Physical Review Letters 108, 013601 (2012)
Thomas-Reiche-Kuhn sum rules are quite important in Quantum Optics, they tell us that the sum of squared dipole moments from a any given energy level is constant in the dipole approximation. They have been useful in providing a validity check in radiation-matter interaction, like no-go theorems in the case of super-radiance in Dicke model. Anyhow, it is a necessary condition for the canonical commutation relation between position and momentum of an atomic electron to hold.
In their article Barnett and Loudon—who, by the way, have written very useful and didactic books in Quantum Optics each—explore the electromagnetic field in lossless magnetodielectric media and show that the equal-time commutation relations for the four electromagnetic field operators deliver four polariton sum rules (for the unity, permittivity, permeability and their product) analogous to the atomic Thomas-Reiche-Kuhn sum rule. They show a proof of these optical TRK sum rules by analysing the case where complex polariton frequencies are restricted to the lower half complex plane; the proof is beautiful and simple, they use complex variable analysis to state the fact that path integration in the half-upper complex planes for the four complex functions is equal to one as both the permittivity and permeability goes to one as the frequency goes to infinity, then, in the lower half, the contour integral is calculated from the zeroes of each of the four quantities corresponding to solutions of the dispersion relation for polaritons in a lossless magnetodielectric mediuml; this delivers a residue equal to one. Finally, by using a relation between residues and derivatives of the dispersion relation, they show that this residue is actually a sum over the polariton phase velocities, a little bit more of algebra and tada! They also sketch the effect of losses but leave it for a following article.
They results indicate that it is not possible to design a medium where all polariton modes are in the negative-index region. Interesting for all the meta-material research. I'm curious how this comes out in the presence of losses.
Man, oh man. I'm really dusted in complex variable. I'm still half-way in the calculations, but the paper is so beautiful written that everything seems so logical and simple.
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