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Casimir interaction between eccentric cylinders
Francisco Diego MazzitelliUniversidad de Buenos Aires
QFEXT-07Leipzig
Plan of the talk
• Motivations
• The exact formula for eccentric cylinders
• Particular cases: concentric cylinders and a cylinder in front of a plane
• Quasi-concentric cylinders: a simplified formula
• Efficient numerical evaluation of the vacuum energy (concentric case)
• Conclusions
REFERENCES:• D. Dalvit, F. Lombardo, F.D.Mazzitelli and R. Onofrio, Europhys Lett 2004
•D. Dalvit, F. Lombardo, F.D.Mazzitelli and R. Onofrio, Phys. Rev A 2006
• F.D. Mazzitelli, D. Dalvit and F. Lombardo, New Journal of Physics, Focus Issue on Casimir Forces (2006)
• F. Lombardo, F.D. Mazzitelli and P. Villar, in preparation
Motivations
• Theoretical interest: geometric dependence of the Casimir force
Motivations
• New experiments with cylinders?
A null experiment: measure the signal to restore the zero eccentricity configurationafter a controlled displacement
Dalvit, Lombardo, FDM, Onofrio, Eur. Phys. Lett 2004
25
3
)(120 Ω−ΩΔΩ −≈
MabcLahπ
Resonator of massM and frecuency Ω
Dalvit, Lombardo, FDM, Onofrio, Eur. Phys. Lett 2004
Motivations
Frequency shift of a resonator
2/7
2/13
2384 dcLaF planecylhπ≈−
a
d
L
Dalvit, Lombardo, FDM, Onofrio, Eur. Phys. Lett 2004; R. Onofrio et al PRA 2005
Motivations
A cylinder in front of a plane
Intermediate between plane-plane and plane-sphere
( ) ( ) ( ) ( )∑ −=Γ∂−Γ∂=Γ∂ ∞a
aa ϖω,
,,00 21
kkkhEEEC
( )=Γ∂CE Γ∂ ∞Γ∂∞ ∞
The exact formula for eccentric cylinders
Vacuum energy:
a = radius of the inner cylinderb = radius of the outer cylinderd = minimum distance between surfaces= eccentricityL >> a,b
THE CONFIGURATION:
Very long cylinders: symmetry in the z-direction
where
Using Cauchy´s theorem:
F = 0 gives the eigenvalues of the two dimensional problemnλ
Defining the interaction energy as
we end with a finite integral ( = 0) along the imaginary axis
We need an explicit expression for M
TM modes: BTM modes: Bz z = 0= 0
Dirichlet b.c.
Eigenvalues in the annular region
And a similar treatment for TE modes…
TE
replace these Bessel functions by their derivatives
Putting everything together, subtracting the configuration corresponding tofar away conductors, and using the asymptotic expansion of Bessel functions:
The exact formula for eccentric cylinders
Each matrix elementis a series of Bessel functions
€
a =b /a
Particular cases I:
CONCENTRIC CYLINDERS
When = 0 the matrix inside the determinant becomes diagonal
€
In−m (0) = δn−m
Large values of α: a wire inside a hollow cylinder
The Casimir energy is dominated by the n=0 TM mode
Logarithmic decay
All modes contribute – use uniform expansions for Bessel functions.
Example:
SMALL DISTANCES: BEYOND THE PROXIMITY APPROXIMATION
After a long calculation….
PFATM
TE
The following correction is probably of order )1( log )1( 3 −− aa
Lombardo, FDM, Villar, in preparation
PFA
The next to leading order approximation coincides with the semiclassicalapproximation based on the use of Periodic Orbit Theory, and is equivalentto the use of the geometric mean of the areas in the PFA
( FDM, Sanchez, Von Stecher and Scoccola, PRA 2003)
Similar property in electrostatics.
next to leading
Particular cases II
A cylinder in front of a plane
b, with H = b - fixed
da
H
Matrix elements for eccentric cylinders:
Using uniform expansion and addition theorem of Bessel functions:
Matrix elements for cylinder-plane(Bordag 2006, Emig et al 2006)
Idem for TE modes
QUASI-CONCENTRIC CYLINDERS
a,b arbitrary
Idea: consider only the matrix elements near the diagonal
Lowest non trivial order: tridiagonal matrix
Main point:
€
δ <<a −1
Recursive relation for the determinant of a tridiagonal matrix
…..a simpler formula….
where
Not a determinant, only a sum
The numerical evaluation is much more easy
Quasi concentric cylinders: the large distance limit (α >> 1)
As expected it is again dominated by the n=0 TM mode
Logarithmic decay:
-Similar to cylinder - plane (Bordag 2006, Emig et al 2006)
-Interesting property for checking PFA
- Analogous property in electrostatics
Quasi concentric cylinders: short distances
Uniform expansions for Bessel functions:
The result coincides with the leading order with the Proximity Force Approximation
Beyond leading order ? Work in progress…
Dalvit, Lombardo, FDM, Onofrio, Eur. Phys. Lett 2004
Efficient numerical evaluation
The numerical calculations are more complex when the distancesbetween the surfaces is small, since it involves more modes (largermatrices).
Idea: use the PFA to improve the convergence
A trivial example: evaluation of a slowly convergent series
€
zM = 1n1.1
n=1
M
∑z1000 = 5.6 z
109 = 9.325 z∞ =10.5844
zM = zM − dxx1.1
1
M
∫ + dxx1.1
1
M
∫ → ΔM + 10
Δ10 = 0.66 Δ100 = 0.587 Δ1000 = 0.5846
Application: concentric cylinders
the same, with Bessel functions replaced by their leading uniform expansion
Analytic expression, it has the correctleading behaviour (but not the subleading)
The numerical evaluation of the difference converges faster
Lombardo, FDM, Villar, in preparation
€
E = A(α −1)3 [1+ B(α −1)]+ akα
k= 0
∞
∑~
Improved calculation
Direct calculation
)]1(5.01[)1( 3 −+
−= a
aAENTL
PFA
NTLPFA
cc
EE12
Numerical fit: 212 )1(286.099997.0 −−≈ aNTLPFA
cc
EE
Numerical data
fit
Expected 1 0.302
We are trying to generalize this procedure to other geometries(non trivial)
Conclusions• We obtained an exact formula for the vacuum energy of a system of
eccentric cylinders
• The formula contains as particular cases the concentric cylinders and the cylinder-plane configurations
• We obtained a simpler formula in the case of quasi concentric cylinders, using a tridiagonal matrix
• In all cases we analyzed the large and small distance cases: at large distances we found a characteristic logarithmic decay. At small distances we recovered the PFA. In the concentric case we obtained an analytic expression up to the next to next to leading order
• We used the leading behaviour at small distances to improve the convergence of the numerical evaluations in the concentric case