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Casimir interaction between eccentric cylinders Francisco Diego Mazzitelli Universidad de Buenos Aires QFEXT-07 Leipzig

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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

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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

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Motivations Theoretical interest: geometric dependence of the Casimir force

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Motivations New experiments with cylinders? A null experiment: measure the signal to restore the zero eccentricity configuration after a controlled displacement Dalvit, Lombardo, FDM, Onofrio, Eur. Phys. Lett 2004

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Resonator of mass M and frecuency Ω Dalvit, Lombardo, FDM, Onofrio, Eur. Phys. Lett 2004 Motivations Frequency shift of a resonator

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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

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The exact formula for eccentric cylinders Vacuum energy :

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a = radius of the inner cylinder b = radius of the outer cylinder d = minimum distance between surfaces e= eccentricity L >> a,b THE CONFIGURATION:

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Very long cylinders: symmetry in the z-direction where Using Cauchy´s theorem: F = 0 gives the eigenvalues of the two dimensional problem

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Defining the interaction energy as we end with a finite integral ( = 0) along the imaginary axis We need an explicit expression for M

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TM modes: B z = 0 Dirichlet b.c. Eigenvalues in the annular region

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And a similar treatment for TE modes… TE replace these Bessel functions by their derivatives Putting everything together, subtracting the configuration corresponding to far away conductors, and using the asymptotic expansion of Bessel functions:

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The exact formula for eccentric cylinders Each matrix element is a series of Bessel functions

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Particular cases I: CONCENTRIC CYLINDERS When = 0 the matrix inside the determinant becomes diagonal

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Large values of α: a wire inside a hollow cylinder The Casimir energy is dominated by the n=0 TM mode Logarithmic decay

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All modes contribute – use uniform expansions for Bessel functions. Example: SMALL DISTANCES: BEYOND THE PROXIMITY APPROXIMATION

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After a long calculation…. PFA TM TE The following correction is probably of order Lombardo, FDM, Villar, in preparation

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PFA The next to leading order approximation coincides with the semiclassical approximation based on the use of Periodic Orbit Theory, and is equivalent to 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

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Particular cases II A cylinder in front of a plane b, with H = b - fixed d a H

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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

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QUASI-CONCENTRIC CYLINDERS a,b arbitrary Idea: consider only the matrix elements near the diagonal Lowest non trivial order: tridiagonal matrix Main point:

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Recursive relation for the determinant of a tridiagonal matrix

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…..a simpler formula…. where Not a determinant, only a sum The numerical evaluation is much more easy

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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

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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

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Efficient numerical evaluation The numerical calculations are more complex when the distances between the surfaces is small, since it involves more modes (larger matrices). Idea: use the PFA to improve the convergence A trivial example: evaluation of a slowly convergent series

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Application: concentric cylinders the same, with Bessel functions replaced by their leading uniform expansion

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Analytic expression, it has the correct leading behaviour (but not the subleading) The numerical evaluation of the difference converges faster Lombardo, FDM, Villar, in preparation ~

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Improved calculation Direct calculation

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Numerical fit: Numerical data fit Expected 1 0.302 We are trying to generalize this procedure to other geometries (non trivial)

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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

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