2 Q$_{ens}$ ≃ 3 Q (1) the single dipole-NP system, that determines system quantum efficiency, Q = Γ $^{r}$/ Γ, where Γ = Γ r + Γ nr is the full decay rate. Indeed, the radiated energy is W = ( ℏ kc/ 2) Q , k and c being wave vector and speed of light, and its distance dependence fol- lows that of Q . Namely, with decreasing d , the emission first increases due to plasmon enhancement, and then, at several nm from metal surface, it is quenched due to sup- pression of Q by non-radiative losses. Both enhancement and quenching were observed in recent experiments on fluorescing molecules attached to a metal NP, 15–19 and, not too close to NP surface, the distance dependence of single-molecule fluorescence 17,18 was found in excellent agreement with single dipole-NP models. 20–23 regardless of the ensemble size. Thus, the total radiated energy of the ensemble, W$_{ens}$ = ( ℏ kc/ 2) Q$_{ens}$ , is reduced to just 3 W . The remaining energy is trapped by N − 3 subradiant states and eventually dissipated in the metal rather than being emitted with a much slower rate, as it would be the case in free space. On the other hand, at several nm from metal surface, the non-radiative cou- pling is dominated by higher l plasmons causing strong mixing of superradiant and subradiant states, i.e., all system eigenstates have comparable quantum efficien- cies and Q$_{ens}$ ∝ N . Therefore, with increasing distance, Q$_{ens}$ , should first exhibit a sharp rise with its slope ∝ N , and then switch to a more slower 3 Q dependence. When radiation takes place from an ensemble of emit- ters near a metal nanostructure, there are two distinct types of plasmon-induced couplings between the emit- ters. The first is plasmon-enhanced radiative coupling , described by radiative decay matrix Γ r $_{jk}$, where indexes j,k = 1 ,...,N refer to emitters, that is a straightforward extension of Dicke radiative coupling obtained by incor- porating SP local field into the common radiation field. Correspondingly, the eigenstates of Γ r jk are superradiant and subradiant states characterized by the strength of their coupling to radiation field. In the ideal case of ”point sample,” i.e., kL ≪ 1, the subradiant decay rates are negligibly small and Γ r jk essentially has just three non-zero eigenvalues, corresponding to superradiant de- cay rates, each scaling with N as ∼ N Γ $^{r}$/ 3. 10 Indications of such behavior were reported in the re- cent experiment by Dulkeith et al. , 16 where a systematic study of the ensemble fluorescence vs. distance to metal surface was performed for Cy5 fluorophores attached to Au NP in water. The distance was controlled by vary- ing fluorophores concentration; with increasing concen- tration, the linker molecules stretched outwards to ac- commodate repulsive dipole-dipole interaction between fluorophores. Therefore, within isolated dipole-NP pic- ture, one would expect that for larger distances, at which fluorophores concentration was higher than average, the measured quantum efficiency, normalized to some aver- age fluorophores number, should have exceeded the cal- culated efficiency in single dipole-NP models. 20 Instead, with increasing distance, the normalized Q , exhibited rapid saturation and, at large distances, was consider- ably smaller in magnitude than the calculated one. The second coupling mechanism is non-radiative en- ergy transfer between dipoles that takes place in two steps: an excited dipole first transfers its energy to plas- mons in nanostructure via its electric field, and then this energy is transferred to another dipole. This process in- volves plasmons with all angular momenta l , and it is described by non-radiative decay matrix, Γ nr $_{jk}$. Impor- tantly, plasmons with l > 1 couple to both superradiant and subradiant states, so that the eigenstates of full decay matrix , Γ$_{jk}$ = Γ r $_{jk}$+Γ nr $_{jk}$, are not superradiant and subra- diant states, but their admixtures. Close to metal surface where non-radiative processes are dominant, Γ nr jk prevails over Γ r $_{jk}$, and no cooperative behavior is expected. How- ever, when dipoles are removed from the surface by more than several nm, the energy transfer occurs primarily via optically active dipole surface SP and therefore no signif- icant mixing of superradiant and subradiant states takes place and superradiance is intact. In the above discussion, we completely ignored inter- actions between the dipoles. In fact, the role of dipole- dipole interactions in cooperative emission is highly non- trivial since they introduce a disorder into the energy spectrum by causing frequency shifts among randomly distributed in space but otherwise identical emitters. 24 Mesoscopic cooperative emission from a disordered sys- tem, i.e., for particular disorder realizations rather than its averaged effect, was considered in Ref. 25 (see also Ref. 4). It was found that frequency shifts due to dipole-dipole interactions lift the degeneracy of subradi- ant states without having significant impact on superra- diant states. It was also shown that interactions between collective eigenstates are much weaker than those be- tween individual dipoles, including typical nearest neigh- bors (i.e., separated by ∼ LN − 1 / $^{3}$), due to cancellations between dipole-dipole terms among individual pairs with their constituents belonging to different eigenstates. 25 On the other hand, for completely random distribution and high concentration of dipoles, the rare instances of ex- tremely close dipoles (i.e., with separation ≪ LN − 1 / $^{3}$) can prevent the formation of superradiant states. 26 The role of interactions in plasmon-mediated coop- erative emission is characterized by several distinctive features. First, since the decay matrices Γ$_{jk}$ contain This observation was confirmed by numerical calcula- tion of eigenvalues of Γ$_{jk}$, i.e., full decay rates of sys- tem eigenstates, for ensemble of N dipoles randomly distributed in a solid angle around a NP. 10 Namely, in a wide range of dipole-NP distances, three eigenvalues corresponding to superradiant states are well separated from the rest and scale with N according to ∼ N Γ / 3. Since the superradiant states are the optically active ones with radiative decay rate also scaling as ∼ N Γ $^{r}$/ 3, their quantum efficiencies essentially coincide with Q of single dipole-NP system. Therefore, in the cooperative regime, the ensemble quantum efficiency, Q$_{ens}$ , is thrice that of