Population of collective modes in light scattering by many atoms

The interaction of light with an atomic sample containing a large number of particles gives rise to many collective (or cooperative) effects, such as multiple scattering, superradiance, and subradiance, even if the atomic density is low and the incident optical intensity weak (linear optics regime). Tracing over the degrees of freedom of the light field, the system can be well described by an effective atomic Hamiltonian, which contains the light-mediated dipole-dipole interaction between atoms. This long-range interaction is at the origin of the various collective effects, or of collective excitation modes of the system. Even though an analysis of the eigenvalues and eigenfunctions of these collective modes does allow distinguishing superradiant modes, for instance, from other collective modes, this is not sufficient to understand the dynamics of a driven system, as not all collective modes are significantly populated. Here, we study how the excitation parameters, i.e., the driving field, determines the population of the collective modes. We investigate in particular the role of the laser detuning from the atomic transition, and demonstrate a simple relation between the detuning and the steady-state population of the modes. This relation allows understanding several properties of cooperative scattering, such as why superradiance and subradiance become independent of the detuning at large enough detuning without vanishing, and why superradiance, but not subradiance, is suppressed near resonance. We also show that the spatial properties of the collective modes allow distinguishing diffusive modes, responsible for radiation trapping, from subradiant modes.

Figure 1

Distribution of the eigenvalues ${\lambda }_k=-{\mathrm{\Gamma }}_k/2+i{E}_k$ in the complex plane (for $\mathrm{\Delta }=0$) with the geometrical factor $|P_k{\left(\mathbf{\Omega }\right)|}^2$ represented in the color scale. The parameters of the atomic sample are $N=3000$ and $kR\simeq 26.3$, yielding $b_0\simeq 8.7$ and ${\rho }_0{k}^{-3}\simeq {10}^{-2}$.

Figure 2

Same as in Fig. 1 but the color scale now shows the spectral factor $1/|{\lambda }_k{|}^2$ (left column) and the populations $p_k$ (right column). The different rows are for different detunings, from top to bottom $\mathrm{\Delta }=0,1,10$. Note the different color scale for each panel, showing that at large detuning, the spectral factor is almost uniform.

Figure 3

Close-up of the populations of the long-lived modes, at resonance (a) and far from resonance (b). Same parameters as in Fig. 1.

Figure 4

Spatial properties of the collective modes. Same parameters are in Fig. 1, but the color scale now shows (a) the rms size ${\sigma }_k$ of the modes (normalized by the sample size $R$), and (b) the participation ratio ${\mathcal{R}}_k^{\mathrm{P}}$ (normalized by the atom number $N$). We show in panels (c)–(e) the average over 120 realizations of the excitation probability (mode intensity divided by atomic density) for atoms located in a slice $\left|z\right|<R/5$, for each kind of mode, selected by the following conditions: (c) $0.9<{\mathrm{\Gamma }}_k<1.1$ and $-0.1<E_k<0.1$; (d) ${\mathrm{\Gamma }}_k<0.2$; (e) $P{R}_k>0.45$.

Figure 5

Study of the weighted averages $\langle E\rangle$ and $\langle \mathrm{\Gamma }\rangle$ [Eq. (13)] as a function of the detuning $\mathrm{\Delta }$ and the on-resonance optical thickness $b_0$. (a) The frequency shift $\langle E\rangle -\mathrm{\Delta }$ is plotted as a function of $\mathrm{\Delta }$ for different $b_0$. (b) $\langle \mathrm{\Gamma }\rangle$ is plotted as a function of $\mathrm{\Delta }$ for different $b_0$. (c) $\langle \mathrm{\Gamma }\rangle$ is plotted as a function of $b_0$ for the two extreme cases, $\mathrm{\Delta }\to \infty$ (red circles) and $\mathrm{\Delta }=0$ (blue diamonds). The dashed lines correspond to the empirical relations given in Eqs. (14) and (15).

Figure 6

Same numerical data as in Figs. 5 and 5 but $\langle \mathrm{\Gamma }\rangle -(1+b_0/4)$ is plotted as a function of $b\left(\langle E\rangle \right)=b_0/(1+4{\langle E\rangle }^2)$ (log scale). All points collapse on a single curve (dashed line), well described by Eq. (17).

Figure 7

Linewidth distribution $P\left({\mathrm{\Gamma }}_k\right)$ without weighting (dashed black line) and with weighting corresponding to the populations $p_k$ computed far from resonance ($\mathrm{\Delta }\to \infty$, solid red line) or at resonance ($\mathrm{\Delta }=0$, dotted blue line). The parameters of the atomic sample are $N=5000,kR\simeq 27.3$ yielding $b_0\simeq 13.4,{\rho }_0{k}^{-3}\simeq 1.55\times {10}^{-2}$, and we used 120 realizations.