Super- and subradiant emission of two-level systems in the near-Dicke limit

We analyze the stability of super- and subradiant states in a system of identical two-level atoms in the near-Dicke limit, i.e., when the atoms are very close to each other compared to the wavelength of resonant light. The dynamics of the system are studied using a renormalized master equation, both with multipolar and minimal-coupling interaction schemes. We show that both models lead to the same result and, in contrast to nonrenormalized models, predict that the relative orientation of the (coaligned) dipoles is unimportant in the Dicke limit. Our master equation is of relevance to any system of dipole-coupled two-level atoms, and gives bounds on the strength of the dipole-dipole interaction for closely spaced atoms. Exact calculations for small atom systems in the near-Dicke limit show the increased emission times resulting from the evolution generated by the strong dipole-dipole interaction. However, for large numbers of atoms in the near-Dicke limit, it is shown that as the number of atoms increases, the effect of the dipole-dipole interaction on collective emission is reduced.

Figure 1

$\hslash {\tilde{\Delta }}_{nm}^{\perp }$ (dotted line), ${10}^4\hslash {\tilde{\Delta }}_{nm}^{\parallel }$ (solid line), and nonregularized ${\Delta }_{nm}^{\perp }+{\Delta }_{nm}^{\parallel }$ (dashed line) in units of the ground-state energy ${\mathcal{E}}_0$ for (a) ${\eta }_{nm}=0$ and (b) ${\eta }_{nm}=1$.

Figure 2

Double-logarithmic plot of the moduli of $\hslash {\tilde{\Delta }}_{nm}^{\perp }$ (dotted line), $\hslash {\tilde{\Delta }}_{nm}^{\parallel }$ (solid line), and nonregularized ${\Delta }_{nm}^{\perp }+{\Delta }_{nm}^{\parallel }$ (dashed line) in units of the ground-state energy ${\mathcal{E}}_0$ for (a) ${\eta }_{nm}=0$ and (b) ${\eta }_{nm}=1$.

Figure 3

Populations $P_{\mathsf{b}}$, $P_{\mathsf{c}}$, $P_{\mathsf{d}}$ of the three levels $\mathsf{b},\mathsf{c},\mathsf{d}$ for ${\xi }_{12}=\xi$, ${\xi }_{23}=\xi$, and ${\xi }_{13}=2\xi$, where $\xi =0.005$, and $\eta =1$. The initial state is $|\mathsf{b}⟩$.

Figure 4

Difference between polar photon counting distributions ${\sigma }^{\prime }={\sigma }_{nd}-{\sigma }_{wd}$ with $\left({\sigma }_{wd}\right)$ and without $\left({\sigma }_{nd}\right)$ dipole-dipole interactions, for atomic orientation ${\eta }_{nm}=0$ relative to the dipoles, and separation $r_{nm}=1\times {10}^{-10}\text{m}$. The distributions are averaged over $100000$ trajectories for $0<t<{10}^4{\gamma }^{-1}$.

Figure 5

Waiting time distribution of the fifth photon, $w_t=⟨\overline{\Psi }\left(t\right)|{\widehat{B}}^{†}\left(t\right)\widehat{B}\left(t\right)|\overline{\Psi }\left(t\right)⟩$ for $|\Psi \left(t=0\right)⟩$ the normalized state at the time of the fourth photon emission, and $|\overline{\Psi }\left(t\right)⟩$ the unnormalized state at time $t$, (i) including and (ii) excluding the dipole-dipole interaction for five atoms in a linear configuration with interatomic separation $r_{nm}=1\times {10}^{-10}\text{m}$. The distributions are averaged over $10000$ trajectories.

Figure 6

$t_{\text{Dicke}}/t_{\text{rate}}$ for different numbers of atoms, $N$, and separations, $r_{nm}$. The smallest separation was $1.7\times {10}^{-4}{\lambda }_0$ and the largest $5.1\times {10}^{-4}{\lambda }_0$. For these separations, ${\tilde{\gamma }}_{nm}\gtrsim 0.99\gamma$, for all $\left(n,m\right)$ validating the emission rate approximation.