Historical research guide · 0747 · 2012
Macroscopic quantum electrodynamics in nonlocal and nonreciprocal media
Stefan Yoshi Buhmann · David T Butcher · Stefan Scheel
In one minute
Stefan Buhmann, David Butcher and Stefan Scheel extend the quantum theory of light inside matter to the strangest materials now being built. Ordinary macroscopic quantum electrodynamics assumes a medium responds at a point to the field at that point, and reciprocally — swap the source and the detector and you get the same answer. Chiral metamaterials, topological insulators, moving media and Tellegen media all break one assumption or both. The authors quantise the electromagnetic field for the most general linear absorbing medium, starting from Ohm’s law written with a conductivity tensor that links two separate points, and show the resulting theory still obeys the free-space commutation relations, the fluctuation–dissipation theorem and Maxwell’s equations. The trick that makes it work is a symmetrised definition of the real and imaginary parts of a tensor. They then find something unexpected: duality, the global exchange of electric and magnetic properties, becomes a continuous symmetry only in media that violate reciprocity. Less time-reversal symmetry buys more duality symmetry.
Connection
Chapter 2 treats the vacuum as a structured medium whose fluctuations do mechanical work, and this is the machinery for calculating that work inside real engineered materials — including the chiral metamaterials and topological insulators proposed for repulsive Casimir forces. Anyone designing a vacuum-fluctuation device in chapter 6 needs exactly this: a quantisation that survives when the medium is spatially dispersive, magnetoelectric, or in motion.
From the source
The electromagnetic field can be quantised in the most general linear absorbing medium — nonlocal, bianisotropic, and violating Onsager reciprocity — starting only from Ohm’s law with a conductivity tensor that links the current at one point to the field at another. The resulting scheme satisfies the canonical commutation relations of free-space quantum electrodynamics, the linear fluctuation–dissipation theorem, and the macroscopic Maxwell equations.Abstract; Section 2, Equations 1 to 26; Section 5, Conclusion
What makes the generalisation work is a symmetrisation: the real and imaginary parts of a two-point tensor field are redefined as the half-sum and the half-difference of the tensor and the Hermitian conjugate of its argument-swapped self. These generalised parts reduce to ordinary real and imaginary parts for reciprocal media, and their appearance in the fluctuation–dissipation theorem is the key to avoiding any restriction on the allowed medium response.Section 2, Equations 6 and 7; Section 5, Conclusion
The standard integral identity of macroscopic quantum electrodynamics survives without reciprocity: the imaginary part of the Green tensor still equals the permeability of free space times the frequency times the Green tensor convolved with the real part of the conductivity and with the Hermitian conjugate Green tensor, a relation the authors derive directly from the Green operator being the two-sided inverse of the Helmholtz operator.Section 2, Equation 16; Appendix, Equations A.1 to A.4
The same medium can be quantised a second way, in terms of permittivity, permeability and the two magnetoelectric susceptibilities, splitting the internal current into polarisation and magnetisation parts. When the magnetoelectric susceptibilities do not vanish, the noise polarisation and the noise magnetisation no longer commute — a coupling absent from every purely dielectric treatment. The split is not unique in spatially dispersive media, because magnetisation can be absorbed into the transverse polarisation.Section 3, Equations 29 to 43 and the closing paragraph
Duality invariance — the global rotation that exchanges electric and magnetic properties — is realised as a full continuous symmetry only when Onsager reciprocity is allowed to fail. For isotropic, anisotropic or reciprocal media the rotation angle is restricted to whole multiples of a quarter turn. The authors state the pattern plainly: a reduction in reciprocity symmetry leads to an enhancement of duality symmetry.Section 4, Duality invariance, Equations 48 to 50 and the list of special cases
The scheme is written to be used: it is the foundation for exact calculations of dispersion forces, Förster energy transfer and molecular transition rates in chiral or nonreciprocal environments, for quantum friction in moving media, and — through the Curie principle — for detecting parity and charge-parity violation in atoms through their interaction with a nonreciprocal surface.Section 5, Conclusion, final paragraph
The way in
https://doi.org/10.1088/1367-2630/14/8/083034How to cite it
Stefan Yoshi Buhmann, David T Butcher, Stefan Scheel (2012) Macroscopic quantum electrodynamics in nonlocal and nonreciprocal media doi:10.1088/1367-2630/14/8/083034