Subradiance in dilute atomic ensembles: Role of pairs and multiple scattering

We study numerically the slow (subradiant) decay of the fluorescence of motionless atoms after a weak pulsed excitation. We show that, in the linear-optics regime and for an excitation detuned by several natural linewidths, the slow decay rate can be dominated by close pairs of atoms (dimers) forming superradiant and subradiant states. However, for a large-enough resonant optical depth and at later time, the dynamics is dominated by collective many-body effects. In this regime, we study the polarization and the spectrum of the emitted light, as well as the spatial distribution of excitation inside the sample, as a function of time during the decay dynamics. The behavior of these observables is consistent with what would be expected for radiation trapping of nearly resonant light. This finding sheds light on subradiance in dilute samples by providing an interpretation based on the light behavior of the system (multiple scattering) which is complementary to the more commonly used picture of the collective atomic Dicke state.

Figure 1

Fluorescence dynamics of atomic ensembles of various sizes for (a) the total normalized radiation intensity in all directions and polarizations $I\left(t\right)/I\left(0\right)$ and (b) the instant trapping time $\tau \left(t\right)$. The density of atoms is $n=0.01k^3$, the duration of excitation is $\gamma T=2000$, and the detuning is $\delta =-4\gamma$. The number of realizations changes from ${10}^4$ ($kL=20$) to 5000 ($kL=70$). The smallest size (green solid line) corresponds to $N=2$ atoms ($4\times {10}^6$ realizations in that case). The horizontal black solid line in (b) corresponds to the computed lifetime of subradiant dimers that are resonant with the excitation.

Figure 2

Same as in Fig. 1 with larger sample sizes $L$. The number of realizations changes from 5000 ($kL=70$) to 300 ($kL=100$).

Figure 3

Instant trapping time $\tau \left(t\right)$ for different detunings $\delta$ of the exciting radiation. The system size is $kL=100$ ($b_0\approx 19$) and the number of realizations is 300. At late time radiation trapping dominates, while at intermediate time and at large detuning, the decay rate due to diatomic clusters is clearly visible.

Figure 4

Cut of the spatial distribution of excited atoms for two time delays after the switching off of the excitation: (a) $\gamma t=30$ and (b) $\gamma t=180$. The parameters are $kL=100$ and $\delta =-4\gamma$ and the number of realizations is 300. The width of the cut is $0.5/k$.

Figure 5

Fluorescence spectrum for two time delays after the switching off of the excitation: (a) $\gamma t=20$ and (b) $\gamma t=180$. The parameters are $kL=100$ and $\delta =-4\gamma$.

Figure 6

(a) Normalized intensities of two orthogonal circularly polarized components of the radiation emitted at an angle $\theta =\pi /4$. The incident radiation is left circularly polarized and the other parameters are $kL=100$ and $\delta =-4\gamma$. (b) Corresponding degree of circular polarization.

Figure 7

Dependence of the characteristic decay time ${\tau }^{*}$ with the resonant optical thickness for different detuning $\delta$ of the driving field.