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when comparing the plots for Δ$_{J}$ and Γ$_{J}$ (curves for same eigenvalue sets have similar patterns). By their d depen- dence, the eigenvalues fall into three main groups. The superradiant states have the largest decay rate Γ$_{J}$ for all d and relatively small mainly positive frequency shift for d ≳ R/ 2; these states are dominated by plasmon- enhanced radiative coupling . The non-degenerate state with large positive energy shift and smallest decay rate is dominated by direct nearest-neighbor dipoles interaction; this state is least affected by the presence of NP and does not participate in the emission. The third group of states with mostly negative Δ$_{J}$ and small Γ$_{J}$ is dominated by nonradiative plasmon coupling . Closer to NP surface, the coupling becomes dominant due to high- l plasmons and all states develop large decay rates and negative en- ergy shifts. Note that down to d ≳ R/ 4, the admix- ture between superradiant and subradiant modes is still relatively weak; below R/ 4, the non-radiative coupling dominates the spectrum and the admixture is strong. 

For larger ensembles, the eigenstates have similar structure, as illustrated in Figs. 7 and 8 which show cal- culated eigenvalues for dipoles in C60 and C80 config- urations, respectively. Importantly, even with decreas- ing distance between the emitters in large ensembles, the dipole-dipole interactions still do not destroy coopera- tive emission. This can be understood from the follow- ing argument. 25 Mixing of superradiant and subradiant states takes place if the interactions between them are sufficiently strong. The latter requires that the electric field of a collective state is strongly inhomogeneous in space since, e.g., subradiant states couple only weakly to homogeneous field. On the other hand, such a field 

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is comprised of individual fields of all the const ituent dipoles so the resulting field’s spatial fluctuation s are weak if no two dipoles approach too close to each o ther, i.e., deviations of nearest-neighbor separations from their average, ¯ s = LN − 1 / $^{3}$, L being characteristic system size, are small. However, if deviations from ¯ s ar e large, i.e., two dipoles can be separated by a much close r dis- tance, s ≪ LN − 1 / $^{3}$, causing a strong spatial field fl uctua- tion, then the eigenstates are no longer superradian t and subradiant states and cooperative emission is destr oyed. This argument was confirmed numerically here by fi nding system eigenstates for both cases – dipoles on a s pher- ical lattice with some fluctuations (see Figs. 4 an d 5), and a completely random angular distribution with fixed dipole-NP distance with no minimal separation bet ween two dipoles (not shown). In the latter case, no sup erra- diant states were formed and the reason was trac ed to configurations with extremely close dipoles. Note, how- ever, that with both radial and angular distribution s be- ing random, these are rare events. In the case of rep ulsive interactions between individual dipoles, considered here, deviations from the average dipole-dipole separat ion ¯ s are exceedingly small and cooperative emission sur vives the interactions. 

Another sharp contrast between plasmonic and pho- tonic Dicke effects is the fate of subradiant states. I n the latter, the energy trapped in subradiant states is ev entu- ally radiated, albeit with a much slower rate, result ing in sharp spectral features of emission spectrum. 2,3 I nstead, in plasmonic systems, the trapped energy is dissipat ed in the NP and only a small fraction of total energy leaves the system via superradoant states. Thus, the net effect 

