Abstract
We give a comprehensive presentation of methods for calculating the Casimir force to arbitrary accuracy, for any number of objects, arbitrary shapes, susceptibility functions, and separations. The technique is applicable to objects immersed in media other than vacuum, to nonzero temperatures, and to spatial arrangements in which one object is enclosed in another. Our method combines each object’s classical electromagnetic scattering amplitude with universal translation matrices, which convert between the bases used to calculate scattering for each object, but are otherwise independent of the details of the individual objects. This approach, which combines methods of statistical physics and scattering theory, is well suited to analyze many diverse phenomena. We illustrate its power and versatility by a number of examples, which show how the interplay of geometry and material properties helps to understand and control Casimir forces. We also examine whether electrodynamic Casimir forces can lead to stable levitation. Neglecting permeabilities, we prove that any equilibrium position of objects subject to such forces is unstable if the permittivities of all objects are higher or lower than that of the enveloping medium; the former being the generic case for ordinary materials in vacuum.
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- 1.
Because of this relationship, these scattering amplitudes are also referred to as elements of the T-matrix. In standard conventions, however, the T-matrix differs from the matrix elements of the \({\mathbb{T}}\)-operator by a basis-dependent constant, so we will use the term “scattering amplitude” to avoid confusion.
- 2.
Alternatively, we can set up asymptotically incoming and outgoing waves on the outside and regular waves inside. The amplitudes of the outgoing waves are then given by the S-matrix, which is related to the scattering amplitude \({\fancyscript{F}}\) by \({\fancyscript{F}}=( S- I)/2.\) Although these two matrices carry equivalent information, the scattering amplitude will be more convenient for our calculation.
- 3.
The sequence of two changes of variables is known as Hubbard-Stratonovich transformation in condensed matter physics.
- 4.
\({\mathbb{G}}_M\) satisfies \(\left({\varvec\nabla} \times\mu_M^{-1}(ic\kappa){\varvec\nabla} \times+\varepsilon_M(ic\kappa)\kappa^2\right){\mathbb{G}}_M(ic\kappa,{\mathbf x},{\mathbf x}^{\prime})=\delta({\mathbf x}-{\mathbf x}^{\prime}){\mathbb{I}},\) and is related to \(G_M,\) the Green’s function of the imaginary frequency Helmholtz equation, by \({\mathbb{G}}_M(ic\kappa,{\mathbf x},{\mathbf x}^{\prime}) = \mu_M(ic\kappa)\left({\mathbb{I}} + (n_M \kappa)^{-2} {\varvec\nabla}\otimes{\varvec\nabla}^{\prime}\right) G_M(icn_M\kappa,{\mathbf x},{\mathbf x}^{\prime})\).~Here, \(n_M(ic\kappa)=\sqrt{\varepsilon_M(ic\kappa) \mu_M(ic\kappa)}\) is the index of refraction of the medium, whose argument is suppressed to simplify the presentation. Thus \({\mathbb{G}}_M,\) in contrast to \({\mathbb{G}}_0,\) takes into account the permittivity and permeability of the medium when they are different from one.
- 5.
To obtain the free energy at finite temperature, in place of the ground state energy \({\fancyscript{E}},\) \(\int {\frac{d\kappa}{2\pi}}\) is replaced by the sum \({\frac{kT}{\hbar c}}\sum^{\prime}_{\kappa_n\geq 0}\) over Matsubara ‘wavenumbers’ \(\kappa_n = 2\pi n k T/\hbar c\) with the \(\kappa_0=0\) mode weighted by \(1/2\).
- 6.
In practice, \({\mathbb{T}}_A\) and \({\mathbb{T}}_R\) suffice to have the same sign over the frequencies, which contribute most to the integral (or the sum) in (5.43)
- 7.
The first curl in the operator \({\mathbb{V}}_J\) results from an integration by parts. It is understood that it acts on the wave function multiplying \({\mathbb{V}}_J\) from the left.
References
Ashkin, A.: Acceleration and trapping of particles by radiation pressure. Phys. Rev. Lett. 24, 156–159 (1970)
Ashkin, A., Gordon, J.P.: Stability of radiation-pressure particle traps: an optical Earnshaw theorem. Opt. Lett. 8, 511–513 (1983)
Bachas, C.P.: Comment on the sign of the Casimir force. J. Phys. A: Math. Theor. 40, 9089–9096 (2007)
Balian, R., Duplantier, B.: Electromagnetic waves near perfect conductors. II. Casimir effect. Ann. Phys., NY 104, 300–335 (1977)
Balian, R., Duplantier, B.: Electromagnetic waves near perfect conductors. I. Multiple scattering expansions. Distribution of modes. Ann. Phys., NY 112, 165–208 (1978)
Birman, M.S., Krein, M.G.: On the theory of wave operators and scattering operators. Sov. Math.-Dokl. 3, 740–744 (1962)
Bordag, M., Robaschik, D., Wieczorek, E.: Quantum field theoretic treatment of the Casimir effect. Ann. Phys., NY 165, 192–213 (1985)
Braunbek, W.: Freies Schweben diamagnetischer Körper im magnetfeld. Z. Phys. 112, 764–769 (1939)
Braunbek, W.: Freischwebende Körper im elektrischen und magnetischen Feld. Z. Phys. 112, 753–763 (1939)
Bulgac, A., Magierski, P., Wirzba, A.: Scalar Casimir effect between Dirichlet spheres or a plate and a sphere. Phys. Rev. D 73, 025007 (2006)
Bulgac, A., Wirzba, A.: Casimir Interaction among objects immersed in a fermionic environment. Phys. Rev. Lett. 87, 120404 (2001)
Büscher, R., Emig, T.: Geometry and spectrum of Casimir forces. Phys. Rev. Lett. 94, 133901 (2005)
Capasso, F., Munday, J.N., Iannuzzi, D., Chan, H.B.: Casimir forces and quantum electrodynamical torques: Physics and Nanomechanics. IEEE J. Sel. Top. Quant. 13, 400–414 (2007)
Casimir, H.B.G.: On the attraction between two perfectly conducting plates. Proc. K. Ned. Akad. Wet. 51, 793–795 (1948)
Casimir, H.B.G., Polder, D.: The Influence of retardation on the London-van der Waals forces. Phys. Rev. 73, 360–372 (1948)
Chan, H.B., Aksyuk, V.A., Kleiman, R.N., Bishop, D.J., Capasso, F.: Quantum mechanical actuation of microelectromechanical systems by the Casimir force. Science 291, 1941–1944 (2001)
Chan, H.B., Bao, Y., Zou, J., Cirelli, R.A., Klemens, F., Mansfield, W.M., Pai, C.S.: Measurement of the Casimir force between a gold sphere and a silicon surface with nanoscale trench arrays. Phys. Rev. Lett. 101, 030401 (2008)
Chen, F., Klimchitskaya, G.L., Mostepanenko, V.M., Mohideen, U.: Demonstration of the difference in the Casimir force for samples with different charge-carrier densities. Phys. Rev. Lett. 97, 170402 (2006)
Chen, F., Klimchitskaya, G.L., Mostepanenko, V.M., Mohideen, U.: Control of the Casimir force by the modification of dielectric properties with light. Phys. Rev. B 76, 035338 (2007)
Chen, F., Mohideen, U., Klimchitskaya, G.L., Mostepanenko, V.M.: Demonstration of the lateral Casimir force. Phys. Rev. Lett. 88, 101801 (2002)
Chew, W.C., Jin, J.M., Michielssen, E., Song, J.M. (eds.): Fast and Efficient Algorithms in Computational Electrodynamics. Artech House, Norwood, MA (2001)
Dalvit, D.A.R., Lombardo, F.C., Mazzitelli, F.D., Onofrio, R.: Exact Casimir interaction between eccentric cylinders. Phys. Rev. A 74, 020101(R) (2006)
Decca, R.S., López, D., Fischbach, E., Klimchitskaya, G.L., Krause, D.E., Mostepanenko, V.M.: Tests of new physics from precise measurements of the Casimir pressure between two gold-coated plates. Phys. Rev. D 75, 077101 (2007)
Druzhinina, V., DeKieviet, M.: Experimental observation of quantum reflection far from threshold. Phys. Rev. Lett. 91, 193202 (2003)
Dzyaloshinskii, I.E., Lifshitz, E.M., Pitaevskii, L.P.: The general theory of van der Waals forces. Adv. Phys. 10, 165–209 (1961)
Earnshaw, S.: On the nature of the molecular forces which regulate the constitution of the luminiferous ether. Trans. Camb. Phil. Soc. 7, 97–112 (1842)
Ederth, T.: Template-stripped gold surfaces with 0.4-nm rms roughness suitable for force measurements: Application to the Casimir force in the 20–100 nm range. Phys. Rev. A 62, 062104 (2000)
Emig, T., Graham, N., Jaffe, R.L., Kardar, M.: Casimir forces between arbitrary compact objects. Phys. Rev. Lett. 99, 170403 (2007)
Emig, T., Graham, N., Jaffe, R.L., Kardar,M.: Casimir forces between compact objects: The scalar objects. Phys. Rev. D 77, 025005 (2008)
Emig, T., Graham, N., Jaffe, R.L., Kardar, M.: Orientation dependence of Casimir forces. Phys. Rev. A 79, 054901 (2009)
Emig, T., Hanke, A., Golestanian, R., Kardar, M.: Probing the strong boundary shape dependence of the Casimir force. Phys. Rev. Lett. 87, 260402 (2001)
Emig, T., Hanke, A., Golestanian, R., Kardar, M.: Normal and lateral Casimir forces between deformed plates. Phys. Rev. A 67, 022114 (2003)
Emig, T., Jaffe, R.L., Kardar, M., Scardicchio, A.: Casimir interaction between a plate and a cylinder. Phys. Rev. Lett. 96, 080403 (2006)
Feinberg, G., Sucher, J.: General form of the retarded van der Waals potential. J. Chem. Phys. 48, 3333–3334 (1698)
Feinberg, G., Sucher, J.: General theory of the van der Waals interaction: A model-independent approach. Phys. Rev. A 2, 2395–2415 (1970)
Feynman, R.P., Hibbs, A.R.: Quantum mechanics and path integrals. McGraw-Hill, New York (1965)
Geim, A.: Everyone’s magnetism. Phys. Today 51(9), 36–39 (1998)
Genet, C., Lambrecht, A., Reynaud, S.: Casimir force and the quantum theory of lossy optical cavities. Phys. Rev. A 67, 043811 (2003)
Gies, H., Klingmüller, K.: Casimir edge effects. Phys. Rev. Lett. 97, 220405 (2006)
Golestanian, R.: Casimir-Lifshitz interaction between dielectrics of arbitrary geometry: A dielectric contrast perturbation theory. Phys. Rev. A 80, 012519 (2009)
Golestanian, R., Kardar, M.: Mechanical response of vacuum. Phys. Rev. Lett. 78, 3421–3425 (1997)
Golestanian, R., Kardar, M.: Path-integral approach to the dynamic Casimir effect with fluctuating boundaries. Phys. Rev. A 58, 1713–1722 (1998)
Graham, N., Jaffe, R.L., Khemani, V., Quandt, M., Scandurra, M., Weigel, H.: Casimir energies in light of quantum field theory. Phys. Lett. B 572, 196–201 (2003)
Graham, N., Quandt, M., Weigel, H.: Spectral methods in quantum field theory. Springer, Berlin (2009)
Graham, N., Shpunt, A., Emig, T., Rahi, S.J., Jaffe, R.L., Kardar, M.: Casimir force at a knife’s edge. Phys. Rev. D 81, 061701(R) (2010)
Harber, D.M., Obrecht, J.M., McGuirk, J.M., Cornell, E.A.: Measurement of the Casimir-Polder force through center-of-mass oscillations of a Bose-Einstein condensate. Phys. Rev. A 72, 033610 (2005)
Henseler, M., Wirzba, A., Guhr, T.: Quantization of HyperbolicN-Sphere scattering systems in three dimensions. Ann. Phys., NY 258, 286–319 (1997)
Jaekel, M.T., Reynaud, S.: Casimir force between partially transmitting mirrors. J. Physique I 1, 1395–1409 (1991)
Jones, T.B.: Electromechanics of Particles. Cambridge University Press, Cambridge (1995)
Kats, E.I.: Influence of nonlocality effects on van der Waals interaction. Sov. Phys. JETP 46, 109 (1997)
Kenneth, O., Klich, I.: Opposites Attract: A theorem about the Casimir force. Phys. Rev. Lett. 97, 160401 (2006)
Kenneth, O., Klich, I.: Casimir forces in a T-operator approach. Phys. Rev. B 78, 014103 (2008)
Kim, W.J., Brown-Hayes, M., Dalvit, D.A.R., Brownell, J.H., Onofrio, R.: Anomalies in electrostatic calibrations for the measurement of the Casimir force in a sphere-plane geometry.Phys. Rev. A 78,020101(2008)
Klimchitskaya, G.L., Mohideen, U., Mostepanenko, V.M.: The Casimir force between real materials: Experiment and theory. Rev. Mod. Phys. 81, 1827–1885 (2009)
Krause, D.E., Decca, R.S., López, D., Fischbach, E.: Experimental investigation of the Casimir force beyond the proximity-force approximation. Phys. Rev. Lett. 98, 050403 (2007)
Krein, M.G.: On the trace formula in perturbation theory. Mat. Sborn. (NS) 33, 597–626 (1953)
Krein, M.G.: Perturbation determinants and a formula for the trace of unitary and selfadjoint operators. Sov. Math.-Dokl. 3, 707–710 (1962)
Lambrecht, A., Neto, P.A.M., Reynaud, S.: The Casimir effect within scattering theory. New J. Phys. 8, 243 (2006)
Lamoreaux, S.K.: Demonstration of the Casimir force in the 0.6 to \(6\, \upmu\hbox {m}\) range. Phys. Rev. Lett. 78, 5–8 (1997)
Landau, L.D., Lifshitz, E.M.: Electrodynamics of continuous media. Pergamon Press, Oxford (1984)
Levin M., McCauley A.P., Rodriguez A.W., Reid M.T.H., Johnson S.G. (2010) Casimir repulsion between metallic objects in vacuum. arXiv:1003.3487
Li, H., Kardar, M.: Fluctuation-induced forces between rough surfaces. Phys. Rev. Lett. 67, 3275–3278 (1991)
Li, H., Kardar, M.: Fluctuation-induced forces between manifolds immersed in correlated fluids. Phys. Rev. A 46, 6490–6500 (1992)
Lifshitz, E.M.: The theory of molecular attractive forces between solids. Sov. Phys. JETP 2, 73–83 (1956)
Lifshitz, E.M., Pitaevskii, L.P.: Statistical physics Part 2. Pergamon Press, New York (1980)
Lippmann, B.A., Schwinger, J.: Variational principles for scattering processes. i. Phys. Rev. 79, 469–480 (1950)
Milton, K.A., Parashar, P., Wagner, J.: Exact results for Casimir interactions between dielectric bodies: The weak-coupling or van der waals limit. Phys. Rev. Lett. 101, 160402 (2008)
Milton K.A., Parashar P., Wagner J. (2008) From multiple scattering to van der waals interactions: exact results for eccentric cylinders. arXiv:0811.0128
Mohideen, U., Roy, A.: Precision measurement of the Casimir force from 0.1–\(0.9\, \mu\hbox{m}\). Phys. Rev. Lett. 81, 4549–4552 (1998)
Morse, P.M., Feshbach, H.: Methods of theoretical physics. McGraw-Hill, New York (1953)
Munday, J.N., Capasso, F.: Precision measurement of the Casimir-Lifshitz force in a fluid. Phys. Rev. A 75, 060102(R) (2007)
Munday, J.N., Capasso, F., Parsegian, V.A.: Measured long-range repulsive Casimir-Lifshitz forces. Nature 457, 170–173 (2009)
Palasantzas, G., van Zwol, P.J., De Hosson, J.T.M.: Transition from Casimir to van der Waals force between macroscopic bodies. Appl. Phys. Lett. 93, 121912 (2008)
Parsegian, V.A.: van der Waals Forces. Cambridge University Press, Cambridge (2005)
Rahi, S.J., Emig, T., Graham, N., Jaffe, R.L., Kardar, M.: Scattering theory approach to electrodynamic Casimir forces. Phys. Rev. D 80, 085021 (2009)
Rahi, S.J., Emig, T., Jaffe, R.L., Kardar, M.: Casimir forces between cylinders and plates. Phys. Rev. A 78, 012104 (2008)
Rahi S.J., Kardar M., Emig T. Constraints on stable equilibria with fluctuation-induced forces. Phys. Rev. Lett. 105, 070404 (2010)
Rahi, S.J., Rodriguez, A.W., Emig, T., Jaffe, R.L., Johnson, S.G., Kardar, M.: Nonmonotonic effects of parallel sidewalls on Casimir forces between cylinders. Phys. Rev. A 77, 030101 (2008)
Rahi, S.J., Zaheer, S.: Stable levitation and alignment of compact objects by Casimir spring forces. Phys. Rev. Lett. 104, 070405 (2010)
Reid, M.T.H., Rodriguez, A.W., White, J., Johnson, S.G.: Efficient computation of Casimir interactions between arbitrary 3D objects. Phys. Rev. Lett. 103, 040401 (2009)
Renne, M.J.: Microscopic theory of retarded Van der Waals forces between macroscopic dielectric bodies. Physica 56, 125–137 (1971)
Robaschik, D., Scharnhorst, K., Wieczorek, E.: Radiative corrections to the Casimir pressure under the influence of temperature and external fields. Ann. Phys., NY 174, 401–429 (1987)
Rodriguez, A., Ibanescu, M., Iannuzzi, D., Capasso, F., Joannopoulos, J.D., Johnson, S.G.: Computation and visualization of Casimir forces in arbitrary geometries: Non-monotonic lateral-wall forces and failure of proximity force approximations. Phys. Rev. Lett. 99, 080401 (2007)
Rodriguez, A.W., Joannopoulos, J.D., Johnson, S.G.: Repulsive, nonmonotonic Casimir forces in a glide-symmetric geometry. Phys. Rev. A 77, 062107 (2008)
Rodriguez-Lopez, P., Rahi, S.J., Emig, T.: Three-body Casimir effects and nonmonotonic forces. Phys. Rev. A 80, 022519 (2009)
Rosa, F.S.S.: On the possibility of Casimir repulsion using metamaterials. J. Phys.: Conf. Ser. 161, 012039 (2009)
Rosa, F.S.S., Dalvit, D.A.R., Milonni, P.W.: Casimir-Lifshitz theory and metamaterials. Phys. Rev. Lett. 100, 183602 (2008)
Roy, A., Lin, C.Y., Mohideen, U.: Improved precision measurement of the Casimir force. Phys. Rev. D 60, 111101(R) (1999)
Schaden, M., Spruch, L.: Infinity-free semiclassical evaluation of Casimir effects. Phys. Rev. A 58, 935–953 (1998)
Schwinger, J.: Casimir effect in source theory. Lett. Math. Phys. 1, 43–47 (1975)
Ttira, C.C., Fosco, C.D., Losada, E.L.: Non-superposition effects in the Dirichlet–Casimir effect. J. Phys. A: Math. Theor. 43, 235402 (2010)
Weber, A., Gies, H.: Interplay between geometry and temperature for inclined Casimir plates. Phys. Rev. D 80, 065033 (2009)
Wirzba, A.: Quantum mechanics and semiclassics of hyperbolic n-disk scattering systems. Phys. Rep. 309, 1–116 (1999)
Wirzba, A.: The Casimir effect as a scattering problem. J. Phys. A: Math. Theor. 41, 164003 (2008)
Zaheer, S., Rahi, S.J., Emig, T., Jaffe, R.L.: Casimir interactions of an object inside a spherical metal shell. Phys. Rev. A 81, 030502 (2010)
Zhao, R., Zhou, J., Koschny, T., Economou, E.N., Soukoulis, C.M.: Repulsive Casimir force in chiral metamaterials. Phys. Rev. Lett. 103, 103602 (2009)
Acknowledgements
The research presented here was conducted together with Noah Graham, Steven G. Johnson, Mehran Kardar, Alejandro W. Rodriguez, Pablo Rodriguez-Lopez, Alexander Shpunt, and Saad Zaheer, whom we thank for their collaboration. This work was supported by the National Science Foundation (NSF) through grant DMR-08-03315 (SJR), by the DFG through grant EM70/3 (TE) and by the U. S. Department of Energy (DOE) under cooperative research agreement #DF-FC02-94ER40818 (RLJ).
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Rahi, S.J., Emig, T., Jaffe, R.L. (2011). Geometry and Material Effects in Casimir Physics-Scattering Theory. In: Dalvit, D., Milonni, P., Roberts, D., da Rosa, F. (eds) Casimir Physics. Lecture Notes in Physics, vol 834. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20288-9_5
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