Γ r jk = Γ r 0 [ ( e$_{j}$ · e$_{k}$ ) − α ′ 1 [ K (1) jk + h.c. ] + | α$_{1}$ | $^{2}$T (1) jk ] , Γ nr jk = 3Γ r 0 2 k 3 ∑ l α ′′ l T ( l ) jk . (22) dW ( ω ) dω = cϵ$_{0}$ 4 π 2 ∫ | E ( r , ω ) | 2 r $^{2}$d Ω , (25) K (1) jk = 1 r 3 j ( e$_{j}$ · e$_{k}$ ) , T (1) jk = 1 r 3 $_{j}$r 3 k ( e$_{j}$ · e$_{k}$ ) . (21) dW dω = 4 r $^{2}$ϵ$_{0}$ µ $^{2}$ω4 0 c 3 (26) × ∑ jk ∫ d Ω v$_{j}$v ∗ k [ G ( r , r$_{j}$ ) · e$_{j}$ ] · [ G $^{∗}$( r , r$_{k}$ ) · e$_{k}$ ] , G$_{µν}$ ( r, r$_{j}$ ) = e ikr 4 πr [ δ$_{µν}$ − 4 π 3 ∑ m ˆ r$_{µ}$ Y$_{1}$$_{m}$ (ˆ r ) χ ν ∗ 1 $_{m}$( r$_{j}$ ) − 4 π 3 ˜ α$_{1}$r ∑ m [ ∇ $_{µ}$Y$_{1}$$_{m}$ (ˆ r ) ] ψ ν ∗ 1 $_{m}$( r$_{j}$ ) ] , (27) Γ r jk = Γ r 0 [ 1 + 2 α ′ 1 ( 1 r 3 j + 1 r 3 k ) + 4 | α$_{1}$ | 2 r 3 $_{j}$r 3 k ] cos γ$_{jk}$, Γ nr jk = 3Γ r 0 2 k 3 ∑ l α ′′ l ( l + 1) 2 r l +2 j r l +2 k P$_{l}$ (cos γ$_{jk}$ ) , (23) 4 πr 2 ∫ d Ω [ G $^{0}$( r , r$_{j}$ ) · e$_{j}$ ] · [ G 0 $^{∗}$( r , r$_{k}$ ) · e$_{k}$ ] = 2 3 ( e$_{j}$ · e$_{k}$ ) . (28) ( 4 πr 3 ) 2 ∑ mm ′ [ ψ ∗ 1 $_{m}$( r$_{j}$ ) · e$_{j}$ ][ ψ$_{1}$$_{m}$ ′ ( r$_{k}$ ) · e$_{k}$ ] (29) × ∫ d Ω 4 π ∇ Y$_{1}$$_{m}$ (ˆ r ) · ∇ Y ∗ 1 m ′ (ˆ r ) = 2 3 T (1) jk , Δ$_{jk}$ = Δ 0 $_{jk}$+Γ r 0 [ α ′′ 1 ( 1 r 3 j + 1 r 3 k ) cos γ$_{jk}$ − 3 4 k 3 ∑ l α ′ $_{l}$( l + 1) 2 r l +2 j r l +2 k P$_{l}$ (cos γ$_{jk}$ ) ] , (24) 4 πr 3 ∑ m ∫ d Ω 4 π [ e$_{k}$ · ∇ Y$_{1}$$_{m}$ (ˆ r ) ][ ψ ∗ 1 $_{m}$( r$_{j}$ ) · e$_{j}$ ] = 2 3 K (1) jk , (30) ∫ d Ω 4 π [ G ( r , r$_{j}$ ) · e$_{j}$ ] · [ G $^{∗}$( r , r$_{k}$ ) · e$_{k}$ ] (31) = 1 (4 πr ) 2 2 3 [ ( e$_{j}$ · e$_{k}$ ) − α$_{1}$K (1) jk − α ∗ $_{1}$K (1) kj + | α$_{1}$ | $^{2}$T (1) jk ] . 5 III. RADIATED ENERGY and for parallel orientation ( e$_{j}$ · ˆr$_{j}$ = 0), we similarly get The decay matrix , Γ$_{jk}$ = − 2ImΣ$_{jk}$, can be decomposed into radiative and nonradiative terms, Γ$_{jk}$ = Γ r jk + Γ nr $_{jk}$, as follows In this section, we derive general expression for total energy radiated by an ensemble of dipoles near a metal nanostructure. The radiated energy in the unit frequency interval is obtained by integrating the far field ( r → ∞ ) spectral intensity over solid angle 29 and then averaging the result over initial random phases of individual dipoles, ϕ$_{j}$ . The electric field E ( r , ω ) is given by Eq. (7), where v$_{j}$ is the solution of Eq. (4). Then the energy density takes the form The diagonal elements Γ r jj and Γ nr jj describe, respectively, plasmon-enhanced radiative decay rate of an isolated dipole near a NP, and nonradiative transfer of its en- ergy to electronic excitations in metal. The non-diagonal elements of Γ r jk describe plasmon-enhanced radiative cou- pling that generalizes the Dicke mechanism responsible for cooperative emission by incorporating local field en- hancement into near field radiative coupling. On the other hand, the non-diagonal terms in Γ nr jk describe non- radiative energy transfer between dipoles mediated by NP plasmons. The latter coupling is absent in the pho- tonic Dicke effect but it plays important role in the plas- monic Dicke effect, as we will see below. with G ( r , r$_{j}$ , ω ) = G $^{0}$( r , r$_{j}$ , ω ) + G $^{s}$( r , r$_{j}$ , ω ) replaced by its far field asymptotics (see Appendix), Since the numerical calculations below are carried for normal dipoles orientations, we provide here the corre- sponding expressions for self-energy matrix Σ$_{jk}$. The de- cay matrix has the form where first two terms come from the free space part, G $^{0}$, and the last term comes from the scattered part, G $^{s}$, of the Green diadic. The angular integral in Eq. (25) can be performed using the relations Eq. (19). The free space contribution yields where P$_{l}$ is Legendre polynomial and γ$_{jk}$ is the angle between positions of dipoles j and k measured from NP center (here cos γ$_{jk}$ = e$_{j}$ · e$_{k}$ ). The real part of self-energy matrix has the form The other integrals in the product Eq. (26) are evaluated using the relations and yielding where the second term describes NP-induced interac- tions. The latter in turn consists of two terms, first coming from plasmon-enhanced radiative coupling and second coming from non-radiative coupling. Note that both NP-induced terms are weaker than their counterparts in Γ$_{jk}$ while having same symmetry and therefore is not expected to significantly alter the eigenstates. On the other hand, the dipole-dipole inter- action term, Eq. (10), has different symmetry and can become large for two dipoles in a close proximity to each other. The effect of interactions on cooperative emission is studied in Sec. IV.