PHOTONS IN FRACTAL STRUCTURES: THERMODYNAMICS, QUANTUM OPTICS AND CASIMIR EFFECT
Fractals define a new and interesting realm for a discussion of basic phenomena in QED and
quantum optics and their implementation. This interest results from specific properties of
fractals, e.g., their dilatation symmetry as opposed to the translation symmetry of Euclidean
space and the corresponding absence of Fourier mode decomposition. Moreover, the
existence of a set of distinct (usually non integer) dimensions characterizing the physical
properties (spatial or spectral) of fractals make them a useful testing ground for
dimensionality dependent physical problems.
We shall start by noting that the absence of Fourier transform on a fractal implies necessarily
different notions of volume in direct and reciprocal spaces and thus the need to modify the
Heisenberg uncertainty principle. Implications for field quantization and the definition of the
notion of photon on a fractal will be further addressed.
These ideas will find interesting applications in quantum optics of fractal cavities. More
specifically, we shall discuss the existence of a strong Purcell effect, the modification of
spontaneous emission, and the Casimir effect.
We shall then turn to the case of massive bosons and discuss the nature of Bose-Einstein
condensation and the onset of superfluidity in fractal structures. The existence of distinct
fractal dimensions characterizing spatial and spectral properties is instrumental in
understanding the dimensionality dependence of the BEC and the existence of a superfluid
order either through the existence of an “Off Diagonal Long Range Order” (ODLRO) or the
generalization of the Mermin-Wagner theorem.