Orientation dependence of Casimir forces
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Emig, T. et al. “Orientation dependence of Casimir forces.”
Physical Review A 79.5 (2009): 054901. © 2009 The American
Physical Society.
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http://dx.doi.org/10.1103/PhysRevA.79.054901
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PHYSICAL REVIEW A 79, 054901 共2009兲
Orientation dependence of Casimir forces
T. Emig,1,2 N. Graham,3 R. L. Jaffe,4 and M. Kardar5
1
Institut für Theoretische Physik, Universität zu Köln, Zülpicher Strasse 77, 50937 Köln, Germany
Laboratoire de Physique Théorique et Modèles Statistiques, Université Paris-Sud, CNRS UMR 8626, 91405 Orsay, France
3
Department of Physics, Middlebury College, Middlebury, Vermont 05753, USA
4
Department of Physics, Center for Theoretical Physics, and Laboratory for Nuclear Science, Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139, USA
5
Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
共Received 10 November 2008; published 14 May 2009兲
2
The Casimir interaction between two objects, or between an object and a plane, depends on their relative
orientations. We make these angular dependences explicit by considering prolate or oblate spheroids. The
variation with orientation is calculated exactly at asymptotically large distances for the electromagnetic field
and at arbitrary separations for a scalar field. For a spheroid in front of a mirror, the leading term is orientation
independent, and we find the optimal orientation from computations at higher order.
DOI: 10.1103/PhysRevA.79.054901
PACS number共s兲: 34.35.⫹a, 03.70.⫹k, 12.20.⫺m, 42.25.Fx
Casimir forces, predicted in 1948 关1兴 and probed in highprecision experiments over the past decade 关2–5兴, are particularly important at micrometer to nanometer length scales.
In constructing and manipulating devices at these length
scales, it is important to have an accurate understanding of
the material, shape, and orientation dependences of these
forces. Since these dependences often originate in the inherent many-body character of the force, they cannot be reliably
obtained by commonly used approximations. In this work we
investigate shape and orientation dependences of the Casimir
force using a recently developed method 关6兴 共see also 关7兴兲
that makes it possible to compute the Casimir interaction for
arbitrary compact objects based on their scattering matrices.
A closely related work is the classic paper of Balian and
Duplantier 关8兴, which provides expressions for the large distance asymptotic force between perfectly conducting objects
of arbitrary shape and orientation. Using our method, we
generalize these results in a form suitable for extension to
arbitrary dielectrics and distances. As tangible examples we
then focus on ellipsoids, computing the orientation dependent force between two spheroids, and between a spheroid
and a plane 关9兴. The latter geometry is particularly interesting, as there is no orientation dependence at leading order
for any object 共metal or dielectric兲 in front of a mirror. We
obtain the preferred orientation of a spheroid from a computation at higher order, in which T matrices are evaluated in an
expansion around a spherical shape. To numerically study a
wider range of eccentricities and separations, we consider a
scalar theory 共in place of electromagnetism兲, for which efficient computations can be performed using a basis of spheroidal harmonics. For the case of Neumann boundary conditions, we find a transition in the preferred orientation as the
spheroid approaches the plane.
We start from a general expression for the Casimir energy
as an integral over imaginary wave number 关6兴 of
Tr ln共1 − N兲, which after expansion in powers of N reads
E=−
បc
2␲
冕 冋
⬁
0
册
1
d␬ Tr共N兲 + Tr共N2兲 + ¯ .
2
E12
1 =−
បc 1
1 2
兵13共␣xx
␣xx + ␣1yy␣2yy + 2␣1xy␣2xy兲
d7 8␲
1 2
1 2
␣zz − 30共␣xz
␣xz + ␣1yz␣2yz兲 + 共␣ → ␤兲
+ 20␣zz
1 2
2
− 7共␣xx
␤yy + ␣1yy␤xx
− 2␣1xy␤2xy兲 + 共1 ↔ 2兲其,
共2兲
which generalizes the results of Feinberg and Sucher 关10兴 for
isotropic magnetically polarizable particles, and of Balian
and Duplantier 关8兴 for perfectly conducting objects at nonzero temperatures. For the case of an ellipsoidal object with
static electric permittivity ⑀ and magnetic permeability ␮, the
polarizability tensors are diagonal in a basis oriented to its
principal axes, with elements 共for i 苸 兵1 , 2 , 3其兲
共1兲
Here N = T1U12T2U21, where T1 and T2 relate incoming and
scattered electromagnetic 共EM兲 fields while the “translation
1050-2947/2009/79共5兲/054901共4兲
matrix” U12 relates the incoming wave at one object to the
outgoing wave at the other. Naturally, evaluating energies
共and forces兲 from these formal expressions require judicious
choice of basis.
A natural basis for the field is 共vector兲 spherical harmonic
multipoles centered at each object, and labeled by 共l , m , ␭兲;
the additional index ␭ 苸 兵M , E其 distinguishes between transverse electric or magnetic modes. Each factor of U decays as
e−␬d so we can obtain an asymptotic series, valid at large
␭␭
␭␴
separations d. Since Tlm,l
⬀ ␬l+l⬘+1, and Tlm,l
⬀ ␬l+l⬘+2 for
⬘m⬘
⬘m⬘
␭ ⫽ ␴, the expansion is dominated by the lowest-order multipoles. Because electromagnetism does not admit monopole
fluctuations, the leading asymptotic behavior comes from p
waves, l = l⬘ = 1. By examining the response of an object to a
uniform electric 共magnetic兲 field, it is straightforward to reEE
MM
共T1m,1m
兲 to elelate elements of the 3 ⫻ 3 matrix T1m,1m
⬘
⬘
ments of the standard electric 共magnetic兲 polarizability matrix ␣ 共␤兲. In terms of the Cartesian components of the latter
共with the ẑ axis pointing from one object to the other兲, the
first asymptotic contribution to the energy is
␣0ii =
⑀−1
V
,
4␲ 1 + 共⑀ − 1兲ni
␤0ii =
␮−1
V
,
4␲ 1 + 共␮ − 1兲ni
共3兲
where V = 4␲r1r2r3 / 3 is the ellipsoid’s volume, and the socalled depolarizing factors are given by
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©2009 The American Physical Society
PHYSICAL REVIEW A 79, 054901 共2009兲
BRIEF REPORTS
π
x
z
0
z‘
冕
⬁
0
0
共s +
冑共s + r21兲共s + r22兲共s + r23兲 ,
1 − n3
,
2
n3 =
冑
冉
冊
1 − e2
1+e
− 2e ,
log
3
2e
1−e
2
共5兲
where the eccentricity e = 1 − 4LR2 is real for a prolate spheroid 共L ⬎ 2R兲 and imaginary for an oblate spheroid 共L ⬍ 2R兲.
The polarizability tensors for an arbitrary orientation are then
obtained as ␣ = R−1␣0R, where R is the matrix that rotates
the principal axis of the spheroid to the Cartesian basis, i.e.,
R共1 , 2 , 3兲 → 共x , y , z兲. The result provides the leading Casimir
energy for spheroids of arbitrary size, orientation, and material. Note that for rarefied media with ⑀ ⯝ 1 and ␮ ⯝ 1 the
polarizabilities are isotropic and proportional to the volume.
Hence, to leading order in ⑀ − 1 the interaction is orientation
independent at asymptotically large separations, as we would
expect, since pairwise summation is valid for ⑀ − 1 Ⰶ 1.
As a special case, we present the explicit formula for
identical perfectly conducting prolate spheroids with the
length L Ⰷ R. To implement the perfect conductor limit, we
take ⑀ → ⬁. In this limit, the full T matrix is independent of
␮. Because we are doing an expansion around zero frequency, however, we must set ␮ = 0 to suppress the magnetic
field that is still allowed at zero frequency. The orientation of
each “needle” relative to the line joining them 共the initial z
axis兲 is parametrized by the two angles 共␪ , ␺兲, as depicted in
Fig. 1. Then
π
0
FIG. 2. 共Color online兲 As in Fig. 1 for oblate 共pancake-shaped兲
spheroids, with a contour plot of energy at large separations.
E12
1 共 ␪ 1, ␪ 2, ␺ 兲 = −
បc
d7
冋
冦
5L6
L
1152␲ ln − 1
R
冉
⫻ cos2 ␪1 cos2 ␪2 +
冊
2
13 2
cos ␺ sin2 ␪1 sin2 ␪2
20
册
冢 冣冧
3
L 4R 2
− cos ␺ sin 2␪1 sin 2␪2 + O
8
L
ln
R
共4兲
in terms of the semiaxis dimensions ri. We can gain further
insights into these results by focusing on the case of spheroids, for which r1 = r2 = R and r3 = L / 2. Then the depolarizing factors can be expressed in terms of elementary functions,
n1 = n2 =
0
π
ds
r2i 兲
z‘
z
FIG. 1. 共Color online兲 Orientation of a prolate 共cigar-shaped兲
spheroid: the symmetry axis 共initially the z axis兲 is rotated by ␪
about the x axis and then by ␺ about the z axis. For two such
spheroids, the energy at large distances is given by Eq. 共6兲. The
latter is depicted at fixed distance d, and for ␺1 = ␺2, by a contour
plot as function of the angles ␪1 and ␪2 for the x-axis rotations.
Minima 共maxima兲 are marked by filled 共open兲 dots.
r 1r 2r 3
2
x
z‘’
z‘’
ni =
π
y
y
, 共6兲
where ␺ ⬅ ␺1 − ␺2.
We note the following features of this result:
共i兲 The energy is minimized for ␪1 = ␪2 = 0, i.e., for two
needles aligned parallel to their separation vector, as we expect since this case is dominated by the fluctuations of electric dipoles along the symmetry axis.
共ii兲 At almost all orientations the energy scales as L6, and
vanishes logarithmically slowly as R → 0.
共iii兲 The dependence vanishes when one needle is orthogonal to ẑ 共i.e., ␪1 = ␲ / 2兲 while the other is either parallel
to ẑ 共␪2 = 0兲 or has an arbitrary ␪2 but differs by an angle ␲ / 2
in its rotation about the z axis 共i.e., ␺1 − ␺2 = ␲ / 2兲. In these
cases the energy comes from the next order term in Eq. 共6兲,
and takes the form
E12
1
冉
冊
␲
␲
L 4R 2
បc
, ␪ 2,
=−
共73 + 7 cos 2␪2兲.
2
2
1152␲d7 L
ln − 1
R
共7兲
共iv兲 From Eq. 共7兲 we see that the least favorable configuration 共␪2 = ␲ / 2兲 corresponds to two needles orthogonal to
each other and to the line joining them.
For perfectly conducting oblate spheroids with R Ⰷ L / 2,
the orientation of each “pancake” is again described by a pair
of angles 共␪ , ␺兲, as depicted in Fig. 2. A contour plot of the
leading angular dependence is also presented in the figure; its
explicit formula will not be reproduced here but we note the
following features:
共i兲 The leading dependence is proportional to R6, and does
not disappear for any choice of orientations. Furthermore,
this dependence remains even as the thickness of the pancake
is taken to zero 共L → 0兲. This is very different from the case
of the needles, where the interaction energy vanishes with
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PHYSICAL REVIEW A 79, 054901 共2009兲
BRIEF REPORTS
thickness as ln−1共L / R兲. We attribute this effect to dimensionality: The limiting linear needle is only marginally visible to
the EM field while the limiting two-dimensional plane remains as an opaque obstacle. The lack of L dependence is
due to the assumed perfectly metallic screening. If the dielectric function remains finite in the static limit, a conventional
scaling with volumes of the objects is expected.
共ii兲 The configuration of minimal energy corresponds to
two pancakes lying on the same plane 共␪1 = ␪2 = ␲ / 2, ␺ = 0兲
and has energy −បc共173/ 18␲3兲R6 / d7. This is due to the
electric-dipole fluctuations in the planes.
共iii兲 When the two pancakes are stacked on top of each
other, the energy is increased to −បc共62/ 9␲3兲R6 / d7.
共iv兲 The least favorable configuration is when the pancakes lie in perpendicular planes, i.e., ␪1 = ␲ / 2 and ␪2 = 0,
with an energy −បc共11/ 3␲3兲R6 / d7.
For an object interacting with a perfectly reflecting mirror,
we can use its image to construct N = −T̃1ŨR,11, where R
stands for the reflected object 关11兴. Alternatively, we can
express the scattering matrix for the mirror in a basis of
plane waves, and use a translation matrix to convert between
this basis and the basis for scattering of the compact object.
At leading order we have
E1m
1 =−
បc 1
Tr共␣ − ␤兲 + O共d−5兲,
d4 8␲
共8兲
translation matrices as straightforward generalizations of the
spherical case, in which spherical harmonics and Bessel
functions are replaced by spheroidal harmonics and spheroidal radial functions. Prolate spheroidal coordinates 关12兴 are
defined by taking ␾ as the usual azimuthal angle and
冑x2 + y2 = a冑共␰2 − 1兲共1 − ␩2兲,
z = a␩␰ ,
共10兲
where the interfocal separation is 2a. The surface of the
spheroid is defined by a constant spheroidal radius
␰ = ␰0 = 1 / e, with L = 2␰0a and R = a冑␰20 − 1. Oblate coordinates
are obtained by analytic continuation of the radial coordinate
with ␰ → −i␰ and k → ik. We use the conventions of Ref. 关13兴
in which the spheroidal functions are normalized identically
to their spherical analogs.
The T matrix is diagonal, with elements which in terms of
radial spheroidal functions of the first and third kinds are
Rm共1兲⬘共i␬a;␰ 兲
S
Tlm
共i␬兲 = − Rlm共3兲⬘共i␬a;␰0兲 , where the prime indicates derivative
l
0
with respect to the second argument. The translation matrix
converts between the spheroidal basis, indexed by l and m,
and the plane-wave basis, indexed by the momentum parallel
to the plane q⬜. The calculation is performed on the imagi2
, which is pure
nary axis; with q⬜ real, and q储 ⬅ i冑␬2 + q⬜
q⬜
imaginary. The translation matrix element Ulm
共i␬兲 at separaeiq L
q⬜,␾q
tion L is Ulm
= il共−1兲m q Y−m
共i
␬
a
,
␩
,
␾
兲,
where
Ym
l is a
l
q cos ␪−q⬜ cos ␾q sin ␪
. Here, ␪ is the
spheroidal harmonic and ␩ =
i␬
angle between the spheroid axis and the normal to the plane,
q sin ␾
and tan ␾ = q sin ␪+q⬜⬜ cos q␪ cos ␾q , where ␾q is the polar angle of
q⬜ around the perpendicular to the plane. To implement the
matrix multiplication needed to compute N, we integrate
over q⬜. Normalizing the integration measure introduces a
factor of 2q储 / 共i␬兲.
The numerical results can be compared to the large distance expansion for an object with Neumann boundary conditions that is placed opposite to a plane 共the latter with
Dirichlet or Neumann boundary conditions兲. The general
form of the energy in this case is
储
储
储
which is clearly independent of orientation. A similar expression for a mirror and an atom is present in the classic work of
Casimir and Polder 关1兴, and implicit in Ref. 关8兴 for perfect
conductors. 共The orientation independence of energy is special to zero temperature; at nonzero temperatures the general
expression in Ref. 关8兴 does contain angular dependence.兲
Orientation dependence in this system thus comes from
higher multipoles. The next order also vanishes so the leading term is the l = 3 contribution to the term linear in N in Eq.
共1兲, in which case the scattering matrix is not known analytically.
Instead, we obtained the preferred orientation by considering a distorted sphere in which the radius R is deformed to
R + ␦ f共␽ , ␸兲. The function f can be expanded into spherical
harmonics Y lm共␽ , ␸兲, and spheroidal symmetry can be mimicked by choosing f = Y 20共␽ , ␸兲. By performing a perturbative expansion of spherical T matrices in ␦, the leading orientation dependent part of the energy is obtained as
E f = − បc
1607
␦R4
6
640冑5␲3/2 d
cos共2␪兲.
共9兲
A prolate spheroid 共␦ ⬎ 0兲 thus minimizes its energy by
pointing toward the mirror while an oblate spheroid 共␦ ⬍ 0兲
prefers to lie in a plane perpendicular to the mirror. 共We
assume that the perturbative results are not changed for large
distortions.兲 These configurations are also preferred at small
distances d since 共at fixed distance to the center兲 the object
reorients to minimize the closest separation.
Interestingly, the above conclusion is not generally true,
and as discussed below, we find that there can be a transition
in preferred orientation as a function of d in the simpler case
of a scalar field. Then we can obtain the T matrices and
储
E= ⫾
冋
册
1
1
បc
共␤xx + ␤yy + 3␤zz兲 ,
4
2V −
d 64␲
16␲
共11兲
for a Dirichlet 共+兲 or Neumann 共−兲 plane, where V is the
volume, and ␤ is similar to the static magnetic polarizability
tensor in Eq. 共3兲 for ␮ = 0. For a spheroid this results in
E= ⫾
បc R2L 9 − 4n3 − n23 + 共3n3 − 1兲cos 2␪
,
d4 96␲
1 − n23
共12兲
where n3 is given by Eq. 共5兲. In contrast to the EM energy of
Eq. 共8兲, the leading term of Eq. 共12兲 is orientation dependent.
For a prolate spheroid with R Ⰶ L we have
E= ⫾
បc R2L
关9 − cos共2␪兲 + O„共R/L兲2…兴,
d4 96␲
and for an oblate spheroid with R Ⰷ L, we have
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PHYSICAL REVIEW A 79, 054901 共2009兲
BRIEF REPORTS
- 0.026
asymptotic
d=17.5R
- 0.028
d=13.1R
- 0.03
d=10.9R
d=9.8R
d=8.7R
- 0.032
d=7.6R
- 0.034
d=6.5R
1
8
0
1
4
3
8
1
2
FIG. 3. 共Color online兲 The orientation dependence of the Casimir energy for a prolate Neumann spheroid with L / R = 4.8 opposite a Neumann plane, as a function of the angle ␪ between the
normal to the plane and the spheroid axis, for various values of d,
which is the distance from the spheroid center to the plane. The
asymptotic result of Eq. 共12兲 is depicted by the top line. For d
ⱗ 8.5R, the spheroid prefers to point toward the plane while at
larger distances it spontaneously breaks the rotational symmetry
and aligns parallel to the plane. The larger the value of ␰0 = 1 / e, the
larger the distance at which this transition occurs.
both a needle and a pancake prefer to be parallel to the plate;
in the former case because of ␤xx and ␤yy, and in the latter
case because of ␤zz. For a Dirichlet plate, the energy is minimal if the needle or the pancake is perpendicular to the plate
as would have been predicted by pairwise summation.
In Fig. 3 we present a sample of the full numerical calculation, using an optimized version 关14兴 of the spheroidal harmonic package of Falloon et al. 关15兴. We truncate the infinite
sum at l = 2 although we have verified for individual cases
that summing through l = 4 does not significantly change
these results. At close separations, the prolate spheroid prefers to point toward the plane, as we expect from the proximity force approximation. At large separations, we have
shown that the configuration parallel to the plane has the
lowest energy. We thus observe a spontaneous breaking of
the symmetry around the perpendicular to the plane; the
separation at which this transition occurs varies with the
spheroid’s eccentricity.
These examples provide a glimpse into the myriad effects
of shape and orientation on the Casimir effect. One can
imagine that with better precision of experiment and theory
such forces can be employed to manipulate small objects at
the micrometer and nanometer scales.
To leading order in R / L and at fixed separation d, the energy
for a Neumann plate is minimized at ␪ = ␲ / 2 for a prolate
spheroid and at ␪ = 0 for an oblate spheroid. This means that
We acknowledge discussions with S. J. Rahi. This research was supported by the DFG through Grant No.
EM70/3 共T.E.兲, NSF Grants No. DMR-08-03315 共M.K.兲 and
No. PHY-0555338 a grant from Research Corporation
共N.G.兲, and the U. S. Department of Energy under cooperative research agreement No. DF-FC02-94ER40818 共R.L.J.兲.
Part of this work was carried out at the Kavli Institute for
Theoretical Physics, with support from NSF Grant No.
PHY05-51164.
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B. G. Casimir, Proc. K. Ned. Akad. Wet. 51, 793 共1948兲.
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关3兴 U. Mohideen and A. Roy, Phys. Rev. Lett. 81, 4549 共1998兲.
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关5兴 H. B. Chan, V. A. Aksyuk, R. N. Kleiman, D. J. Bishop, and F.
Capasso, Science 291, 1941 共2001兲.
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Lett. 99, 170403 共2007兲; Phys. Rev. D 77, 025005 共2008兲.
关7兴 O. Kenneth and I. Klich, Phys. Rev. B 78, 014103 共2008兲.
关8兴 R. Balian and B. Duplantier, Ann. Phys. 共N.Y.兲 104, 300
共1977兲; 112, 165 共1978兲.
关9兴 A nonretarded spheroid-plane interaction at fixed orientation
has been considered in C. E. Roman-Velazquez and C.
Noguez, J. Phys. A 39, 6695 共2006兲.
关10兴 G. Feinberg and J. Sucher, Phys. Rev. A 2, 2395 共1970兲.
关11兴 T. Emig, J. Stat. Mech. 共2008兲, P04007.
关12兴 C. Flammer, Spheroidal Wave Functions 共Stanford University
Press, Stanford, 1957兲.
关13兴 N. Graham and K. D. Olum, Phys. Rev. D 72, 025013 共2005兲.
关14兴 See http://community.middlebury.edu/~ngraham.
关15兴 P. E. Falloon, P. C. Abbott, and J. B. Wang, J. Phys. A 36,
5477 共2003兲.
E= ⫾
បc R3
关2 + cos共2␪兲 + O„共L/R兲…兴.
d4 24␲2
共14兲
054901-4