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# Plasmon-mediated superradiance near metal nanostructures 

$^{1}$Department of Physics, Jackson State University, Jackson, MS 39217, USA and 

Vitaliy N. Pustovit 1 , 2 and Tigran V. Shahbazyan 1 

$^{2}$Centre de Recherche Paul Pascal, CNRS, Avenue A. Schweitzer, 33600 Pessac, France 

We develop a theory of cooperative emission of light by an ensemble of emitters, such as fluo- rescing molecules or semiconductor quantum dots, located near a metal nanostructure supporting surface plasmon. The primary mechanism of cooperative emission in such systems is resonant energy transfer between emitters and plasmons rather than the Dicke radiative coupling between emitters. We identify two types of plasmonic coupling between the emitters, (i) plasmon-enhanced radiative coupling and (ii) plasmon-assisted nonradiative energy transfer, the competition between them gov- erning the structure of system eigenstates. Specifically, when emitters are removed by more than several nm from the metal surface, the emission is dominated by three superradiant states with the same quantum yield as a single emitter, resulting in a drastic reduction of ensemble radiated en- ergy, while at smaller distances cooperative behavior is destroyed by nonradiative transitions. The crossover between two regimes can be observed in distance dependence of ensemble quantum effi- ciency. Our numerical calculations incorporating direct and plasmon-assisted interactions between the emitters indicate that they do not destroy the plasmonic Dicke effect. 

Superradiance of an ensemble of dipoles confined within a limited region in space has been discovered in the pioneering work by Dicke. 1 The underlying physical mechanism can be described as follows. Suppose that a large number, N , of dipoles with frequency ω$_{0}$ are con- fined in a volume with characteristic size L much smaller than the radiation wavelength λ$_{0}$ = 2 π/ω$_{0}$ . Then radia- tion of an ensemble is a cooperative process in which the emission of a photon is accompanied by virtual photon exchange between individual emitters. This near field ra- diative coupling between the dipoles leads to formation of new system eigenstates, each comprised of all individual dipoles. The eigenstates with angular momentum l = 1 are superradiant , i.e., their radiative lifetimes are very short, ∼ τ/N , where τ is radiative lifetime of an indi- vidual dipole; the remaining states are subradiant with much longer decay times, ∼ τ ( λ$_{0}$/L ) 2 ≫ τ . Since the appearance of Dicke paper, cooperative ef- 

fects based on Dicke radiative coupling mechanism have been extensively studied in atomic and semiconductor systems (see, e.g., reviews in Refs. 2–4). Two differ- ent decay times corresponding to superradiant and sub- radiant states were observed in a system of two laser- trapped ions 5 and, more recently, in laterally arranged quantum dots. 6 Other examples of cooperative behavior analogous to the Dicke effect include, e.g., electron tun- neling through a system of quantum dots 7,8 and sponta- neous phonon emission by coupled quantum dots. 4,9 

Recently, we extended the Dicke effect to plasmonic systems comprised of N dipoles located in the vicinity of a metal nanostructure, e.g., metal nanoparticle (NP), supporting localized surface plasmon (SP). 10 In such sys- tems, the dominant coupling mechanism between dipoles is plasmonic rather than radiative, i.e., it is based on virtual plasmon exchange (see Fig. 1). This plasmonic coupling leads to formation of collective states, simi- lar to Dicke superradiant states, which dominate pho- 

## I. INTRODUCTION 

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ton emission. Furthermore, the nanostructure acts as a hub that couples nearby and remote dipoles with about equal strength and hence provides a more efficient hy- bridization of dipoles compared to radiative coupling. In general, as dipoles orientations in space are non- uniform, there are three superradiant states with total angular momentum l = 1, each having radiative decay rate ∼ N Γ $^{r}$/ 3, where Γ r is radiative decay rate of a single dipole near a nanostructure (i.e., with plasmon enhancement). 10 

The principal difference between plasmonic and usual (photonic) Dicke effects stems from non-radiative energy transfer between the dipoles and the nanostructure. Let us first outline its role for the case of a single dipole near metal NP. When an excited emitter is located close to metal surface, its energy can be transferred to op- tically inactive excitations in the metal and eventually dissipated (Ohmic losses). This is described by the non- radiative decay rate, Γ nr ∝ d − $^{3}$, where d is the dipole– surface separation. 11 Note that very close to metal sur- face ( ∼ 1 nm), this dependence changes to ∝ d − 4 due to surface-assisted generation of electron-hole pairs out of the Fermi sea. 12,13 As a result, the radiation of a cou- pled dipole-NP system is governed by a competition be- tween non-radiative losses and plasmon enhancement 14 

