Abstract
We review several different approaches for computing Casimir forces and related fluctuation-induced interactions between bodies of arbitrary shapes and materials. The relationships between this problem and well known computational techniques from classical electromagnetism are emphasized. We also review the basic principles of standard computational methods, categorizing them according to three criteria—choice of problem, basis, and solution technique—that can be used to classify proposals for the Casimir problem as well. In this way, mature classical methods can be exploited to model Casimir physics, with a few important modifications.
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- 1.
The name method of moments is also commonly applied to BEM techniques for EM. However, this terminology is somewhat ambiguous, and can refer more generally to Galerkin or other weighted-residual methods (and historically referred to monomial test functions, yielding statistical “moments”) [53].
- 2.
This applies equally well, if somewhat indirectly, to the path-integral expressions of Sect. 6.6 where one evaluates a log determinant or a trace of an inverse, since this is done using either eigenvalues or the same matrix factorizations that are used to solve \(A{{\mathbf{x}}}={{\mathbf{b}}}\).
- 3.
Technically, all eigensolvers for \(N>4\) are necessarily iterative, but modern dense-eigensolver techniques employ direct factorizations as steps of the process [65].
- 4.
The electric field \({\mathbf{E}}({\mathbf{x}})\) from a dipole current \({\mathbf{J}}=\delta^{3}({\mathbf{x}}-{\mathbf{x}}^{\prime})\hat{e}_{k}e^{-i\omega t}\) is \({\mathbf{E}}({\mathbf{x}})=i\omega{\mathbf{G}}_{k}^{E}(\omega,{\mathbf{x}},{\mathbf{x}}^{\prime})e^{-i\omega t}\).
- 5.
This can be seen more explicity by substituting \({\mathbf {G}}^{H}=\frac{1} {\omega^{2}}\frac{1}{\mu}\nabla\times{\mathbf {G}}^{E}\times\nabla^{\prime}\frac{1} {\mu^{\prime}}-\frac{1} {\omega^{2}\mu^{\prime}}\delta\) into (6.6), with \(\delta\) denoting \(\delta({\mathbf{x}}-{\mathbf{x}}^{\prime})I\) and \(\mu\) or \(\mu^{\prime}\) denoting \(\mu({\mathbf{x}})\) or \(\mu({\mathbf{x}}^{\prime})\), respectively. In particular, \([\nabla\times\frac{1} {\varepsilon}\nabla\times-\omega^{2}\mu](\frac{1} {\omega^{2}}\frac{1}{\mu}\nabla\times{\mathbf G}^{E}\times\nabla^{\prime}\frac{1} {\mu^{\prime}}-\frac{1} {\omega^{2}\mu^{\prime}}\delta)\) yields \(\nabla\times[\frac{1} {\omega^{2}\varepsilon}\nabla\times\frac{1} {\mu}\nabla\times{\mathbf G}^{E}-{\mathbf G}^{E}]\times\nabla^{\prime}\frac{1}{\mu^{\prime}}-\nabla\times\frac{1} {\omega^{2}\mu^{\prime}\varepsilon}\nabla\times\delta+\delta\), which via (6.4) gives \(+\nabla\times\frac{1} {\omega^{2}\varepsilon}\delta\times\nabla^{\prime}\frac{1} {\mu^{\prime}}-\nabla\times\frac{1} {\omega^{2}\mu^{\prime}\varepsilon}\nabla\times\delta+\delta=\delta\) as desired, where in the last step we have used the fact that \(\delta\times\nabla^{\prime}=\nabla\times\delta\) [since \(\nabla\times\) is antisymmetric under transposition and \(\nabla^{\prime}\delta(\mathbf {x}-{\mathbf{x}}^{\prime})=-\nabla\delta({\mathbf{x}}-{\mathbf{x}}^{\prime})\)].
- 6.
This follows from the standard identity that the limit \(\operatorname{Im}[(x+i0^{+})^{-1}]\), viewed as a distribution, yields \(-\pi\delta(x)\) [86].
- 7.
Lest the application of this field identity appear too glib, we can also obtain the same equality directly from the Green’s functions in the correlation functions. We have \(\int\mu\langle|{\mathbf{H}}|^{2}\rangle=\frac{\hbar} {\pi}\operatorname{tr}\int\xi^{2}\mu{\mathbf G}^{H}({\mathbf{x}},{\mathbf{x}})\), and from the identity after (6.6) we know that \(\xi^{2}\mu{\mathbf G}^{H}=-\nabla\times{\mathbf G}\times\nabla^{\prime}\frac{1}{\mu^{\prime}}+\delta\). However, because \(\nabla\times\) is self-adjoint [6], we can integrate by parts to move \(\nabla\times\) from the first argument/index of \({\mathbf G}^{E}\) to the second, obtaining \(-{\mathbf G}^{E}\times\nabla^{\prime}\frac{1} {\mu^{\prime}}\times\nabla^{\prime}=\xi^{2}\varepsilon^{\prime}{\mathbf G}^{E}-\delta\) from the first term under the integral. (Here, we employ the fact that \({\mathbf G}^{E}\) is real-symmetric at imaginary \(\omega=i\xi\), from Sect. 6.5.1.2, to apply (6.10) to the second index/argument instead of the first.) This cancels the other delta from \(\xi^{2}\mu{\mathbf G}^{H}\) and leaves \(\xi^{2}\varepsilon{\mathbf G}^{E}\), giving \(\varepsilon\langle|{\mathbf{E}}|^{2}\rangle\) as desired.
- 8.
- 9.
Even if the \(\varepsilon\) discontinuities are dealt with in this way, however, one may still fail to obtain second-order accuracy if the geometry contains sharp corners, which limit the accuracy to \(O(\Updelta x^{p})\) for some \(1<p<2\) [101]. This is an instance of Darboux’s principle: the convergence rate of a numerical method is generally limited by the strongest singularity in the solution that has not been explicitly compensated for [53].
- 10.
This is known as an electric-field integral equation (EFIE); one can also express the equations for perfect conductors in terms of boundary conditions enforced on magnetic fields (MFIE) or some linear combination of the two (CFIE), and the most effective formulation is still a matter of debate [90].
- 11.
- 12.
Technically, only currents from surfaces bordering the medium of \({\mathbf{x}}\) contribute to this sum.
- 13.
Numeric integration (quadrature) approximates an integral \(\int f(x)dx\) by a sum \(\sum_{i}f(x_{i})w_{i}\) over quadrature points \(x_{i}\) with weights \(w_{i}\). There are many techniques for the selection of these points and weights, and in general one can obtain an error that decreases exponentially fast with the number of points for analytic integrands [53, 83, 84, 113]. Multidimensional quadrature, sometimes called cubature, should be used to integrate the stress tensor over a 2d surface, and numerous schemes have been developed for low-dimensional cubature [114, 115] (including methods that adaptively place more quadrature points where they are most needed [116]). For spherical integration surfaces (or surfaces that can be smoothly mapped to spheres), specialized methods are available [117, 118].
- 14.
Alternatively, the path integral can be performed directly in \({\mathbf{A}}\), resulting in an expression equivalent to the sum over energy density in Sect. 6.5 [87] and which in an FD discretization reduces in the same way to repeated solution of the Green’s-function diagonal at every point in space [23].
- 15.
See also the Chap. 4 by A. Lambrecht et al. in this volume for additional discussion of Casimir interactions among periodic structures.
- 16.
If \(C(\xi)\) has zero slope at \(\xi=0^{+}\), then the trapezoidal rule differs from the integral by \(O(\Updelta\xi^{4})\) or less, depending upon which derivative is nonzero at \(\xi=0^{+}\) [53].
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Acknowledgements
This work was supported in part by the Army Research Office through the ISN under contract W911NF-07-D-0004, by the MIT Ferry Fund, and by the Defense Advanced Research Projects Agency (DARPA) under contract N66001-09-1-2070-DOD. We are especially grateful to our students, A. W. Rodriguez, A. P. McCauley, and H. Reid for their creativity and energy in pursuing Casimir simulations. We are also grateful to our colleagues F. Capasso, D. Dalvit, T. Emig, R. Jaffe, J. D. Joannopoulos, M. Kardar, M. Levin, M. Lončar, J. Munday, S. J. Rahi, and J. White, for their many suggestions over the years.
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Johnson, S.G. (2011). Numerical Methods for Computing Casimir Interactions. 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_6
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