STM-D-0418Paper2009Published and peer-reviewed

Casimir forces in the time domain: Theory

Alejandro W. Rodriguez · Alexander P. McCauley · John D. Joannopoulos · Steven G. Johnson

Abstract and summary · read the original at the source

In one page

Alejandro Rodriguez, Alexander McCauley, John Joannopoulos and Steven Johnson give the Casimir force a general-purpose calculator. Until this paper most Casimir numbers came from shapes simple enough to solve on paper — two flat plates, a sphere above a plate. Real hardware is not shaped like that. The MIT team shows you can get the force in any geometry, out of any material, by doing what engineers already do every day: run Maxwell’s equations forward in time on a grid, the finite-difference time-domain method. The trick is a change of variable. The usual route evaluates the force at imaginary frequencies, which makes a time-stepping code blow up; instead they map the problem exactly onto an identical one in a medium with a plain frequency-independent conductivity, where everything decays instead. You place current pulses on a surface drawn around the object, watch the electric and magnetic fields that come back, and integrate over the surface and over time. What you get is the exact Casimir force, computed by free, unmodified, off-the-shelf software.

Why it matters hereChapter 2 says the vacuum is a real medium you can push on; chapter 6 says you can build hardware that does the pushing. Both need the force predicted in the layered, material-specific, distinctly un-flat geometries a real device has, and this is the method that delivers it — the same computational road taken by Fabrizio Pinto’s finite-difference Casimir work on this site at /library/stm-e01aa49628.

What it claims

01A finite-difference time-domain scheme can compute the Casimir force in arbitrary geometries and for arbitrary materials, exploiting existing FDTD software without any modification to it.Abstract; Section 1, introduction
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02Passing to imaginary frequencies, the standard step in most Casimir calculations, gives rise to exponentially growing solutions in a time-stepping code and is therefore unsuitable for the time domain.Section 2, opening; Section 2.B, complex frequency domain
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03The system can be mapped exactly onto a transformed problem in which every material property is modified to include dissipation, so that a deformation of the frequency contour becomes an ordinary dissipative medium evaluated at a real frequency.Section 2.B and Section 2.C, time domain approach
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04The whole recipe is three steps — map to the dissipative problem, measure the electric and magnetic fields produced by current pulses placed at each point of a surface enclosing the body, then integrate those fields over the surface and over time against a known kernel — and the result is the exact Casimir force in the limit of sufficient resolution.Section 2, opening list of steps; Equation 29
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05Because the transformed medium dissipates, the error caused by stopping the simulation at a finite time falls exponentially rather than as one over the elapsed time, which is how it would fall with no dissipation.Section 3.A, fields in real time; Figure 3 and its inset
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06The conductivity has an optimum: too small and the field response oscillates rapidly and amplifies numerical error in more than one dimension, too large and the cavity mode decays slowly and the run takes longer — for the geometries here the value is of order one in units of 2 pi c over the plate spacing.Section 3.C, convergence of the force
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Read it · abstract

Abstract

We introduce a method to compute Casimir forces in arbitrary geometries and for arbitrary materials based on the finite-difference time-domain (FDTD) scheme. The method involves the time-evolution of electric and magnetic fields in response to a set of current sources, in a modified medium with frequency-independent conductivity. The advantage of this approach is that it allows one to exploit existing FDTD software, without modification, to compute Casimir forces. In this manuscript, part I, we focus on the derivation, implementation choices, and essential properties of the time-domain algorithm, both considered analytically and illustrated in the simplest parallel-plate geometry. Part II presents results for more complex two- and three-dimensional geometries.

The way in

https://doi.org/10.1103/PhysRevA.80.012115Licence checked directly. The preprint is arXiv:0904.0267, version 2 filed 21 April 2009 under the arXiv.org perpetual non-exclusive distribution licence, which does not grant redistribution, and the published article carries the APS default licence; no Creative Commons statement appears in the article, on the arXiv record or on the publisher page. Unpaywall labels the green copy in MIT’s DSpace repository CC BY-NC, but that is a repository-level guess rather than a statement in the article, so this page carries the summary, the claims and the authors’ own abstract and sends the reader to the source. Published as Physical Review A 80, 012115 (2009), from the Department of Physics and the Department of Mathematics, Massachusetts Institute of Technology; the article’s own first page carries the title ‘Casimir forces in the time domain: I. Theory’. The work was supported by the Army Research Office through the ISN, the MIT Ferry Fund and US DOE grant DE-FG02-97ER25308. The summary, the claims and the locators below were written from the full author text.

How to cite it

Alejandro W. Rodriguez, Alexander P. McCauley, John D. Joannopoulos, Steven G. Johnson (2009) Casimir forces in the time domain: Theory. doi:10.1103/PhysRevA.80.012115

Where it sits in the curriculum

What the vacuum is Energy from the vacuum