Leading order contribution to lateral Casimir force
between corrugated dielectric slabs.
Prachi Parashar
Oklahoma Center for High Energy Physics and Homer L. Dodge Department of Physics and
Astronomy, University of Oklahoma, Norman, OK 73019, USA
Collaborators: Inés Cavero-Peláez, Kimball. A. Milton, and K. V. Shajesh
Date: July 8-9, 2010
Event: Quantum Vacuum Meeting
Venue: Texas A & M University, College Station, Texas.
Prachi Parashar (University of Oklahoma)
Non-contact gears III:EM Case
QV: July 9, 2010
1 / 26
Outline
1
Lateral Casimir force
2
Motivation
Scalar case: Review of Noncontact Gears I and II
3
Formalism
Electromagnetic case: Statement of the problem
4
Leading order contribution
Evaluation of the reduced Green’s dyadic
General result for the leading order term
5
Limits
Conductor limit
Thin plate limit
6
Conclusions and things to do
Prachi Parashar (University of Oklahoma)
Non-contact gears III:EM Case
QV: July 9, 2010
2 / 26
Lateral Casimir force
Consider potential
Vi (z, y ) = λi δ(z − ai − hi (y )),
i = 1, 2,
a = a2 − a1 > 0,
where, the functions hi (y ) describe the corrugations on the plate.
Casimir energy for a configuration when one of the plate is laterally shifted
by an amount y0 ,
h1
h1 (y + y0 ),
h2 (y ),
can be written in the form
a
h2
y0
∆E (a, hi , y0 ) = E − E
(0)
d=
2π
k0
(a) = E1 (a, h1 ) + E2 (a, h2 ) + E12 (a, hi , y0 ),
where E12 isolates the interaction energy due to the lateral shift.
Lateral Casimir force is defined as the negative change in energy due to
the lateral shift:
∂E12
∂
∆E = −
.
FLat (a, hi , y0 ) = −
∂y0
∂y0
Prachi Parashar (University of Oklahoma)
Non-contact gears III:EM Case
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3 / 26
Motivation
1997-2010: Starting with first theoretical study by Golestanian et. al. in
1997, various theoretical and experimental analysis of lateral Casimir force
has been done till now. One remarkable theoretical calculation for lateral
Casimir force between dielectric gratings was done exactly by Lambrecht
and Marachevsky, which shows 0.1% accuracy between theory and
experiment.
In noncontact Gears I and II (PRD 78, 065018, 065019) we presented
next-to-leading order contibution to the lateral Casimir force in planar
geometry and leading order contribution in cylinderical geometry due to
scalar field interacting with semi-transparent delta potential.
h1
a
a
h2
y0
d=
a1
θ0
2π
k0
a2
Prachi Parashar (University of Oklahoma)
Non-contact gears III:EM Case
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Leading order perturbation-Dirichlet and Weak
Leading order contribution to the lateral Casimir force in Dirichlet limit
and weak limit for sinusoidal corrugations are
(2)
h1 h2 (1,1)
(k0 a),
A
a a D
(0) h1 h2 (2)
A (k0 a).
= k0 a sin(k0 y0 ) FCas,W a a W
(0)
FLat,D = 2k0 a sin(k0 y0 ) |FCas,D |
(2)
FLat,W
(1,1)
AD
(k0 a)
e−k0 a e2 (k0 a)
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
2
4
6
Prachi Parashar (University of Oklahoma)
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10 k0 a
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Next-to-leading order perturbation - Dirichlet
Next-to-Leading order contribution to the lateral Casimir force in the
Dirichlet limit for sinusoidal corrugations is
(0) h1 h2 15
(4)
FLat,D = 2 k0 a sin(k0 y0 ) FCas,D a a 4
2
h1
h22
h1 h2 (2,2)
(3,1)
×
+ 2 AD (k0 a) − 2 cos(k0 y0 )
A
(k0 a) .
a2
a
a a D
(3,1)
AD
(2,2)
(k0 a)
AD
(k0 a)
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
2
4
Prachi Parashar (University of Oklahoma)
6
8
10 k0 a
Non-contact gears III:EM Case
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Next-to-leading order perturbation - Weak
The next correction to the lateral Casimir force for the the weak coupling
case is
(0) h1 h2 3
(4)
FLat,W = k0 a sin(k0 y0 ) FCas,W a a 2
2
h1 h2 (4)
h22
h1
(4)
+ 2 AW (k0 a) − 2 cos(k0 y0 )
A (2k0 a) ,
×
a2
a
a a W
where
(n)
AW (t0 ) = e −t0
n
X
en (t0 )
t0m
=
.
m!
e t0
m=0
e−k0 a e4 (k0 a)
e−2k0 a e4 (2k0 a)
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
2
4
Prachi Parashar (University of Oklahoma)
6
8
10 k0 a
Non-contact gears III:EM Case
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10 k0 a
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Analysis
Weak coupling limit - Non-perturbative results
Z
(k0 a)3 1 ∞ dt
sin(t + k0 y0 )
[∞,∞]
FW
=−
Re
3 ,
sin(k0 y0 ) π −∞ t
[(t + ik0 a)2 + {k0 r (t)}2 ] 2
for 2h < a.
[∞,∞]
FW
[∞,4]
FW
[∞,∞]
1.4
k0 y0 =
FW
π
4
−1
2
4
6
10 k0 a
8
1.2
k0 h = 0.3
−0.01
1.0
0.8
−0.02
0.6
−0.03
0.4
k0 y0 =
π
4
k0 h = 0.1, 0.3, 0.5
0.2
−0.04
0.5
1.0
1.5
2.0
k0 a
We observe that the perturbative results, when the next-to-leading order is
included, compares with the exact result remarkably well for k0 h ≪ 1 and
2h < a. Similar results should hold for the Dirichlet limit also.
Prachi Parashar (University of Oklahoma)
Non-contact gears III:EM Case
QV: July 9, 2010
8 / 26
Electromagnetic case: Statement of the problem
We consider two dielectric slabs
Vi (z, y ; ω) = ω 2 (ǫi (ω) − 1) [θ(z − ai − hi (y )) − θ(z − bi − hi (y ))] ,
where i = 1, 2
y0
ε1
ε2
d1
d2
d=
2π
k0
a
h1
h2
where, the thickness of the individual slabs is di = bi − ai , such that
a = a2 − b1 > 0 represents the distance between the slabs.
Prachi Parashar (University of Oklahoma)
Non-contact gears III:EM Case
QV: July 9, 2010
9 / 26
Lateral Casimir force
The vacuum energy arising from the fluctuation of the electromagnetic
field in presence of the potential is given by
Z
dω
i
Tr ln Γ Γ−1
E =−
0 ,
2
2π
where Γ is the Green’s dyadic which obeys
−∇ × ∇ × + ω 2 1 + V1 (x) + V2 (x) Γ(x, x′ ; ω) = −ω 2 1 δ(x − x′ ),
and Γ0 is the Green’s dyadic for the free space.
As shown earlier Casimir energy for a configuration when one of the slab is
laterally shifted by an amount y0 is
∆E (a, hi , y0 ) = E − E (0) (a) = E1 (a, h1 ) + E2 (a, h2 ) + E12 (a, hi , y0 ).
Lateral Casimir force is defined as the negative change in energy due to
the lateral shift
∂
∂E12
FLat (a, hi , y0 ) = −
∆E = −
.
∂y0
∂y0
Prachi Parashar (University of Oklahoma)
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10 / 26
Formalism(contd.)
Interaction energy E12 is given by
Z
dω
i
Tr ln 1 − Γ1 ∆V 1 Γ2 ∆V 2 .
E12 =
2
2π
where Γi is the Green’s dyadic when one slab has corrugations.
h
i−1
Γ(0) ,
Γi = 1 − Γ(0) ∆V i
(0)
where ∆V i = V i − V i
(0)
V i given by
is the deviation from the background potential
(0)
V i (z; ω) = (ǫi (ω) − 1) [θ(z − ai ) − θ(z − bi )] .
Γ(0) is the Green’s dyadic for no corrugation configuration.
Prachi Parashar (University of Oklahoma)
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11 / 26
Leading order contribution - perturbative expansion
Formally expanding the log and keeping terms to second order in
corrugation amplitude (which requires expanding potential for small
corrugation amplitude) we get
Z
1
dζ
(1)
(1)
(2)
Tr ln Γ(0) ∆V 1 Γ(0) ∆V 2 ,
E12 = −
2
2π
where
(1)
∆ V i (z, y ; iζ) = −hi (y ) (ǫi (iζ) − 1) [δ(z − ai ) − δ(z − bi )] ,
is the first term in expansion of potential for small corrugation amplitude.
Γ(0) is translationally invariant in x and y direction. So we can Fourier
transform x and y directions
Z ∞
Z
(2)
E12
dky ∞ dky′
h̃1 (ky − ky′ ) h̃2 (ky′ − ky ) L(2) (ky , ky′ ),
=
Lx
−∞ 2π −∞ 2π
where Lx is the length in x direction and h̃i (ky ) are the Fourier transforms
of the corrugation amplitude functions hi (y ).
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Leading order contribution(cont.)
Kernel L(2) (ky , ky′ ) is given by
L
(2)
(ky , ky′ )
1
=−
2
Z
dζ
2π
Z
dkx (2)
I (kx , ζ, ky , ky′ ),
2π
where,
I (2) (kx , ζ, ky , ky′ ) = (ǫ1 (iζ) − 1) (ǫ2 (iζ) − 1) ×
h
(0)
(0)
Γ̃ij (a2 , a1 ; kx , ky , ω) Γ̃ji (a1 , a2 ; kx , ky′ , ω)
(0)
(0)
(0)
(0)
−Γ̃ij (b2 , a1 ; kx , ky , ω) Γ̃ji (a1 , b2 ; kx , ky′ , ω)
−Γ̃ij (a2 , b1 ; kx , ky , ω) Γ̃ji (b1 , a2 ; kx , ky′ , ω)
i
(0)
(0)
+Γ̃ij (b2 , b1 ; kx , ky , ω) Γ̃ji (b1 , b2 ; kx , ky′ , ω) ,
So the quantity we need to evaluate is the Green’s dyadic for the
background potential, which is the slabs without corrugation.
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Evaluation of the reduced Green’s dyadic
The Green’s dyadic Γ(0) obeys
h
i
(0)
(0)
−∇ × ∇ × + ω 2 1 + ω 2 V1 (x) + ω 2 V2 (x) Γ(0) (x, x′ ; ω) = −ω 2 1 δ(x − x′ ).
For planar geometry the problem reduces to solving for two scalar Green’s
function g E (z, z ′ ) and g H (z, z ′ ) which satisfies
∂2
− 2 + k 2 + ζ 2 ǫ(z) g E (z, z ′ ) = δ(z − z ′ ),
∂z
k2
∂ 1 ∂
2
+
+ζ
g H (z, z ′ ) = δ(z − z ′ ),
−
∂z ǫ(z) ∂z
ǫ(z)
where ǫ(z) = 1 + V (z) and k = kx since we can choose ky = 0.
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Evaluation of the reduced Green’s dyadic(cont.)
The reduced Green’s dyadic in terms of g E (z, z ′ ) and g H (z, z ′ ) is
Γ(0) (z, z ′ ; k, 0, ζ) =
2
6
6
6
6
6
4
1
∂
1
∂
ǫ(z ′ ) ∂z ′ ǫ(z) ∂z
g H (z, z ′ )
−ζ 2 g E (z, z ′ )
0
∂
ik
ǫ(z) ǫ(z ′ ) ∂z
0
g H (z, z ′ )
0
3
g H (z, z ′ )
7
7
7
0
7.
7
5
H
′
k2
g
(z,
z
)
ǫ(z) ǫ(z ′ )
ik
∂
ǫ(z) ǫ(z ′ ) ∂z
where we have dropped the delta functions, which do not contribute.
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Evaluation of the reduced Green’s dyadic(cont.)
Denoting Γ matrix components as Γij , we can write the general form as

kx2
Γ
k 2 11
+
ky2
Γ
k 2 22



(0)
′
k k
k k
Γ (z, z ; kx , ky , ζ) = 
 kx 2 y Γ11 − kx 2 y Γ22


kx
k Γ31
Prachi Parashar (University of Oklahoma)
kx ky
Γ
k 2 11
Non-contact gears III:EM Case
ky2
Γ
k 2 11
−
+

kx ky
Γ kx Γ
k 2 22 k 13 
kx2
k2
ky
k Γ31
Γ22



ky
Γ

13
k


Γ33
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Green’s function regions
5
Q
6
R
3
N
O
P
B
Y
4
M
2
X
9
L
A
U
V
W
1
S
7
T
8
z ′ = b2
z ′ = a2
z ′ = b1
Prachi Parashar (University of Oklahoma)
z = b2
z = a2
z = b1
z = a1
z
=
z′
z ′ = a1
Non-contact gears III:EM Case
QV: July 9, 2010
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General result for the leading order term
Using solutions to the electric and the magnetic Green’s function we can evaluate
I (2) (kx , ζ, k1y , k2y ) as
"
−
1 1 1 1
k 2 k ′2 2κ 2κ′
1 1
M(−α1 , −α1′ )M(−α2 , −α2′ )(kx2
∆ ∆′
+ ky ky′ )2 ζ 4
+ ∆1 ∆¯1′ M(−α1 , ᾱ1′ )M(−α2 , ᾱ2′ )kx2 (ky − ky′ )2 ζ 2 κ′
2
+ ∆1¯ ∆1′ M(ᾱ1 , −α1′ )M(ᾱ2 , −α2′ )kx2 (ky − ky′ )2 ζ 2 κ2
o
n
2
+ ∆¯1 ∆¯1′ M(ᾱ1 , ᾱ1′ )(kx2 + ky ky′ )κκ′ + M(−ᾱ1 , −ᾱ1′ )k 2 k ′ ε11
#
n
o
2
× M(ᾱ2 , ᾱ2′ )(kx2 + ky ky′ )κκ′ + M(−ᾱ2 , −ᾱ2′ )k 2 k ′ ε12
,
where
ˆ
∆ = (1 − α12 e −2κ1 d1 )(1 − α22 e −2κ2 d2 ) e κa
˜
−α1 α2 (1 − e −2κ1 d1 )(1 − e −2κ2 d2 ) e −κa ,
ˆ
′
2
M(αi , αi′ ) = (εi − 1) (1 − αi2 ) e −κi di (1 − αi′ ) e −κi di
˜
′
−(1 + αi )(1 − αi e −2κi di )(1 + αi′ )(1 − αi′ e −2κi di ) ,
Prachi Parashar (University of Oklahoma)
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General result for the leading order term(cont.)
κ2i = k 2 + ζ 2 εi
κ̄i = κi /εi
αi = (κi − κ)/(κi + κ)
Quantities with primes are obtained by replacing ky → ky′ everywhere.
Quatities with bar are defined in similar way with κ replaced with
κi → κi =
Prachi Parashar (University of Oklahoma)
κi
ǫi
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Conductor limit
In the conductor limit ǫ → ∞
#
"
2 + κ′ 2 − (k − k ′ )2 }2
′
{κ
κ
κ
y
y
(2)
.
Iε→∞
(κ, κ′ , ky − ky′ ) = −
1+
sinh κa sinh κ′ a
4 κ2 κ′ 2
For the case of sinusoidal corrugations described by
h1 (y ) = h1 sin[k0 (y + y0 )]
h2 (y ) = h2 sin[k0 y ]
the lateral force is evaluated to be
(0) h1 h2 (1,1)
(2)
Fε→∞
= 2k0 a sin(k0 y0 ) FCas A
(k0 a),
a a ε→∞
Prachi Parashar (University of Oklahoma)
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QV: July 9, 2010
20 / 26
Conductor limit cont.
A(1,1)
ε→∞ (t0 )
s2
15
= 4
π
s̄ 2
Z
∞
dt
−∞
+ t2
Z
2
s+
where
=
and
the Dirichlet scalar case.
∞
0
2 − t 2 )2 s
s+
1 (s 2 + s+
0
s̄ds̄
+
,
2
sinh s sinh s+ 2
8 s 2 s+
= s̄ 2 + (t + t0 )2 . The first term corresponds to
(1,1)
Aε→∞ (k0 a)
1.0
0.8
0.6
0.4
0.2
2
(1,1)
4
6
8
10 k0 a
Figure: Plot of Aε→∞ (k0 a) versus k0 a. The dotted curve represents 2 times the
Dirichlet
Prachi
Parashar case.
(University of Oklahoma)
Non-contact gears III:EM Case
QV: July 9, 2010
21 / 26
Conductor limit cont.
We can evaluate one of the integrals to get
"
Z
sin(2t0 u/π) sinh2 u 7
15 ∞
2
(1,1)
du
− sinh u
Aε→∞ (t0 ) =
4 0
(2t0 u/π) cosh6 u 2
#
1 2t0 2 sinh2 u
1 2t0 4 sinh2 u
−
,
+
2 π
cosh4 u 16 π
cosh2 u
(2)
which reproduces the result in Emig et. al. apart from an overall factor of
2.
The double integral representation is more useful for numerical evaluation
because of the oscillatory nature of the function sin x/x the above
expression.
Prachi Parashar (University of Oklahoma)
Non-contact gears III:EM Case
QV: July 9, 2010
22 / 26
Thin plate limit
It is under construction.......
It is interesting to analyze di → 0 limit. If we set
ω 2 (ǫi − 1) = ωp2i
λi = ωp2i di = fixed as di → 0,
then we can write our potential as
Vi = λi δ(z − ai − hi (y )).
This should model 2D sheets. However, if a ∼ 10−7 m and ωp ∼ 1015 Hz,
then for conductor case
ωp2i da ≫ 1 ⇒ d ≫ 10−7 m,
which is not exactly a single atom thin layer. This limit can work for weak
case.
Prachi Parashar (University of Oklahoma)
Non-contact gears III:EM Case
QV: July 9, 2010
23 / 26
Conclusion and things to do
1
We have obtained result for leading order contribution to the lateral
Casimir force for corrugated dielectric slabs and obtained conductor
limit for the same.
2
Obtain thin plate limit and get exact results in weak case.
3
Calculate next-to-leading order contribution and compare results with
experiment as well as various numerical/analytical exact results
(Mohideen, Reynaud, Emig, Marachevsky).
Prachi Parashar (University of Oklahoma)
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QV: July 9, 2010
24 / 26
1
Thanks to Jef Wagner, Elom Albalo and Nima Pourtolami for useful
comments and discussion.
2
Thank you all for listening.
Prachi Parashar (University of Oklahoma)
Non-contact gears III:EM Case
QV: July 9, 2010
25 / 26
References
I. Cavero-Pelaez, K. A. Milton, P. Parashar and K. V. Shajesh, Phys.
Rev. D 78, 065018 (2008) [arXiv:0805.2776 [hep-th]].
I. Cavero-Pelaez, K. A. Milton, P. Parashar and K. V. Shajesh, Phys.
Rev. D 78, 065019 (2008) [arXiv:0805.2777 [hep-th]].
T. Emig, A. Hanke, R. Golestanian and M. Kardar, Phys. Rev. A 67,
022114 (2003).
K. A. Milton, P. Parashar and J. Wagner, Phys. Rev. Lett. 101,
160402 (2008) [arXiv:0806.2880 [hep-th]].
K. A. Milton, P. Parashar and J. Wagner, arXiv:0811.0128 [math-ph].
J. S. Schwinger, L. L. . DeRaad and K. A. Milton, Annals Phys. 115,
1 (1979).
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