Historical research guide · 0771 · 2009
Macroscopic Quantum Electrodynamics and Duality
Stefan Yoshi Buhmann · Stefan Scheel
In one minute
Casimir and van der Waals forces are forces the vacuum exerts, and their size and sign depend on what the surrounding materials do to the electromagnetic field. Until metamaterials arrived, only the electric response was adjustable; now the magnetic response is too, and the design space for shaping vacuum forces has become very large. Stefan Buhmann and Stefan Scheel, then at Imperial College London, show that a large part of that space is redundant. Maxwell’s equations with no free charges are unchanged when electric and magnetic quantities are swapped for one another, and although the force operators themselves are not, the quantities you actually measure are: the Casimir force on a body in free space, van der Waals potentials, and atomic decay rates all come out the same. Swap every electric property for its magnetic partner and the answer is already known, so a result established for one arrangement gives you its mirror image for free — and the search for optimal geometries and materials is effectively halved.
Connection
Chapter 2 says the vacuum is a real medium whose forces can be measured, and chapter 6 turns that into hardware — Casimir-based devices depend on knowing which geometries and materials give which sign and strength of vacuum force. This paper is a designer’s tool for exactly that search: it proves an exact symmetry of macroscopic quantum electrodynamics, states the conditions it holds under, and shows how to read off a magnetic result from an electric one without redoing the calculation. It also draws a precise line — the symmetry lives in the measurable quantities, not in the underlying force operators — which is the kind of care that makes vacuum engineering claims checkable.
From the source
Maxwell’s equations in the absence of free charges and currents satisfy duality invariance on an operator level, once the fields are grouped into dual pairs of electric and magnetic quantities and transformed together with the medium’s polarisation and magnetisation.Eqs. (1) to (4) and Table I
Duality is a continuous symmetry only where the permittivity equals the permeability — which includes free space and the perfect lens, where both are minus one. A medium of non-trivial impedance reduces the continuous symmetry to a discrete one with four members, whose group structure is that of the cyclic group of order four.Eqs. (5) to (8)
Neither the Lorentz forces on bodies and atoms nor the atom-field couplings are duality invariant at the operator level, even for atoms and bodies at rest in time-independent fields; the invariance is prohibited by the noise polarisation and magnetisation that the constitutive relations of an absorbing medium necessarily carry.Eqs. (11) to (13)
The quantities actually measured are invariant. Casimir forces, van der Waals potentials and local-field corrected decay rates are unchanged under a global exchange of electric and magnetic properties — the Casimir force provided the body sits in free space, the atomic quantities provided local-field corrections are included through the real-cavity model when the atom is embedded in a medium.Abstract; Eqs. (14) to (17) and the transformation rules (23) to (26)
The symmetry is a working shortcut. From the known retarded van der Waals potential of two polarisable atoms, exchanging polarisability for magnetisability divided by the speed of light squared and swapping permittivity for permeability gives the potential of two magnetisable atoms immediately; and because two purely electric mirror-symmetric bodies always attract, two purely magnetic ones must as well.The paragraph following Eq. (26)
The authors state the practical payoff: the parameter space to be considered in the search for optimal geometries and materials is effectively halved, and the invariance extends to other effective quantities of macroscopic quantum electrodynamics such as frequency shifts, heating rates and energy transfer rates.Introduction, final paragraph; Conclusion
The way in
https://doi.org/10.1103/PhysRevLett.102.140404How to cite it
Stefan Yoshi Buhmann, Stefan Scheel (2009) Macroscopic Quantum Electrodynamics and Duality doi:10.1103/PhysRevLett.102.140404