The Quantum Vacuum of Complex Media. A Unified Approach to the Dielectric Constant, the Spontaneous Emission and the Zero-Temperature Electromagnetic Pressure
aa r X i v : . [ qu a n t - ph ] D ec The Quantum Vacuum of Complex Media. A Unified Approachto the Dielectric Constant, the Spontaneous Emission and theZero-Temperature Electromagnetic Pressure
M. Donaire
1, 2, ∗ Centro de F´ısica do Porto, Faculdade de Ciˆencias da Universidade do Porto,Rua do Campo Alegre 687, 4169-007 Porto, Portugal. Departamento de F´ısica de la Materia Condensada,Universidad Aut´onoma de Madrid, E-28049 Madrid, Spain.
Abstract
We study from a critical perspective several quantum-electrodynamic phenomena commonly re-lated to vacuum electromagnetic (EM) fluctuations in complex media. We compute the resonance-shift, the spontaneous emission rate, the local density of states and the van-der-Waals-Casimirpressure in a dielectric medium using a microscopic diagrammatic approach. We find, in agree-ment with some recent works, that these effects cannot be attributed to variations on the energyof the EM vacuum but to variations of the dielectric self-energy. This energy is the result of theinteraction of the bare polarizability of the dielectric constituents with the EM fluctuations of an actually polarized vacuum . We have found an exact expression for the spectrum of these fluctua-tions in a statistically homogeneous dielectric. Those fluctuations turn out to be different to theones of normal radiative modes. It is the latter that carry the zero-point-energy (ZPE). Concern-ing spontaneous emission, we clarify the nature of the radiation and the origin of the so-calledlocal field factors. Essential discrepancies are found with respect to previous works. We performa detailed analysis of the phenomenon of radiative and non-radiative energy transfer. Analyticalformulae are given for the decay rate of an interstitial impurity in a Maxwell-Garnett dielectric andfor the decay rate of a substantial impurity sited in a large cavity. The construction of the effectivedielectric constant is found to be a self-consistency problem. The van-der-Waals pressure in a com-plex medium is computed in terms of variations of the dielectric self-energy at zero-temperature.An additional radiative pressure appears associated to variations of the EM vacuum energy. ∗ Electronic address: [email protected] ontents I. Introduction II. The spontaneous emission process in a dielectric medium. The role ofthe electromagnetic vacuum fluctuations
III. Spontaneous vs. Stimulated Emission W ω α αµ IV. The virtual cavity scenario γ -factors 33 V. The real-cavity scenario
VI. The nature of dipole emission and the coherent vacuum
VII. Analytical calculations in Maxwell-Garnett dielectrics
VIII. Spontaneous decay of a low-polarizability substantial impurity IX. Comparison with previous approaches
X. The zero-temperature electromagnetic pressure XI. Conclusions References . INTRODUCTION
The concepts of radiative fluctuations and virtual particles are inherent to the natureof quantum mechanical processes in the framework of Quantum Field Theory (QFT). Theactual existence of fluctuations is however questionable as the manner they enter quantumphenomena depends on the formalism employed. For instance, one can say that radiativecorrections renormalize the ’bare’ mass of a free electron as long as one can postulate theexistence of a ’bare’ electron in absence of coupling to the electromagnetic (EM) field. Like-wise, one can say that zero-point quantum fluctuations of the EM field give rise to a shifton the energy of stationary atomic states provided that the actual existence of those statescould be postulated in the absence of coupling to radiation. However, quantum fields coupleto each other and such a coupling cannot be switched off. Bare masses and stationary energystates are actually unobservable. In addition, if the electromagnetic field appears coupled toa stochastic system, eg. a thermal reservoir or a gas of (actual) charged particles randomlydistributed, additional fluctuations appear and add up to the once previously considered.The crucial difference of these with respect to the zero-point fluctuations is that they canbe modified or even switched off by tuning some external parameter. It is implicit in thispicture that the spectrum of fluctuations can be expanded as a power series in the free-space electromagnetic propagator and the coupling to actual charges, in application of theLippman-Schwinger equation. Many electromagnetic phenomena are explained appealingto the actual existence of zero-point fluctuations using quantum or semiclassical approaches[46]. That is the case of the Casimir-Polder effect, the Lamb shift, the van der Waalsforces and the spontaneous decay of excited atoms. Recently [2], it has been pointed outthat the Casimir effect as formulated in Quantum Electrodynamics (QED) is not actually amanifestation of the zero-point vacuum fluctuations of quantum fields but the result of theinteraction between actual charges and currents –see also [3].In this paper, we are concerned with all the above phenomena in the context of randommedia. We will show that they all can be addressed employing a unified formalism. Ourstudy includes the computation of the modified decay rate of an excited emitter and thecalculation of the dielectric constant of a homogeneous medium. Closely related, we willcompute the local density of states and the EM energy density of a dielectric. Van-der-Waals-Casimir forces and radiative pressure manifest as a response to virtual variations on4he total EM energy density due to changes in the spacial distribution of dipoles. To explainall these phenomena we will make use of both Classical Optics and QFT formalisms.To what Optics concerns, it is a general issue the characterization of a material systemwhich couples to radiation by means of its coherent transport properties and the study of thedecay of unstable local states. With respect to the latter, it is known since the work of Purcell[4] that the spontaneous emission rate of an atom, Γ, in a dielectric host medium depends onthe interaction of the emitter with the material environment. This is so because the mediumdetermines the spectrum of the EM fluctuations which mediate the atom self-interactionsand hence, its self-energy. In the first place the host medium modifies the density of channelsinto which the atom can radiate –i.e. the Local Density of States (LDOS)– and hence thevalue of Γ. Second, the integration of the associated self-energy gives rise to a shift in theresonance frequency of the atom. Third, the dipole-transition-amplitude gets also modified.As a result, the medium is said to renormalize the polarizability of the emitter.In addition and complementarily, the medium polarizes the EM vacuum. This reflects inthe fact that the dispersion relations that EM normal modes satisfy are determined by theeffective susceptibility of the medium. Complementarity can be seen in that LDOS and Γdepend as well on those parameters which determine the coherent transport features of themedium. That is, on the refractive index and the mean free path. As a matter of fact, bothnull transmittance and inhibition of spontaneous emission are expected to occur in photonicband gap materials [5].For practical purposes, the understanding of life-times in random media is relevant in thecontext of fluorescence biological imaging [6] and nano-antennas [7]. Also, understanding ofunconventional coherent transport properties is essential in engineering metamaterials forelectromagnetic and acoustic waves [8].To what QFT concerns, we will postulate the existence of two distinguishable EM vacuaattending to the the existence of two different spectra of fluctuations. These are, a source-less vacuum in which normal modes propagate and a self-polarization vacuum in which boththe radiative emission and the photons mediating the self-polarization of the emitter prop-agate. While the fluctuations which carry the divergent ZPE live on the former vacuum,those which gives rise to the van-der-Waals-Casimir forces live in the latter. In addition, a coherent-vacuum in which coherent emission propagates will be identified.The paper is organized as follows, 5n Section II we first analyze the role of electromagnetic fluctuations in the paradigmaticquantum process of spontaneous dipole emission. That section serves also to describe thefeatures of the framework in which our approach fits.Next, we develop our approach in several steps. It is based on a microscopical diagram-matic treatment which explodes the seminal works carried out by Foldy [9], Lax [10], Frisch[11], Bullough and Hynne [12–14] and Felderhof et al. [15, 16]. Our main goal will be todevelop a general formalism suitable for treating in the same footing any emission-relatedprocess in any particular scenario within the framework of linear random media. Thus, thenext sections are organized in a manner that allows us to address systematically and sepa-rately each of the features which characterize a given scenario and a given process. Thesefeatures are,A) The nature of the emission. That is, it can be spontaneous emission from an iso-lated atom; stimulated emission by an external exciting field on a polarizable particle; acombination of both the spontaneous emission of an atom and the emission induced on thepolarizable molecule which hosts the atom. This matter is addressed in Section III.B) The nature of the embedding of the emitter in the host medium. When the emitter isitself a host particle or it resides in a host particle we talk of virtual cavity as the medium isin fact unaltered by the presence of the emitter. In any other case, the emitter sites withina real cavity of radius R . This matter is addressed in Sections IV and V.C) The nature of the host medium. For the sake of simplicity, we will restrict ourselvesto statistically homogeneous and isotropic three dimensional host media. Further classifi-cations are made attending to the relation between the wavelength of the radiation, λ , thecorrelation length between the host medium constituents, ξ , and, if it applies, the radius ofthe emitter cavity, R . In the virtual cavity scenario, ξ = R and the topology is refereed toas cermet topology as the medium is composed by disconnected point dipoles. An effectivedielectric medium can be defined as long as λ ≫ ξ . In the latter case the propagation ofcoherent radiation is fully characterized by an effective dielectric constant, ǫ eff . On thecontrary, any real cavity breaks manifestly translation invariance and no effective mediumcan be defined generally. Nevertheless, if R ≫ ξ , the cavity is said large and the medium isseen by the emitter as a continuum. Further, if λ ≫ ξ , the medium behaves as an almost-effective one w.r.t. the propagation of the coherent radiation emitted from the center of thecavity. It is in that case that the topology is that of a simply-connected-non-contractible6anifold in which the emitter is placed at the center of a cavity of radius R surrounded bya continuous effective medium. All these matters are addressed in Sections V and VI B 3.D) The nature of the radiation. In the first place, radiation can be either transverse orlongitudinal attending to its direction with respect to the propagation wave vector. Second,radiation can be coherent and incoherent. The coherent part has two contributions. Thefirst one is that emitted directly by the source dipole into the host medium. The secondone comes from the coherent interference of the latter with the radiation induced in the sur-rounding dipoles. Coherent radiation propagates through a coherent-vacuum. Incoherentradiation is that which gets dispersed or absorbed. These matters are addressed in SectionVI together with a study of the process of radiative/non-radiative energy transfer.Equipped with the above detailed analysis, we address explicit computations which can beperformed analytically. In Section VII we concentrate on the virtual cavity scenario. First,in Section VII A we compute the spontaneous decay rate of a weakly-polarizable intersti-tial impurity within a Maxwell-Garnett dielectric. In Section VII B we compute formallythe dielectric constant of a Maxwell-Garnett atomic gas for the case that the single-atom-polarizabilities are known in free space and non-radiative effects can be ignored. We analyzeold experiments and propose suitable modifications in order to test our approach.We address the real cavity scenario in Section VIII. There, we compute the spontaneousdecay rate of a weakly-polarizable substantial impurity placed in a large real cavity.In Section IX we compare our results with previous ones. Several arguments are givento explain why those fail. In particular, we concentrate on explaining why of the erroneoususe of the so-called local field factors.Finally, we explain in Section X the computation of the zero-temperature EM pressurein a gas. Both a contribution coming from variations on the matter self-energy and anotherone coming from normal mode fluctuations are found. Only the former is identified with thevan-der-Waals pressure in agreement with recent interpretations of van-der-Waals-Casimirforces.We summarize our results in the Conclusions, Section XI.7 I. THE SPONTANEOUS EMISSION PROCESS IN A DIELECTRIC MEDIUM.THE ROLE OF THE ELECTROMAGNETIC VACUUM FLUCTUATIONS
This section has a double purpose. On the one hand, it offers a comprehensive overviewof the features which characterize the physical scenarios to which the approach developed inthe next sections applies. We will clarify to which extend our approach is really microscopical and to which extend quantization is taken into account. On the other hand, it describesqualitatively some of the subtle points related to the role of the EM vacuum fluctuations. Tothese aims, we analyze qualitatively the phenomenon of spontaneous emission which, sincethe pioneering work by Einstein [17], has been considered paradigmatic in the understandingof the EM quantum vacuum. Elements of both Quantum Field Theory and Classical Opticswill be used. There is little new in this section and our arguments base on well-known resultsof standard textbooks –eg. [18]. Nevertheless, we feel it is necessary in order to appreciatethe subtle points of the next sections. Reference will be made, with no further explanationat this stage, to concepts and results which will appear later.
A. Fermi’s golden rule
The formula for Γ is given by Fermi’s golden rule. According to it, in second orderof perturbation theory an excited state of matter localized at ~r decays via electric dipoletransition with a rate given byΓ = 2 πǫ ~ X I,γ |h A | ˆ H int ( ~r, t ) | I, γ ω IA ih I, γ ω IA | ˆ H int ( ~r, t ) | A i| δ (cid:16) ω IA − ( ω A − ω I ) (cid:17) , (1)Γ ≃ πǫ ~ X I,γ IA |h A | ˆ ~µ · ˆ ~E ( ~r, t ) | I, γ ω IA ih I, γ ω IA | ˆ ~µ · ˆ ~E ( ~r, t ) | A i| δ (cid:16) ω IA − ( ω A − ω I ) (cid:17) , (2)where the dipole approximation has been taken in Eq.(2). | A i is the initial state in whichthe emitter is in an excited atomic stationary state of energy ~ ω A with wave function ψ A ( x )and the electromagnetic field is in its ground state in which there are no actual photons. {| I, γ ω IA i} is the set of intermediate states in which the emitter is in a lower energy stationarystate of wave function ψ I ( x ) after the release of a quantum of EM energy, which equals ~ ω IA = ~ ( ω A − ω I ) according to the delta function of energy conservation. In free space, | γ ω IA i is a transverse photon. ˆ ~µ is the transition dipole operator and ˆ ~E ( ~r, t ) is the electricfield operator in the Schr¨odinger picture. Following the introduction by Barnett et al. in819], by Fourier-transforming the time dependence of ˆ ~E ( ~r, t ) into ˆ ~E ω ( ~r ), decomposing thelatter in terms of creation/annihilation operators and using the completeness of the spaceof EM states, we can eliminate the time dependence in Eq.(2) in favor of ω IA and arrive atΓ = 2 πǫ ~ X I | ~µ AI | s.p. h Ω | ˆ ~E ω IA ( ~r ) · ˆ ~E † ω IA ( ~r ) | Ω i s.p. , (3)where † denotes hermitian conjugate and | Ω i s.p. is the self-polarization vacuum state whichis left for a more precise definition. From now on we will simplify matters assuming theemitter is a two-level atom –with state labels A, B – with a unique transition dipole matrixelement ~µ AB = h A | ˆ ~µ | B i = R d x ψ A ( ~x ) ~µ ( ~x ) ψ ∗ B ( ~x ). It is important to bear in mind that thestationary atomic states so far considered are bound states of actual charges. Those are,the atomic nuclei and the atomic electrons. Therefore, they are eigenstates of some effectiveHamiltonian which already contain electromagnetic interactions as a result of the partialintegration of ’virtual’ photons. No actual photons (i.e. propagating photons) are involvedin those atomic states and the interaction integrated is essentially electrostatic. Hence, theresultant ~µ AB is directly related to the so-called electrostatic polarizability –see Section III B.Therefore, once knowing µ AB the problem becomes one of computing the radiative vacuumfluctuations of energy ~ ω AB which couple to the otherwise stationary dipole.Applying next the fluctuation-dissipation theorem [20] we can express the electric fieldfluctuations of frequency ω IA in terms of the imaginary part of a Green’s tensor, s.p. h Ω | ˆ ~E ω AB ( ~r ) ˆ ~ † E ω AB ( ~r ′ ) | Ω i s.p. = − ~ ω AB ǫ c π ℑ{ ¯ G ( ~r, ~r ′ ; ω AB ) } . (4)¯ G ( ~r, ~r ; ω AB ) is the Green’s tensor which propagates the electric field from the point dipolesource which oscillates with frequency ω AB at ~r back to itself. That field mediates the in-teraction of the dipole with itself. That is, it is the self-polarization field which propagatesin | Ω i s.p. . Alternatively, ¯ G is interpreted as the propagator of virtual photons which areemitted and reabsorbed at the dipole source.Up to know, we have followed a quite standard procedure. Alternatively, other micro-scopical approaches base on generalized optical Bloch equations in the context of second-quantization formalism [21–23]. Throughout this paper we will stick to a Green’s functionbased formalism. 9 IG. 1: ( a ) Feynman rules of the QED field theory. The wavy line denotes the propagator of trans-verse ( A ⊥ ) and longitudinal ( A k ) photons. The thick dashed line denotes the propagator of chargedparticles. On the right hand side, tree level interaction diagram from Dirac’s lagrangian [29]. ( b )One-loop mass renormalization diagram for charged particles. ( c ) One-loop vertex renormalizationdiagram. ( d ) One-loop polarized photon propagator. B. Zero-point and in-free-space vacuum fluctuations
We give here a non-rigorous overview on the derivation of Eqs.(1-3) in free space in termsof the interaction between actual charges and EM fluctuations.At zero temperature, with no actual charges at all and disregarding virtual ones, photonsare non-interacting species. They are radiative and hence transverse. Virtual photons aredepicted as closed wavy lines which stand for frequency modes of the propagator of ~A ⊥ with origin and end at the same point. The ’bare’ EM vacuum of sourceless modes at zerotemperature will be denoted by | i . As noted in [24], the propagator of ~A ω equals that for ~E ω modulo a prefactor ( ω/c ) − . Hence, ~E = ∂∂t ~A . In application of the fluctuation-dissipationtheorem we can write, h | ˆ ~E ω ⊥ ( ~r ) ˆ ~E ω †⊥ ( ~r ) | i = ω c h | ˆ ~A ω ⊥ ( ~r ) ˆ ~A ω †⊥ ( ~r ) | i and (5)10 | ˆ ~E ( ~r ) ˆ ~E † ( ~r ) | i = Z h | ˆ ~E ω ⊥ ( ~r ) ˆ ~E ω †⊥ ( ~r ) | i d ω = − ~ ǫ πc Z ω ℑ{ ¯ G (0) ⊥ ( ~r, ~r, ω ) } d ω = ~ ǫ π c ¯ I Z d ω ω . (6)Only sourceless fluctuations satisfying the transversality condition ∇ · ~A = 0 contribute inthe above equations. Thus, the Coulomb gauge ∇ · ~A = 0 is a natural choice in this context.Longitudinal photons cannot propagate energy as such propagation needs of charged mattersupport. The Green’s function in Eq.(6) is that which propagates the electric field of anisolated oscillating stationary dipole, h ~ ∇ × ~ ∇ × − ω c I i ¯ G (0) ( ~r, ~r ′ , ω ) = − δ (3) ( ~r − ~r ′ ) . (7)The left hand side of Eq.(7) is also the Maxwell’s equation for the ω -mode of the electricfield in free space. Although the real part of ¯ G (0) ( ~r, ~r ′ , ω ) diverges for ~r → ~r ′ , its imaginarypart yields the desired result in Eq.(6) [13].Further, ℑ{ ¯ G (0) ⊥ ( ~r, ~r, ω ) } relates to the LDOS of the electric field which propagate in thebare vacuum through [25, 26]LDOS ω ] E = − ω πc Tr n ℑ{ ¯ G (0) ⊥ ( ~r, ~r, ω ) } o . (8)Same expression holds for the LDOS of the electromagnetic field ~A ω , LDOS ω [24]. It followsthat the spectrum of fluctuations of ~E ω and LDOS ω are proportional,LDOS ω = ǫ c ~ ω Tr {h | ˆ ~E ω ⊥ ( ~r ) ˆ ~E ω †⊥ ( ~r ) | i} , (9)and the total EM energy of the bare vacuum per unit volume reads E = c − Z ~ ω LDOS ω d ω = ǫ Z Tr {h | ˆ ~E ω ⊥ ( ~r ) ˆ ~E ω †⊥ ( ~r ) | i} d ω = ~ π c Z d ω ω . (10)The above integral has no cut-off and diverges as ∼ ω . That is the so-called zero-point-energy (ZPE) and the corresponding fluctuations are referred to as zero-point vacuum fluc-tuations.When the electromagnetic interaction turns on (i.e. e = 0), the theory is that of QED.The Feynman rules are the tree-level diagrams in Fig.1( a ). Loop corrections give rise to11ermion mass renormalization, interaction vertex renormalization and electric charge renor-malization – Figs.1( b, c, d ) respectively. The latter is also responsible for the (virtual) polar-ization of the EM vacuum due to charged-field fluctuations. In Fig.2 we depict a number ofdiagrams contributing to the ZPE of QED. The diagrams in Fig.2( a ) and Fig.2( d ) refer tothe zero-point fluctuations of the non-interacting fields. In the rest, vacuum fluctuations ofboth fields combine. Diagrams ( b ) and ( c ) refer to fluctuations in a polarized EM vacuum.The fermionic loop of Fig.1( d ) enters the diagrams of the upper row as a one-loop polariza-tion function. The pairs of diagrams ( b ) and ( e ), ( c ) and ( f ) are actually identical. Thatis, from the point of view of the EM field, fermionic loops polarize the EM vacuum. Fromthe point of view of charged fields, radiative corrections renormalize their charge. In anycase, the inclusion of these fluctuations does not make any better the convergence of theZPE. In addition, at finite temperature, the spectrum of thermal fluctuations is Planck’sand additional photon interaction vertices show up –see [27].Next, let us introduce actual charges. Let us consider an isolated atomic dipole source FIG. 2: QED vacuum diagrams contributing to the ZPE. Diagrams ( a ) and (d) refer to the non-interacting theory. The diagrams ( b ) and ( e ), ( c ) and ( f ) are topologically identical. They giverise to a polarization of the EM vacuum as seen by photons (upper row) or to a renormalizationof the electric charge e as seen by charged fields (lower row). made of a positively charged heavy nucleus and a number of negatively charged electrons in12otion around it. In this scenario longitudinal photons are also relevant for they mediate theelectrostatic interaction between charges. The primitive stationary atomic states incorporateactual charges and longitudinal photons. They are eigenstates of the Hamiltonianˆ H = Z X i =1 m e | ˆ ~p i | + X i>j ˜ e \ | ~r i − ~r j | + Z X i =1 ˜ eQ c | ~r i | , (11)where Z is the atomic number; Q is the total charge of the nucleus placed at the center ofthe atom; the subscripts i, j label each electron with renormalized mass and charge m e and˜ e respectively; and ˆ ~p i ( x i , t ) is the ordinary linear momentum operator of the i th electron.The radiative interactions which are not yet integrated in stationary states (i.e. transversephotons) enter the (non-relativistic) Hamiltonian of interaction [18],ˆ H int = X i − ˜ em e c ˆ ~A ⊥ ( r i , t ) · ˆ ~p i ( r i , t ) + ˜ e m e c ˆ ~A ⊥ ( r i , t ) · ˆ ~A ⊥ ( r i , t ) , (12)where the magnetic interaction due to spin coupling has been neglected. The dominantinteraction is given by the first term on the r.h.s. At leading order in ˜ e , the representationof the self-energy of state A is the second diagram on the l.h.s. of the approx. symbol inFig.5( d ). The dotted line between the points of emission and reabsorbtion there depictsthe virtual transition state B. For typical frequency transition energies, the wavelength ofthe emitted photons is much longer than the radius of the atoms. In such a case, the firstterm in Eq.(12) reduces to − ˜ em e c ˆ ~A ⊥ ( x i , t ) · ˆ ~p i ( x i , t ) ≃ − ˜ e ˆ ~x · ˆ ~E ⊥ ( x i , t ) = − ˆ ~µ i · ˆ ~E ⊥ ( x i , t ) , (13)where ˆ ~µ i is the electrostatic dipole moment operator associated to the i th electron. This isthe electric dipole approximation, restricted to transverse modes in free space, that is usedin Eq.(2). This way, the propagator of virtual photons in the self-energy diagrams can bereduced to the electric field propagator. That propagator is also the scattering amplitude,computed at second-order of perturbation theory in QED, of the process through which anexcited two-level point dipole transfers a quantum of energy (not necessarily by means ofradiation) to a non-excited dipole. One can verify for instance in [28] that such a scatteringamplitude equals the Green’s function of Eq.(7),¯ G (0) ( R, ω AB ) = [ ~µ AB ~µ BA ] − X γ ~q,κ h h A, B | ⊗ h | ˆ H int | γ ~q,κ i ⊗ | A, A ih A, A | ⊗ h γ ~q,κ | ˆ H int | i ⊗ | A, B i ~ ω AB − ~ cq + h A, B | ⊗ h | ˆ H int | γ ~q,κ i ⊗ | B, B ih B, B | ⊗ h γ ~q,κ | ˆ H int | i ⊗ | A, B i− ~ ω AB − ~ cq i , (14)13here ˆ H int is given by Eq.(13), R is the distance between the dipoles and the sum runs overall possible values of the momentum ~q and polarization states κ of the only photon in theintermediate states, | γ ~q,κ i ⊗ | A, A i and | γ ~q,κ i ⊗ | B, B i .In application of the fluctuation-dissipation theorem in-free-space we can writeΓ = 2 πǫ ~ | ~µ AB | s.p. D h Ω | ˆ ~E ω AB ⊥ ( ~r ) · ˆ ~E † ω AB ⊥ ( ~r ′ ) | Ω i s.p. D = − ω AB ǫ ~ c | µ AB | ℑ n Tr − { ¯ G (0) ⊥ ( ~r, ~r ′ ; ω AB ) } o , | ~r − ~r ′ | ≪ a, where | Ω i s.p. D stands for the EM self-polarization vacuum seen from the unique dipole in-free-space. Because ¯ G (0) ⊥ ( ~r, ~r ′ ; ω AB ) above equals that in Eq.(7) for photons propagating throughthe bare zero-point vacuum, we conclude that the spectrum of fluctuations of | Ω i s.p. D formspart of the spectrum of fluctuations of | i .In a sense, it can be interpreted that zero-point EM fluctuations interact with the dipoleto make it emit a photon. However, there is no ’borrowing’ of vacuum energy as suggestedin the literature (eg. [28]). The zero-point EM fluctuations and hence the associated ZPEremain as depicted in Figs.5( c , d ). Polarized fluctuations add up on top of those whichyield the ZPE. On the one hand, fluctuations of the self-polarization vacuum ’interact’ withthe actual dipole making it fluctuate and acquire an additional self-energy. Reciprocally, thedipole randomly localized yields an additional effective vertex of interaction which enterspolarizing the EM fluctuations of the sourceless vacuum | Ω i s.l. D . In Fourier space, the vertexread − ( ω/c ) V − α , where α is the electrostatic bare polarizability of the dipole and V thetotal volume of the sample (eventually infinite). Its diagrammatic representation is that inFig.3( a ). The reason why we depict the bare dipole there as a closed loop is that it is abound state. Momentum and energy can only leak out of it through the coupling to EMmodes. It behaves w.r.t. the EM fluctuations in | Ω i s.l. D as one of the loops of fermionicfluctuations. In the process of joining the external fermionic legs of the QED diagrams ofFig.3( a ) in order to get an α loop there is an implicit integration of high frequency degreesof freedom. Those are, all the electrostatic interactions which yield the stationary atomicstate plus high frequency transverse modes. This implies that there is a frequency cut-offin all the processes in which α plays the role of an effective vertex. If the radius of thedipole is a , the cut-off must be of the order of c/a to preserve consistency with the dipoleapproximation. An analogous reasoning may apply to restrict the interval of frequencieswhich are integrated in the Casimir energy of a system formed by two perfectly conducting14arallel plates. In that case, the restriction is imposed to preserve consistency with theperfect conductivity approximation [1, 30].So far, we have distinguished between the EM vacuum seen by sourceless photons (i.e. FIG. 3: ( a ) Formal integration of electrostatic and rest of high frequency modes in the stationaryatomic state A and bare electrostatic polarizability α . ( b ) Effective interaction vertex in thedipole approximation after integration of high frequency modes. ( c ) Vacuum polarization diagramin presence of a unique dipole. normal EM modes), | Ω i s.l. D , and that seen by the photons whose source is the dipole, | Ω i s.p. D .This distinction is based on the double role played by loops of actual charged-fields andloops of virtual photons. Those are, the renormalization of the bare polarizability and thepolarization of the EM vacuum. The same as shown in Fig.2 for the ZP QED vacuum, botheffects are complementary. The EM vacuum seen from the emitter was identified above,at least partially, with the bare zero-point vacuum. That is, the radiation emitted by thedipole sees empty space in its way. Likewise, self-polarization photons see empty space fromand back to the dipole. However, the restrictions ω < c/a must apply in order to considerthe emitter as a point dipole. Hence, the equivalence ~r = ~r ′ implicit in Eqs.(5-10) has beensubstituted by the limit | ~r − ~r ′ | ≪ a in Eq.(15) to set explicitly the limitation in spatialresolution. Therefore we will write | Ω i s.p. D = | i| ω LDOS sourceless D . The two terms on the r.h.s. correspondto the integrands of the two terms on the r.h.s. of Eq.(16) respectively. The geometrical series ofthe second term stand for the states multiply-polarized by the presence of the unique dipole. counted for in the second term on the r.h.s of Eq.(16).At leading order, the counterpart of the addition of the diagrams of Figs.2( a ) , ( b ) of thevirtually polarized vacuum is the sum on the r.h.s of the approx. symbol in Fig.5( d ) ofthe actually polarized vacuum. Likewise, the counterpart of the addition of the diagramsin Figs.2( d ) , ( e ) for the renormalization of the fine structure constant, is the sum of the16 IG. 5: Interpretation of the ’recombination’ process between a point dipole and zero-point EMvacuum fluctuations. ( a ) Zero-point vacuum fluctuation loop of the electromagnetic field. Theequivalence in Eq.(5) has been depicted. ( b ) Feynman diagrams contributing to a stationary atomicstate and its equivalence after integration of high frequency modes as in Fig.3. ( c ) Microscopicrepresentation of the recombination process leading to polarized EM vacuum fluctuations in thepresence of an actual dipole. ( c ) Microscopic representation of the recombination process leadingto an actual fluctuating dipole. Equivalent interpretation: renormalization of the dipole polariz-ability through radiative corrections. In ( d , ), same phenomena as in ( d , ) using the effectivenomenclature of Fig3. diagrams on the r.h.s of the approx. symbol in Fig.5( d ) for the renormalization of thepolarizability. However, the equivalence between the last diagram on Fig.5( d ) and the lastdiagram on Fig.5( d ) does not hold because the ’fermionic loop’ in Figs.2( d ) , ( e ) turn intoan actual dipole in Fig.5. Although both sourceless and sourced photons see statisticallyhomogeneous vacua, | Ω i s.p. is attached to actual dipoles while | Ω i s.l. is translation invariant.17s a matter of fact, the series in Fig.5( d ) stands for the self-energy of the actual dipole.We next overview the renormalization of the polarizability in-free-space. As mentionedabove, the imaginary part of ¯ G (0) ( ~r, ~r ) yields the right result consistent with the quantumradiative corrections which dress up the electrostatic polarizability [13]. However, it containsboth longitudinal and real transverse parts which diverge. The phenomenological regular-ization scheme that we use in Section III B follows that of [35]. It allows us to attributea physical interpretation to the divergences. The longitudinal divergence is associated tothe longitudinal modes which are integrated out in the original stationary state. Therefore,it relates to µ AB and ω AB . The associated electrostatic polarizability α is proportional to µ AB in a two-level atom. The real transverse divergence is interpreted in terms of a shiftin the transition energy with respect to the stationary original state. That gives rise to astatic shifted polarizability, α stat. ( ω ) = α ω res ω res − ω , α being real. The shift in energy is theanalog to the Lamb shift in the Hydrogen atom. That scheme of renormalization yields theLorentzian-type polarizability in free space α ( ω ) ≈ α ω res ω res − ω + i Γ . There have been proposedhowever several schemes to accomplish the computation of α in-free-space for idealized twoand three-level atoms. We refer to the reader to the works of Berman, Boyd and Milonni [31]for a computation based on Heisenberg and Schr¨odinger’s pictures; to that of Barnett andLoudon [32] for a Green function approach; and to that of Bialynicki-Birula and Sowinski foran approach based on Feynman diagrams [33]. Even in the latter work, the authors beginwith a priori modeled electronic bound states and perform a dimensional reduction fromthree to zero spatial dimensions. It has been pointed out in [31] that a Lorentzian profileis an approximation not fully justified and it has been found in [34] that other more com-plicated form can be more suitable. We will not attempt here to analyze those approachesand we will stick to the phenomenological Lorentzian profile. Also, we will adapt in SectionIII a phenomenological treatment appealing to the existence of a classical effective dielec-tric constant within the scatterers. That procedure is in all equivalent to the actual QEDtreatment once divergences in ¯ G (0) ( ~r, ~r, ω ) are conveniently regularized. C. In-random-medium vacuum fluctuations In a random medium, on top of the in-free-space fluctuations, additional fluctuations areinduced by the multiple scattering of light with the host scatterers and the stochasticity of18 IG. 6: Diagrammatical representation of the geometrical series which give rise to the radiativerenormalization of the electrostatic polarizability of a two-level atom in free space. In ( a, b ) it iswritten as a succession of coupled virtual states. In ( c ), the effective rules in Fig.3 are used. In( b, c ), the dipole approximation is implicit. the medium itself. The additional fluctuations affect both the polarization of the sourcelessEM vacuum and the self-energy of matter.The ondulatory nature of light implies that fluctuations appear as considering the in-terference between all the different paths that a virtual photon can follow in its way. Theparticularity of a random medium with respect to the free space is that the different pathsare not statistically equivalent as a result of the spatial correlations between host scatterers.Therefore, each path has a statistical weight and fluctuations are affected by the randomnessof the ensemble of scatterer configurations. In turn, this gives rise to a stochastic equationwhose Green function incorporates the EM fluctuations we seek for. Purely thermal fluc-tuations of the electromagnetic field will not be considered in this paper. However, it canbe intended that the fluctuations induced by spatial correlations have an indirect thermalorigin. That is, it is the classical degrees of freedom which characterize the configurationof host scatterers, vector positions, velocities and angular momenta, that obey a thermaldistribution. In our approach, atomic orbitals do not overlap and the separation between theemitter and the host scatterers and the separation between host scatterers themselves aredetermined by the range and strength of the cohesive forces –but for the dipole interaction-as well as the temperature. In fluids such atomic gases, one can think of that force as the19hort-ranged repulsive force between the external electrons of the atoms. In experiments,the range of this force can be tuned by ionazing the atoms or molecules [71]. For practicaluses, throughout this paper we will consider an effective rigid exclusion volume around boththe emitter and the host scatterers. In the construction of the dielectric constant in SectionVII B we will only assume knowledge of the in-free-space polarizabilities of the atomic spacesconsidered as isolated. In crystals and tight-binding models in general, atomic orbitals nat-urally overlap and the electronic band structure of the system is completely different tothat of the atomic species. The assignment of a ’new’ single polarizability to each atomicsite might be still possible provided that the only remaining interaction between individualatoms is effectively dipole-like. Nevertheless, for the case that the emitter is a distinguish-able dipole, the only relevant information for studying the influence of the medium in itsdecay rate is that of the spatial embedding of the emitter into the medium provided themedium is correctly described by an effective dielectric constant –see Section IV B.We summarize here the basis of our approach. In the process of dressing up the elec-trostatic polarizability α , all the virtual photons running inside each dipole are assumedintegrated out. The interaction between atoms is dipole-like provided atomic orbitals do notoverlap. The ’bare’ propagator of such an interaction is the one given by Eq.(14) and the’bare’ interaction vertex is given by the strength coupling − k ρα . This way, in-free-spaceradiative fluctuations and those induced by the host medium can be incorporated in the samefooting. Quantization enters at the level of atomic polarizabilities, dipole-dipole interactionand EM fluctuations. In our approach, matter fluctuations enter classically through theimplementation of the correlation functions between scatterers. These correlation functionsare affected neither by the dipole-dipole nor by the radiation-dipole interaction. Thus, thecoupling between matter and radiation is assumed weak unless specified otherwise. Explicittreatment of quantum long wavelength excitons, typical of tight-binding models, is disre-garded. The reader is referred to the pioneer works by Fano and Hopfield [36, 37] for moreconvenient approaches to highly ordered systems. As long as collective quantum excitonscan be neglected, and in the weak-coupling regime between medium and radiation, thereis no need to develop any further macroscopic quantization scheme. In particular, there isno need to postulate any additional noise polarization operator –compare with [38]. Onlyfor the case that the emitter is a substantial impurity we will assume any other fluctuationincorporated in the given dielectric function of the host medium without further explicit20omputations –see Sections V,VIII. Additional non-radiative effects, such as collisional ef-fects, can be incorporated through convenient renormalization of the vertex − k ρα .Next, let us consider formally the host medium as an stochastic configuration of dipoles,where stochasticity concerns the classical variables which determine the configuration. Inorder to treat the EM fluctuations induced by the classical matter fluctuations in the samefooting as the rest, it is necessary for them to fulfill some conditions. The Green’s func-tion which appears in the fluctuation-dissipation theorem of Eq.(4) refers to an effectiveand stationary configuration. That implies conditions over the dynamics of the ensemble ofscatterers with respect to that of the emission process and photon propagation. In the firstplace the host medium must pass through all the accessible configurations in a time scalemuch less than Γ − . That is, the relaxation time of the ensemble, τ , satisfies τ ≪ Γ − . Thisguarantees that the degrees of freedom of the host medium are the fast variables which canbe integrated out. Second, stationarity demands that τ ≫ ω − . In addition, it is implicit inthe derivation of Fermi’s Golden rule that Γ − ≫ ω − , which is a weak-coupling conditionon the coupling radiation-emitter. That way, the spontaneous emission process in such arandom medium adjusts to the Born-Markov approximation [39]. We can also appeal tothe ergodic character of the electric field so that ensemble averages are equivalent to timeaverages [40, 41].We follow next a similar treatment to that of [41] in the introduction of the matrix den-sity operator. | Ω i s.p. is not | i| ω As mentioned in Subsection II B, it is only in free-space that ¯ G ( ~r, ~r ; ω ) equals the propaga-tor of Maxwell’s equation for the electric field in free space, ¯ G (0) ( ~r, ~r ; ω ). ¯ G (0) ( ~r ; ω ) consistsof an electrostatic (Coulombian) dipole field propagator,¯ G (0) stat. ( r ) = h k ~ ∇ ⊗ ~ ∇ i(cid:16) − π r (cid:17) (24)plus a radiation field propagator,¯ G (0) rad. ( r ) = e i k r − πr I + (cid:2) k ~ ∇ ⊗ ~ ∇ (cid:3) e i k r − − πr , (25)with k = ω/c . In reciprocal space and for isotropic systems, any tensor can be decomposedinto a longitudinal and a transverse part with respect to the propagation direction of thewave vector ~k . That is, ¯ G ( ~k ) = G ⊥ ( k )(¯ I − ˆ k ⊗ ˆ k ) + G k ( k )ˆ k ⊗ ˆ k , where ˆ k is the unitary vectorparallel to ~k . In free space,¯ G (0) ( ~k ) = Z d r e i~k · ~r ¯ G (0) ( r ) = Z d r e i~k · ~r h ¯ G (0) rad. ( r ) · (¯ I − ˆ k ⊗ ˆ k ) + ¯ G (0) stat. ( r ) · ˆ k ⊗ ˆ k i = G (0) ⊥ ( k )(¯ I − ˆ k ⊗ ˆ k ) + G (0) k ( k )ˆ k ⊗ ˆ k, (26)with G (0) ⊥ ( k ) = 1 k − k , G (0) k ( k ) = 1 k . (27)While the radiative component is fully transverse, the electrostatic one is fully longitudinal.In most of this work we will deal with the Fourier transform of ¯ G (0) . The reason being thatit is easier to keep track of the longitudinal and transverse contributions to the emission bydirect computation of¯ G ω ( ~r, ~r ) = 13 Tr − { ¯ G ω ( ~r, ~r ) } ¯ I = 13 Tr − nZ d k (2 π ) ¯ G ( ~k ) o ¯ I = 13 hZ d k (2 π ) G ⊥ ( k ) + Z d k (2 π ) G k ( k ) i ¯ I , (28)23here the factor 2 in front of G ⊥ ( k ) comes from Tr − { ¯ I − ˆ k ⊗ ˆ k } and stands for the twotransverse modes.In a generic medium, there does not exist direct identification between radiative andtransverse emission and neither does it between electrostatic and longitudinal emission, butfor in free space. In free space the whole radiation consists of transverse modes which are allcoherent propagating modes as they satisfy the free-space (on-shell) dispersion relation k = k . That is, they are poles of G (0) ⊥ ( k ). Because G (0) k does not contain poles and longitudinalmodes do not couple to transverse radiative ones in free space, the above identificationsare possible. On the contrary, the presence of a polarizable environment gives rise to non-coherent radiation as well as absorbtion which contain both longitudinal and transversemodes. Unless a very specific configuration of scatterers holds –eg. a dipole chain [42]– thecoherent propagation is entirely transverse and corresponds to the far-field coherent signalwhich would be received by a distant antenna in the medium. The incoherent radiation is thenoise signal received by that antenna. It has its origin in the coupling of ’bare’ radiative-transverse modes to ’bare’ electrostatic-longitudinal modes and contains both transverseand longitudinal effective modes. The coupling takes place both at the host dipoles andthe cavity surface. Note also that, differently to the self-polarization field in free space thatappears in Eq.(15) which is purely transverse, the one in Eq.(20) and thereafter containsboth transverse and longitudinal modes. The reason being that while all the longitudinalmodes have been integrated out in the bare polarizability of the unique dipole in free space,still long wavelength longitudinal modes mediate the interaction (i.e. the induction) betweendistant dipoles in a random medium. Nevertheless, the fact that the emission be longitudinaldoes not imply that it is not radiative as erroneously interpreted in some works.In free space, ℑ n Tr − { ¯ G ω ( ~r, ~r ) } o | free = ℑ n Tr − { ¯ G (0) rad. ( ~r, ~r ) } o = 2 ℑ nZ d k (2 π ) k − k o = − k / π, (29)the emission is entirely radiative and, in application of Eq.(15), we can write Γ = ω AB πǫ ~ c | µ AB | , where the 0 script refers to the use of in-free-space parameters. The real partsof both γ (0) ⊥ ≡ R d k (2 π ) G (0) ⊥ ( k ) and γ (0) k ≡ R d k (2 π ) G k ( k ) diverge and are needed of regulariza-tion. Because they are the zero-order terms of the perturbative expansion of Tr { ¯ G ω ( ~r, ~r ) } ,24e can write Tr { ¯ G ω ( ~r, ~r ) } = Z d k (2 π ) G ⊥ ( k ) + Z d k (2 π ) G k ( k ) (30) ≡ γ T ot. ⊥ + γ T ot. k = [2 γ (0) ⊥ + γ (0) k ] + 2 γ ⊥ + γ k , (31)where 2 γ ⊥ and γ k are the divergenceless pieces. Physically, the γ -factors account for thedipole self-energy associated to intermediate EM states of frequency ω . The self-polarizationfield propagated with ¯ G ω is also referred to as local field . The imaginary part of the γ -factorsrelates to LDOS emissionω throughLDOS emissionω = − ω πc ℑ{ γ (0) ⊥ + γ ⊥ ) + γ k } . (32)It is also possible to relate Γ with the power emitted in the decay process. That is,because such power is due to the self-interaction of the dipole with its own field, we canwrite W µ = ω res ℑ{ ~µ · ~E ∗ exc } = ω res c ǫ ℑ{ ~µ · ¯ G ∗ ω res ( ~r, ~r ) · ~µ ∗ } = − ω res c ǫ | µ | ℑ n Tr { ¯ G ω res ( ~r, ~r ) } o , (33)where in the second equality the exciting field ~E ∗ exc has been identified with the self-polarization field, ω res c ǫ ¯ G ∗ ω res ( ~r, ~r ) · ~µ ∗ . The values of ω res and ~µ are those of the resonancefrequency and the transition dipole in the medium (which may differ from the in-free-spacevalues, ω AB , ~µ AB ). Comparing Eq.(23) with Eq.(33) we can write Γ µ = ω res ~ W µ . III. SPONTANEOUS VS. STIMULATED EMISSIONA. Stimulated emission, W ω Let us considered first that the emitter is a dipole stimulated by an stationary externalfield. The exciting field, ~E exc ( ~r ) = ~E ω ( ~r ), oscillates in time with frequency ω far from anyresonance frequency of the dipole. In this case, the processes of absorbtion and emissionof radiation by the emitter become stationary after a relaxation time of the order of Γ − ,which is supposed much greater than the relaxation time of the host medium. Therefore,the process through which the absorbtion and emission become stationary is Markovianwith respect to the dynamics of the host medium. The stochastic computation of G is so25ustified as in the spontaneous emission process. In order to regularize the divergences in¯ G (0) we consider the dipole as a spherical scatterer of radius a and relative dielectric constant ǫ e . This model implies a classical regularization scheme. Again, the dipole approximationrequires a ≪ k − , with k = ω/c , λ = 2 π/k . Formally, the emitted power reads, W ω = ω ℑ{ Z d r Θ( r − a ) ~p ω ( ~r ) · ~E ω ∗ ( ~r ) } , (34)where ~p ω ( ~r ) is the density of dipole moment induced, which is proportional to ~E ω ( ~r ) inour linear and small particle approximation and is affected by self-polarization effects. Weintroduce the self-polarizing field through the insertion of appropriate Green’s functions inthe above expression, ~p ω ( ~r ) = Z d r ′′ Θ( r − a ) ǫ χ e Z d r ′ ¯ G ω ( ~r, ~r ′ ) · [ ¯ G (0) ] − ( ~r ′ , ~r ′′ ) · ~E ω ( ~r ′′ ) , (35) W ω = ω ℑ nZ d r χ e Θ( r − a ) Z d r ′ d r ′′ ¯ G ω ( ~r, ~r ′ ) · [ ¯ G (0) ] − ( ~r ′ , ~r ′′ ) · ~E ω ( ~r ′′ ) · ~E ω ∗ ( ~r ) o . (36)In these expressions, χ e = ( ǫ e − 1) is the relative electrostatic susceptibility of the emitter–not to be confused with the susceptibility of the random medium– and¯ G ω ( ~r ) ≈ ¯ G (0) ( ~r ) ∞ X m =0 h − k χ e Z Θ( v − a ) ¯ G ω ( v )d v i m (37)is the propagator which makes account of the infinite number of self-polarization cycles whichgive rise to radiative corrections. ¯ G ω ( ~r − ~r ′ ) propagates virtual photons from a point ~r ′ insidethe emitter back to another point ~r also within the emitter. All the equations above becomesimple in the small particle limit, a ≪ k − , for the electric field is nearly uniform within theemitter and so are the density of dipole moment and the propagator ¯ G ω ( ~r, ~r ′ ). The n -pointsirreducible diagrams which enter the computation of ¯ G ω ( ~r ) can be approximated by theseries of Fig.7( b ) in which the two-point correlation functions Θ( r − a ) appear consecutivelyas factors of a product. That way the corresponding integrals appear disentangled and thecorresponding series becomes geometrical. The underlying approximation is R d r Θ( r − a ) ¯ G ω ( r ) ≃ π a ¯ G ω (0) = π a h γ (0) ⊥ + γ (0) k + 2 γ ⊥ + γ k i ¯ I . We already mentioned that 2 γ (0) ⊥ and γ (0) k are divergent. Those divergences are cured in our classical model by the presenceof the finite radius a . Since the limit of the integral Lim { R d r Θ( r − a ) ¯ G (0) stat. ( r ) } = k I as26 a → r − a ) of the integrandwhich we used to model the spherical shape of the dipole which yields the finite value k I [43]. Any other geometry would give a different numerical value. By equating that resultwith π a γ (0) k ¯ I (leaving ℜ{ γ (0) ⊥ } still free) we obtain γ (0) k = ( π a k ) − . The net effectof this regularization procedure is the dressing up of the single particle susceptibility withall the in-free-space electrostatic corrections. This procedure is depicted in Fig.7( c ). Itsquantum counterpart is the integration of the electrostatic interactions which give rise toan atomic bound state in a two-level atom. That way we can define ˜ χ e ≡ ǫ e +2 χ e and obtainthe bare electrostatic polarizability α ≡ πa ǫ e − ǫ e +2 . We emphasize that it is the electrostaticpolarizability α which really has physical meaning regardless of the regularization schemeapplied in its computation. We refer to section III of [35] for a comprehensive summaryof regularization methods. With the above definitions we can rewrite Eq.(36) in terms of FIG. 7: ( a ) Feynman’s rules for the classical regularization scheme of Section III A. ( b ) Dia-grammatic representation of Eq.(37). ( c ) Diagrammatic representation of the dressing up of χ e leading to α . Approximation symbols denote that the field within the emitter is taken uniform.( d ) Diagrammatic representation of a the action of a self-polarization cycle on the electrostaticpolarizability. electrostatically renormalized operators, W ω = ωǫ ℑ nZ d r ˜ χ e Θ( r − a ) Z d r ′ d r ′′ ¯˜ G ω ( ~r, ~r ′ ) · [ ¯ G (0) ] − ( ~r ′ , ~r ′′ ) · ~E ω ( ~r ′′ ) · ~E ω ∗ ( ~r ) o , (38)27here ¯˜ G ω ( ~r, ~r ′ ) ≡ ¯ G (0) ( ~r, ~r ′ ) ∞ X m =0 ( − k α ) m − m (cid:16) γ (0) ⊥ + 2 γ ⊥ + γ k (cid:17) m . Note that longitudinal and transverse modes couple to each other in the series of Eq.(39)as scattering takes place off the dipole surface. This is the classic phenomenological analogto the quantum coupling of longitudinal and transverse modes of the diagrams of Fig.5( c ).Finally, in function of the γ -factors the power emitted and absorbed by the induced dipolereads W ω = ωǫ ℑ n α k α [2 γ (0) ⊥ + 2 γ ⊥ + γ k ] o | E ω | (39)= − ω ǫ c n | α | | k α [2 γ (0) ⊥ + 2 γ ⊥ + γ k ] | ℑ{ γ T ot. ⊥ + γ T ot. k } (40) − k ℑ{ α }| k α [2 γ (0) ⊥ + 2 γ ⊥ + γ k ] | o | E ω | . (41)The term in Eq.(41) corresponds to the power absorbed within the emitter. The term inEq.(40) corresponds to the power radiated into the medium. The former is non-zero onlyif ǫ e contains an imaginary part [96]. We can write Eq.(39) in terms of a renormalizedpolarizability ˜ α as W ω = ωǫ ℑ{ ˜ α ~E ω · ( ~E ω ) ∗ } = ωǫ | E ω | ℑ{ ˜ α } with˜ α = α k α γ (0) ⊥ + k α [2 γ ⊥ + γ k ] , (42)so that Eqs.(40,41) can be written also as W ω = − ω ǫ c | ˜ αE ω | h ℑ{ γ T ot. ⊥ + γ T ot. k } − k ℑ{ α }| α | i . (43) B. Relation between induced and spontaneous emission for the Lorentz dipolemodel, Γ α The expression for ˜ α still contains a divergent term in the denominator, k α γ (0) ⊥ . Whileits imaginary part is convergent and gives rise to the usual radiative corrections for an off-resonance point dipole in vacuum, α ≃ α [1 − i π k α ] − , its real part diverges and needs ofregularization. Following the regularization scheme of [35] we can take advantage of this andincorporate the divergence into a resonance in α . The procedure must be compatible with theregularization scheme used for α and it is our choice that it reproduces the phenomenological28orentzian model (L) for the polarizability of a two-level atom, α L (˜ k ) = f α k res [ k res − ˜ k − i Γ α ˜ k / ( ck res )] − , (44)where k res is the resonance wave number and f α is the ’renormalized’ value of the electrostaticpolarizability of the emitter, both evaluated within the medium. We proceed by equatingEq.(42) and Eq.(44) in absence of absorbtion within the emitter and with all the parametersevaluated in free-space – which is denoted with script 0, α [1 − i π α ˜ k + 13 α ˜ k ℜ{ γ (0) ⊥ } ] − = α [1 − ˜ k /k − i Γ ˜ k ck ] − . (45)From Eq.(45) we identify ℜ{ γ (0) ⊥ } = − k α and Γ = cα k / π . We refer to α stat = α [1 − ˜ k /k ] − as the ’shifted’ electrostatic polarizability at frequency ω = c ˜ k . By consistencywith the decay rate of a dipole with transition amplitude µ in free-space according toFermi’s Golden rule, Γ = k ǫ π ~ | µ | , we find the relation α = | µ | ǫ ~ ck . The latter canbe found in textbooks (eg. [44, 45]). It is important to emphasize that, although theparametrization of α in Eq.(44) is phenomenological and questionable on the basis of sometwo-level atom quantum models [34], the renormalization procedure after regularization ofdivergences is independent of such parametrization. Our renormalization scheme bases onthe renormalization of the photon propagator in which, after regularization, − ˜ k ρα stat entersas a point vertex. Hence, although the computed values of all the renormalized parametersdepend on the functional form attributed to the in-free-space polarizability, the local densityof states computed through Eq.(23) is independent of such arbitrary choice provided thatthe value of α stat is the correct one.In the host medium, the values of Γ, k res and f α get renormalized according to theequations, Γ α = − c f α ˜ k ℑ{ γ T ot. ⊥ + γ T ot. k }| ˜ k = k res = − Γ πk ˜ k ℑ{ γ T ot. ⊥ + γ T ot. k }| ˜ k = k res , (46)where k res is a real non-negative root of the equation(˜ k/k ) − α ˜ k ℜ{ γ ⊥ + γ k }| ˜ k = k res , (47)and f α = α ( k /k res ) . (48)29q.(47) is the Lamb-shift of the resonance frequency, which is as a result of the variationof the real part of the dipole self-energy [47]. In addition, Eqs.(46-48) together with therequirement of consistency with Fermi’s Golden rule imply that µ gets renormalized withrespect to µ . C. Combination of spontaneous and induced emission, Γ αµ Finally, consider the spontaneous emission of a point dipole like that in Eq.(22), butnow with an additional bare electrostatic polarizability α . The situation is analogous tothat of a fluorescent atom with transition dipole amplitude µ within a complex molecule.Let us assume that the value of µ is fixed so that µ gets effectively regularized only by theself-polarization cycles due to the presence of polarizable atoms within the host molecule,being α the total bare polarizability of the molecule. The net effect is that the spontaneousfield emitted by the atom in the decay process gives rise to an induced dipole moment in themolecule which modifies the decay rate. If there existed other kinds of interactions betweenthe fluorescent atom and the host particle, µ and k res might be modified by additional non-radiative effects. In the following, we will refer to the atom as emitter, to the moleculeof polarizability α as host particle and to the surrounding medium as host-medium. Thedipole moment of the system emitter-host-particle reads ~p = ~µ + ω c ˜ α ¯ G ω ( ~r, ~r ) ~µ , ˜ α being therenormalized polarizbility of the host particle. The perturbative series in Fig.8 allows us tobuild up the analytical expression for Γ αµ straight away from that of W ω . The integration ofthe electrostatic part which yields the term proportional to α I must be removed with respectto the series in Fig.8( a ) for in the present case the source is the spontaneous emitter withtransition amplitude µ . The rest of the terms remain unaltered but for the substitution ofthe induced non-radiative dipole moments ǫ α ~E ω and ǫ α ~E ∗ ω by the fixed dipole moments ~µ and ~µ ∗ at the emission and reception sites respectively,Γ αµ = 2 ǫ ~ | µ | ℑ n ( ǫ α ) − h α − i π k α + k α [2 γ ⊥ + γ k ] − α io = − ω ǫ c ~ | µ | | − i π k α + k α [2 γ ⊥ + γ k ] | h ℑ{ γ T ot. ⊥ + γ T ot. k } (49) − k ℑ{ α }| − i k π + 2 γ ⊥ + γ k | i , (50)30e recognize again the power absorbed within the host particle in the last term and thepower radiated into the medium in the remaining. Because any possible resonance of thehost particle is assumed far from that of the emitter, we have ignored the regularization of ℜ{ γ (0) ⊥ } . FIG. 8: ( a ) Diagrammatic representation of Eq.(39). ( b ) Diagrammatic representation ofEqs.(49,50). IV. THE VIRTUAL CAVITY SCENARIOA. Virtual cavity vs. real cavity The γ -factors are the traces of the solution of the Lippmann-Schwinger stochastic equa-tion for the propagator of the self-polarization field of a point source, G ( ~r, ~r ′ ), G ij ( ~r, ~r ′ ) = G (0) ij ( ~r, ~r ′ ) − k Z d r G (0) ik ( ~r, ~r ) hD ǫ km ( ~r − ~r ) − δ mj E(cid:12)(cid:12)(cid:12) ~r,~r ′ i G mj ( ~r , ~r ′ ) . (51)See [97] for an explanation about the notation. The brackets of the stochastic kernel hD ǫ km ( ~r − ~r ) − δ mj E(cid:12)(cid:12)(cid:12) ~r,~r ′ i denote average taking over all possible configurations of the sur-rounding host scatterers and both ~r and ~r ′ are inside the emitter such that | ~r ′ − ~r | < a ≪ k − .In turn, the distinction between ~r and ~r ′ is just formal as it cannot be resolved. The explicitdependence of ǫ ( ~r − ~r ) on the emitter position ~r and the restriction symbolized by | ~r,~r ′ signal the fact that ~r, ~r ′ are kept fixed inside the emitter as performing the average over allpossible configurations of host scatterers. This way, any scattering process of the virtualphotons in their way from and towards the emitter correlates to the emitter position. Incase the emitter is in all equivalent to the rest of host scatterers within the host medium,such a correlation is the same as that among the host scatterers themselves. It is in thissense that translation invariance is only virtually broken and the exclusion volume around31he dipole emitter is referred to as virtual cavity . In case the emitter is distinguishable w.r.t.any other scatterer, that correlation differs. Hence, translation invariance is actually brokenand the cavity is not virtual but a real cavity . The break of translation invariance, eithervirtually or actually, makes the computation of ¯ G ( ~r, ~r ′ ) different to that of the ordinaryDyson propagator. This is the reason why the propagator appearing in Eq.(22) for Fermi’sGolden rule is not Dyson propagator as erroneously suggested in [35, 49]. In any case, it isimportant to emphasize that a translation invariant Dyson propagator does exist for normalmodes in the virtual cavity scenario. In contrast, a strictly translation invariant propagatorfor normal modes does not exist in the real cavity scenario.This section and the next one are devoted to the computation of ¯ G and the γ -factors inboth the virtual cavity (VC) and the real cavity (RC) scenarios. Notice that not only theequivalence/inequivalence of the spatial correlation of the emitter matters in order to defineeach scenario but also the equivalence/inequivalence of the polarizability of the emitter w.r.t.that of the host scatterers. That is so because the γ -factors also depend on the polarizabilityof the emitter. In particular, the VC scenario requires that the dipole transition amplitudeof the emitter satisfies the relation, α = | µ | ǫ ~ ck . α being the electrostatic bare polarizabilityof the host particles. As pointed out in [50], the bare polarizability of a two-level excitedemitter has the opposite sign with respect to that of the same particle in its ground state[44]. More importantly, the polarizability during the decay process is undefined. Therefore, stricto sensu only in the case of induced emission with all the scatterers of the medium intheir ground state and for the case of a fixed dipole on top of a ground state host parti-cle, emission in a virtual cavity scenario holds. Hence, it is only in these scenarios that atranslation-invariant actually polarized sourceless vacuum can be defined.The case of emission from a real cavity is much more generic and its treatment, at leastformally, more complicated. There are nevertheless two situations where calculations sim-plify. These are, in the first case the excited emitter is weakly-polarizable and occupies aninterstitial position in the host medium. In the second case, the interaction emitter-hostmedium is such that the emitter expels the surrounding host particles at a distance R muchgreater than the the typical distance between scatterers.32 . Computation of the virtual-cavity γ -factors Rather than solving the stochastic integral equation Eq.(51) directly, we will computefirst the translation invariant bulk propagator ¯ G ( ~r − ~r ′ ) which obey the Dyson equation, G ij ( ~r − ~r ′ ) = G (0) ij ( ~r, ~r ′ ) − k Z d r G (0) ik ( ~r, ~r ) hD ǫ km ( ~r − ~r ) − δ mj Ei G mj ( ~r , ~r ′ ) . (52)In contrast to Eq.(51), the average has no-constraint in this case. Hence, ¯ G ( ~r − ~r ′ ) is a func-tion of ~r − ~r ′ for any two given points in the bulk. The field so propagated is the propagatingcoherent field or Dyson field, ~E D ( ~r ). That is, the field obtained by averaging point-wise thevalue of the classical Maxwell electric field ~E Max. ( ~r ) in the ensemble of spatial host scatterersconfigurations, ~E D ( ~r ) = D { ~E mMax. ( ~r ) } E ensemble –see Section VI B 1 too. The photons whichpropagate with ¯ G ( ~r, ~r ′ ) are said on-shell or normal modes as they obey certain dispersionrelations –see below. Because the set of fields { ~E mMax. ( ~r ) } are in one-to-one correspondencewith the set of scatterer configurations and since each one obeys a linear Maxwell equationwith certain dielectric function ¯ ǫ m ( r ) specific of each particular configuration, ~E D ( ~r ) obeysthe same functional equation but with an averaged dielectric function D { ¯ ǫ m ( r ) } E ensemble .Hence, Eq.(52) [11]. There exists a relation between the propagators ¯ G of Eq.(52) and ¯ G ofEq.(51) which we proceed to find out.In Fourier space, isotropy allows to split the Dyson equation for ¯ G ( k ) in two uncoupledand mutually orthogonal algebraic equations, G ⊥ ( k ) = G (0) ⊥ ( k ) − k G (0) ⊥ ( k ) χ ⊥ ( k ) G ⊥ ( k ) , (53) G k ( k ) = G (0) k ( k ) − k G (0) k ( k ) χ k ( k ) G k ( k ) , (54)where ¯ G (0) ⊥ ( k ) and ¯ G (0) k ( k ) are given in Eq.(27) and ¯ χ ( ~k ) is the electric susceptibility tensorsuch that ¯ ǫ ( ~k ) = ¯ I + ¯ χ ( ~k ) is the effective dielectric tensor. Statistical isotropy allows todecompose ¯ ǫ and ¯ χ in longitudinal and transverse components, ǫ ⊥ , k ( k ), χ ⊥ , k ( k ). For amedium made of a collection of disconnected dipoles, χ ⊥ , k ( k ) can be expanded as a series ofone-particle-irreducible (1PI) multiple-scattering terms of order n , χ ⊥ , k ( k ) = ∞ X n =1 X ( n ) ⊥ , k ( k ) ρ n ˜ α n . (55)The functions χ ⊥ , k are named 1PI as they contain integrals over undefined photon mo-menta which cannot be disentangled [11, 29]. In the above formula, ρ is the average33umerical volume density of scatterers and ˜ α is the renormalized polarizability of singlescatterers. The functions X ( n ) ⊥ , k ( k ) incorporate the spatial dispersion due to the spatialcorrelation within clusters of n scatterers. In particular, X (1) ⊥ = X (1) k = 1 incorporatesall the self-correlation factors. In field theory terminology, ¯ χ ⊥ , k are proportional to thephoton self-energy functions, Σ ⊥ , k ( ~k ) = − k χ ⊥ , k ( ~k ) –not to be confused with the dipoleself-energy. Alternatively, Eqs.(53,54) can be written in terms of the so-called T -matrix, T ⊥ , k ( k ) ≡ Σ ⊥ , k ( k ) + Σ ⊥ , k ( k ) G ⊥ , k ( k )Σ ⊥ , k ( k ), as G ⊥ , k ( k ) = G (0) ⊥ , k ( k ) + G (0) ⊥ , k ( k ) T ⊥ , k ( k ) G (0) ⊥ , k ( k ) . (56)Two remarks are in order at this point. The first one concerns the topology of the hostmedium. Because the emitter is treated as a point emitter and it is equivalent to the hostscatterers, a cermet topology is inherent to the virtual cavity scenario. In the expression ofEq.(55), spatial dispersion in ¯ χ cannot be disregarded as there must be at least a minimumdistance ξ between scatterers which determines their exclusion volume. Hence, this distanceis the virtual cavity radius. For the sake of consistency with the linear medium approxi-mation, ξ must be greater than the dipole radius to avoid overlapping between orbitals andso that the dipole approximation be valid. Second, in view of Eqs.(53,54), longitudinal andtransverse modes of the Dyson field do not couple to each other as traveling throughout arandom medium the same as it occurs with photons in free space. Eqs.(53,54) can be solvedindependently yielding the Dyson propagator components, G ⊥ ( k ) = 1 k [1 + χ ⊥ ( k )] − k ,G k ( k ) = 1 k [1 + χ k ( k )] . (57)In terms of free propagators and self-energy functions they can be depicted perturbatively asin Fig.9. A more detailed examen in Section VI will show that longitudinal and transversebare photons –i.e., the ones with propagator ¯ G (0) – do couple necessarily when they experiencemultiple scattering processes. In Eqs.(53,54), longitudinal and transverse bare photons enterboth χ ⊥ ( k ) and χ k ( k ) by means of the spatial correlations among scatterers.Getting back to the computation of ¯ G and the γ -factors, we first observe that 2 γ ⊥ and γ k are made of 1PI diagrams in which the starting and ending points coincide at the emitterlocation. Those diagrams belong to ¯ χ ( k ) and amount to the so-called recurrent scattering.As a matter of fact, the emitter itself enters the bulk propagator of Eq.(57) as an ordinary34 IG. 9: Diagrammatic representation of the Dyson propagator ¯ G . scatterer in the VC scenario. The second observation is that, as the averaging process inEq.(51) is subject to the fixed location of the emitter, every scattering event in the 1PIdiagrams contributing to γ is correlated to the emitter either at the near end or at therare end of each diagram, indistinguishably. This is a consequence of reciprocity. Takingadvantage of this feature in every multiple-scattering diagram like that in Fig.10( b ), wecan attribute all the irreducible correlations of the intermediate scattering events to theemitter on the left. By so proceeding, we end up with an effective separation of all thosepieces irreducibly correlated to the emitter on the left completely disentangled from thosenon irreducibly correlated which form non-1PI pieces on the right. The sum of the 1PIpieces on the left amounts to ¯ χ/ρ ˜ α , where the factor 1 /ρ ˜ α stands for amputation of the thefirst random scatter which enters the diagrams of ¯ χ in favor of the ’virtually’ fixed emitterlocation. The sum of the non-1PI pieces on the right amounts to the bulk propagator ¯ G .Therefore, we end up with the formulae [48] –Figs.10( c, d ), G V C ⊥ ( k ) = 1 ρ ˜ α χ ⊥ ( k ) G ⊥ ( k ) = 1 ρ ˜ α χ ⊥ ( k ) k [1 + χ ⊥ ( k )] − k , (58) G V C k ( k ) = 1 ρ ˜ α χ k ( k ) G k ( k ) = 1 ρ ˜ α χ k ( k ) k [1 + χ k ( k )] . (59)The above expressions contain both G (0) ⊥ and G (0) k which carry divergences. The expressions2 γ V C ⊥ = Z d k (2 π ) h χ ⊥ ( k ) / ( ρ ˜ α ) k [1 + χ ⊥ ( k )] − k − G (0) ⊥ ( k ) i , (60) γ V C k = Z d k (2 π ) h ρ ˜ α χ k ( k ) k [1 + χ k ( k )] − G (0) k ( k ) i , (61)are however fully convergent. We emphasize that the above expressions for the virtual-cavity γ -factors are exact under the assumption that the emitter can be treated as a point35 IG. 10: ( a ) Feynman rules. Only two-point irreducible correlation functions have been usedfor the sake of simplicity. ( b ) Diagrammatic representation of the equivalence between multiple-scattering processes amounting to ¯ G . ( c ),( d ) Diagrammatic representations of Eq.(60) and Eq.(61)respectively. dipole equivalent in all to the rest of host scatterers. The explicit inclusion of ˜ α –as givenby Eq.(42)– in the computation of G V C ⊥ , k through Eqs.(58,59) makes it depend implicitly ontheir integral quantities γ ⊥ , k given in Eqs.(60,61). Later on we will see that the form of theabove expressions for 2 γ V C ⊥ and γ V C k will allow us to classify the nature of the emission in atransparent manner.Alternatively, we can write ¯ G V C ( k ) in other forms making use of Dyson’s equation. Infunction of the bulk propagator of Eqs.(53,54) it reads G V C ⊥ , k ( k ) = 1 k ρ ˜ α h − G ⊥ , k G (0) ⊥ , k i . (62)In function of the T -matrix of Eq.(56) it reads G V C ⊥ , k ( k ) = − k ρ ˜ α T ⊥ , k ( k ) G (0) ⊥ , k ( k ) . (63)Finally, we write the Lippmann-Schwinger equation of Eq.(51) for the self-polarizationlocal field propagator in Fourier space, G V C ⊥ ( k ) = G (0) ⊥ ( k ) + G (0) ⊥ ( k ) Ξ V C ⊥ ( k ) G V C ⊥ ( k ) , (64) G V C k ( k ) = G (0) k ( k ) + G (0) k ( k ) Ξ V C k ( k ) G V C k ( k ) , (65)36here Ξ V C ⊥ , k ( k ) = − ρ ˜ αχ ⊥ , k ( k ) G (0) ⊥ , k ( k ) h − χ ⊥ , k ( k ) ρ ˜ α + k χ ⊥ , k ( k ) G (0) ⊥ , k ( k ) i (66)is the stochastic kernel expressed as a function of ¯ χ , ¯ G (0) and ρ ˜ α .Because the transference matrix (¯ t -matrix in brief) formalism is profusely used in nu-merical simulations which involve discrete configurations of point dipoles (see eg.[51, 52]),we will next show that analogous formulae to the ones above are obtained for a fixed con-figuration of point dipoles. Let us take the m th configuration of the statistical ensembledescribed in Section II C. Its corresponding self-polarization vacuum state there was de-noted by | φ m i s.p. . Let us assume the configuration consists of N + 1 host scatterers withfixed position vectors denoted by { ~r i } with i = 0 , ..., N . We then decide to excite the dipoleat ~r with an external monochromatic field of frequency ω = k c such that ~E exc ( ~r ) = ~E ω iff ~r = ~r and ~E exc ( ~r ) = 0 otherwise, as in Section III A. All the dipoles are equivalent andhave renormalized polarizability ˜ α . Therefore, the dipole moment of the emitter reads ~p ( ~r ) = ǫ ˜ α ~E ω . (67)The total field at the emitter location is the sum of the exciting field plus the self-polarizationfield which the dipole moment ~p ( ~r ) itself creates at ~r . That is, ~E ( ~r ) = ~E ω + k ǫ ¯ G ( ~r , ~r ) · ~p ( ~r )= ~E ω + k ˜ α ¯ G ( ~r , ~r ) · ~E ω . (68)The second term on the r.h.s. of Eq.(68) is the field which propagates freely in space fromall the induced dipoles to ~r , k ˜ α ¯ G ( ~r , ~r ) · ~E ω = k ǫ X { ~r i } ¯ G (0) ( ~r , ~r i ) · ~p ( ~r i ) . (69)Except for the emitter, all the rest dipoles are only induced by their mutual interactions.The ¯ t -matrix, with components ¯ t ( ~r i , ~r j ), in the same spirit as that in Eq.(56), yields thedipole induced at some point ~r i as a result of its interaction with the dipole excited by anexternal source at some other point ~r j . That is, ~p ( ~r i ) = (cid:16) k ǫ (cid:17) − X { ~r j } − ¯ t ( ~r i , ~r j ) · ~E exc ( ~r j ) . (70)37nalogously to the susceptibility function in Eq.(55), ¯ t ( n ) ( ~r i , ~r j ) can be expanded in powersof − k ˜ α , ¯ t ( ~r i , ~r j ) = P n =1 ¯ t ( n ) ( ~r i , ~r j ). Each term in the sum contains n factors − k ˜ α and n − G (0) ( ~r m , ~r l ) which propagate the field though all possible paths connecting thepoints ~r i and ~r j . As an example,¯ t (4) ( ~r i , ~r j ) = ( − k ˜ α ) X { ~r l ,~r m } ¯ G (0) ( ~r i , ~r m ) · ¯ G (0) ( ~r m , ~r l ) · ¯ G (0) ( ~r l , ~r j ) , (71)where restrictions apply over the indices of the sums to incorporate correlations. The analogto the ¯ T ( k ) matrix of Eq.(55) is just¯ T ( ~k, ~k ′ ) = X { ~r i ,~r j } ¯ t ( ~r i , ~r j ) e i [ ~k · ~r i − ~k ′ · ~r j ] . (72)Because in our case the only externally excited dipole is that at ~r , ~p i = − (cid:16) k ǫ (cid:17) − ¯ t ( ~r i , ~r ) · ~E ω .Inserting this formula into Eq.(69) we obtain¯ G ( ~r , ~r ) = 1 − k ˜ α X { ~r i } ¯ G (0) ( ~r , ~r i ) · ¯ t ( ~r i , ~r ) , (73)which is the analogous expression to that in Eq.(63) but for a fixed configuration of scatterers. V. THE REAL-CAVITY SCENARIO The computation of the γ -factors in the real cavity scenario is generally quite more com-plicated as an effective medium cannot be strictly speaking defined and translation invariantbulk propagator and suceptibility tensors do not exist. There are two main differences withrespect to the virtual cavity scenario. The first one is that the emitter does not behaveas an ordinary scatterer of the host medium regarding photon propagation. The second isthat the presence of the cavity induces additional correlations among the scattering eventsexperienced by photons near the cavity surface. That is, because scattering events are corre-lated to the emitter through the stochastic kernel of Eq.(51), they indirectly correlate amongthemselves beside the inherent spatial correlation of scatterers in absence of cavity.There are nevertheless two situations in which approximate solutions can be found. Thefirst one corresponds to the case in which the emitter hardly alters the host medium. For thisto be the case, the polarizability of the emitter must be much weaker than that of the hostscatterers and its cavity radius R much less than the typical distance between scatterers,38 ≪ ρ − / . In his situation the foreign emitter is said weakly polarizable and interstitial.The second situation which can be treated analytically is that in which the cavity is large, R ≫ ρ − / . The density function of host scatterers is altered in a large patch but remainsunaltered locally within distances of the order of ξ where correlations matter. In this case,the foreign emitter is said to be substantial. A. Interstitial weakly-polarizable impurity within a small cavity The two-point correlation of the host scatterers to the foreign emitter reflects in thiscase on a slight modification of their average density function in the neighborhood of theimpurity. The radial distribution density function with respect to the emitter site reads¯ ρ ( r i ) = ρ [1 − Θ( r i − R )] , R ≪ ρ − / , (74)where ~r i is the position vector of a generic scatterer with origin at the emitter locationand ρ is the ’would-be’ uniform numerical density of host scatterers in absence of emitter.As an example, we depict in Fig.11( a ) the diagrammatic representation of the scatteringamplitude for the process in which a photon emitted by the foreign emitter is scattered bya host scatterer and absorbed by another one, F (2) ( k ), F (2) ( k ) = µ ( ρ ˜ α ) Z d r d r e i~k · ~r h Tr { ¯ G (0) ( ~r ) · ¯ G (0) ( ~r − ~r ) }× [1 − Θ( r − R )][1 − Θ( r − R )][1 − h ( | ~r − ~r | ) i , (75)where h ( | ~r − ~r | ) is the two-point correlation function between host scatterers with supportwithin a correlation volume ∼ ξ . It is clear from Fig.11( a ) that in general it is not possibleto disentangle the correlation of the emitter to each scatterer event. However, for the casethat k ξ ≪ k R ≪ 1, one can apply the overlap approximation [15] for those modes k ≪ ξ − , R − . This is so because those modes cannot distinguish between the exclusionvolume of the host scatterers and the real cavity. That is what Fig.11( b ) depicts. In principle,the overlap approximation in this scenario demands that the only relevant correlation in themedium be given by the Heviside function of exclusion volume. This is clearly the caseof a weakly correlated host medium (eg. a diluted gas like that of Fig.12( a )) for which h ( | ~r − ~r | ) ≃ − Θ( | ~r − ~r | − ξ ) with ξ ≪ ρ − / . For a highly correlated medium with ξ ∼ ρ − / (eg. a solid like that of Fig.12( b )) a term proportional to δ (1) ( r − ξ ) amounting39or first neighbors must be included in h ( | ~r − ~r | ). Nevertheless, for k ξ ≪ k ≪ ξ − is equally valid. Therefore, the condition k ξ ≪ χ eff the constant effective susceptibility valid forthose k -modes with kξ ≪ 1. The contribution of transverse modes with kξ ≪ ⊥ and W ⊥ is well approximated by the formulae of the VC scenario which we will computein Section VI B 3. The corresponding γ -factor will be found to be2 γ ⊥ | k,k ≪ ξ − ≈ Z d k (2 π ) χ eff ρ ˜ α G eff ⊥ ( k ) . (76)For those modes k & ξ − , the equivalence with the virtual cavity formulae Eqs.(60,61) isonly possible for the case that R ≈ ξ –see Fig.11( c ). This is only applicable to a weakly-correlated host medium as otherwise R would not satisfy the interstitial condition R ≪ ρ − .That is the case of the Maxwell-Garnett dielectric which will be studied in Section VII A.This has the nice implication that, if the impurity is an excited host scatterer of a diluted gas,the γ V C factors computed in Section IV B are applicable in good approximation providedthat the polarizability of the excited atom during its decay process is weak in comparisonto that of the rest of scatterers. B. Substantial impurity within a large cavity As mentioned above, the presence of an emitter which substitutes one or several hostscatterers can modify sensitively the homogeneity of the host medium. In particular, thisis the case of a host scatterer which is promoted to an excited atomic state in a stronglycorrelated medium. In comparison to the scenario of the previous Subsection, this is thesame situation but with ξ ≃ ρ − / . Because the polarizability of the excited scatterer isdifferent to that of the rest, namely α exc. , it behaves as a foreign emitter which occupies aneffective volume of the order of ρ − . Thus, the approximate average polarizability density ata point ~r away from the emitter is given by ρ ˜ α [1 − Θ( r − ρ − / )] + ρα exc. Θ( r − ρ − / ). Thismakes the electrical susceptibility to be clearly not a homogeneous function. For instance,40 IG. 11: ( a ) Diagrammatic representation of Eq.(75). ( b ) Overlap approximation restricted tothose modes k ≪ ξ − subject to k ξ ≪ 1. ( c ) Overlap approximation for shorter wavelengths validonly for R ≈ ξ . the two-scattering term would read¯ χ (2) ( ~r , ~r ) ≈ ρ h ˜ α [1 − Θ( r − ρ − / )] + ρα exc. Θ( r − ρ − / ) i × h ˜ α [1 − Θ( r − ρ − / )] + ρα exc. Θ( r − ρ − / ) i ¯ G (0) ( ~r − ~r ) h ( | ~r − ~r | ) . (77)That implies that the susceptibility function evaluated at ~r , ~r close to the emitter can bevery different to that at ~r , ~r far from it.However, whatever the nature of the foreign emitter is, if the cavity radius is large incomparison to the mean distance between scatterers, R ≫ ρ − / , the medium will lookstatistically homogeneous at scales of the order of ρ − / as seen from the emitter. Thisis so because the electric field which propagates from the emitter is statistically uniformwithin the interval [ R, R + ρ − / ] for R ≫ ρ − / . Thus, the average value of the numericaldensity of host scatterers at a point ~r i away from the emitter reads in good approximation,¯ ρ ( ~r i ) ≃ ρ [1 − Θ( r i − R )]. Because the typical correlation length between host scatterers is ξ . ρ − / ≪ R , the cavity exclusion volume factor − Θ( r − R ) is common to all the scattererswhich are correlated among themselves and enter the susceptibility function. This has the41ice implication that solids with well-defined susceptibility function ¯ χ ( k ) can be treatedthis way with no-need to know the individual polarizabilities of their constituents. Anyfluctuation not yet considered for the case of point-host-dipoles (eg. excitons) is assumedintegrated out in ¯ χ ( k ), provided that the wavelength of those fluctuations is much shorterthan R . In the following we develop a formalism which bases on the definition of quantitiesanalogous to ¯ χ ( k ) and ¯ G ( k ) but for a non-translation invariant medium.Let us formulate mathematically the above approximation. Let g ( r ) be the full two-point FIG. 12: Two dimensional sketch of the real cavity scenarios which can be treated analytically insome approximation. ( a ) Weakly correlated host medium (eg. gas) with an interstitial impurity µ . ( b ) Strongly correlated host medium (eg. glass) with an interstitial impurity. ( c ) Large cavityenclosing a substantial impurity. correlation function of the emitter with the host scatterers, g ( r ) = 1 + h C ( r ) , (78)where h C ( r ) is the irreducible piece. In general, because the cavity is macroscopical, it candistort the homogenous distribution of host scatterers in the surrounding medium. In sucha case h C ( r ) must contain a maximum at r ≃ ξ that makes account of the overdensity offirst neighbors. For the sake of simplicity, we will restrict h C to a step function which standsfor the cavity exclusion volume, h C ( r ) ≈ − Θ( r − R ). The Fourier transform of g ( r ) reads g ( k ) = (2 π ) δ (3) ( ~k ) + h C ( k ) . (79)Because h C ( r ) has support in r ≤ R , it is therefore expected that its Fourier transform getssupport in k . /R . On the other hand, in case that spatial dispersion in ¯ ǫ ( k ) be relevant,42uch a dispersion must be of the order of 1 /ξ . Therefore, any convolution of g ( | ~k − ~k ′ | ) withthe self-energy operator ¯Σ( k ′ ) and any generic function f ( k ′ ) can be approximated by Z d k (2 π ) g ( | ~k ′ − ~k | ) ¯Σ( k ) · ¯ f ( k ) ≈ ¯Σ( k ′ ) · Z d k (2 π ) g ( | ~k ′ − ~k | ) ¯ f ( k ) . (80)In Eq.(80), the difference between ~k and ~k ′ is negligible in comparison to 1 /ξ in the rangeof momenta where h C ( | ~k − ~k ′ | ) takes nearly constant value and ¯Σ( k ) is not zero. In turn,that implies that the correlation of the emitter with any 1PI (multiple)scattering processcan be approximated by the correlation of the emitter to any of the scatterers involved insuch a process. In particular, things get mathematically simpler if the correlation functionsconnect the impurity either to the first or to the last scatterers within the 1PI diagrams of¯Σ( k ). This allows to write the series for the RC propagator, ¯ G RC , as two apparently differentexpansions, ( a ) and ( b ), as depicted in Figs.13( a, b ) respectively. The corresponding seriesread G RC ⊥ = ∞ X n =0 G a ( n ) ⊥ , G RC k = ∞ X n =0 G a ( n ) k , (81) G RC ⊥ = G (0) ⊥ + ∞ X n =1 G b ( n ) ⊥ , G RC k = G (0) k + ∞ X n =1 G b ( n ) k . (82)In each series, the n th terms read respectively, FIG. 13: Diagrammatic representation of the self-polarization propagator G according toEqs.(81,83,84) –series ( a )– and Eqs.(82,85,86)) –series ( b )– in the large cavity scenario. G a ( n ) ⊥ ( k ) = κ ( n ) ⊥ ( k )Σ ⊥ ( k ) G (0) ⊥ ( k ) , (83) G a ( n ) k ( k ) = κ ( n ) k ( k )Σ k ( k ) G (0) k ( k ) , (84)43nd G b ( n ) ⊥ ( k ) = κ ( n − ⊥ ( k )Σ ⊥ ( k ) G (0) ⊥ ( k )Σ ⊥ ( k ) κ (1) ⊥ ( k ) , (85) G b ( n ) k ( k ) = κ ( n − k ( k )Σ k ( k ) G (0) k ( k )Σ k ( k ) κ (1) k ( k ) . (86)The recurrent formulae for the partial pseudo-susceptibilities κ ( n ) ⊥ , k –the reason for this nomen-clature will get clear later on– read κ ( n ) ⊥ ( k ) = 12 Z d k ′ (2 π ) g ( | ~k ′ − ~k | ) h (1 + cos θ ) × κ ( n − ⊥ ( k ′ )Σ ⊥ ( k ′ ) G (0) ⊥ ( k ′ )+ sin θ κ ( n − k ( k ′ )Σ k ( k ′ ) G (0) k ( k ′ ) i for n ≥ , (87) κ ( n ) k ( k ) = Z d k ′ (2 π ) g ( | ~k ′ − ~k | ) h cos θ × κ ( n − k ( k ′ )Σ k ( k ′ ) G (0) k ( k ′ )+ sin θ κ ( n − ⊥ ( k ′ )Σ ⊥ ( k ′ ) G (0) ⊥ ( k ′ ) i for n ≥ , (88)and κ (0) ⊥ , k ( k ) ≡ − ˜ k Σ ⊥ , k ( k ) − = 1 /χ ⊥ , k ( k ) for n = 0 . (89)In Eqs.(87,88), Σ ⊥ , k ( k ′ ) can be factored out of the integrals as Σ ⊥ , k ( k ) in application of theapproximation in Eq.(80). Therefore, the building blocks of the series are G (0) ⊥ , k ( k ) and theirreducible pieces of κ (1) ⊥ , k ( k ). Hereafter we will refer to the 1PI parts of κ (1) ⊥ , k ( k ) as cavityfactors , C ⊥ ( k ) ≡ Z d r e i~k · ~r h C ( r )Tr { ¯ G (0) ( r )[ I − ˆ k ⊗ ˆ k ] } = 12 Z d k ′ (2 π ) h C ( | ~k ′ − ~k | ) h G (0) ⊥ ( k ′ )+ G (0) ⊥ ( k ′ ) cos θ + G (0) k ( k ′ ) sin θ i , (90) C k ( k ) ≡ Z d r e i~k · ~r h C ( r )Tr { ¯ G (0) ( r )[ˆ k ⊗ ˆ k ] } = Z d k ′ (2 π ) h C ( | ~k ′ − ~k | ) × h G (0) k ( k ′ ) cos θ + G (0) ⊥ ( k ′ ) sin θ i , (91)and so κ (1) ⊥ , k ( k ) = − k [ G (0) ⊥ , k ( k ) + C ⊥ , k ( k )].Also, Eqs.(85,86) resemble the formulae of the γ -factors for the virtual cavity scenario–Eqs.(58,59). That is, if we define the total pseudo-susceptibility as κ ⊥ , k ( k ) ≡ P n =0 κ ( n ) ⊥ , k ( k )44e can write G RC ⊥ ( k ) = κ ⊥ ( k ) χ ⊥ ( k ) G (0) ⊥ ( k ) , (92) G RC k ( k ) = κ k ( k ) χ k ( k ) G (0) k ( k ) . (93)However, the above equations are not yet quite similar to those for the VC scenario. Inparticular, the propagator on the r.h.s. of Eqs.(92,93) is that in free-space whereas it isthe bulk propagator in the virtual cavity formulae. The reason being that κ ⊥ , k ( k ) containboth 1PI and non-1PI processes. We can separate κ ⊥ , k ( k ) into 1PI and non-1PI (N1PI)pieces, κ ⊥ , k ( k ) = κ P I ⊥ , k ( k ) + κ N P I ⊥ , k ( k ) according to the following decomposition in partialpseudo-susceptibility functions, κ N P I ( n ) ⊥ , k ( k ) = κ ( n − ⊥ , k ( k )Σ ⊥ , k ( k ) G (0) ⊥ ( k ) , n ≥ ,κ P I ( n ) ⊥ , k ( k ) = κ ( n ) ⊥ , k ( k ) − κ N P I ( n ) ⊥ , k ( k ) , n ≥ ,κ P I (0) ⊥ , k ( k ) = κ (0) ⊥ , k ( k ) , κ N P I (0) ⊥ , k ( k ) = 0 . (94)Using this decomposition, we can write Eqs.(92,93) in the form, G RC ⊥ ( k ) = κ P I ⊥ ( k ) χ ⊥ ( k ) G ⊥ ( k ) , (95) G RC k ( k ) = κ P I k ( k ) χ k ( k ) G k ( k ) , (96)where G ⊥ , k ( k ) is the Dyson propagator given by Eqs.(53,54) for a ’would-be’ homogeneousmedium in absence of the real cavity. As a matter of fact, in passing from Eqs.(92,93) toEqs.(95,96) one can use the same arguments as those employed in the virtual cavity scenarioto push the effective bulk propagator to the right as in the diagrams of Fig.10( b ). Thus,the above equations resemble those expressions for the VC scenario in Eqs.(58,59) with thereplacement of χ ⊥ , k ( k ) ρ ˜ α by κ P I ⊥ , k ( k ) χ ⊥ , k ( k ). Therefore, the relation between the propagators G RC ⊥ , k ( k ) and G V C ⊥ , k ( k ) for two identical random media which differ just by the presence/absenceof a real cavity enclosing the emitter is given by G RC ⊥ , k ( k ) G V C ⊥ , k ( k ) = ρ ˜ ακ P I ⊥ , k ( k ) . (97)Finally, we write the Lippmann-Schwinger equations for the large real cavity scenario, G RC ⊥ ( k ) = G (0) ⊥ ( k ) + G (0) ⊥ ( k ) Ξ RC ⊥ ( k ) G RC ⊥ ( k ) , (98) G RC k ( k ) = G (0) k ( k ) + G (0) k ( k ) Ξ RC k ( k ) G RC k ( k ) , (99)45here the stochastic kernel isΞ RC ⊥ , k ( k ) = − κ P I ⊥ , k ( k ) χ ⊥ , k ( k ) G (0) ⊥ , k ( k ) h − κ P I ⊥ , k ( k ) χ ⊥ , k ( k ) + k χ ⊥ , k ( k ) G (0) ⊥ , k ( k ) i . (100) VI. THE NATURE OF DIPOLE EMISSION AND THE COHERENT VACUUM In this section we first study a simple model in which dipole emission can be studiedmicroscopically in full detail. We compare the results with the usual Lorentz-Lorenz (LL)and Onsager-B¨ottcher (OB) formulae and other more recent works. Next, the same decom-position is performed over the general formulae of the previous sections. Our microscopicalapproach allows us to distinguish between the transverse and the longitudinal, the coher-ent and the incoherent, the direct-coherent and the induced-coherent, and the dispersed-incoherent and the absorptive components of the dipole emission. Special attention is paidto the phenomenon of radiative/non-radiative energy transfer. Our study reveals why thenature of dipole emission was erroneously interpreted in previous works.In connection with QFT formalism, we define a coherent vacuum associated to coherentemission. We find its relation to the sourceless vacuum in a random medium, which is itselfassociated to direct emission. A. Decomposition of dipole emission in the single scattering approximation Our simplified model consists of a stimulated emitter embedded in a diluted host mediummade of spherical inclusions of electrostatic bare polarizability α . The scatterers present aminimum spherical exclusion volume of radius ξ and density ρ ≪ ξ − . The emitter is excitedby an external field ~E ω whose frequency, ω = ck is such that k ξ ≪ 1. A similar modelwas studied in [23, 53, 54]. The single scattering approximation implies that virtual photonsexperience, at the most, a single scattering event with the scatterers in the host medium.This way, the only relevant correlation is that given by the two-point function h ( r ) = − Θ( r − R ). If the cavity radius R satisfies R = ξ and the electrostatic polarizability of the emitteris the same as that of the host scatters, α , this set-up corresponds to the VC scenario. Itcorresponds to the RC scenario otherwise. Let us assume without loss of generality that R = ξ and let us take α e as the electrostatic polarizability of the emitter which, eventually,can be taken equal to α . This implies that, in the single scattering approximation both46 IG. 14: ( a ) Feynman’s rules . ( b ) Diagrammatic representation of W [1] . The first three diagramsstand for those terms in Eq.(103). The next three diagrams take account of the self-polarizationterms of Eqs.(119,120). scenarios are equivalent with just exchanging the cavity factors appearing in the RC scenarioof Section V B with the two-scattering term of the susceptibility in the VC scenario, ¯ χ (2) .They relate through χ (2) ⊥ , k = − k ( ρ ˜ α ) C ⊥ , k | R = ξ , where ˜ α is given by Eq.(42) and C ⊥ , k ( k ) aregiven in Eqs.(90,91). The susceptibility of this model reads χ [1] k , ⊥ ( k ) = ρ ˜ α + χ (2) k , ⊥ ( k ). In thefollowing, the superscript [1] will denote single-scattering approximation.The quantity to compute is W ω = − ω ǫ c | α e | | k α e [2 γ (0) ⊥ + 2 γ ⊥ + γ k ] | ℑ{ γ T ot ⊥ + γ T ot k }| E ω | . (101)Pictorially, γ [1] is given by the second and third diagrams of Fig.14( b ). It reads, γ [1] = Tr nZ d r ¯ G (0) ( ~r )( − k ρ ˜ α ) ¯ G (0) ( ~r ) [1 − Θ( r − ξ )] o . (102)Let us disregard for now self-polarization cycles on the emitter. In doing so, the emittercavity is said empty. In the limit k ξ ≪ 1, up to O (0) terms in k ξ we get W [1] ,empω ≃ W ω n ℜ{ ρ ˜ α } + ℑ{ ρ ˜ α } [ 1( k ξ ) + 1 k ξ ] o , (103)47here W ω = ω α e ǫ πc | E ω | is the power emitted in free space. The O (0) terms of Eq.(103)equal the usual Lorentz-Lorenz (LL) and Onsager–B¨ottcher (OB) formulae in absence ofback-reaction effects [55–57] with ǫ [1] = 1 + ρ ˜ α , W [1] LL = W ω (cid:16) ℜ{ ǫ [1] } + 23 (cid:17) ℜ{√ ǫ [1] } , W [1] ,empOB = W ω (cid:16) ℜ{ ǫ [1] } ℜ{ ǫ [1] } + 1 (cid:17) ℜ{√ ǫ [1] } , (104) W [1] LL = W [1] ,empOB = W ω (1 + 76 ℜ{ ρ ˜ α } ) . (105)Same results as those in Eqs.(102-105) were obtained by the authors in [23] using a QuantumOptics formalism. The factors (cid:16) ℜ{ ǫ } +23 (cid:17) and (cid:16) ℜ{ ǫ } ℜ{ ǫ } +1 (cid:17) are referred to in the literatureas virtual cavity and empty cavity local field factors respectively. The factor p ℜ{ ǫ } issometimes refereed to as bulk factor. Both W LL and W empOB are commonly attributed in theliterature to transverse emission. Our next task is to clarify the actual nature of the termsin Eqs.(102,103) through a rigorous microscopical analysis. In Fourier space, Eq.(102) canbe decomposed into transverse and longitudinal parts,2 γ [1] ⊥ = 2 ρ ˜ α Z d k (2 π ) h G (0) ⊥ ( k ) + C ⊥ ( k ) i G (0) ⊥ ( k ) = 2 ρ ˜ α Z d k (2 π ) Z d k ′ (2 π ) × h G (0) ⊥ ( k ′ ) δ (3) ( ~k ′ − ~k ) G (0) ⊥ ( k ) (106)+ 12 G (0) ⊥ ( k ′ ) h ( | ~k ′ − ~k | )(1 + cos θ ) G (0) ⊥ ( k ) (107)+ 12 G (0) k ( k ′ ) h ( | ~k ′ − ~k | ) sin θG (0) ⊥ ( k ) i , (108) γ [1] k = ρ ˜ α Z d k (2 π ) h G (0) k ( k ) + C k ( k ) i G (0) ⊥ ( k ) = ρ ˜ α Z d k (2 π ) Z d k ′ (2 π ) × h G (0) k ( k ′ ) δ (3) ( ~k ′ − ~k ) G (0) k ( k ) (109)+ G (0) k ( k ′ ) h ( | ~k ′ − ~k | ) cos θ G (0) k ( k ) (110)+ G (0) ⊥ ( k ′ ) h ( | ~k ′ − ~k | ) sin θ G (0) k ( k ) i , (111)where h ( q ) = πq [sin ( qξ ) − qξ cos ( qξ )] and we can write, W [1] ⊥ = W ω [1 − πcω ℑ{ γ [1] ⊥ } ] , W [1] k = W ω [ − πcω ℑ{ γ [1] k } ] . (112)In the integrals of Eqs.(106-111), the only imaginary pieces are those associated to thepoles of G (0) ⊥ ( k ) factors. Therefore, ℑ{ γ [1] ⊥ , k } is composed of terms of the form ℜ{ ρ ˜ α } times48esidues computed at k = k plus terms of the form ℑ{ ρ ˜ α } times the real part of theintegrals. Alternatively, we can write 2 γ [1] ⊥ and γ [1] k in terms of spatial integrals,2 γ [1] ⊥ = ρ ˜ α Tr nZ d r [ ¯ G (0) rad. ( r ) + ¯ G (0) stat. ( r )] · ¯ G (0) rad. ( r ) [1 − Θ( r − ξ )] o = ρ ˜ α Z d r × Tr n ¯ G (0) rad. ( r ) · ¯ G (0) rad. ( r ) (113) − ¯ G (0) rad. ( r ) · ¯ G (0) rad. ( r )Θ( r − ξ ) (114) − ¯ G (0) stat. ( r ) · ¯ G (0) rad. ( r )Θ( r − ξ ) o , (115) γ [1] k = ρ ˜ α Tr nZ d r [ ¯ G (0) rad. ( r ) + ¯ G (0) stat. ( r )] · ¯ G (0) stat. ( r ) [1 − Θ( r − ξ )] o = ρ ˜ α Z d r × Tr n ¯ G (0) stat. ( r ) · ¯ G (0) stat. ( r ) (116) − ¯ G (0) stat. ( r ) · ¯ G (0) stat. ( r )Θ( r − ξ ) (117) − ¯ G (0) rad. ( r ) · ¯ G (0) stat. ( r )Θ( r − ξ ) o . (118)The correspondence between the integrals in Fourier space and those in spatial space isobvious. However, such an obvious correspondence is only possible in the single scatteringapproximation. This is so because the bulk propagator is identical to that in free spacebut for the substitution k → ǫ [1] k so that it does not present spatial dispersion andthere exists a direct identification between the radiative and electrostatic bulk propagatorsin spatial space and transverse and longitudinal propagators in Fourier space respectively.Note also that, in absence of the correlation function, there is no coupling between radiative(transverse) and electrostatic (longitudinal) modes.Next, we search for the terms in Eqs.(106-118) which amount to those of Eq.(105). Thefree space term 2 γ (0) ⊥ together with that in Eq.(106) (or Eq.(113)) yield W ω (1 + ℜ{ ρ ˜ α } ),which corresponds to the bulk factor W ω ℜ{√ ǫ [1] } . This is the emission directly radiated bythe dipole into the medium –see Fig.16( a ). Eq.(106) (or Eq.(113)) corresponds also to thediagram of Fig.5( d ), which explains why of the two transverse propagators there. Eq.(107)(or Eq.(114)) yields higher order terms. The common terms to 2 γ [1] ⊥ and γ [1] k are those inEq.(108) (or Eq.(115)) and Eq.(111) (or Eq.(118)) which amount to W ω ℜ{ ρ ˜ α } each. Theyequal one local field factor each, (cid:16) ℜ{ ǫ [1] } +23 (cid:17) or (cid:16) ℜ{ ǫ [1] } ℜ{ ǫ [1] } +1 (cid:17) . The first thing we learn fromthis analysis is that, contrarily to the common assumption, one of the local field factors isassociated to longitudinal emission while the other one together with the bulk factor belongto transverse emission. The latter is associated to the poles of the bulk transverse propagator49f the general expression in Eq.(58) and form part of the coherent radiative emission –seenext subsection. On the contrary, the former is associated to the imaginary part of χ k whichcontains a transverse bare propagator (see Eq.(91)). It is part of the incoherent emissionwhich is dispersed. The existence of the common term in Eqs.(108,111) is a consequenceof reciprocity. However, while it gives rise to coherent emission when the coupling reads G k − G ⊥ , it gives rise to dispersion as read in opposite direction as in Fig.15( b ).Regarding the ξ -dependent terms of Eq.(103), they both are proportional to ℑ{ ρ ˜ α } . The FIG. 15: ( a ) Feynman’s rules. ( b ) Common vertex derived from Eqs.(108,111) which coupleslongitudinal to transverse modes, with | ˆ k × ˆ k ′ | = sin θ . ( c ) Longitudinal-longitudinal vertex fromEq.(110), with | ˆ k · ˆ k ′ | = cos θ . ( d ) Diagrammatic representation of the origin of the coupling ofsingle scattering radiative corrections to longitudinal external modes. leading order one, ℑ{ ρ ˜ α } ( k ξ ) , is given by Eq.(109) plus Eq.(110) (or Eq.(116) plus Eq.(117)) andis associated to the imaginary parts of both the longitudinal bulk propagator –Eq.(109)– andthe longitudinal susceptibility –Eq.(110). The term ℑ{ ρ ˜ α } k ξ has two identical contributionsof value ℑ{ ρ ˜ α } k ξ coming from the common term to 2 γ [1] ⊥ and γ [1] k in Eq.(108) and Eq.(111)respectively. They are associated to the imaginary parts of χ ⊥ ( k ) and χ k ( k ) respectively.All these terms proportional to ℑ{ ˜ α } are related to absorbtion in the host scatterers.Finally we consider the back-reaction of the host medium on the emitter with electrostaticpolarizability α e . To this aim and at first order in ρ we have to compute the self-polarizationdiagrams in the second row of Fig.14( b ). For the sake of simplicity, in the following we50isregard absorbtion and radiative corrections in ˜ α and keep leading order terms in α and k ξ . That is, ǫ [1] ≃ ρα is fully real and γ [1] k ≃ − k π ρα [ i k ξ ) ] , γ [1] ⊥ ≃ − k π [ i + ρα i . (119)Plugging Eq.(119) into Eq.(101), we obtain W [1] ,selfpω ≃ W [1] ,empOB [1 + 49 α e V ξ ρα ] , (120)where V ξ is the volume of the cavity. One can verify the agreement of the above expressionwith the OB formula which incorporate self-polarization terms at leading order in α , α e [14, 57], W [1] ,selfpOB = W ω (cid:16) ℜ{ ǫ [1] } Re { ǫ [1] } +1 − αeVξ ( Re { ǫ [1] }− (cid:17) ℜ{√ ǫ [1] } . B. General decomposition of dipole emission Next, we go beyond the single scattering approximation. Again, let us consider Eq.(101)in absence of absorbtion in the emitter. In order to examine the nature of the emission, itis convenient to write the γ -factors in terms of the propagators of both the virtual and thereal cavity scenarios using Eqs.(58-59) and Eqs.(95-96) respectively, W V Cw ∝ Z d k (2 π ) h ℑ{ χ ⊥ ( k ) ρ ˜ α G ⊥ ( k ) } + ℑ{ χ k ( k ) ρ ˜ α G k ( k ) } i , (121) W RCw ∝ Z d k (2 π ) h ℑ{ κ P I ⊥ χ ⊥ ( k ) G ⊥ ( k ) } + ℑ{ κ P I k χ k ( k ) G k ( k ) } i . (122)From now on, we will just work with W V Cw bearing in mind that the computations are inall equivalent to those for W RCw but for the replacement χ ⊥ , k → ρ ˜ ακ P I ⊥ , k χ ⊥ , k . For the sake ofsimplicity we will drop the scripts V C and ω and we will denote the proportionality constantomitted in the above equations by W o .The first obvious decomposition is that between transverse, 2 W ⊥ , and longitudinal emis-sion, W k , 2 W ⊥ = W o Z d k (2 π ) ℑ{ χ ⊥ ( k ) ρ ˜ α G ⊥ ( k ) } , (123) W k = W o Z d k (2 π ) ℑ{ χ k ( k ) ρ ˜ α G k ( k ) } . (124)51 . Coherent and incoherent emission; direct and induced coherent emission Coherent emission is that associated to the coherent propagating field. That is, theemission whose modes satisfy the same dispersion relations as the normal modes whichpropagate through the bulk [12, 36, 37], k ǫ ⊥ ( k ) − k | k = k prop. ⊥ = 0 for transverse modes (125)and ǫ k ( k ) | k = k prop. k = 0 for longitudinal modes. (126)These are the poles of G ⊥ ( k ) and G k in the integrands of Eqs.(127,129). Therefore, weidentify from Eqs.(121,122),2 W Coh. ⊥ = W o Z d k (2 π ) ℜ{ χ ⊥ ( k ) ρ ˜ α }ℑ{ G ⊥ ( k ) } , (127)2 W Incoh. ⊥ = W o Z d k (2 π ) ℑ{ χ ⊥ ( k ) ρ ˜ α }ℜ{ G ⊥ ( k ) } , (128) W Coh. k = W o Z d k (2 π ) ℜ{ χ k ( k ) ρ ˜ α }ℑ{ G k ( k ) } , (129) W Incoh. k = W o Z d k (2 π ) ℑ{ χ k ( k ) ρ ˜ α }ℜ{ G k ( k ) } . (130)Because longitudinal normal modes need of material support to propagate and are subjectto very specific arrangements of scatterers [42], W Coh. k can be ignored for practical purposes.Let us examine firstly W Coh. in the framework of Classical Optics. It is given by W Coh. = − ω c ǫ | ~p | Z d r ′ Tr − n ℜ{ ¯ χ ( ~r − ~r ′ ) /ρ ˜ α } · ℑ{ ¯ G ( ~r ′ , ~r ; ω ) } o , (131)where both ¯ χ and ¯ G are written in the spatial space representation for convenience and ~p = ǫ ˜ α ~E ω ( ~r ) is the dipole moment induced by the external field ~E ω ( ~r ) on the emitter.Considering the fields classically, the fluctuation-dissipation relation reads ℑ{ ¯ G ω ( ~r ′ , ~r ) } = − πǫ ~ k h ~E ωD ( ~r ′ ) ~E ω ∗ D ( ~r ) i , (132)where ~E ωD ( ~r ) is the ω -mode of the coherent-Dyson field and the script D stands bothfor Dyson and for direct emission. Using Eq.(132) and writing ℜ{ ¯ χ ( ~r − ~r ′ ) /ρ ˜ α } =[ ℜ{ ¯ χ ( ~r − ~r ′ ) /ρ ˜ α } − δ (3) ( ~r − ~r ′ ) I ] + δ (3) ( ~r − ~r ′ ) I in Eq.(131), we separate explicitly the fieldemitted directly by the source dipole from that which is emitted by the induced surrounding52ipoles, W Coh. = πω ~ | ~p | h ~E ωD ( ~r ) · ~E ω ∗ D ( ~r ) i (133)+ πω ~ | ~p | Tr − nZ d r ′ D [ ℜ{ ¯ χ ( ~r − ~r ′ ) /ρ ˜ α } − δ (3) ( ~r − ~r ′ ) I ] · ~E ωD ( ~r ′ ) ~E ω ∗ D ( ~r ) Eo . (134)The term in Eq.(133) is the coherent power carried by the field directly emitted by thesource dipole into the bulk as if it were an external source. In Eq.(134) we can identify thefield emitted at ~r ′ by the induced dipoles sited around the source which propagates towardsthe source located at ~r , E ωI ( ~r ) = Z d r ′ [ ℜ{ ¯ χ ( ~r − ~r ′ ) /ρ ˜ α } − δ (3) ( ~r − ~r ′ ) I ] · ~E ωD ( ~r ′ ) , (135)where the subscript I stands for either induced or indirect . Therefore, we can write W Coh. = W Coh.D + W Coh.I = πω ~ | ~p | h h| ~E ωD ( ~r ) | i + ℜ{h ~E ωI ( ~r ) · ~E ω ∗ D ( ~r ) i} i . (136)As expected, the first term is the coherent power emitted directly by the dipole source whilethe second term is the coherent power associated to the interference of the field emitted bythe source and that emitted by the induced dipoles.As an example, let us consider W [1] ,Coh.I in the single-scattering model of Section VI A.It is given by Eq.(115) –modulo the appropriate prefactors– and its diagram is that ofFig.16( b ). Making the identification [ ℜ{ ¯ χ [1] ( ~r − ~r ′ ) /ρ ˜ α } − δ (3) ( ~r − ~r ′ ) I ] ≃ k ρα Θ( | ~r − ~r ′ | − ξ ) ℜ{ ¯ G (0) stat. ( ~r − ~r ′ ) } , and applying the fluctuation-dissipation relation, it reads W [1] ,Coh.I = W ω π k ~ Tr nZ d r ′ ¯ G (0) stat. ( ~r − ~r ′ ) · [Θ( | ~r − ~r ′ | − ξ ) ǫ ρα h ~E ωD ( ~r ′ )] ~E ω ∗ D ( ~r ) i o , (137)where the Dyson field is fully transverse. The quantity within square brackets is thepolarization density induced by the coherent field on the surrounding host scatterers, ~p ω ( ~r ′ ) = Θ( | ~r − ~r ′ | − ξ ) ǫ ρα ~E ωD ( ~r ′ ). The electrostatic components of the fields emittedby the induced dipoles within the sphere of radius ξ add up at the emitter location, ~r , inthe form of and induced effective field, ~E ′ ωI ( ~r ) = k ǫ Z d r ′ ¯ G (0) stat. ( ~r − ~r ′ ) · ~p ω ( ~r ′ ) . (138)It is the transverse component of the above field that interferes with ~E D ( ~r ), W [1] ,Coh.I = W ω π ǫ k ~ h ~E ′ ωI ( ~r ) · ~E ω ∗ D ( ~r ) i . (139)53ecause ~E ′ ωI ( ~r ) has its source in ~p ω ( ~r ′ ) and this is excited by ~E ωD ( ~r ′ ), the interference iscoherent.In experiments, it is possible in principle to measure both the total and the coherentintensity. One can think for instance of a medium made of dipole antennas in which oneof them is excited by some external fixed field of frequency ω = ck . Part of the powersupplied is absorbed directly by the source antenna located at ~r , which is given by the termof Eq.(41). The rest is radiated. Again, part of the radiation will be extinguished, but W Coh. ( ~r ) as given in Eq.(131) will propagate coherently through the medium and can becollected, eventually, by some distant receiver. If the medium can be treated as an effectivemedium, the coherent intensity collected by a receiver located at ~r ′ will be given by W Coh. ( | ~r ′ − ~r | ) = W Coh. ( ~r ) exp [ − k | ~r ′ − ~r | ¯ κ ] , (140)where ¯ κ ≡ ℑ{ p χ eff } is the extinction coefficient which contains both dispersion and ab-sorbtion by the host scatterers.It is also instructive to compute W Coh. for the stimulated emission of one of the atoms of amonoatomic dielectric close to the resonance. Let us assume the polarizability of the atomsadjusts to the Lorentzian function of Eq.(44). Using Eq.(40), we can write the coherentpower emitted by the stimulated atom sited at ~r as W Coh.atom = − ω ǫ c ρ | ˜ α (˜ k ) | | ~E ω | Tr − nZ d r ′ ℜ{ ¯ χ ( ~r, ~r ′ )˜ α }ℑ{ ¯ G ( ~r ′ , ~r ; ω ) } o(cid:12)(cid:12)(cid:12) ˜ k = ω/c (141)Close to the resonance, ˜ α ( k res ) ≈ i ck res α / Γ is nearly pure imaginary and the total powersupplied reads W ω res T ot. = ω res ǫ | ~E ω res | ℑ{ ˜ α ( k res ) } ≈ ω res ǫ | ~E ω res | | ˜ α | . (142)We can write ℑ{ ¯ G ( ~r ′ , ~r ; ω ) } in terms of ˆ ~E ωD using the fluctuation-dissipation relation ofEq.(132) and, making use of the constitutive relation for coherent fields, write the polariza-tion vector as ~P ω ( ~r ) = ǫ Z d r ′ ¯ χ ( ~r, ~r ′ ) · ~E ωD ( ~r ′ ) . (143)Finally, by inserting the above results into Eq.(141) we obtain the more familiar expression W Coh.ω res = − π ~ W ω res T ot. ℑ{h ρ ~P ∗ ω res ( ~r ) · ~E ω res D ( ~r ) i} , (144)where ρ ~P ∗ ω res ( ~r ) is the average molecular dipole moment. Therefore, we see that in the VCscenario there is a well defined ratio between the coherent emission of a stimulated dipole54lose to resonance and the total power supplied. That ratio being given as a function ofcoherent fields. 2. The coherent vacuum and the additional polarization of the sourceless vacuum In the following we interpret the coherent emission in QFT terms. From Eqs.(132,136), weconclude that the states accessible to direct coherent emission are the normal modes whichsatisfy Eqs.(125,126). Therefore, in an effective manner, the vacuum though which thesemodes propagate is the sourceless vacuum, | Ω i s.l. . This effective equivalence is analogous tothat found in Section II between | Ω i s.p. D and | i for a unique dipole in free space. | Ω i s.l. ismade of the superposition of EM vacua, {| φ m i s.l. } , in which the normal modes of Maxwell’sequations with permitivities { ǫ m ( ~r ) } propagate. Following the formalism of Section II C,the m th pair of vacuum state | φ m i s.l. and dielectric permitivity ǫ m ( ~r ) is associated in aone-to-one correspondence to the m th configuration of host scatterers such that | Ω i s.l. = P m √ M mm | φ m i s.l. , X m,n s.l. h φ m | ˆ M ˆ ~E ω ( ~r ) ˆ ~E ω † ( ~r ) | φ n i s.l. = s.l. h Ω | ˆ ~E ω ( ~r ) ˆ ~E ω † ( ~r ) | Ω i s.l. = − ~ ω ǫ πc ℑ{ ¯ G ( ~r, ~r ′ ; ω ) } . (145)On the other hand, it is plain from Eqs.(127,129) that the modes which contribute to W Coh. are the same as those which propagate in | Ω i s.l. . However, the amplitude of thespectrum of fluctuations which enter W Coh. differs w.r.t. that of the fluctuations in | Ω i s.l. by a multiplicative factor ℜ{ ¯ χ/ρ ˜ α } . In other words, the EM vacuum in which coherentemission propagates, | Ω i Coh , is additionally polarized w.r.t. | Ω i s.l. . The renormalizationfunction is given by Z ω ⊥ , k = ℜ n χ ω ⊥ , k ρ ˜ α o , which takes account of the polarization due to theclosest scatters surrounding the emitter. We will see later on that Z ω ⊥ , k is the so-called localfield factor in the theory of the effective medium. In the framework of QFT, Z ω ⊥ , k can beinterpreted as the field-strength renormalization factors. In a scalar QFT it is the analyticalstructure of G ( k ) together with Z in the complex plane that determine the amplitude ofthe vacuum fluctuations and the mass spectrum of the particles propagating in space-time[29]. Correspondingly, it is the analytical structure of G ⊥ , k ( k ) together with ℜ{ χ ⊥ , k ( k ) ρ ˜ α } thatdetermine the amplitude of the EM vacuum fluctuations and the k prop. ⊥ , k -spectrum of the55oherent photons propagating in a random medium. Following up the QFT interpretation, Z ω ⊥ , k ( k ) yields the probability density for creating a photon of frequency ω from vacuum atthe emitter site, Z ω ( k ) = | Coh h Ω | E ⊥ , k ( ~r ) | γ ωk i| . Out of those photons, the coherent ones areselected and propagated through the bulk by G ω ⊥ . Also, the electric field gets renormalizedand the field so renormalized is nothing but the Dyson field, ~E ωD | ⊥ , k = Z − / ⊥ , k ~E ω ⊥ , k . This leadsto the equivalence relation, s.l. h Ω | ˆ ~E ω ( ~r ) ˆ ~E ω † ( ~r ) | Ω i s.l. = Coh h Ω | ˆ ~E ωD ( ~r ) ˆ ~E ω † D ( ~r ) | Ω i Coh = − ~ ω ǫ πc ℑ{ ¯ G ( ~r, ~r ′ ; ω ) } . (146)Thus, in passing from ~E ωD to ~E ω , any correlation function between vacuum states picks upa factor [ Z ω ] / per field operator.All this suggests that an effective electromagnetic theory in random media can be formu-lated in terms of the renormalized quantities –see eg. Ch.7-10 of [29]. The host scatterersplay the role of bare vertices of photon interactions as depicted in the diagrams of Fig.15.The correlations, depicted by dashed lines in our diagrams, play an analogous role to thatof the gauge fields which mediate interactions between charged fields in QFTs. If corre-lations are assumed not to have dynamics in accordance to the weak-coupling-Markovianapproximation, they must be intended as a back-ground field. 3. Radiation in an effective medium In this section we deal with the effective medium theory, which is implicitly used in themajority of works on dipole emission. In this context, and only in this context, an analogcontribution to the radiative emission in free space can be defined.The effective medium theory is defined such that, for some range of frequencies and for kξ ≪ 1, an effective homogeneous complex dielectric constant can be defined as ǫ eff ≡ Lim. { ǫ ⊥ , k ( k ) } as kξ → 0, so that the Dyson propagator G ⊥ ( k ) can be approximated by theeffective propagator G eff ⊥ ( k ) for some range of frequencies, G eff ⊥ ( k ) ≡ ǫ eff k − k , G eff k ( k ) ≡ ǫ eff k . (147)56he field with propagator ¯ G eff will be referred to as Dyson-effective field or macroscopic field, ~E effD . Correspondingly, the field with propagators G eff ⊥ ( k ) ≡ χ eff ρ ˜ α ǫ eff k − k , G eff k ( k ) ≡ χ eff ρ ˜ α ǫ eff k , (148)will be referred to as local-effective field ~E effloc . Further on, in order for G eff ⊥ ( k ) to be theGreen function of a propagating field of wave number k it is required k/k & kξ ≪ 1, which implies k ξ ≪ & /ξ . The neglectedmodes correspond to the near field dispersion and absorbtion carried out by the scattererssurrounding the source –see Fig.16 and also [15]. In turn, this implies the neglect of alllongitudinal emission and the restriction of the integration domain in Eq.(123) for transverseemission, 2 W eff ⊥ ≡ W ⊥ | k ξ ≪ k ≪ ξ − ≃ W o Z d k (2 π ) ℑ n χ eff ρ ˜ α ǫ eff k − k o (149)= 2 ˜ W o ℑ n χ eff ρ ˜ α √ ǫ eff o , ˜ W o ≡ − k π W o . (150)The coherent component of the above formula, 2 W eff ⊥ | Coh. can be written in an analogousmanner to the radiative emission in free space,2 W eff ⊥ | Coh. = 2 ˜ W o ℜ n χ eff ρ ˜ α o ℜ{√ ǫ eff } . (151)2 W eff ⊥ | Coh. can be given in closed form for a Maxwell-Garnett (MG) fluid. In such a fluidthe only relevant correlation between host particles is the negative correlation due to theexclusion volume ∼ (4 π/ ξ which prevents scatterers from overlap. Provided k ξ ≪ ǫ eff obeys the MG formula [58], ρ ˜ α = ǫ eff − ǫ eff +2) / . It was firstly proved in [15] and then in[59] following a diagrammatic approach that the MG formula is correct at all orders in ρ ˜ α .Note however that the usage of the renormalized polarizability ˜ α incorporates in our casethe effects of recurrent scattering. Using the MG formula, Eq.(150) reads2 W MG ⊥ | Coh. = 2 ˜ W o ℜ n ǫ eff + 23 o ℜ{√ ǫ eff } . (152)It is in order to make a couple of comments regarding the use of effective fields in thecomputation of LDOS, W and Γ. First, in Eq.(149) the symbol ≃ stands for the approxi-mation R 1. That implies57hat the integral of Eq.(149), which is the inverse Fourier transform of G eff ⊥ ( k ) evaluated at r = 0, does not actually carry information about the fluctuations of the microscopical localself-polarization field evaluated around the emitter location, r ≤ ξ . Instead, the fluctuationsthere are those of the field ~E effloc , which is the local field spatially-averaged in volumes ofthe order of ξ over which also the spatial average of ǫ ( r ) is performed in order to get ǫ eff .Therefore, because the contribution to W k comes from high frequency modes in the range k & /ξ , it is not possible to take the effective dielectric constant limit kξ ≪ χ k ( k ).Differently to normal propagating modes, the electric field of the source dipole is not ap-proximately uniform within a cavity volume of the order of ∼ ξ and the emitted photonsdo not see an effective (uniform) dielectric constant there. In the real large-cavity scenariohowever, it might still happen that the field at distances 1 /R is approximately the same asthat at 1 / ( R + ξ ), ξ being the typical dispersion length scale of ¯ χ in the host medium. It isonly in that case and for k ξ ≪ χ uniform outside the realcavity.Second, let us assume for a moment that the effective field approximation is good enough.It is known long ago that the effective field which appears in Fermi’s Golden rule in the formof the equal-point correlation functional, h ~E ( r ) ~E ( r ) i , is not ~E effD ( ~r ) but ~E effloc ( ~r ). Furtheron, assuming the medium is passive, emission restricts to 2 W eff ⊥ | Coh. . In application of thefluctuation-dissipation theorem,2 W eff ⊥ | Coh. = W o Z d k (2 π ) ℜ{ χ eff ρ ˜ α }ℑ{ G eff ⊥ ( k ) } = − πǫ ~ k W o ℜ{ χ eff ρ ˜ α }h ~E eff ⊥ D ( r ) ~E eff ⊥ D ( r ) i . (153)Notice that it contains just a factor ℜ{ χ eff ρ ˜ α } , which equals a Lorentz-Lorentz local fieldfactor for the case that MG formula applies, L LL = ℜ{ ǫ eff +23 } . It is a common mistake tothink that L LL should appear as squared in front of h ~E effD ( r ) ~E effD ( r ) i in Fermi’s Golden rule.The reason being that while the propagator of ~E effloc ( r ) is that of ~E effD ( r ) multiplied by L LL , G eff ∼ L LL G eff , the fluctuation-dissipation theorem applied over G eff yields a contributionquadratic in ~E D but linear in L LL . This contradicts the paradigm which states that theeffect of a dielectric host is to multiply each occurrence of the dipole moment of a a two-levelatom by a local-field factor [22] –see Section IX for further discussion.58 . On the radiative and non-radiative energy transfer Taking advantage of the above study, we can interpret the radiative and non-radiativenature of the processes involved in the dipole emission of the single scattering model. Letus consider the diagrams in Fig.16. They represent either the stimulated emission or thespontaneous emission of an emitter in the single-scattering approximation –Eq.(103). In thecase of stimulated emission, µ stands for the polarizability assigned to the emitter. Thepropagators on the right hand side of the emitter are part of the bulk propagator G . Actualemission is denoted by wavy lines while induction is mediated by propagators depicted bysolid lines. Curved propagators on the left hand side of the emitter indicate being con-strained by spatial correlations within χ . Emission with origin at the emitter is direct. It isinduced emission otherwise.Since the work of Andrews [28, 60], it is well appreciated that the radiative and non-radiative energy transfer between a donor and an acceptor correspond to a unique quantummechanical process. When no induction exists, the emission is direct and there is no trans-fer of energy. It corresponds to the diagrams ( a ), for in-free-space emission; and ( b ), whichstands for single scattering in the bulk –Eq.(113).Induced emission processes, either coherent or incoherent, are characterized by two fea-tures. First, there exists some correlation between the donor and the acceptor. Second, itis one of the transverse virtual photons of frequency ω res which mediated the interactiondonor-acceptor, that is made actual when satisfying the dispersion relation of Eq.(125). Thepair donor-acceptor behaves as an effective dipole. That in diagram ( c ) is coherent and sois its (transverse) radiation, as correlations apply to the electrostatic induction –Eq.(115).On the contrary, that in diagram ( d ) is incoherent, as correlations apply over the transverseinduction whose photon is the one becoming actual –Eq.(118). As explained above, theprocesses in ( c ), ( d ) carry one local field factor each in the single scattering approximation.They are the processes which can be considered to amount to radiative transfer.The non-radiative transfer processes are those in diagrams ( e, f, g ). They are character-ized by two features. First, the actual photons have their origin either in radiative correc-tions over the acceptor bare polarizability or on the imaginary part of the bare polarizability.Hence, the corresponding terms are all proportional to ℑ{ ˜ α } . Only radiative corrections aredepicted in Fig.16. Second, the transference of energy takes place through either purely59diagrams in ( e )– or partially –diagrams ( f, g )– longitudinal loops which remain closedduring the re-emission process. In general the spectrum of energy levels of the acceptorpresents a number of channels into which the quantum of energy absorbed, ~ ω res , can bere-emitted. Therefore, re-emission takes place in a variety of frequencies and the associateddecay rate is said non-radiative. Note however that this implies a slight inconsistency inthe case that donor and acceptor be identical two-level atoms. That is so because in sucha case the acceptor can only re-emit in the same frequency as that of the direct radiation.Nevertheless, because two-level dipoles are unrealistic idealizations, we will consider thoseprocesses as non-radiative without loss of generality. The diagrams in ( e ) stand for theterms in Eqs.(116,117). They would diverge if considered separately. They represent thephenomenon commonly referred to as F¨oster Resonant Energy Transfer (FRET) [61]. Thediagrams ( f ) and ( g ) correspond to the absorptive longitudinal and transverse terms ofEq.(118) and Eq.(115) respectively. VII. ANALYTICAL CALCULATIONS IN MAXWELL-GARNETT DI-ELECTRICS An MG dielectric is one made of point dipoles with well defined single particle polarizabili-ties. The correlation length ξ among dipoles satisfies k ξ ≪ ρ ˜ α = ǫ eff − ǫ eff + 2) / . (154)Hereafter we will denote ǫ eff with ǫ MG instead. In this scenario it is possible to obtain ana-lytical formulae for the virtual cavity γ -factors. We apply those formulae to two situations ofexperimental interest. In the first one, we compute the decay rate of an excited atom whichcan be either an interstitial optical center or one of the molecules of the dielectric itself. Inthe second situation, we compute the effective dielectric constant for an MG dielectric closeto the resonance frequency. 60 IG. 16: ( a ) Diagrammatic representation of the stimulated emission/spontaneous emission of anemitter in the single-scattering approximation –Eq.(103). Explanation is given within the text. A. Spontaneous decay rate of a weakly-polarizable excited atom Let us consider the explicit computation of Γ for an excited atom. This can be either aforeign ion placed within one of the molecules of the dielectric or a molecule of the dielectricitself. In the former case, the manner the emission rate of an optic center gets modified whenplaced within the ligand is a difficult issue as, in general, one should consider the internalatomic structure of the host molecule [62–64]. To simplify matters, we assume here that thebare dipole transition amplitude of the ion and its resonance frequency get renormalized toknown values µ , ω res as a result of the interactions with other atoms within the ligand. Thisscenario is that of Section III C. The electrostatic polarizability of the doped molecule, α ,is carried out by some other polarizable atoms, being α equivalent to the bare polarizabilityof the rest of the host scatterers. The total dipole moment associated to the particle which61osts the ion is the combination of a fixed and an induced dipole, ~p host = ~µ + ω c ˜ α ¯ G ω ( ~r, ~r ) ~µ ,˜ α being the renormalized polarizbility of the host particle.The case of an excited host particle can be treated in the same fashion provided thatthe additional condition ξ ≪ ρ − / is satisfied and the effective polarizability of the excitedmolecule is negligible –see Section V A. That corresponds to the setup of Section III C with α = 0 and assuming previous knowledge of µ and ω res .In this scenario, the only correlation function which matters is the two-point functionwhich amounts to the exclusion volume of each atom, h ( r ) = − Θ( r − ξ ). For higher corre-lation orders and following [15] we will make use of the overlap approximation. Ultimately,because Eq.(154) establishes a relationship between χ MG and ˜ α , and the γ -factors can beexpressed as power series of ρ ˜ α , a relationship can be given between Γ, LDOS and the ef-fective parameters which determine the transport of coherent light. Those are the index ofrefraction, ¯ n , and the extinction mean free path, l ext , which relate to χ MG through¯ n = ℜ{ p χ MG } , l − ext = 2 ωc ℑ{ p χ MG } . (155)First, we compute G ⊥ . In the overlap approximation, χ ⊥ ( k ) = ρ ˜ α − χ (2) ⊥ ( k ) / ( ρ ˜ α ) , (156)where at resonance, χ (2) ⊥ ( k ) = − ( ˜ αρ ) k res Z d r e i~k · ~r h ( r )Tr { ¯ G (0) ( r )[ I − ˆ k ⊗ ˆ k ] } = − ( ˜ αρ ) k res Z d k ′ (2 π ) h ( | ~k ′ − ~k | ) h G (0) ⊥ ( k ′ )+ G (0) ⊥ ( k ′ ) cos θ + G (0) k ( k ′ ) sin θ i . (157)For an MG dielectric, h ( Q ) = − πξ j ( Q ) /Q = 4 πξ Q cos Q − sin QQ with Q ≡ | ~k ′ − ~k | ξ and j thespherical Bessel function of first order. Applying next Eq.(58) at the resonance frequency,we find G ⊥ ( k ) = G (0) ⊥ ( k ) h ρ ˜ α [ k res G (0) ⊥ ( k ) − χ (2) ⊥ ( k ) / ( ρ ˜ α ) ] i − . (158)From the above expression it is not straightforward to differentiate between the coherentand the incoherent components of 2Γ ⊥ as given in Eqs.(127,128) for the power emission.However, it is possible to go around this problem by first computing Eq.(158) order by order62n ζ ≡ k res ξ and then find out the origin of those terms in view of Eqs.(127,128). First, wewrite 2 γ T ot. ⊥ as2 γ T ot. ⊥ = k res π ζ nZ ∞ d Q Q ζ − Q ∞ X n =0 ( − ρ ˜ α ) n [ ζ ζ − Q − ( ρ ˜ α ) − χ (2) ⊥ ( ζ , Q )] n o , (159)where Q ≡ kξ . Careful examination of the integrand of Eq.(159) yields the following de-composition which includes all the terms up to O (0) in ζ ,2 γ T ot. ⊥ ≃ k res π ζ n ∞ X n =0 ( − ρ ˜ α ) n Z ∞ d Q Q ζ − Q [ ζ ζ − Q − / n (160)+ ∞ X n =1 ( − ρ ˜ α ) n Z ∞ d Q [ j ( Q ) /Q ] n o (161)= 2 γ T ot.A ⊥ + 2 γ T ot.B ⊥ . (162)The integrand of Eq.(160) contains the poles of G ⊥ which satisfy the dispersion relations ofcoherent modes. Eq.(161) carries those very high frequency modes, k ≫ k res , which containthe near field contribution. By taking the limit ζ → 0, it is immediate to identify in Eq.(160)the factor − / ρ ˜ α ) with the zero mode of − χ (2) ⊥ ( k ). In the overlap approximation, thatleads to the MG formula for the effective susceptibility. Hence, it is χ (2) ⊥ ( k = 0) = 1 / ρ ˜ α ) that enters in the place of χ (2) ⊥ ( k ) in Eq.(158) for G eff ⊥ . We take advantage of this to arrangethe leading order terms of Eq.(160) (which are those of O (0) in ζ ). We obtain,2Γ A ⊥ = 2˜Γ o ℑ{− γ T ot.A ⊥ } πk res ≈ o ℜ n ǫ MG + 23 √ ǫ MG o , (163)˜Γ o = k res ǫ π ~ | µ | (cid:12)(cid:12)(cid:12) − i π k α + 13 k α [2 γ ⊥ + γ k ] (cid:12)(cid:12)(cid:12) − , (164)where ˜Γ o contains the γ -dependent factor which renormalizes | µ | due to the polarizationof the host particle. It is also immediate to verify that the coherent contribution of 2Γ A ⊥ isprecisely 2Γ MG ⊥ | Coh. as given in Eq.(152) (modulo prefactors),2Γ A,Coh. ⊥ = 2Γ MG ⊥ | Coh. = 2˜Γ o ℜ n ǫ MG + 23 o ℜ{√ ǫ MG } , (165)while 2Γ A,Incoh. ⊥ = − o ℑ n ǫ MG + 23 o ℑ{√ ǫ MG } . (166)Therefore, at leading order in ζ = k res ξ , we conclude that the coherent emission –and only!the coherent emission– can be computed in the framework of the effective dielectric constant63pproximation.On the contrary, incoherent emission does not only contains the term 2Γ A,Incoh. ⊥ , but alsothe terms in 2Γ B ⊥ which are proportional to ℑ{ γ T ot.B ⊥ } given in Eq.(161). 2 γ T ot.B ⊥ takesaccount of those non-radiative high frequency modes which scale as 1 /ζ . Eq.(161) containsterms of the form I (1) p,q ≡ Z ∞ d Q sin ( pQ ) Q q +1 with p < q + 1 , n ≤ p ≤ n, (167) I (2) p,q ≡ Z ∞ d Q cos ( pQ ) Q q with p < q, n ≤ p ≤ n. (168)With increasing value of n ≫ I (1) p,q = π − q p q (2 q )! (169) I (2) p,q = π − q p q − (2 q − . (170)Therefore, for any finite value of ρ ˜ α , the corresponding series in equation Eq.(161) is rapidlyconvergent. We give below the first five terms of that series,2 γ B ⊥ = − k res π ζ h ρ ˜ α − 215 ( ρ ˜ α ) + 471280 ( ρ ˜ α ) − ρ ˜ α ) + 68916232145927168( ρ ˜ α ) + ... i . (171)Let us consider next the computation of γ k . The Green function for longitudinal modesreads G k ( k ) = 1 k res χ k ( k ) ρ ˜ α [1 + χ k ( k )] − , (172)where in the overlap approximation, χ k ( k ) = ρ ˜ α − χ (2) k ( k ) / ( ρ ˜ α ) , (173)with χ (2) k ( k ) = − ( αρ ) k Z d r e i~k · ~r h ( r )Tr { ¯ G (0) ( r )[ˆ k ⊗ ˆ k ] } = − ( αρ ) k Z d k ′ (2 π ) h ( | ~k ′ − ~k | ) × h G (0) k ( k ′ ) cos θ + G (0) ⊥ ( k ′ ) sin θ i . (174)The above equation can be computed in closed form, χ (2) k ( Q ) = ( ρ ˜ α ) [1 + j ( Q ) Q f ( ζ )] , f ( ζ ) = 2 i e iζ ( i + ζ ) . (175)64ikewise, it is possible to give a closed expression for G k ( Q ) using the MG relationship for ǫ MG in function of ρ ˜ α , G k ( Q ) = 1 k res [1 + 6 ǫ MG − ǫ MG + 2 j ( Q ) Q ] − . (176)The use of this expression in the computation of Γ k implies the computation of an infiniteseries of roots of 1 + 6 ǫ MG − ǫ MG +2 j ( Q ) Q [15]. Instead of that, we rather expand Eq.(172) in powersof ρ ˜ α as this allows to keep control over the convergence of the series, G k ( Q ) = 1 k res ∞ X n =0 ( ρ ˜ α ) n ( − n f n ( ζ )[ j ( Q ) /Q ] n . (177)Finally, we arrive at γ k = k res π ζ ∞ X n =1 ( − n +1 ( ρ ˜ α ) n f n ( ζ ) Z ∞ d Q Q [ j ( Q ) /Q ] n . (178)Note the similarity between the above equation and that for 2 γ T ot.B ⊥ in Eq.(161). This is atthe root of the coincidence of the terms of order ρ ˜ α/ζ in the single-scattering model studiedpreviously. The convergence the series in Eq.(178) can be verified using similar argumentsto those employed for Eq.(161). γ k contains however additional terms of order 1 /ζ andorder zero in ζ as a result of exponenzing the function f ( ζ ) in Eq.(175). We give below thefirst five terms of the orders O ( ζ − ), O ( ζ − ) and O ( ζ ), γ k ≃ − k res π ζ [ ρ ˜ α + 23 ( ρ ˜ α ) + 524 ( ρ ˜ α ) + 2722835 ( ρ ˜ α ) + 40949870912 ( ρ ˜ α ) + ... ] (179) − k res π ζ [ 12 ρ ˜ α + 23 ( ρ ˜ α ) + 516 ( ρ ˜ α ) + 5442835 ( ρ ˜ α ) + 2047451741824( ρ ˜ α ) + ... ] (180) − i k res π [ 13 ρ ˜ α + 49 ( ρ ˜ α ) + 524 ( ρ ˜ α ) + 10888505 ( ρ ˜ α ) + 2047452612736 ( ρ ˜ α ) + ... ] . (181)For an MG dielectric, it is possible to distinguish between dispersion and absorbtion in thesame manner we did for the simple single-scattering model in Section VI B 4. In particular,the terms in Eq.(179) are the analog of a FRET process for the case that the transfer ofenergy takes place between the emitter and an MG dielectric. Below, we write the completeformulae for the different contributions to the decay rate (up to renormalization prefactors)65n function of the effective dielectric constant,Γ Coh.MG = 2˜Γ o ℜ n ǫ MG + 23 o ℜ{√ ǫ MG } , (182)Γ k ,Disper.MG = 2˜Γ o ℜ{ χ MG + 13 χ MG − . χ MG + 0 . χ MG − . χ MG + ... } (183)Γ Absorb.MG = − o ℑ n ǫ MG + 23 o ℑ{√ ǫ MG } (184)+ 2˜Γ o ζ ℑ{ χ MG + 12 χ MG − . χ MG + 0 . χ MG − . χ MG + ... } (185)+ 2˜Γ o ζ ℑ{ χ MG + 13 χ MG − χ MG + 0 . χ MG − . χ MG + ... } (186)Note however that when multiple-scattering processes are involved, it might well be thatpart of the decay attributed to absorbtion is actually due to dispersion by clusters of cor-related host dipoles. A microscopical analysis of the origin of ℑ{ χ MG } would be needed todiscriminate between both extinction processes.At this point, it is interesting to compare the order O ( ζ ) term of Γ MGrad. to that usuallyemployed in the literature which includes the square of the Lorentz-Lorenz local field factor,Γ LL = 2˜Γ o (cid:16) ǫ MG +23 (cid:17) √ ǫ MG . In our computation, Γ MGrad. = Γ Coh.MG + Γ k ,Disper.MG . These terms arethe relevant ones for emission in absence of absorbtion, ℑ{ ǫ MG } ≪ ℜ{ ǫ MG } . Expanding inpowers of χ MG = ǫ MG − 1, we obtain up to order five,Γ MGrad. = 2˜Γ o ℜ n χ MG + 38 χ MG − . χ MG + 0 . χ MG − . χ MG + ... o (187)Γ LL = 2˜Γ o ℜ n χ MG + 2372 χ MG + 0 . χ MG − . χ MG + 0 . χ MG + ... o . (188)Therefore, in absence of extinction, the agreement is almost perfect up to order χ MG . Aswe will argue in the next section, this agreement is just accidental. Therefore, in orderto differentiate between both formulae in non-absorptive media, it is necessary that higherorder terms be relevant. That is, the index of refraction has to satisfy ¯ n & √ 2. Never-theless, we must point out that it has been assumed implicitly in Eqs.(187,188), by usinga unique value for ˜Γ o , that the renormalization factor which accompanies µ is the same inboth equations. For the decay of a weakly-polarizable host molecule this is not a problem.However, for the decay of a foreign particle within a host molecule this might not be so.Even for a non-dissipative medium the terms of orders O ( ζ − ) and O ( ζ − ) are relevant (seeEq.(49)) and they do not appear in the Lorentz-Lorenz approach. Therefore, any try todifferentiate between both approaches must necessarily assume some ad hoc common value66or the renormalization factor. We give below the γ -factors which enter the renormalizationfactor in Eq.(164) bearing in mind that such a renormalization factor must be set to 1 forthe case of a weakly-polarizable excited host molecule,2 γ ⊥ + γ k = γ ζ + γ ζ − + γ ζ − (189)= − i k res π [ 76 χ MG + 38 χ MG − . χ MG + 0 . χ MG − . χ MG + ... ](190) − k res π ζ [ χ MG + 15 χ MG + 0 . χ MG − . χ MG + 0 . χ MG + ... ] (191) − k res π ζ [ χ MG + 13 χ MG − χ MG + 0 . χ MG − . χ MG + ... ] . (192)The above formulae are thought to fit the experimental results analyzed quantitatively in[65]. There, Ce +3 ions of low polarizability replace low-polarizability cations of several hostmedia. The life-time of the transition 5 d → f is measured in several hosts and the dataare fitted in [64] to the usual VC and RC formulae. The authors find ’relative’ agreementwith the VC model. However, in their fits they leave the value of µ to be fitted as well,introducing this way an additional degree of freedom which improves artificially the fits.Therefore, their results are far from conclusive. B. The dielectric constant of a Maxwell-Garnett dielectric The problem in this case is one of self-consistency. On the one hand, we must computethe renormalized single-particle polarizability of the molecules, ˜ α , due to their embedding inthe dielectric. On the other hand, the renormalization of ˜ α gives rise to a renormalization ofthe dielectric constant itself with respect to the MG formula involving in-free-space polariz-abilities. Following the renormalization scheme developed in Section III B, beside the valueof α and the in-free-space resonance wave number, k , knowledge of the exclusion volume(i.e. the correlation length ξ ) and of non-radiative effects (i.e. the collisional shift ∆ k coll. and collisional line broadening Γ coll. ) are needed. While the latter effect are thought to berelevant at high temperature, radiative effects are expected to dominate at low temperatureand for frequencies close to the resonance.A possible setup which adjusts to this scenario corresponds to that in the experimentcarried out in [66]. There, selective-reflection techniques are employed to measure the fre-quency shift in a high temperature potassium gas. At the frequencies of interest, the bare67lectrostatic polarizability reads α = f πr e k − where r e = e πǫ m e c = 2 . 82 10 − nm is theelectron radius, the resonance wavelength is λ = 2 πk − = 770 . nm and the strength factoris f = 0 . 339 for the transition 4 S / ⇄ P / [67]. The range of atomic densities inthe experiment is 10 m − . ρ . · m − . We can keep the renormalization scheme ofSection III B by just replacing α with a renormalized static polarizability which includesboth the regularization of 2 ℜ{ γ (0) ⊥ } and collision factors, α coll.stat. (˜ k ) = α k [ k + ∆ k coll. − i Γ coll. c ˜ k − ˜ k ] − . (193)This leads to the Lorentzian renormalized single-particle polarizability,˜ α (˜ k ) = α k h k + ∆ k coll. − i Γ coll. c ˜ k − ˜ k + 13 α k ˜ k ( i ℑ{ γ (0) ⊥ } + 2 γ ⊥ + γ k ) ˜ k i − . (194)Next, in application of MG formula, the effective dielectric constant reads˜ ǫ MG (˜ k ) = 1 + ρ ˜ α (˜ k )1 − ρ ˜ α (˜ k ) = 1 + χ MG (˜ k ) = 1 (195)+ ρα k h k + ∆ k coll. − i Γ coll. ˜ k/c − ˜ k + 13 α k ˜ k ( i ℑ{ γ (0) ⊥ } + 2 γ ⊥ + γ k ) ˜ k − α k ρ i − . The γ -factors are those of Eqs.(189-192). Because they depend on χ MG , the problem be-comes one of self-consistency. Self-consistency can be seen also as a consequence of includingrecurrent scattering in the derivation of the MG formula. In physical grounds, it reflects thecomplementarity of the double role played by the actual dipoles. That is, on the one handthey polarize the sourceless EM vacuum characterized by ǫ MG . On the other hand, theyrenormalize their own polarizability, ˜ α .Once the self-consistency problem has been solved in terms of χ MG , the γ -factors canbe computed and the renormalized values of Γ, ω res and α follow the equations of SectionIII B. Note that the resonance frequency of ˜ ǫ MG differs w.r.t. to that of ˜ α in the so-calledLorentz-shift, ∆ k L = − α k ρ .The authors of [66] however did not include all the possible radiative effects in their anal-ysis. They restricted themselves to the Lorentz-shift, ∆ k L . Because ∆ k L does not dependon χ MG , no self-consistency was demanded. In the experimental setup of [66] no furtherradiative effects are relevant despite the fact that the medium is highly opaque. The reasonbeing that collisional effects dominate. A detailed analysis of the problem –see [68]– revealsthat this is the case. The main reason is that those ζ dependent terms in Eqs.(191,192),68hich would be expected to become large for | χ MG | ∼ k ξ ≪ 1, turn out to be irrelevantin comparison to the collisional contributions. For a cold gas of potassium, a estimate for ξ would be the van-der-Waals radius of K atoms, which is roughly 0 . nm . However, athigh temperature ξ is rather determined by the collision cross section between atoms, whichturns out to be much greater. It would still be possible to increase the relative weight of theradiative terms with respect to the collisional ones by lowering the atomic density. However,for a precise calculation we would still need a good estimation for ξ .To the opposite extreme belongs the scenario in which the medium is an atomic glass. Insuch a case, there is the advantage that ξ can be accurately determined. In fact, ξ ∼ ρ − / .However, as pointed out in Section II C, atomic orbitals overlap in such a way that theelectronic band configurations can be very different to those of the individual atoms iso-lated. On top of that, there might be contributions of free electrons, which introduce anadditional source of uncertainty in computing that component of the susceptibility due toplasma oscillations [69].In view of the difficulties that high temperature atomic gases and solids present in evalu-ating the radiative contributions to the renormalization of the dielectric constant, we proposealternative setups. They must be such that undesired dynamical and collisional effects bereduced to a minimum. In addition, the electronic structure of the dipole-constituents mustbe hardly altered by their cohesive interactions. A cold atomic gas offers a scenario whichadjusts to these requirements [70]. The other possibility is the preparation of colloidal liq-uids [71]. There, particles of well defined polarizability are in suspension on a backgroundsolvent. The particles are ionized in such a way that the spatial correlation among themis determined by the Coulombian long-ranged mutual repulsion and crystallization is sup-pressed. The degree of order and the value of ξ can be tuned to satisfy the conditions of anMG dielectric. VIII. SPONTANEOUS DECAY OF A LOW-POLARIZABILITY SUBSTANTIALIMPURITY In this section we compute the decay rate, Γ, of a fluorescent weakly-polarizable emitterwith transition dipole moment µ and resonance frequency ω res = ck . The emitter locatesat the center of a large cavity of radius R much larger than the correlation length ξ among69 IG. 17: ( a ) Cermet topology scenario. The emitter with polarizability α is surrounded by adisconnected net of scatterers with correlation length ξ . ( b ) Simply-connected-non-contractibletopology in which the emitter is placed at the center of a cavity of radius R surrounded by acontinuous medium of dielectric constant ǫ . host scatterers. The polarizability of the emitter is low in comparison to that of the hostscatterers. The long-wavelength limit applies on the host medium, k ξ ≪ 1, such that itbehaves locally as an effective medium of uniform dielectric constant ǫ = 1 + χ . The setupcorresponds to that of Section V B. The topology associated to the embedding of the emitteris that of a simply-connected-non-contractible manifold –see Fig.17( b ). Its experimentalcounter part corresponds to that of Eu ions in random media [72]. Our master formula isΓ = − ω res ǫ c ~ | µ | ℑ{ γ RC ⊥ + γ RC k } , (196)where the γ -factors are those of the RC scenario given by Eqs.(81-96). Following thoseequations, we will compute the γ -factors as power series of χ . Our computation is exactup to order two in χ . In order to compare with the conventional empty-cavity modelformulation, we will restrict ourselves to the case k R ≪ 1, although exact formulae at allthe orders in k R can be obtained beyond this approximation.70he γ -factors are given by2 γ RC ⊥ (˜ k ) = − i ˜ k π − k Z d k (2 π ) [ C ⊥ + G (0) ⊥ ] χ ⊥ G (0) ⊥ ( k )+ 2˜ k Z d k (2 π ) [ C ⊥ + G (0) ⊥ ] G ⊥ χ ⊥ ( k ) , (197) γ RC k (˜ k ) = − ˜ k Z d k (2 π ) [ C k + G (0) k ] χ k G (0) k ( k )+ ˜ k Z d k (2 π ) [ C k + G (0) k ] G k χ k ( k ) , (198)where the cavity factors are those of Eqs.(90,91). Considering that the polarizability of the FIG. 18: ( a ) Feynman’s rules . ( b ) Diagrammatic representation of the transverse and longitudinal γ -factors, 2 γ RC ⊥ and γ RC k , as given by Eqs.(197,198). emitter is negligible and that the interaction of the emitter with the surrounding particlesdoes not modify its energy levels, neither changes on µ nor in the resonance frequency hold.71hat way, we can perform the substitution of ˜ k with k in the above formulae. Up to zeroorder in k R we obtain,2 γ RC ⊥ + γ RC k ≃ − i π k h(cid:16) ǫ + 23 (cid:17) √ ǫ − 49 ( ǫ − ǫ i − k π h(cid:16) k R ) + 1 k R (cid:17) [( ǫ − ǫ + 23 ǫ ] i . (199)Normalizing by the in-free-space decay rate, Γ = ω res πǫ c ~ | µ | , Γ readsΓ = Γ h ℜ{ (cid:16) ǫ + 23 (cid:17) √ ǫ } − ℜ{ ( ǫ − ǫ } (200) − (cid:16) k R ) + 1 k R (cid:17) ℑ{ ǫ − ǫ − ( ǫ − } i . (201)The terms in Eq.(201) are associated to absorbtion in the host medium [73]. We recognizein the first term of the r.h.s of Eq.(200) the usual bulk term corrected by Lorentz-Lorenz(LL) local field factors, Γ LL = Γ ℜ{ (cid:16) ǫ +23 (cid:17) √ ǫ } . The second term there includes correctionsof order & χ . It is also remarkable that if the empty-cavity Onsager–B¨ottcher (OB) localfield factors are used instead, Γ empOB = Γ ℜ{ (cid:16) ǫ ǫ +1 (cid:17) √ ǫ } does agree with the two radiativeterms of Eq.(200) up to order χ ,Γ empOB ≃ Γ (1 + 76 χ − χ + ... ) . (202)Our computation is in fact exact up to O (2). However, the nature of the emission associatedto the terms of Eq.(200) is different to the one attributed in the conventional empty-cavitymodel. In the latter, Γ empOB is considered as fully transverse. We show next that transverse-coherent emission is just part of it, being the rest incoherent longitudinal emission associatedto dispersion. We can read the contribution of coherent emission from the diagrams in Fig.18,Γ Coh. = Γ − πk ℑ{ γ P ⊥ } = Γ − πk h Z ℑ{ G ⊥ ( k ) } d k (2 π ) +2˜ k ℜ{ χC ⊥ ( k ) } Z ℑ{ G (0) ⊥ ( k ) } d k (2 π ) +2 k ℜ{ [ χC ⊥ ( √ ǫk )] } Z ℑ{ G ⊥ ( k ) } d k (2 π ) − k ℜ{ χC ⊥ ( √ ǫk ) } Z ℑ{ G ⊥ ( k ) } d k (2 π ) i , (203)where the first term on the r.h.s of Eq.(203) corresponds to direct-coherent emission –transverse components of diagrams ( b ), ( b ), ( b ) in Fig.18. In the above expressions,72 ⊥ ( √ ǫk ) stands for C ⊥ ( k ) | k = √ ǫk . Γ Coh. can be written in a more compact form asΓ Coh. = Γ − πk h ℜ{ [1 − k χC ⊥ ( √ ǫk )] } Z ℑ{ G ⊥ ( k ) } d k (2 π ) (204)+ 2 ℜ{ k χC ⊥ ( √ ǫk ) } Z ℑ{ G (0) ⊥ ( k ) } d k (2 π ) i , (205)which immediately leads toΓ Coh.RC = Γ h ℜ{√ ǫ }ℜ n(cid:16) ǫ + 23 (cid:17) o − ℜ{ ǫ − } i , (206)at leading order in k R . For the dispersive and absorptive terms we get,Γ k ,Disper.RC = Γ ℜ n ǫ − ( ǫ − ǫ o , (207)Γ Absorb.RC = − Γ h ℑ{√ ǫ }ℑ n(cid:16) ǫ + 23 (cid:17) o (208)+ (cid:16) k R ) + 1 k R (cid:17) ℑ{ ǫ − ǫ − ( ǫ − } i (209)Once more, the same reserves after Eq.(185) about the distinction between absorbtion anddispersion apply. IX. COMPARISON WITH PREVIOUS APPROACHES We have proved that our computation of LDOS emissionω of Eq.(32) with the γ -factorsgiven by Eqs.(60,61) is exact for a microscopically linear and statistically homogeneousdielectric in which the electrical susceptibility tensor can be expanded as in Eq.(55). Herewe review on previous works, compare them with ours and emphasize where the errors inthose approaches are. Because the literature about complex media concentrates in setupswhere an effective medium can be defined, we will restrict ourselves to such a particularscenario. Our analytical study of an MG dielectric of Section VII belongs to that kind. A. The erroneous inclusion of local field factors The majority of approaches on non-dissipative media agree in giving an effective densityof states of the form LDOS emissionω = ǫ c ~ ω h ~E ωLoc. ( ~r ) · ~E ω † Loc. ( ~r ) i ≈ | k ≪ ξ − L f ¯ n LDOS ω , (210)73here L f is a local field factor. The approximation symbol denotes the restriction to long-wavelength modes with k ≪ ξ − which is implicit in the use of the effective mediumtheory. h ~E ωLoc. ( ~r ) i is interpreted as the electric field felt by the emitter located at ~r and h ~E ωLoc. ( ~r ) ~E ω † Loc. ( ~r ) i is the spectrum of its vacuum fluctuations. As pointed out in SectionVI B 3, because modes with k > ξ − are not included in Eq.(210), such a field is actuallynot microscopic but assumed uniform within the cavity of radius ξ . Hereafter we use thenomenclature of Section VI B 3 where ~E effloc. ( ~r ) stands for ~E ωLoc. ( ~r ) in the effective mediumtheory. Likewise, ~E effD ( ~r ) denotes the macroscopic field in the bulk. The usual procedurein the literature is to compute ~E effloc. ( ~r ) classically by imposing boundary conditions at thecavity surface. Those boundary conditions depend on whether the medium inside the cavityis equivalent or not to the medium in the bulk. In both cases, effective media are assumed toexist on both sides and the local and macroscopic fields are related via the induced polariza-tion field [74] –see also [14] for a microscopic rigorous approach. In either case one obtains ~E effloc. = L f ~E effD , in the small cavity limit. In the case the emitter is a dipole equivalent toall the rest, the cavity is virtual and L f = L LL = ǫ MG +23 is the Lorentz-Lorenz local fieldfactor [55]. For the case that the cavity contains an emitter of much weaker polarizabilitythan that of the host scatterers, the cavity is assumed to be empty and L f = L emp = ǫ ǫ +1 [56]. Same approach is followed in more complicated dielectric configurations [75].It is obvious that the above procedure cannot be the correct one as it ignores the modes k & ξ − and hence the ζ -dependent terms in LDOS emissionω . Nevertheless, let us assumefor the moment that the above procedure is legitimum. The first erroneous assumptionwhich appears implicitly in Eq.(210) is that ...the local field factor appears squared, as Γ (orLDOS) can be expressed in terms of and expectation value of the product of the two electricfield operators ( ~E Loc. ) [76] – also in [22, 50, 77, 78]. The point being that the fact that theclassical values of ~E effloc and ~E effD are related by a constant of proportion L f , does not implyby any means that there is a constant of proportion L f between their quadratic fluctuationsin vacuum, ~E effloc ( ~r ) = L f ~E effD ( ~r ) ; h ~E effloc ( ~r ) · ~E eff † loc ( ~r ) i = L f h ~E effD ( ~r ) · ~E eff † D ( ~r ) i . (211)The proof of Eq.(211) was given in Section VI B 3. We found there that the propagator of ~E effloc reads G eff k⊥ ( k ) = Lim. {G k , ⊥ }| kξ → = χ eff k⊥ ρ ˜ α G eff k , ⊥ ( k ) , (212)74ith χ eff ⊥ ρ ˜ α = L LL = ǫ MG +23 , while G eff ⊥ defined in Eq.(147) is the propagator of ~E effD . Ne-glecting extinction, ℑ{ ǫ MG } ≪ ℜ{ ǫ MG } , and in application of the fluctuation-dissipationtheorem, h ~E effloc ( ~r ) · ~E eff † loc ( ~r ) i = − ~ ω ǫ πc L LL Z d k (2 π ) ℑ{ G eff ⊥ ( k ) } = L LL h ~E effD ( ~r ) · ~E eff † D ( ~r ) i ∝ L LL ¯ n, (213)which is proportional to the MG coherent emission computed in Eq.(152) instead. An alter-native manner to understand where the confusion resides consists of interpreting the macro-scopic vector fields as operators acting on the coherent vacuum defined in Section VI B 1. Inthat framework, two factors Z / = L / LL appear in the two-field vacuum expectation valuein passing from h ~E effD ( ~r ) · ~E eff † D ( ~r ) i to h ~E effloc ( ~r ) · ~E eff † loc ( ~r ) i . That is, h ~E effloc ( ~r ) · ~E eff † loc ( ~r ) i = Coh h Ω | ˆ ~E eff ( ~r ) · ˆ ~E eff † ( ~r ) | Ω i Coh = χ eff ρ ˜ α s.l. h| Ω | ˆ ~E eff ( ~r ) · ˆ ~E eff † ( ~r ) | Ω i s.l. = L LL h ˆ ~E effD ( ~r ) · ˆ ~E eff † D ( ~r ) i . (214)The non-necessary implication expressed in Eq.(211) was first appreciated by the authors of[38]. The key argument given there being the lack of noise polarization. The authors of thatwork introduced an operator to account for such an effect and then required consistency.In our approach, we compute exactly the Green’s function of the self-polarization field.Application of the fluctuation-dissipation theorem takes care of the polarization noise in anexact manner.In some old works it is claimed that it is only the bulk density of states, ∼ ¯ n , the oneresponsible for the spontaneous decay which enters Fermi’s Golden rule. That is based on themacroscopic quantization carried out for the first time in [79]. Such interpretation has beenmodified since then as further studies on the role and origin of the local field factors have beenperformed. However, still recent works [35, 49, 80] appeal to the erroneous interpretationthat the density of states accessible to the photons emitted according to Fermi’s Goldenrule is just given by the bulk term. The latter corresponds to density of bulk normal modesinstead which we have denoted by LDOS sourcelessω . The relation between the coherent (andonly coherent!) emission spectrum and LDOS sourcelessω was made clear in Section VI B 2.75 . The actual nature of radiation and the need to go beyond the effective mediumapproximation It is a common error in the literature to identify the total radiative emission with trans-verse emission. As a matter of fact, the expression in Eq.(210) is intended as proportionalto transverse spontaneous emission, eg. [19, 38, 81]. This error is related to the use ofmacroscopic fields and so to the neglect of short-wavelengths.In the first place there is a problem in imposing continuity conditions at the cavity sur-face. This is particularly problematic in the VC scenario. On the one hand, one assumesthat an effective medium exists on both sides of the cavity, which implies some length scaleover which the susceptibility is averaged in space. On the other hand, by imposing some’matching’ conditions at the cavity surface one is assuming that the width of the surfaceis negligible in comparison to the length scale over which the spatial average has been per-formed on each side. However, that width cannot be less than the typical correlation lengthbetween the scatterers, ξ . That is precisely the radius of the inner cavity in which the spa-tial average is performed. Therefore, matching conditions are expected to fail in general butfor those coherent modes which propagate in the ’effective’ medium. Those are, the onesin W Coh. ⊥ and Γ Coh. ⊥ on Eqs.(165,152). Because the rest of the transverse emission contains ζ -dependent terms, it is clear that Eq.(210) cannot be fully transverse. However, we haveseen in Section VII A that Eq.(210) yields almost the right result for the ζ -independenttotal decay rate up to order two in χ –Eqs.(187,188). This accidental coincidence is due theaddition of the longitudinal dispersive terms of Eq.(183). We showed in the single-scatteringmodel that, at leading order, the ζ -independent longitudinal term equals the contributionof a local field factor. The resultant radiation was proved to be incoherent –see Section VI.Nevertheless, it is still possible to distinguish between ours and the usual formula withoutgoing to higher orders in χ . If instead of using an integration sphere to collect the totalradiation in far field the coherent field were measured, the difference could be verified eitherin experiments or in numerical simulations.One possible way to go around the problem with the boundary conditions would be toimpose them not over the classical fields but over the Green’s functions. In the VC modelthis is not a solution as one would face again the problem with the length scales. In theRC model things can be more promising provided R ≫ ξ . However, one has still to assume76ome form for the Green’s functions on both sides as done in [82]. Unfortunately, in doing soall the intermediate correlations between the emitter and the medium are neglected. Onlythe first scattering after the emission of a virtual photon and the last scattering before itsabsorbtion get correlated to the emitter this way.It is also an error to attribute to the whole absorptive ζ -dependent decay rate a longitu-dinal nature. This was already noticed in [82]. In the overlap approximation, some of theabsorptive terms of Eq.(185) correspond to the transverse γ -factor of Eq.(171). C. Why the approach of [57] is not microscopic enough We comment on the approach of [57] specifically because it is referred by some authors asa microscopical proof of the usual formulae of the RC and VC models based on Eq.(210). Inthe following and for the sake of brevity we will not write all the equations in [57] but onlythose essential for our arguments. To avoid confusion, we will quote them within doublebrackets.The same as ours, the approach there bases on the computation of the Green functionof a system which consists of an impurity placed within a host medium. They calculate therenormalized polarizability of the impurity following a similar procedure to that we usedin Section III B. The host medium considered in [57] is a cubic lattice with lattice spacing ξ ≈ ρ − / . The medium is therefore strongly correlated. However, because for the frequen-cies of interest k ξ ≪ 1, an effective medium can be defined with χ eff ≃ χ MG for the reasonsargued in Section V A. This is used by the authors to extrapolate their results to randommedia. The propagator of the local field is denoted by G m in Eq.[(13)]. The authors firstcompute a transference matrix and a propagator denoted by G , in the long-wavelength limit kξ ≪ 1. The resultant propagator G ( ~R, ~R ′ ) of Eq.[(11)] links any two distant lattice sitesand contains only long-wavelength modes. Afterwards, the authors proceed to compute the t -matrices of impurities which are embedded in the host medium. To do so, they calculate G m ( ~r m , ~r m ), where ~r m denotes the position vector of the impurity. In that calculation, thepropagator of Eq.[(11)] is attached to the impurities in two different manners, dependingon whether the impurity occupies a lattice site –in which case it is said substantial, withsubscript s – or not –in which case it is said interstitial, with subscript i . That is done inEqs.[(16,17)] through the computation of the T m -matrices.77n the first place, the approach is not microscopical enough as the propagator in Eq.[(11)]does not contain the necessary resolution to probe distances shorter than ξ , and the distancefrom either an interstitial or substantial impurity to the first neighbors in the lattice is ≤ ξ .Therefore, only transverse long-wavelength modes can be accurately accounted for.Nevertheless, for the case of an interstitial impurity their approach may be a good approx-imation under some conditions, as the impurity is connected through free space propagatorsto the T -matrix of the lattice and in the resultant function G i ( ~r i , ~r i ), ~r i is not a lattice vector.However, the correlation of the impurity to the lattice that way reduces to the correlation tothe nearest lattice scatterers (the ones at the extremes of the T -matrix). An exact treatmentwould require the implementation of additional correlations as we did in Section V B.The situation is worse for the case of a substantial impurity. This can be seen by com-parison with the exact result we obtained in the virtual cavity scenario. That is, for thecase that the polarizability of the substantial ”impurity” is equivalent to that of the restof particles of the lattice, the resultant setup would correspond to the strict virtual cavityscenario. Therefore, in the long-wave length limit, one would expect to obtain a result pro-portional to Eq.(152) for the renormalized long-wavelength decay rate –which is identifiedin [57] with the radiative decay rate after Eq.[(15)]. Instead, they obtain an additional localfield factor which supports the erroneous introduction of one local field factor per electricfield operator. The reason for the mistake is that the authors make an erroneous usage of the T -matrix and the propagator G ( ~R, ~R ′ ) of Eq.[(11)]. The T -matrix in Eq.[(6)] just includesterms which link any two dipoles in the lattice. However, if one is to compute G ( ~R, ~R ) outof it, self-correlations must be explicitly considered. That is, negative correlations are cor-rectly implemented in Eq.[(6)] by excluding the vector ~R = ~ G ( ~R, ~R ′ ) at ~R ′ = ~R does not incorporate self-correlation inthe right manner and its use in Eqs.[(14,16)] is incorrect.In the following, we analyze the formulae of [57], compare them with ours and spot theerrors. We have illustrated the failure in Fig.19. Eq.[(6)] with the approximation of Eq.[(7)]78 IG. 19: ( a ) Diagrammatic representation of the T -matrix as computed in [57], Eq.[(6)], in thelong-wavelength limit of Eq.[(7)], kξ ≪ 1. That limit is effectively equivalent to taking the overlapapproximation. ( b ) Diagrammatic representation of two ’apparently’ different diagrams in ¯ G ( ~r, ~r ).They come to be equivalent when self-correlations are implemented. In the overlap approximationof ( a ), because the 1PI piece underlined on the r.h.s. is not included in the effective self-energyfunction under the overlap approximation, such an equivalence disappears in the approach of [57].( c ) Feynman rules. is equivalent to ¯ T ( kξ → 0) = − k α h k ρα [ ¯ G (0) ( k ) + ¯ C ( kξ → i − , (215)where ¯ C ( kξ → 0) is the cavity tensor with components given in Eqs.(90,91). In the long-wavelength limit, ¯ C ( kξ → 0) = − ρ − k I . α = α [1 + i k π α ] is the in-free-space radiativelycorrected single particle polarizability –the authors of [57] denote it by ˜ α instead. Onlythe two-point correlation function, h C ( r ) = − Θ( r − ξ ) with ξ ≈ ρ − / , enters the cavityfactors. That implies that, effectively, the same negative correlation function which entersthe computation of χ MG in a random medium does so for the computation of the T -matrix ina cubic lattice in the limit kξ ≪ 1. Expanding next the denominator of ¯ T as a perturbativeseries in k ρα [ ¯ G (0) ( k ) + ¯ C ( kξ → ξ → 0. Thus, ¯ T eff ( k ) ≡ ¯ T ( k ) | kξ ≪ reads,¯ T eff ( k ) = ρ − [ χ MG + χ MG ¯ G eff ( k ) χ MG ] , (216)where ¯ G eff ( k ) is given in Eq.(147) and is denoted by G diel in [57]. Hence, for any two distantlattice points ~R = ~R ′ , Eq.[(11)] can be otained in the long-wavelength limit by removing thebare vertices α , ρα attached to the extremes of Eq.(216),¯ G eff~R = ~R ′ ( ~R, ~R ′ ) = ( α/α ) L LL G diel ( ~R, ~R ′ ) . (217)If one is to compute ¯ G eff ( ~R, ~R ) using the general expression Eq.[(4)], the ¯ T -matrix there isnot that in Eq.[(6)], which corresponds to the one which enters the Dyson equation Eq.(56)for a cubic lattice. The point being that that in Eq.[(6)] was computed without consideringthe possibility that the first and the last dipoles in the diagrams were the same. As a matterof fact, application of the overlap approximation from the very beginning prevents fromcorrelating two distant scattering events –see Fig.19– and hence the diagrammatic ’trick’ weused in Section IV B to deduce the exact formulae turns inapplicable. Hence, if one aims towrite the equation for ¯ G ( ~R, ~R ) in the form of Eq.[(4)], one should use a modified ¯ T -matrixwhose form derives from our Eq.(63), T ⊥ , k ( k ) = − [ G (0) ⊥ , k ] − [1 + 1 k ρ ˜ α T ⊥ , k ] . (218)Because a lattice is a strong correlated system, the exact formulation of the problem interms of the ¯ t -matrix is more convenient as the correlation length can be extended eventuallyfurther than ξ . The formula to compute the propagator ¯ G ( ~R, ~R ) is that in Eq.(73),¯ G ( ~R, ~R ) = − k ˜ α X { ~r i } ¯ G (0) ( ~r i ) · ¯ t ( ~r i ) , (219)Note also that, for the sake of consistency, the fully renormalized ˜ α polarizabilities enterthis formula, and neither the in-free-space radiatively corrected one, α , nor the static one, α . The components of ¯ T ~k of Eq.[(6)] in [57] relate to ¯ t ( ~r i ) in our nomenclature through¯ T ~k = P { ~r i } ¯ t ( ~r i ) e i~k · ~r i ,¯ T ~k = − k ˜ α h I + k ˜ α X { ~r i } ¯ G (0) ( ~r i ) e i~k · ~r i i − , { ~r i } = ξ ( a ˆ i + b ˆ j + c ˆ k ) , a, b, c ∈ Z , ~r i = ~ . (220)80riting the ¯ t -matrix in the alternative form, ¯ T ( ~r ) = P { ~r i } ¯ t ( ~r i ) δ (3) ( ~r − ~r i ), Eq.(219) reads G ( ~R, ~R ) = − k ˜ α Z d r ¯ G (0) ( ~r ) · ¯ T ( ~r ) = − k ˜ α Z d k (2 π ) ¯ G (0) ( k ) · ¯ T ~k . (221)For the long-wavelength modes, the same result as in an MG dielectric holds at leading orderin k ξ , ¯ G ( ~R, ~R ) | kξ ≪ = − ik π I ǫ MG + 23 √ ǫ MG . (222)For higher frequency modes, it is unavoidable to compute the first terms of ¯ T ~k . X. THE ZERO-TEMPERATURE ELECTROMAGNETIC PRESSURE The role of the zero-point-energy (ZPE) in a variety of physical phenomena has receivedrenewal attention in the last years. In particular, a strong motivation is to estimate itscontribution to the cosmological constant, Λ, in connection with the possibility of beingmeasured in the laboratory. The need for the existence of the cosmological constant is itselfa controversial matter which is out of the scope of this paper –see eg. [83–85]. Standardcosmology predicts a small value for the current Λ, which needs of an apparently unnaturalfine-tuning. Leaving aside some exotic contributions to dark energy, at least two maincontributions to Λ must be considered. The first one comes from the Standard Modelof particles in the form of condensates of quarks and gluons and the Higgs itself. Thesecond one comes from the ZPE of quantum fields. Eventually, these energies give rise tonegative pressure which sources the repulsive gravity force that leads the expansion of theuniverse. The main difficulty in dealing with ZPE computations is that the integrals involveultraviolate divergences. One could think of the need of being regulated by some physical cut-off. However, this is not yet fully justified [86, 87]. Nevertheless, it is customary –as quotedin [2]– to present genuine quantum electromagnetic effects like the Casimir effect, the van derWaals forces, the Lamb-shift, etc. as evidences of the existence of the ZPE fluctuations of theEM field. Further on, some papers have suggested the possibility of measuring the ZPE of theelectromagnetic field in Josephson junctions [88]. As pointed out by Jetzer and Straumann[89, 90], the only physical observable which can be measured in absence of coupling to gravityis the variation on the vacuum energy as a response to external couplings. Any physicalobservable measured this way remains invariable to any prescription on the normalizationor regularization of the vacuum energy. Hence, the scheme we followed to regularize the81hifted-electrostatic polarizability, α stat , was merely phenomenological. Infinite quantitieswere swept under the carpet and remained unaffected by the rest of the renormalizationprocedure. That regularization scheme only served to us to accommodate in the samefooting further finite contributions. It is analogous to the regularization of the divergencesin the electron self-energy which appear in QED in computing the experimental mass of theelectron [29, 30]. Figs.5( a, b, c , d ) depicts how, after formal integration of those EM modeswhich give rise to the bound atomic state A , one can interpret that the ’remaining’ vacuumfluctuations which amount to the electromagnetic ZPE interact with such a state giving riseto radiative corrections. However, this is just a result of the order in which fluctuations areintegrated out. It is plain that there is no transfer of energy from the ZPE into the dipolesself-energy.Jaffe pointed out in [2] that the Casimir force can be computed out of the derivation of thedensity of electromagnetic states with respect to some parameter on which the background ofmatter fields depend. In the case of the force between two metallic plates, the backgroundfields are the electric currents confined on the plates separated by some distance. Thediagrams involved contain external legs representing the fields on the plates –those arethe current loops depicted on the left plate of Fig.20( c ). This way, the diagrams withphotons attached to the plates cannot be interpreted as vacuum-diagrams. An alternativeinterpretation is given by Milton et al. in [91]. The authors identify the interaction energybetween the plates with the self-energy which renormalizes their (divergent) bare masses. InFig.20( c ) the photon propagators carry the radiative corrections which give rise to the platesself-energy. In the interpretation of [2], it is the variation of the density of those photonstates which yield the Casimir force. In addition, in the context of Lifschitz formalism [92]Milton identifies [1, 93] a bulk energy which depends on the dielectric medium between theplates and is proportional to the volume.We proceed next to describe where energies and forces reside in our formalism. It wasemphasized throughout Section II that actual dipoles play a double role as polarizing theEM vacuum and renormalizing their own polarizability. In energetic terms, this gives riseto a change on the energy carried out by the fluctuations of sourceless EM modes and to ashift on the resonance frequency of the dipoles. The former is the analog of the bulk energyof [93] while the latter is the analog of the self-energy of the metallic plates of [91]. We willdenote the energy density on sourceless EM modes by F rad and that stored in the dipoles82y F mat . The corresponding bare energy densities are the divergent ZPE of Eq.(10) and F = ρ ~ ω A for dipoles in the atomic state A . While the latter, associated to actual matter,can be detected in EM phenomena, the former eventually needs of coupling to gravity to bedetected [94].Once the dipoles are considered bound states, statistical translation invariance holdsat length scales greater than a . It is in this sense that we can talk of a proper vacuum, | Ω i s.l. , in contrast to the scenario of the parallel metallic plates. As it was illustrated inSection II B, the diagrams contributing to the energy of the polarized EM vacuum aretopologically identical to those contributing to the dipole self-energy. The only differencebeing a change on the reference frame. Fig.20( a ) shows a typical diagram of both F rad and F mat . It contributes to F rad when all the points on the diagram are equivalent. Itcontributes to F mat when the origin and end of the diagram is chosen to be a point scattereras in Fig.20( b ). The main difference between F rad and F mat is that while the former can becomputed at zero-order in perturbation theory together with F , the remaining of F mat is asecond-order contribution. That is, F rad + F = D Ω , N X j =1 ψ jA ( ~r ) (cid:12)(cid:12)(cid:12) V − Z d r ǫ | ˆ ~E ( ~r ) | + V − ˆ H ( ~r ) (cid:12)(cid:12)(cid:12) N X j =1 ψ jA ( ~r ) , Ω E (223)= ǫ s.l. h Ω || ˆ ~E ( ~r ) | | Ω i s.l. + F = − ~ πc Z ω Tr n ℑ{ G ⊥ ( ~r, ~r ) } o d ω + F (224)= c − Z ~ ω LDOS sourcelessω d ω + F , (225)where ˆ H is given in Eq.(11) and V is the volume occupied by the dielectric and N isthe number of dipoles. The atomic wave functions of the dipoles in state A have beenexplicitly separated so that their contributions to F are additive. The radiative term onEq.(224) assumes knowledge of χ ω ⊥ for any frequency. However, we have already argued thatour approach has a natural frequency cut-off at c/a . Also, in general, the dipoles contain anumber of atomic resonances so that F should contain a sum over those resonances instead.For the sake of simplicity and in order to give a closed formula, we will restrict ourselvesto the computation of F rad up to frequencies where an effective medium exists, ω < c/ξ ,yielding, F rad | ω We have studied the phenomenon of electric dipole emission in statistically homogeneouscomplex media. Based on the computation of the spectrum of sourceless normal modes andthe spectrum of the EM fluctuations which yield the self-energy of the dielectric matter,we postulate the distinction between a sourceless vacuum, | Ω i s.l. , and a self-polarizationvacuum, | Ω i s.p. . The former is a proper vacuum as it is translation-invariant. The latteris defined upon each dipole. In physical grounds, the distinction bases on the observationthat the constituents of a dielectric play a double role. In the first place, they polarize thezero-point vacuum, | i . Complementarily, they renormalize their own polarizability, α , forthe multiple-dipole interactions generate a self-energy. The EM energy in | Ω i s.l. is computednon-perturbatively. The dielectric self-energy enters at second-order of perturbation theoryfollowing Fermi’s Golden rule.In application of the fluctuation-dissipation theorem, the spectrum of EM fluctuations of | Ω i s.l. is determined by he Dyson propagator, ¯ G . The spectrum of fluctuations of the self-polarization field in | Ω i s.p. is determined by ¯ G . The latter satisfies a Lippmann-Schwinger87 IG. 20: ( a ) Typical 1PI diagram contributing to both F dielrad and F selfEmat . It presents a characteristicfractal structure. ( b ) Diagrammatic representation of LDOS emissionω . Translation invariance aroundthe loop is broken at the position of the emitter. ( c ) Sketch of the Casimir self-energy whichrenormalizes the mass of two perfect conducting plates. On the l.h.s. , current-loops are depicted.On the r.h.s. the representation mimics the formalism we used for the radiative corrections whichrenormalize the bare polarizability of small dipoles. equation with a stochastic kernel. In a homogeneous dielectric the kernel is given byΞ V C ⊥ , k ( k ) = − ρ ˜ αχ ⊥ , k ( k ) G (0) ⊥ , k ( k ) h − χ ⊥ , k ( k ) ρ ˜ α + k χ ⊥ , k ( k ) G (0) ⊥ , k ( k ) i . As a result, G V C ⊥ ( k ) = 1 ρ ˜ α χ ⊥ ( k ) G ⊥ ( k ) = 1 ρ ˜ α χ ⊥ ( k ) k [1 + χ ⊥ ( k )] − k , G V C k ( k ) = 1 ρ ˜ α χ k ( k ) G k ( k ) = 1 ρ ˜ α χ k ( k ) k [1 + χ k ( k )] . The above equation is exact in the strictly virtual cavity scenario (VC). For the self-polarization of an impurity, the computation of G is model-dependent. The model is gener-88cally referred to as real-cavity model (RC). It is possible to compute the kernel of theequation for G in closed form for the case the impurity is placed in a large cavity with R ≫ ξ ,Ξ RC ⊥ , k ( k ) = − κ P I ⊥ , k ( k ) χ ⊥ , k ( k ) G (0) ⊥ , k ( k ) h − κ P I ⊥ , k ( k ) χ ⊥ , k ( k ) + k χ ⊥ , k ( k ) G (0) ⊥ , k ( k ) i , where κ P I ⊥ , k ( k ) is derived formally in Section V B. For a medium in which dipoles remainfixed G can be expressed in function of the transference matrix as,¯ G ( ~r , ~r ) = 1 − k ˜ α X { ~r i } ¯ G (0) ( ~r , ~r i ) · ¯ t ( ~r i , ~r ) . The local density of EM states in | Ω i s.p. is denoted by LDOS emissionω for it correspondsto the density of channels accessible to the emitted photons. The power emission, W ω , thedecay rate, Γ and LDOS emisssionω are decomposed according to their transverse/longitudinaland coherent/incoherent nature as,2 W Coh. ⊥ = W o Z d k (2 π ) ℜ{ χ ⊥ ( k ) ρ ˜ α }ℑ{ G ⊥ ( k ) } , W Incoh. ⊥ = W o Z d k (2 π ) ℑ{ χ ⊥ ( k ) ρ ˜ α }ℜ{ G ⊥ ( k ) } ,W Coh. k = W o Z d k (2 π ) ℜ{ χ k ( k ) ρ ˜ α }ℑ{ G k ( k ) } ,W Incoh. k = W o Z d k (2 π ) ℑ{ χ k ( k ) ρ ˜ α }ℜ{ G k ( k ) } , where W o must be substituted by the appropriate constants in the case of Γ andLDOS emisssionω . We found that the vacuum in which coherent emission propagates, | Ω i Coh , isadditionally polarized w.r.t. that of bulk normal modes, | Ω i s.l. . The renormalization func-tion being, Z ⊥ , k = ℜ{ χ ⊥ , k ρ ˜ α } . In a Maxwell-Garnett dielectric, it equals a local field factor.We have computed the decay rate of a weakly-polarizable interstitial atom in an MGdielectric. The results are meant to fit the experimental data on the life-time of Ce +3 ions.89e obtained,Γ Coh.MG = 2˜Γ o ℜ n ǫ MG + 23 o ℜ{√ ǫ MG } , Γ k ,Disper.MG = 2˜Γ o ℜ{ χ MG + 13 χ MG − . χ MG + 0 . χ MG − . χ MG + ... } Γ Absorb.MG = − o ℑ n ǫ MG + 23 o ℑ{√ ǫ MG } + 2˜Γ o ζ ℑ{ χ MG + 12 χ MG − . χ MG + 0 . χ MG − . χ MG + ... } + 2˜Γ o ζ ℑ{ χ MG + 13 χ MG − χ MG + 0 . χ MG − . χ MG + ... } , where ˜Γ o contains the renormalization of µ . In the single-scattering approximation, a clas-sification of the decay rate in terms of radiative and non-radiative energy transfer has beengiven –see Fig.16.The computation of the dielectric constant of an MG dielectric is a problem of self-consistency due the double polarization role played by point dipoles.We have computed the decay rate of a low-polarizable substantial impurity within alarge cavity. The results are meant to fit the experimental data on the life-time of Eu +3 ions. Formulae are exact up to order O ( χ ),Γ Coh.RC = Γ h ℜ{√ ǫ }ℜ n(cid:16) ǫ + 23 (cid:17) o − ℜ{ ǫ − } i , Γ k ,Disper.RC = Γ ℜ n ǫ − ( ǫ − ǫ o , Γ Absorb.RC = − Γ h ℑ{√ ǫ }ℑ n(cid:16) ǫ + 23 (cid:17) o + (cid:16) k R ) + 1 k R (cid:17) ℑ{ ǫ − ǫ − ( ǫ − } i . In comparison to previous works, we have found that the usual formulae for the decayrate in the virtual and the real cavity models are erroneous in physical grounds. The mainreason being that the inclusion of one local field factor per field operator entering Fermi’sGolden rule is an erroneous assumption. Also, only coherent emission can be accuratelycomputed using macroscopic fields and incoherent radiative emission is longitudinal.The total EM energy density is the sum of the energy of the EM fluctuations in | Ω i s.l. , F rad = − ~ πc R ω Tr n ℑ{ G ⊥ ( ~r, ~r ) } o d ω , plus the total self-energy stored in the dielectric, F mat = F + ρ ǫ | µ | ˜ k ℜ{ γ ˜ k ⊥ + γ ˜ k k }| ˜ k = k res . It is the variation of the latter w.r.t. externalparameters, ξ and ρ , that gives rise to generalized van-der-Waals-Casimir forces in a90omplex media. Neither zero-point EM vacuum energy nor F contribute. The variationof F rad + F mat w.r.t ρ yields an isotropic pressure. The zero-temperature pressure of avan-der-Waals gas is found to be P vdW = − ~ ω π α ρ /ξ . Higher order terms of the virialexpansion have been found together with additional contributions from long-ranged forces.The radiative pressure at zero temperature is estimated to be at least twelve orders ofmagnitude weaker at leading order in ρ .We have argued on the possibility of testing our formulae for ǫ eff , Γ and resonanceshifts in experimental setups. Cold atoms and colloidal liquids are suggested to be suitablecandidates. In order to distinguish ours from the usual formulae of the virtual cavity andthe real cavity models, it has been argued that either high refractive index or optically thickmedia are needed. Alternatively, measuring the coherent radiation instead of integratingthe total one may do.The detection of the zero-temperature radiative pressure, which is estimated inviablein gasses, might be possible in highly correlated cold fluids which present structuralresonances. However, we need a more precise dielectric function than that for the two-levelmono-atomic dielectric used in the derivation of Eq.(237). The dielectric constant shouldbe valid for a much wider range of frequencies. To this respect, Kramers-Kronig relations[95] might help in performing the integration of F dielrad . More investigation is needed on thispoint. Finally, predictions at cosmological scales need of the introduction in the presentformalism of curvature, retardation and magnetic effects. Work on this matter is in progress.We thank S.Albaladejo, L.Froufe, J.J. Saenz, R. Carminati and I. Suarez for fruitfuldiscussions and suggestions. This work has been supported by the Spanish integratedproject Consolider-NanoLight CSD2007-00046, the EU project NanoMagMa EU FP7-NMP-2007-SMALL-1 and the Scholarships Program ’Ciencias de la Naturaleza’ of RamonAreces Foundation. [1] K.A. Milton, The Casimir effect, physical manifestations of zero-point energy World Scientific,New Jersey (2001).[2] R.L. Jaffe, Phys.Rev. D , 021301 (2005). 3] M.Bordag, U. Mohideen, V.M. Mostepanenko, e-print arXiv:quant-ph/0106045v1.[4] E.M. Purcell, Phys. Rev. , 681 (1946).[5] E. Yablonovitch, Phys. Rev. Lett. , 2059 (1987); S. John, Phys. Rev. Lett. , 2486 (1987);S. John and T. Quang, Phys. Rev. A , 1764 (1994).[6] K. Suhling, P.M. W. French and D. Phillips, Photochem. Photobiol. Sci. , 13 (2005).[7] V.V. Protasencko, A.C. Gallagher, Nano. Lett. , 1329 (2004); J.N. Farahani, D.W. Pohl,H.J. Eisler and B. Hecht, Phys. Rev. Lett. , 017402 (2005); R. Carminati et al. , Opt. Com. (368).[8] Z. Liu et al. Science , 1734 (2000); D.R. Smith, J.B. Pendry and M.C.K. Wiltshire, Science , 788 (2004).[9] L.L. Foldy, Phys. Rev. , 107 (1945).[10] M. Lax, Rev. Mod. Phys. , 287 (1951); Phys. Rev. , 621 (1952); Rev. Mod. Phys. ,359 (1966).[11] U. Frisch, Ann. Astroph. , 645 (1966); , 565 (1967).[12] R.K. Bullough, J. Phys. A , 409 (1968); , 477 (1969); , 708 (1970); , 726 (1970); , 751(1970).[13] F. Hynne, R.K. Bullough, Phil. Trans. R. Soc. Lond. A , 251 (1984); , 305 (1987); , 253 (1990).[14] F. Hynne, R.K. Bullough J. Phys. A , 1272 (1972).[15] B.U. Felderhof, G.W. Ford and E.G.D. Cohen, J. Stat. Phys. , 241 (1983).[16] B.U. Felderhof and B. Chichocki, J. Stat. Phys. , 1157 (1988); B.U. Felderhof and R.BJones, Phys. Rev. A, , 5669 (1989); B.U. Felderhof and B. Chichocki, J. Chem. Phys. ,6104 (1990).[17] A. Einstein, Z. Phys. , 121 (1917).[18] J.J. Sakurai, Advanced Quantum Mechanics Addison-Wesley (1994).[19] S.M. Barnett, B. Huttner, R. Loudon and R. Matloob, J. Phys. B Statistical Mechanics John Wiley and Sons, (1987).[21] M.E. Crenshaw, C.M. Bowden, Phys.Rev.Lett. , 1851 (2000).[22] M.E. Crenshaw, Phys. Rev. A , 053827 (2008).[23] P.R. Berman, P.W. Milonni, Phys.Rev.Lett. , 053601 (2004).[24] G.S. Agarwal, Phys. Rev. A , 253 (1975). 25] C.R. Balian and C. Bloch, Ann. Phys. , 271 (1971).[26] E.N. Economou, Green’s Functions in Quantum Physics Springer-Verlag, Berlin (1983).[27] X. Kong and I. Ravndal, Nuc. Phys. B , 627 (1998).[28] D.L. Andrews and D.S. Bradshaw, Eur. J. Phys. , 845 (2004).[29] M.E. Peskin and D.V. Schroeder, An Introduction to Quantum Field Theory Addison-Wesley(1995).[30] W. Greiner, Quantum Mechanics. Special Chapters Springer-Verlag, Berlin Heidelberg (1998).[31] P.R. Berman, R.W. Boyd and P.W. Milonni, Phys. Rev. A , 053816 (2006).[32] R. Loudon and S.M. Barnett, J. Phys. B , S555 (2006).[33] I. Bialynicki-Birula and T. Sowinski, Phys. Rev. A , 062106 (2007).[34] P.W. Milonni, R. Loudon, P.R. Berman and S.M. Barnett, Phys. Rev. A , 043835 (2008).[35] P. de Vries, D.V. van Coevorden, A.Lagendijk, Rev. Mod. Phys. , 447 (1998).[36] F. Fano, Phys. Rev. , 1202 (1956).[37] J. Hopfield, Proc. Roy. Soc. (London) A68 , 441 (1955); J. Hopfield, Phys. Rev. , 1555(1958).[38] S. Scheel, L. Kn¨oll, D.-G. Welsch and S.M. Barnett, Phys. Rev. A , 1590 (1999).[39] H.P. Breuer, F. Petruccione The Theory of Open Quantum Systems Clarendon Press, Oxford(2006).[40] M. Born and E. Wolf, Principles of Optics Pergamon Press, Beijing (1980).[41] R.J. Glauber, Phys. Rev. , 2529 (1963); , 2766 (1963).[42] D.S. Citrin, Nano Lett. (9), 1561 (2004).[43] A.D. Yaghjian, Proc. IEEE, , 248 (1980).[44] A.S. Davydov, Quantum Mechanics Pergamon Press, New York (1976).[45] Y.B. Band, Light and Matter Wiley, West Sussex UK (2006).[46] P.W. Milonni, The Quantum Vacuum. An Introduction to Quantum Electrodynamics Aca-demic Press, Inc. (1994).[47] J.M.Wylie, J.E. Sipe, Phys. Rev. A , 1185 (1984).[48] M.Donaire, e-print arXiv:0811.0323.[49] R. Carminati and J.J. Saenz, Phys.Rev. Lett. , 093902 (2009).[50] D. Toptygin, J. Fluoresc. , 201 (2003).[51] A. Lakhtakia, Astrophys. J. , 494 (1992). 52] B. Draine and P. Flatau, J. Opt. Soc. Am. A , 1491 (1994).[53] A.Lagendijk, B.van Tiggelen, Phys.Rep. , 143 (1996).[54] L.S. Froufe-Perez, R. Carminati and J.J. Saenz, Phys. Rev. A , 013835 (2007).[55] H.A.Lorentz, Wiedem. Ann. , 641 (1880); L.Lorenz, Wiedem. Ann. , 70 (1881).[56] L.Onsager, J. Am. Chem. Soc. , 1486 (1936).[57] P.de Vries, A.Lagendijk, Phys.Rev.Lett. , 1381(1998).[58] J.C. Maxwell-Garnett, Phil. Trans. R. Soc. Lond. A , 385 (1904).[59] A. Lagendijk, B. Nienhuis, B. van Tiggelen and P. de Vries, Phys. Rev. Lett. , 657 (1997).[60] D.L. Andrews, Chem. Phys. , 195 (1989).[61] T. F¨orster, Discussion Faraday Soc. , 7 (1959).[62] H. Chew, Phys. Rev. A , 3410 (1988).[63] K.K. Pukhov, T.T. Basiev, Yu.V. Orlovskii, JETP Lett. , 12 (2008).[64] C.K. Duan, M.F. Reid, Curr. App. Phys. , 348 (2006).[65] C.K. Duan, M.F. Reid and Z. Wang, Phys. Lett. A , 474 (2005).[66] J.J. Maki, M.S. Malcuit, J.E. Sipe and R.W. Boyd, Phys. Rev. Lett. , 972 (1991).[67] R.C. Hilborn, Am. J. Phys. , 982 (1982).[68] M. Donaire, In preparation.[69] M.R. Potter and G.W. Green, J. Phys. F , 1426 (1975).[70] L. Froufe-Perez, W. Guerin, R. Caminati and R. Kaiser, Phys. Rev. Lett. , 073903 (2004).[72] J.P. Schuurmans et al. , Phys. Rev. Lett. Classical Electrodynamics John Wiley and Sons, (1962).[75] P. Lavallard, M. Rosenbauer and T. Gacoin, Phys. Rev. A , 5450 (1996).[76] F.J.P. Schuurmans, A. Lagendijk, J. Chem. Phys. , 3310 (2000).[77] F.J.P. Schuurmans, P. de Vries, A. Lagendijk, Phys. Lett. A , 472 (2000).[78] S.M. Barnett, B. Huttner, R. Loudon and R. Matloob, Phys. Rev. Lett. , 181 (1976).[80] C.A. Gu´erin, B. Gralak and A. Tip, Phys. Rev. E , 056601 (2007).[81] G. Juzeli¯unas, J. Lumin. , 666 (1998). 82] S. Scheel, L. Kn¨oll and D.-G. Welsch, Phys. Rev. A , 4094 (1999).[83] S. Weinberg, Rev. Mod. Phys. , 1 (1989).[84] P.J.E. Peebles, Nucl. Phys. B , 5 (2005).[85] N. Straumann, Mod. Phys. Lett. A , 1083 (2006).[86] P.J.E. Peebles, B. Ratra, Rev. Mod. Phys. , 559 (2003).[87] S.J. Brodsky, R. Shrock, Pre-print:arXiv:0803.2554v1 (2008).[88] C. Beck, M.C. Mackey, Phys. Lett. B , 295 (2005).[89] P. Jetzer, N. Straumann, Phys. Lett. B , 77 (2005).[90] P. Jetzer, N. Straumann, Phys. Lett. B , 57 (2006).[91] K.A. Milton, P. Parashar, K.V. Shajesh and J. Wagner, J. Phys. A , 10935 (2007).[92] E.M. Lifschitz, Soviet. Phys. JTEP , 73 (1956).[93] K.A. Milton, J. Phys. A , R209 (2004).[94] K.A. Milton et al. , J. Phys. A , 164052 (2008).[95] J.S. Toll, Phys. Rev. , 1760 (1956).[96] In metals, ǫ e rather contains an imaginary part in the denominator which accounts for dissi-pative damping in agreement with Drude’s model.[97] In all the calculations of this Section we will evaluate the γ -factors at frequency ω = ck . Itmust be intended as the frequency of the exciting field for the case of an induced dipole likethat of Eqs.(39-41). It must be intended as the resonance frequency for the case of spontaneousemission by a dipole with given transition amplitude µ like those of Eqs.(22,49-50). It must beintended as the frequency c ˜ k of intermediate photons for the self-energy of a Lorentzian-typedipole like that of Eq.(46).of intermediate photons for the self-energy of a Lorentzian-typedipole like that of Eq.(46).