9 GLYPH<s48> GLYPH<s49> GLYPH<s48> GLYPH<s48> GLYPH<s48> GLYPH<s50> GLYPH<s48> GLYPH<s48> GLYPH<s48> GLYPH<s51> GLYPH<s48> GLYPH<s48> GLYPH<s48> GLYPH<s52> GLYPH<s48> GLYPH<s48> GLYPH<s48> GLYPH<s65>GLYPH<s103> GLYPH<s82>GLYPH<s32>GLYPH<s61> GLYPH<s32>GLYPH<s50> GLYPH<s48> GLYPH<s32>GLYPH<s110> GLYPH<s109> GLYPH<s78>GLYPH<s32>GLYPH<s61> GLYPH<s32>GLYPH<s56> GLYPH<s48> GLYPH<s32> GLYPH<s47> GLYPH<s48> GLYPH<s114> GLYPH<s32> GLYPH<s32> GLYPH<s40> GLYPH<s97> GLYPH<s41> GLYPH<s48> GLYPH<s49> GLYPH<s48> GLYPH<s50> GLYPH<s48> GLYPH<s51> GLYPH<s48> GLYPH<s52> GLYPH<s48> GLYPH<s45> GLYPH<s52> GLYPH<s48> GLYPH<s48> GLYPH<s48> GLYPH<s45> GLYPH<s50> GLYPH<s48> GLYPH<s48> GLYPH<s48> GLYPH<s48> GLYPH<s50> GLYPH<s48> GLYPH<s48> GLYPH<s48> GLYPH<s52> GLYPH<s48> GLYPH<s48> GLYPH<s48> GLYPH<s47> GLYPH<s48> GLYPH<s114> GLYPH<s32> GLYPH<s100> GLYPH<s32> GLYPH<s40> GLYPH<s110> GLYPH<s109> GLYPH<s41> GLYPH<s40> GLYPH<s98> GLYPH<s41> GLYPH<s48> GLYPH<s49> GLYPH<s48> GLYPH<s48> GLYPH<s48> GLYPH<s50> GLYPH<s48> GLYPH<s48> GLYPH<s48> GLYPH<s51> GLYPH<s48> GLYPH<s48> GLYPH<s48> GLYPH<s65>GLYPH<s103> GLYPH<s82>GLYPH<s32>GLYPH<s61> GLYPH<s32>GLYPH<s50> GLYPH<s48> GLYPH<s32>GLYPH<s110> GLYPH<s109> GLYPH<s78>GLYPH<s32>GLYPH<s61> GLYPH<s32>GLYPH<s54> GLYPH<s48> GLYPH<s47> GLYPH<s48> GLYPH<s114> GLYPH<s32> GLYPH<s32> GLYPH<s40> GLYPH<s97> GLYPH<s41> GLYPH<s48> GLYPH<s49> GLYPH<s48> GLYPH<s50> GLYPH<s48> GLYPH<s51> GLYPH<s48> GLYPH<s52> GLYPH<s48> GLYPH<s45> GLYPH<s51> GLYPH<s48> GLYPH<s48> GLYPH<s48> GLYPH<s45> GLYPH<s50> GLYPH<s48> GLYPH<s48> GLYPH<s48> GLYPH<s45> GLYPH<s49> GLYPH<s48> GLYPH<s48> GLYPH<s48> GLYPH<s48> GLYPH<s49> GLYPH<s48> GLYPH<s48> GLYPH<s48> GLYPH<s50> GLYPH<s48> GLYPH<s48> GLYPH<s48> GLYPH<s47> GLYPH<s48> GLYPH<s114> GLYPH<s32> GLYPH<s100> GLYPH<s32> GLYPH<s40> GLYPH<s110> GLYPH<s109> GLYPH<s41> GLYPH<s40> GLYPH<s98> GLYPH<s41> FIG. 8: (Color online) Same as in Fig. 6, but for 80 di poles in C80 configuration. FIG. 7: (Color online) Same as in Fig. 6, but for 60 dipoles in C60 configuration. 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. 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. 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 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