content_image/440658.jpg content_image/440657.jpg content_image/440659.jpg
<img src='content_image/440658.jpg'>

inary parts of complex eigenvalues of Σ$_{jk}$ for N = 20 molecules at the sites of C20 fullerene at three different molecule-surface distances. The system spectrum repre- sents several sets of degenerate eigenvalues indicating a high degree of lattice symmetry. For all distances, the histograms show a single eigenvalue with a large posi- tive energy shift (Fig. 2), which corresponds to the di- rect dipole-dipole interaction between nearest-neighbor molecules. On the other hand, there are three degenerate eigenvalues with the largest decay rate, corresponding to predominantly superradiant states while the smaller de- cay rates are those of predominantly subradiant states (Fig. 3). With increasing distance, the mixing between superradiant and subradiant states decreases and, for d = 30 nm, decay rates of all but three eigenstates nearly vanish; note that in our approximation, pure subradiant states should have zero decay rate. Such behavior is due to diminishing contribution of higher- l plasmons at larger distances [see Eqs. ((23) and 24)]. Importantly, direct in- teractions between close molecules result only in energy shift of subradiant states, without affecting superradiant states. We therefore conclude that dipole-dipole interac- tions do not destroy cooperative emission in plasmonic systems. 

This main conclusion remains unchanged when fluctu- ations (up to 10%) of molecules positions in radial direc- tion are included into simulations (Figs. 4 and 5). The spatial disorder lifts lattice symmetry, so that superradi- ant states now have different, however close, decay rates. Note that without interactions, i.e., when molecules an- gular positions are completely random, the spread of su- perradiant decay rates is considerably higher. 10 

To elucidate the structure of collective states, we cal- culate the distance dependences of complex eigenvalues, Δ$_{J}$ − i $_{2}$Γ$_{J}$ , for C20, C60, and C80 configurations of dyes, as shown in Figs. 6, 7, and 8, respectively. For C20 con- figuration (Fig. 6), there are five sets of eigenvalues with 3, 4, 4, 7, and 1-fold degeneracies, in descending order 

<img src='content_image/440657.jpg'>

<img src='content_image/440659.jpg'>

of Γ$_{J}$ magnitudes. Down to the distance of d = 5 nm, the largest decay rates, corresponding to three predom- inantly superradiant states, are well separated from the rest. The steep rise of Γ$_{J}$ at small distances is due to increasing contribution of high- l plasmon modes close to NP surface [see Eqs. (23) and (24)]. The interplay be- tween various coupling mechanisms is especially revealing 

