We study photon escape rates from two-dimensional atomic gases while taking into account cooperative effects between the scatterers. Based on the spectrum of the random Euclidean matrix
Uij=J0(xij), where
xij is the dimensionless random distance between any two atoms, a scaling behavior for the escape rates is obtained. This behavior is similar to the one obtained in the three-dimensional case, but qualitatively different from the result obtained in one dimension.
Photon propagation in a gas of N atoms is studied using an effective Hamiltonian describing photon-mediated atomic dipolar interactions. The density P(Gamma) of photon escape rates is determined from the spectrum of the NxN random matrix Gamma_{ij}=sin(x_{ij})/x_{ij}, where x_{ij} is the dimensionless random distance between any two atoms. Varying disorder and system size, a scaling behavior is observed for the escape rates. It is explained using microscopic calculations and a stochastic model which emphasizes the role of cooperative effects in photon localization and provides an interesting relation with statistical properties of "small world networks."
We study numerically the spectrum of the non-Hermitian effective Hamiltonian that describes the dipolar interaction of a gas of $N\ensuremath{\gg}1$ atoms with the radiation field. We analyze the interplay between cooperative effects and disorder for both scalar and vectorial radiation fields. We show that for dense gases, the resonance width distribution follows, both in the scalar and vectorial cases, a power law $P(\ensuremath{\Gamma})\ensuremath{\sim}{\ensuremath{\Gamma}}^{\ensuremath{-}4/3}$ that originates from cooperative effects between more than two atoms. This power law is different from the $P(\ensuremath{\Gamma})\ensuremath{\sim}{\ensuremath{\Gamma}}^{\ensuremath{-}1}$ behavior, which has been considered as a signature of Anderson localization of light in random systems. We show that in dilute clouds, the center of the energy distribution is described by Wigner's semicircle law in the scalar and vectorial cases. For dense gases, this law is replaced in the vectorial case by the Laplace distribution. Finally, we show that in the scalar case the degree of resonance overlap increases as a power law of the system size for dilute gases, but decays exponentially with the system size for dense clouds.
In the collective photon emission from atomic clouds both the atomic transition frequency and the decay rate are modified compared to a single isolated atom, leading to the effects of superradiance and subradiance. In this article, we analyze the properties of the Euclidean random matrix associated with the radiative dynamics of a cold atomic cloud, previously investigated in the contexts of photon localization and Dicke super- and subradiance. We present evidence of a phase transition, surprisingly controlled by the cooperativeness parameter, rather than the spatial density or the diagonal disorder. The numerical results corroborate the occurrence of such a phase transition at a critical value of the cooperativeness parameter, above which the lower edge of the spectrum vanishes, exhibiting a macroscopic accumulation of eigenvalues. Independent evaluations based on the two phenomena provide the same value for the critical cooperativeness parameter.
We investigate the properties of the cooperative decay modes of a cold atomic cloud, characterized by a Gaussian distribution in three dimensions, initially excited by a laser in the linear regime. We study the properties of the decay rate matrix
S, whose dimension coincides with the number of atoms in the cloud, in order to get a deeper insight into properties of cooperative photon emission. Since the atomic positions are random,
S is a Euclidean random matrix whose entries are a function of the atom distances. We show that in the limit of a large number of atoms in the cloud, the eigenvalue distribution of
S depends on a single parameter
b0, called the cooperativeness parameter, which can be viewed as a quantifier of the number of atoms that are coherently involved in an emission process. For very small values of
b0, we find that the limit eigenvalue density is approximately triangular. We also study the nearest-neighbor spacing distribution and the eigenvector statistics, finding that although the decay rate matrices are Euclidean, the bulk of their spectra mostly behaves according to the expectations of classical random matrix theory. In particular, in the bulk, there is level repulsion and the eigenvectors are delocalized, therefore exhibiting the universal behavior of chaotic quantum systems.
In order to study the cooperative emission of radiation from an extended system of
N two-level atoms, we use the master-equation approach within the Born and Markov approximations. All the properties of the emission of radiation can be deduced from the diagonalization of the coupling matrix $\ensuremath{\Gamma}$, where an element ${\ensuremath{\Gamma}}_{\mathrm{jk}}$ gives the coupling between the atoms labeled
j and
k via the electromagnetic field. We choose a pencil-shaped volume and study analytically the coupling matrix in the limit for small and large Fresnel numbers. For Fresnel numbers of order unity numerical methods are used. Thus we prove that, for the pencil-shaped geometry, any elaborated model must take into account the interaction of the atoms with the infinity of electromagnetic field modes lying inside two small solid angles about the axis of the largest linear dimension, which we label the
z axis. In other words, the intensity of radiation emitted by any atom of the system depends on its position with respect to the others and this "propagation effect" cannot be neglected when interpreting superradiance experiments. For small Fresnel numbers the
z dependence predominates, in the opposite case the (
x,y) dependence does, while for Fresnel numbers of order unity the dependence of all three components must be taken into account. Preliminary calculations are given for which only the
z dependence is considered; for Fresnel numbers of magnitude of order unity or smaller, the effect of propagation along the
z axis displays large delays for the appearance of superradiance with respect to the time at which the superradiant pulse is maximum when the "propagation effects" are neglected (one-mode model).