Photon Localization and Dicke Superradiance in Atomic Gases

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 $N\times N$ 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.”

Figure 1

Behavior of $P(\Gamma )$ for different values of disorder $W$ and size $a=L/\lambda$, with $N=216$. (a) Low disorder, (b),(c) larger disorder, and (d) Dicke limit.

Figure 2

Behavior of $C$ as a function of the size $a=L/\lambda \ge 1$, for different disorder strengths $W$.

Figure 3

All points represented in Fig. 2 collapse on a single curve as expected from (10). The solid line is a fit of (14), and the dashed line is the asymptotic behavior obtained from (12).

Figure 4

Behavior of $C(N)$ in the Dicke limit.

Figure 5

Comparison between cooperative effects determined by the parameter $N/N_{\perp }$ and disorder described by $g=N_{\perp }^2/N$.