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.