Photon localization and Dicke superradiance in atomic gases: crossover to a ``small world'' network

We study photon propagation in a gas of $N$ atoms, using an effective Hamiltonian that accounts for photon mediated atomic dipolar interactions. The configuration average density $P(\Gamma)$ of photon escape rates is obtained from the spectrum of the $N \times N$ random matrix $\Gamma_{ij} = \sin (x_{ij}) / x_{ij}$, where $x_{ij}$ is the dimensionless random distance between any two atoms expressed in units of the photon wavelength. A scaling function is defined to study photons escape rates as a function of disorder and system size. We show that for a strong enough disorder, photons do not escape the gas. This localization is described using a mapping of this problem onto statistical properties of random networks. We show that there is no localization phase transition as expected in disordered systems without correlation, but rather a cross-over between localized and delocalized photons. The mean field solution of this problem displays a ``small world'' behavior. In the Dicke limit, we recover localization associated to cooperative effects.