Historical research guide · 0418 · 2009
Casimir forces in the time domain: Theory
Alejandro W. Rodriguez · Alexander P. McCauley · John D. Joannopoulos · Steven G. Johnson
In one minute
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.
Connection
Chapter 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.
From the source
A 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
Passing 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
The 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
The 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
Because 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
The 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
The way in
https://doi.org/10.1103/PhysRevA.80.012115How 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