like experiments [15] : Should an hypothetical new force reliable theoretical prediction for the Casimir force [21].
5
haveaYukawa-likeform,itsstrengthcouldnotbelarger The most precise experiments are performed with
7
3 than that of gravity if the range is larger than 56µm. metallic mirrors which are good reflectors only at fre-
3 For scales of the order of the micrometer, similar tests quencies smaller than their plasma frequency ωP. Their
. are performed by comparing with theory the results of opticalresponseisdescribedbyareduceddielectricfunc-
1
Casimir force measurements [16, 17]. At even shorter tion usually written at imaginary frequencies ω =iξ as
0
0 scales, the same can be done with atomic [18] or nuclear σ[iξ] ω2
1 [19] force measurements. ε[iξ]=εˆ[iξ]+ , σ[iξ]= P (2)
ξ ξ+γ
: Finally, the Casimir force and closely related Van
v
i der Waals force are dominant at micron or sub-micron The function εˆ[iξ] represents the contribution of inter-
X
distances, which entails that they have strong connec- band transitions and it is regular at the limit ξ → 0.
r tions with various important domains, such as atomic Meanwhile σ[iξ] is the reduced conductivity (σ is mea-
a
and molecular physics, condensed matter and surface suredasafrequencyandtheSIconductivityisǫ σ)which
0
physics, chemical and biological physics, micro- and describes the contribution of the conduction electrons.
nano-technology [20]. A simplified description corresponds to the lossless
limitγ →0often calledthe plasmamodel. As γ ismuch
smaller than ω for a metal such as Gold, this simple
P
Comparison of the Casimir force measurements with model captures the main effect of imperfect reflection.
theory However it cannot be considered as an accurate descrip-
tion since a much better fit of tabulated optical data is
obtainedwithanonnullvalueofγ[21]. Furthermore,the
In short-range gravity tests, the new force would ap-
Drude model meets the important property of ordinary
pearasadifferencebetweentheexperimentalresultF
andtheoreticalpredictionFth. ThisimpliesthatFth aenxdp metals which have a finite static conductivity σ0 = ωγP2,
2
in contrast to the lossless limit which corresponds to an [43], thus opening the way to a comparison with experi-
infinite value for σ . mentalstudiesofPFAintheplane-spheregeometry[44].
0
Another correction to the Casimir expressions is as- Another specific geometry of great interest is that of
sociated with the effect of thermal fluctuations [22, 23] surfaces with periodic corrugations. As lateral transla-
which is correlated to the effect of imperfect reflection tion symmetry is broken, the Casimir force contains a
[24]. BostromandSerneliushaveremarkedthatthesmall lateralcomponentwhich is smaller thanthe normalone,
non zero value of γ had a significant effect on the force but has nevertheless been measured in dedicated experi-
evaluation at T 6= 0 [25]. This remark has led to a blos- ments [45]. Calculationsbeyondthe PFAhavefirstbeen
soming of contradictory papers (see references in [26– performed with the simplifying assumptions of perfect
28]). The currentstatus of Casimir experiments appears reflection [46] or shallow corrugations [47–49]. As ex-
to favor predictions obtained with γ = 0 rather than pected, the PFA was found to be accurate only at the
those corresponding to the expected γ 6= 0 (see Fig.1 limit of large corrugation wavelengths. Very recently,
in [29]). Note that the ratio between the prediction at experiments have been able to probe the beyond-PFA
γ = 0 with that at γ 6= 0 reaches a factor 2 at the limit regime [50, 51] and it also became possible to calculate
of large temperatures or large distances, although it is the forces between real mirrors with deep corrugations
not possible to test this striking prediction with current [52]. More discussions on these topics will be presented
experiments which do not explore this domain. below.
At this point, it is worth emphasizing that micro-
scopic descriptions of the Casimir interaction between
two metallic bulks lead to predictions agreeing with the
Introduction to the scattering approach
lossyDrudemodelratherthanthe losslessplasmamodel
at the limit of largetemperatures or large distances [30–
32]. At the end of this discussion, we thus have to face The best tool available for addressing these questions
a worrying situation with a lasting discrepancy between is the scattering approach. We begin the review of this
theory and experiment. This discrepancy may have var- approach by an introduction considering the two simple
ious origins, in particular artefacts in the experiments cases of the Casimir force between 2 scatterers on a 1-
or inaccuracies in the calculations. A more subtle but dimensionallineandbetweentwoplaneandparallelmir-
maybe more probable possibility is that there exist yet rorscoupledthroughspecularscatteringto3-dimensional
unmastereddifferences betweenthe situations studied in electromagnetic fields [53].
theory and the experimental realizations. The first case corresponds to the quantum theory of
a scalar field with two counterpropagating components.
A mirror is thus described by a 2x2 S−matrix contain-
The role of geometry ing the reflection and transmission amplitudes r and t.
Two mirrors form a Fabry-Perot cavity described by a
global S−matrix which can be evaluated from the el-
The geometry of Casimir experiments might play an
ementary matrices S and S associated with the two
important role in this context. Precise experiments are 1 2
mirrors. All S−matrices are unitary and their determi-
indeed performed between a plane and a sphere whereas
nants are shown to obey the simple relation
calculations are often devoted to the geometry of two
parallelplanes. The estimationof the force in the plane-
lndetS =lndetS +lndetS +i∆ (3)
sphere geometry involves the so-called Proximity Force 1 2
Approximation (PFA) [33] which amounts to averaging d 2iωL
∆=iln , d(ω)=1−r r exp
over the distribution of local inter-plate distances the d∗ 1 2 (cid:18) c (cid:19)
force calculated in the two-planes geometry, the latter
being deduced from the Lifshitz formula [34, 35]. The phaseshift ∆ associatedwith the cavity is expressed
This trivial treatment of geometry cannot reproduce in terms of the denominator d describing the resonance
the rich interconnection expected to take place between effect. The sum of all these phaseshifts over the field
theCasimireffectandgeometry[36]. Intheplane-sphere modes leads to the following expression of the Casimir
geometry in particular, the PFA can only be valid when free energy F
the radius R is much larger than the separation L [37].
But even if this limit is met in experiments, the PFA dω
F =−~ N(ω)∆(ω) (4)
does not tell one what is its accuracy for a given value Z 2π
of L/R or whether this accuracy depends on the mate- 1 1
rial properties of the mirror. Answers to these questions N(ω)= +
exp ~ω −1 2
can only be obtained by pushing the theory beyond the kBT
(cid:16) (cid:17)
PFA,which has beendone inthe pastfew years(see ref-
erences in [38–42]). In fact, it is only very recently that HereN isthemeannumberofthermalphotonspermode,
these calculations have been done with plane and spher- givenbythePlancklaw,augmentedbytheterm 1 which
2
ical metallic plates coupled to electromagnetic vacuum represents the contribution of vacuum [53].
3
This phaseshiftformulacanbe givenalternativeinter- transparency properties. It has been demonstrated with
pretations. In particular, the Casimir force an increasing range of validity in [53], [54] and [7]. The
expression is valid not only for lossless mirrors but also
∂F(L,T) for lossy ones. In the latter case, it accounts for the ad-
F = (5)
∂L ditionalfluctuations accompanyinglossesinside the mir-
rors.
can be seen as resulting from the difference of radiation
It can thus be used for calculating the Casimir force
pressures exerted onto the inner and outer sides of the
between arbitrary mirrors, as soon as the reflection am-
mirrors by the field fluctuations [53]. Using the analytic
plitudes are specified. These amplitudes are commonly
properties of the scattering amplitudes, the free energy
deducedfrommodels of mirrors,the simplest ofwhichis
may be written as the following expression after a Wick
thewellknownLifshitzmodel[34,35]whichcorresponds
rotation (ω =iξ are imaginary frequencies)
to semi-infinite bulk mirrors characterized by a local di-
dξ ~ξ electric response function ε(ω) and reflection amplitudes
F =~ cot lnd(iξ) (6) deduced from the Fresnel law.
Z 2π (cid:18)2kBT(cid:19) In the most general case, the optical response of the
mirrorscannotbedescribedbyalocaldielectricresponse
Using the pole decomposition of the cotangent function
function. The expression (8) of the free energy is still
and the analytic properties of lnd, this can finally be
′ valid in this case with some reflection amplitudes to be
expressed as the Matsubara sum ( is the sum over
positiveintegersmwithm=0counPtemdwithaweight 1) determined from microscopic models of mirrors. Recent
2 attempts in this direction can be found for example in
2πmk T [56–58].
′ B
F =k T lnd(iξ ) , ξ ≡ (7)
B m m ~
Xm
The non-specular scattering formula
The same lines of reasoning can be followed when
studying the geometry of two plane and parallel mir-
rors aligned along the axis x and y. Due to the sym- We now present a more general scattering formula al-
metry of this configuration, the frequency ω, transverse lowingonetocalculatetheCasimirforcebetweenstation-
vector k ≡ (k ,k ) and polarization p = TE,TM are ary objects with arbitrary non planar shapes. The main
x y
preserved by all scattering processes. The two mirrors generalizationwithrespecttothealreadydiscussedcases
are described by reflection and transmission amplitudes isthatthescatteringmatrixS isnowalargermatrixac-
which depend on frequency, incidence angle and polar- counting for non-specular reflection and mixing different
ization p. We assume thermal equilibrium for the whole wavevectorsandpolarizationswhilepreservingfrequency
“cavity + fields” system, and calculate as in the sim- [7, 47]. Of course, the non-specular scattering formula is
pler case of a 1-dimensional space. Care has however to the generic one while specular reflection can only be an
be taken to account for the contribution of evanescent idealization.
waves besides that of ordinary modes freely propagating As previously, the Casimir free energy can be written
outside and inside the cavity [7, 54]. The properties of as the sum of all the phaseshifts contained in the scat-
the evanescent waves are described through an analyti- tering matrix S
calcontinuationof those of ordinaryones,using the well
∞
dω
defined analytic behavior of the scattering amplitudes. F = i~ N(ω)lndetS
At the end of this derivation, the free energy has the Z0 2π
∞
following form as a Matsubara sum [55] dω
= i~ N(ω)TrlnS (9)
Z 2π
F = k T ′lnd(iξ ,k,p) (8) 0
B m
Xk Xp Xm The symbols det and Tr refer to determinant and trace
d(iξ,k,p)=1−r (iξ,k,p)r (iξ,k,p)exp−2κL over the modes of the matrix S. As previously, the for-
1 2
mulacanalsobe writtenafteraWick rotationasa Mat-
ξ ≡ 2πmkBT , κ ≡ k2+ ξ2 subara sum
m ~ r c2
′
F =k T TrlnD(iξ ) (10)
B m
k ≡ A d4π2k2 is the sum over transverse wavevectors Xm
wPith AtheRareaofthe plates, p the sumoverpolariza- D =1−R1exp−KLR2exp−KL
tions and ′ the MatsubaraPsum.
m
This expPression reproduces the Casimir ideal formula The matrix D is the denominator containing all the res-
in the limits of perfect reflection r r →1 and null tem- onance properties of the cavity formed by the two ob-
1 2
perature T → 0. But it is valid and regular at ther- jects 1 and 2 here written for imaginary frequencies. It
mal equilibrium at any temperature and for any optical is expressed in terms of the matrices R and R which
1 2
model of mirrors obeying causality and high frequency represent reflection on the two objects 1 and 2 and of
4
propagation factors exp−KL. Note that the matrices D, δR and δR are the first-order variation of the reflec-
1 2
R and R , which were diagonal on the basis of plane tion matrices R and R induced by the corrugations;
1 2 1 2
waves when they described specular scattering, are no D is the matrix D evaluated at zeroth order in the cor-
0
longer diagonal in the general case of non specular scat- rugation; it is diagonal on the basis of plane waves and
tering. The propagation factors remain diagonal in this commutes with K.
basiswiththeirdiagonalvalueswrittenasin(8). Clearly Explicit calculations of (14) have been done for the
the expression (10) does not depend on the choice of a simplest case of experimental interest, with two corru-
specificbasis. Remarkalsothat(10)takesasimplerform gated metallic plates described by the plasma dielectric
at the limit of null temperature (note the change of no- function. These calculations have led to the following
tation from the free energy F to the ordinary energy E) expression of the lateral energy
dE ∞ dξ A
F = , E =~ lndetD(iξ) (11) δE = GC(kC)a1a2cos(kCb) (14)
dL Z 2π 2
0
with the function G (k ) given in [49]. It has also been
Formula (11) has been used to evaluate the effect of C C
shown that the PFA was recovered for long corrugation
roughness or corrugationof the mirrors[47–49] in a per-
wavelengths,when G (k ) is replaced by G (0) in (14).
turbative manner with respect to the roughness or cor- C C C
This important argument can be considered as a prop-
rugationamplitudes(seethenextsection). Ithasclearly
erly formulated “Proximity Force Theorem” [49]. It has
a larger domain of applicability, not limited to the per-
tobedistinguishedfromtheapproximation(PFA)which
turbative regime, as soon as techniques are available for
consists in an identification of G (k ) with its limit
computing the large matrices involved in its evaluation. C C
G (0). For arbitrary corrugation wavevectors, the de-
Ithasalsobeenusedinthepastyearsbydifferentgroups C
viation from the PFA is described by the ratio
usingdifferentnotations[39–41,59,60]. Therelationbe-
tween these approaches is reviewed for example in [61].
G (k )
C C
ρ (k )= (15)
C C
G (0)
C
The lateral Casimir force between corrugated plates The variation of this ratio ρ with the parameters of
C
interest has been described in a detailed manner in [48,
As already stated, the lateral Casimir force between 49]. Curves are drawn as examples in the Fig. 1 of [48]
corrugated plates is a topic of particular interest. This with λP = 137nm chosen to fit the case of gold covered
configuration is more favorable to theory/experiment plates. An important feature is that ρC is smaller than
comparison than that met when studying the normal unity as soon as kC significantly deviates from 0. For
Casimir force. It could thus allow for a new test of large values of kC, it even decays exponentially to zero,
Quantum ElectroDynamics, through the dependence of leading to an extreme deviation from the PFA.
the lateral force to the corrugation wavevector [48, 49]. Other situations of interest have also been studied.
Here, we consider two plane mirrors, M1 and M2, with When the corrugationplates are rotated with respect to
corrugatedsurfaces describedby uniaxialsinusoidalpro- each other, a torque appears to be induced by vacuum
files (see Fig. 1 in [49]). We denote h and h the local fluctuations, tending to align the corrugation directions
1 2
heights with respect to mean planes z =0 and z =L [62]. In contrast with the similar torque appearing be-
1 2
tween misaligned birefringent plates [63], the torque is
h =a cos(k x) , h =a cos(k (x−b)) (12) here coupled to the lateral force. The advantage of the
1 1 C 2 2 C
configuration with corrugated plates is that the torque
h and h have null spatial averages and L is the mean has a largermagnitude. Another case of interestmay be
1 2
distance between the two surfaces; h1 and h2 are both designed by using the possibilities offered by cold atoms
counted as positive when they correspond to separation techniques. Nontrivialeffectsofgeometryshouldbevis-
decreases;λC isthecorrugationwavelength,kC =2π/λC ible in particularwhen using aBose-Einsteincondensate
the corresponding wave vector, and b the spatial mis- as a local probe of vacuum above a nano-grooved plate
match between the corrugationcrests. [64, 65].
At lowest order in the corrugation amplitudes, when These results suggested that non trivial effects of ge-
a ,a ≪λ ,λ ,L, the Casimir energy may be obtained ometry, i.e. effects beyond the PFA, could be observed
1 2 C P
by expanding up to second order the general formula withdedicatedlateralforceexperiments. Itwashowever
(11). The part of the Casimir energy able to produce difficult toachievethis goalwithcorrugationamplitudes
a lateral force is thus found to be a ,a meeting the conditions of validity of the pertur-
1 2
bative expansion. As already stated, recent experiments
∂δE
Flat =− (13) have been able to probe the beyond-PFA regime with
∂b deep corrugations [50, 51] and it also became possible
∞ dξ exp−KL exp−KL to calculate the forces between real mirrors without the
δE =−~ Tr δR δR
Z0 2π (cid:18) 1 D0 2 D0 (cid:19) perturbative assumption. In particular,an exactexpres-
5
sion has been obtained for the force between two nanos- theoreticalevaluations of ρ revealsa striking difference
G
tructured surfaces made of real materials with arbitrary between the cases of perfect and plasma mirrors. The
corrugationdepth, corrugationwidth and distance [52]. slope βperf obtained for perfect mirrors is larger than
G
that βGold obtained for gold mirrors by a factor larger
G
than 2
The plane-sphere geometry beyond PFA
βperf ∼−0.48 , βGold ∼−0.21 (18)
In the plane-sphere geometry, it is also possible to use G G
Meanwhile, βGold is compatible with the experimental
thegeneralscatteringformula(11)toobtainexpliciteval- G
uationsoftheCasimirforce. Thereflectionmatricesmay bound obtained in [44] (see [43]) whereas βGperf lies out-
here be written in terms of Fresnel amplitudes on the side this bound (see also [59]).
planemirrorandofMieamplitudesonthesphericalone. Thelessontobelearnedfromtheseresultsisthatmore
The scattering formula is then obtained by writing also work is needed to reach a reliable comparison of exper-
transformation formulas from the plane waves basis to iment and theory on the Casimir effect. Experiments
thesphericalwavesbasisandconversely. Theresulttakes are performed with large spheres for which the parame-
the form of a multipolar expansion with spherical waves ter L/R is smaller than 0.01, and efforts are devoted to
labeled by quantum numbers ℓ and m (|m| ≤ ℓ). For calculations pushed towards this regime [67].
doing the numerics, the expansion is truncated at some
Meanwhile, the effect of temperature should also be
maximum value ℓ , which restricts accurate evalua-
max
correlatedwiththe plane-spheregeometry. The firstcal-
tions to a domain x ≡ L/R > x with x propor-
min min
culations accounting simultaneously for plane-sphere ge-
tional to 1/ℓ .
max
ometry,temperatureanddissipationhavebeenpublished
Such calculations have first been performed for per-
veryrecently[68]andtheyshowseveralstrikingfeatures.
fectly reflecting mirrors [59, 66]. It was thus found that
Thefactorof2betweenthelongdistanceforcesinDrude
the Casimir energy was smaller than expected from the
and plasma models is reduced to a factor below 3/2 in
PFA and, furthermore, than the result for electromag-
the plane-sphere geometry. Then, PFA underestimates
netic fields was departing from PFA more rapidly than
the Casimir force within the Drude model at short dis-
wasexpectedfrompreviouslyexistingscalarcalculations
tances, while it overestimates it at all distances for the
[40, 41]. It is only very recently that the same calcula-
perfect reflector and plasma model. If the latter fea-
tionshavebeendoneforthemorerealisticcaseofmetallic
ture were conserved for the experimental parameter re-
mirrors described by a plasma model dielectric function
gion R/L (>102), the actual values of the Casimir force
[43]. Results of these evaluations are expressed in terms
calculated within plasma and Drude model could turn
of reduction factors defined for the force F or force gra-
dient G with respect to the PFA expectations FPFA and out to be closer than what PFA suggests, which would
diminish the discrepancy between experimental results
GPFA respectively
and predictions of the thermal Casimir force using the
F G Drude model.
ρ = , ρ = (16)
F FPFA G GPFA
Examples of curves for ρ and ρ are shown on Fig.2 of
F G
[43] for perfect and plasma mirrors.
Using these results, it is possible to compare the the- Acknowledgments
oreticalevaluations to the experimental study of PFA in
the plane-sphere geometry [44]. In this experiment, the
The authors thank I. Cavero-Pelaez, D. Dalvit, G.L.
force gradientis measuredfor various radii of the sphere
Ingold, M.-T. Jaekel and I. Pirozenkho for fruitful dis-
andtheresultsareusedtoobtainaconstraint|β |<0.4
G
cussions. A.C. and R.M. acknowledge support from
on the slope at origin β of the function
G
the ESF Research Networking Programme CASIMIR
ρ (x)=1+β x+O(x2) (17) (www.casimir-network.com). P.A.M.N. thanks CNPq,
G G
CAPES and Faperj for financial support. A.L. acknowl-
Now the comparisonof this experimental information to edges support from the French Contract ANR-06-Nano-
theslopeobtainedbyinterpolatingatlowvaluesofxthe 062.
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