Cooperative effects and disorder: A scaling analysis of the spectrum of the effective atomic Hamiltonian

We study numerically the spectrum of the non-Hermitian effective Hamiltonian that describes the dipolar interaction of a gas of $N\gg 1$ 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 $P\left(\Gamma \right)\sim {\Gamma }^{-4/3}$ that originates from cooperative effects between more than two atoms. This power law is different from the $P\left(\Gamma \right)\sim {\Gamma }^{-1}$ 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.

Figure 1

Complex-valued spectrum of $H_{\text{eff}}$ (4) in the scalar case (top) and the vectorial case (bottom) for $N=500$ and $\rho {\lambda }^3=0.013$. The spiral branches in the scalar case (magenta curves) represent the eigenvalues of cooperative pairs (12), while the branches in the vectorial case [green (light gray) and blue (dark gray) curves] represent the eigenvalues of cooperative pairs (13) and (14).

Figure 2

Complex-valued spectrum of $H_{\text{eff}}$ (4) in the scalar (black points) and vectorial [red (dark gray) points] cases for $N=500$ and $\rho {\lambda }^3=131.6$. Inset: vectorial [red (dark gray) points] and scalar (black points) eigenvalues in linear scale.

Figure 3

Resonance width distribution in the scalar case (top) and the vectorial case (bottom) for $\rho {\lambda }^3=13.16$. Insets: the resonance width distribution for $\rho {\lambda }^3=131.6$ and $\rho {\lambda }^3=0.44$.

Figure 4

Behavior of ${\Gamma }_{\text{max}}$ in the scalar case (top) and in the vectorial case (bottom). The solid line is given respectively by (18) and (19) in the scalar and vectorial cases.

Figure 5

Behavior of ${\Gamma }_{\text{min}}$ in the scalar case (top) and in the vectorial case (bottom). The solid line in both cases is given by (20).

Figure 6

Energy distribution $P\left(E\right)$ of $N=2$ atoms in the scalar case (top) and in the vectorial case (bottom). The solid line is given respectively by (21) and (22) in the scalar and vectorial cases.

Figure 7

Energy distribution of $N=700$ atoms in the scalar case (top) and in the vectorial case (bottom). Dilute gases are described by (23) in both cases. Dense clouds are described by (25) in the scalar case and by (24) in the vectorial case.

Figure 8

Complex-valued spectrum of $H_{\text{eff}}$ in (4) in the scalar case for $N=500$ and $\rho {\lambda }^3=0.013$ (top) and $\rho {\lambda }^3=131.6$ (bottom). Eigenvalues in green (light gray) (marked “Selected”) are, under our assumption, related to configurations of more than two atoms. Eigenvalues in black (marked “Not selected”) are related to cooperative pairs.

Figure 9

Vectorial case where cooperative pairs are excluded. Top: resonance width distribution. Bottom: energy distribution.

Figure 10

Degree of resonance overlap, $g$, as a function of system size for the scalar case (top) and the vectorial case (bottom) when cooperative pairs are excluded.

Figure 11

Degree of resonance overlap, $g$, as a function of the Ioffe-Regel number for the scalar case (top) and the vectorial case (bottom) when cooperative pairs are excluded.

Figure 12

$\beta \left(g\right)\equiv dlng/dlnL/\lambda$ as a function of $lng$ in the scalar case (top) and the vectorial case (bottom). Assuming that $\beta \left(g\right)$ is a continuous and monotonic function of $L$, in the scalar case there is a value $g_c$ at which $\beta \left(g_c\right)=0$, indicating that $g_c$ becomes independent of the system size $L$.