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