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Collective eigenstates of emission in an N-entity heterostructure and the evaluation of its Green tensors and self-energy components

Our understanding of emission from a collection of emitters strongly interacting among them and also with other polarizable matter in proximity has been approximated by independent emission from the emitters. This is primarily due to our inability to evaluate the self-energy matrices and the collective eigenstates of emitters in heterogeneous ensembles. A method to evaluate the self-energy matrices that is not limited by the geometry and the material composition is presented here to understand and exploit such collective excitations. Numerical evaluations using this method are used to highlight the significant differences between independent and the collective modes of emission in heterostructures. A set of n emitters driving each other and m other polarizable entities, where N=m+n, is used to represent the coupled system of a generalized geometry in a volume integral approach. Closed form relations between the Green tensors of entity pairs in free space and their correspondents in a heterostructure are derived concisely. This is made possible for general geometries because the global matrices consisting of all free-space Green dyads are subject to conservation laws. The self-energy matrix of the emitters can then be assembled using the evaluated Green tensors of the heterostructure, but a decomposition of its components into their radiative and non-radiative decay contributions is non-trivial. This is accomplished using matrix decomposition identities applied to the global matrices containing all free-space dyads. The relations to compute the observables of the eigenstates (such as quantum efficiency, power/energy of emission, radiative and non-radiative decay rates) are presented. We conclude with a note on extension of this method to collective excitations that also include strong interactions with a surface in the near-field.

non-radiative decay rates) are presented. We conclude with a note on extension of this method to collective excitations that also include strong interactions with a surface in the near-field. The interaction of atoms, fluorescing molecules and quantum dots with metal particles/surfaces have been extensively studied for their effect on the decay rates and radiated power of spontaneous emission. [1][2][3][4][5][6][7][8] But such understanding of the effects of surrounding matter on spontaneous emission has been limited to an independent emitter interacting with a particle, 4 surface, 3 or a set of spherical particles; 8 generalized as Purcell effect. Recently, this was extended by a theoretical study to a set of many emitters interacting collectively with a single metal particle at the center. 9, 10 Strongly interacting emitters present collective excitations mediated by virtual photons, and in the presence of other matter can include other virtual excitations as well; plasmons of a metal particle for example. We should expect that when the interactions between a single emitter and a body are not strong relative to the possible multilateral modes, the collective effects can manifest robustly to be observed and exploited. The impediment in understanding collective heterogeneous systems has been the difficulty of evaluating the self-energy components that include all the possible virtual interactions among the coupled emitters. This requires the evaluation of Green tensors coupling emitters in the presence of other polarizable matter in proximity. The diagonal terms of the self-energy matrices of such a heterogeneous system represent the independent emitters interacting with other bodies; they are typically computed using just the selffield of a single emitter due to the other polarizable bodies. Whereas, the off-diagonal terms of the self-energy matrix include the interaction among any two emitters directly and through other matter in proximity, and here we require these Green tensors explicitly. Note that analytical expressions for Green tensors are difficult in general; 11 and hence a numerical evaluation of Green tensors is a preferred approach for complex geometries. Mode decomposition and eigen function based Green tensor representation is suitable for regular geometries, and that is seldom a possibility in realistic nanostructures. We start with a set of n Lorentz dipole oscillators which when coupled can be used to elucidate the conventional Dicke effect of collective emission and the radiative coupling among them.
In addition, here we have m other polarizable entities which can be sub-volumes of a larger body in a volume integral approach. Thus this method is general and precise for arbitrary shapes, properties and a number of emitters/bodies, even in the limits where long wavelength approximations are not valid. It is shown that a problem posed by dipole sources and null field conditions at polarizable volumes can be used to evaluate the modified Green tensors coupling the n emitters in presence of the m other polarizable entities. We use matrix algebra to derive closed form relations between these modified Green tensors and the set of Green dyads coupling pairs of the N entities in free space (or a homogeneous background). The free-space Green dyads of dipolar entities can be trivially derived by the dyadic operations on scalar Green functions of point dipoles. Use of free space dyads of multipoles to avoid a fine discretization of a body into dipoles is also possible for very large problems. The modified Green tensors coupling the emitters are then used to assemble their selfenergy matrix and evaluate their eigenstates. Determining the characteristics of emission from the ensemble involves the additional exercise of the decomposing the imaginary part of the self-energy matrices into their radiative and non-radiative parts. This allows us to evaluate the quantum efficiency, decay rates and the radiated energy/power in the ensemble, given the quantum efficiency and non-radiative relaxation of an isolated emitter. The microscopic aspects in dipole representations of certain types of emitters, 12,13 and an explicit quantization of the emitter/bodies, [14][15][16][17] are not the subject of this work, and these may be introduced in this method without loss of generality for any special cases.
Finally, we conclude with a few numerical results highlighting that self-energy matrices of such interacting systems can reveal collective effects hitherto unexplained; some of our recent experimental observations and their analysis will be described elsewhere. Additions to this method that can include strong interactions of all the N entities with a surface are presented as an addendum. This section has three parts; first we introduce self-energy matrices of a collection of coupled Lorentz oscillators and their significance in studying collective behavior in an emission process. Then we extend this approach to evaluation of the self-energy matrices for cases where the emitters also interact strongly with other matter. This primarily involves evaluating the Green tensors coupling the emitters in the presence of other polarizable matter in a generalized geometry. Finally we present a technique to decompose the radiative and non-radiative contributions to each self-energy component of the self-energy matrix representing a heterogeneous mixture of emitters with other polarizable bodies. This is essential because the radiative and non-radiative decay matrices determine the observables and measurements of the collective emission.

Self-energy matrix of coupled Lorentz oscillators:
The emission from n dipole oscillators interacting with each other is well represented by a coupled system with n eigenstates, some of them super-radiant, and this collective phenomenon of emitters interacting directly is known as the conventional Dicke effect. 18 In the above, the interaction between dipoles are described in terms of the free-space Green By substitution of Δ jk and / 2 Γ jk i for the real and imaginary parts of the dipole cross-interaction terms ( ≠ j k ) in Eq. (3), we get Eq. (4) using a phase conjugation. The system of equations reduces to a form where the energy shifts and modified decay rates of a dipole due to its interaction with other dipoles can be readily interpreted.
It is convenient to represent the above coupled system of equations by the matrix eigenvalue problem in Eq. (5). The unknown cooperative phases φ j for the known oscillating dipole moments Further, the excitation in the dipole oscillators can be given the total energy of a quantum of radiation.
The interaction with another emitter results in energy shifts given by the real parts of the offdiagonal terms of the self-energy matrix, and the corresponding change in decay rates are given by the imaginary parts. The diagonal terms of the self-energy matrix represents the case of the independent or the uncoupled emitter and contains only its radiative decay rate. Energy shifts (or a change in resonant frequency toω ω In the theoretical approach so far, internal non-radiative relaxations of the dipole emitters have not been included, and these can be explicitly added in the diagonal terms of ∑ just as the radiative decay rates of the independent emitter (see Eq. (23)). This involves the assumption that internal non-radiative relaxation of an emitter is independent of its radiative decay. Thus using a harmonic oscillator to represent only the single quantum of emission is the typical approach.
Alternately, the Lorentz oscillators can be assigned an energy that includes both the radiative and non-radiative decay of the emission. This allows for the non-radiative coupling of two emitters in proximity. The total mechanical energy of the oscillator in this case increases by a factor 1/ Q o while the radiative decay includes only a quantum of energy as in Eq. (10). The method presented in this paper is amenable to both definitions of the oscillator. The latter are useful in cases when emitters are also coupled non-radiatively and where dipole-dipole interactions are sufficient to represent this nonradiative coupling.
and , Further, the above description of the collective emission process is restricted by only certain cases where the number of excitations participating in the collective emission process are not negligible compared to the density of optical states (DOS) available at ~ω h o , which is typically large. This situation is possible inside cavities and photonic crystals when specifically the emission energies are near the edges of a band gap, 22 and a strong excitation includes a sufficient density of such emitters in this emission process. Also, a set of emitters can collectively share fewer excitations resulting in emissions of a higher decay rate. 23 Such special cases of non-classical emission may require a more explicit quantization of the field, and fortunately do not include a wide variety of heterogeneous optical materials and their emission.

Evaluation of Green tensors and self-energy components of a heterostructure:
When emitters are neither in vacuum nor in a homogeneous dielectric medium and interact with other polarizable bodies in proximity, evaluation of the Green tensors coupling them is nontrivial. Using a volume integral approach, a heterogeneous volume can be represented by subvolumes with corresponding permittivity. In the presence of m other polarizable volumes much smaller than the wavelength in dimensions, the Green dyads are to be determined using the following Maxwell's equation for a heterostructure in a homogeneous background medium Here j α and ε are polarizability tensor of volume j and the permittivity tensor of the homogeneous background respectively. An analytical solution of this discrete problem, with or without solving for these modified Green dyads, is an intractable problem in general. But given The above can be rewritten into a problem of N oscillators coupled by ( , ; ) | : ... : .. and E ... and Writing such global matrices containing the Green dyads coupling each pair of entities gives us a concise representation, but more importantly, helps unravel the implicit relations between Green dyads required to satisfy the conservation laws. The global matrix of Green dyads o G can be decomposed further into four parts:  G has zeros in diagonal dyads correspondingly for the self-interaction terms (radiation reaction and non-radiative loss) of the independent emitter. o Γ can be conveniently included in the self-energy matrices as in Eq. (22,23) shown later. The polarizability tensorα can be corrected for sub-volumes of a contiguous large body using lattice dispersion relations if required. 24 Such finer volume discretization of a body is needed if its dimensions are not much smaller than the wavelength of emission. This limit becomes more stringent if two distinct bodies are closely spaced; for example when distance between centers →2a, where a is radius of a spherical particle. We will revisit multipolar representations of a body later in the paper. Else, polarizability of a distinct particle much smaller than the wavelength is well approximated by its dipole polarizability; in case of an isotropic material this reduces to In the above problem, the polarizations of the m bodies b P and the collective self-fields at the n emitters e E are unknown; but these do not have to explicitly solved for. The required Green dyads can be implicitly derived by rewriting the problem in terms of the required global matrix of green dyads ee G , coupling pairs of the n emitters in this heterostructure and resulting in these collective Using matrix block multiplications, from Eq. (15) we get the following two relations; first is the momentum conservation relation that has to be satisfied by self-fields e E of point emitters as an optical theorem, 25 and the second is the null-field condition for the polarizable bodies.
where p m represents a polynomial of degree m and the full set of such possible polynomials is P m .

Radiative and non-radiative contributions to self energy components:
The radiative and non-radiative parts of this collective emission have to be determined for a comparison with experimental measurements of the radiative properties and decay rates.    rank. [28][29][30] Similarly matrix deflating techniques are also possible when the real or imaginary part of the polarizability of some of the bodies is zero, and these methods can be found elsewhere. 31 The total rate of decay from an ensemble of heterostructures as observed by lifetime measurements can be traced using the energy radiated by all eigenstates. In determining the power and tracing the decay as in Eq. (33,34) note that we do not evaluate the decay of a specific initial state in a particular geometry of heterostructure. That requires expanding the prepared initial state using a weighted sum of the eigenstates, and this is not of relevance in experiments where initial states of the system are unknown and random (due to the non-radiative process that accompanies).
Alternately, we can determine the eigenstates for all possible random permutations in the geometry of a heterostructure along with random orientations of its dipole emitters. Once the collective eigenstates of a particular random geometry are determined, we sum the emitted power and the decay of all its eigenstates. These quantities represent an average of these observables over the full phase space of possible initial states in the geometry, which are further summed over all random geometries.
These results can be directly compared to experimental measurements involving an ensemble of such structures. ( Before we present numerical results of this evaluation, we conclude this section with a note on few limiting cases and the corrections required. The limits of the dyadics of point dipoles used and the effects of discretization of a body have to be discussed. This method is general enough to include arbitrary local volume discretization of a body in multiple scales for problems with any special cases. One such case is when two small bodies are closely spaced and a finer discretization of the body is required to include multipole interactions; for example when distance between centers → 2a; where a is radius of a spherical particle. To include l-pole effects sufficiently, a finer discretization of a body in dipolar representation scales the global matrices by l 3 in general, and thus computation by ( ) l l = polarizable volume are explicitly included in the global matrix in Eq. (15), the matrix dimensions increase by a factor of only l and this will result in an increase in computation by a factor of only l 3 .
Also, one should expect that when the distance between the surface of a body and an emitter is on the order of charge separations / = d p q, an evaluation using the dyads of point dipoles in Eq. (13) may be not accurate. At these small separations ~ 1 nm, charge screening in the body may not be complete, local inhomogeneity of ε may not be negligible and electron-hole pairs can be created.
Such energy transfer mechanisms can result in coupling on the order~r 4 and higher which may not be represented sufficiently by finer discretization of the body alone. However, studies show that these local deviations are significantly suppressed by both non-linear effects and quantization, resulting in a freezing of higher order classical modes. 32 These effects can result in a domination of the radiative terms on the interaction with the polarizable matter even at these close separations. 33

NOTE ON SURFACE INTERACTIONS:
The interaction of emitters and other bodies with a substrate/surface has been difficult to study because separable co-ordinates for analytical solutions do not exist, and Fresnel coefficients for near-field radiating sources do not have closed form relations. 35,36 The determination of the reflected components of the surface on a radiating dipole in the near field involves computing the Sommerfeld relations; using numerical calculations of these integral relations, the field from a radiating dipole over a surface can be decomposed into cylindrical components parallel to the surface and a plane wave perpendicular to the surface. For any two dipoles k and j at any distance z from the surface and distance ρ from each other along the surface, the integral in Eq. (35) has to converge over the complex domain of wave vector magnitudes k ρ with appropriate branch cuts as the required reflection coefficient R s involves its quadratic roots. 37,38 The reflected component of the electric field due to a radiating dipole in the near field of a surface can then be represented as in Eq. (36). 39 where jk S is a 3 x 3 matrix containing Sommerfeld integral terms of the field for dipoles k and j, and The self-energy matrix of an ensemble of dipole emitters interacting with a single spherical metal particle was described by Pustovit and Shahbazyan, and an analytical solution of the green tensors was derived under long-wavelength approximation. 10 There, the phase of the oscillators was fixed while the amplitudes were modified due to the common field; as described by the self-energy matrix of the ensemble.