QFEXT-07 Leipzig

1. Casimir interaction between eccentric cylindersFrancisco Diego MazzitelliUniversidad de Buenos Aires QFEXT-07 Leipzig

2. 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

3. 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

4. Motivations • Theoretical interest: geometric dependence of the Casimir force

5. 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

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

7. MotivationsA cylinder in front of a plane L a d Intermediate between plane-plane and plane-sphere Dalvit, Lombardo, FDM, Onofrio, Eur. Phys. Lett 2004; R. Onofrio et al PRA 2005

8. The exact formula for eccentric cylinders Vacuum energy:

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

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

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

12. Eigenvalues in the annular region TM modes: Bz = 0 Dirichlet b.c.

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

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

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

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

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

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

21. PFA next to leading 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.

22. Particular cases II A cylinder in front of a plane d a b,   with H = b -  fixed H

23. Using uniform expansion and addition theorem of Bessel functions: Matrix elements for cylinder-plane (Bordag 2006, Emig et al 2006) Idem for TE modes Matrix elements for eccentric cylinders:

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

25. Recursive relation for the determinant of a tridiagonal matrix

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

28. 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

29. 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

30. 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

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

32. 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

33. Improved calculation Direct calculation

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

35. 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