Johannes Feist*, Antonio I. Fernández-Domínguez, and Francisco J. García-Vidal*

Macroscopic QED for quantum nanophotonics:Emitter-centered modes as a minimal basis formulti-emitter problemsAbstract: We present an overview of the framework ofmacroscopic quantum electrodynamics from a quantumnanophotonics perspective. Particularly, we focus our at-tention on three aspects of the theory which are crucial forthe description of quantum optical phenomena in nanopho-tonic structures. First, we review the light-matter inter-action Hamiltonian itself, with special emphasis on itsgauge independence and the minimal and multipolar cou-pling schemes. Second, we discuss the treatment of theexternal pumping of quantum-optical systems by classi-cal electromagnetic fields. Third, we introduce an exact,complete and minimal basis for the field quantization inmulti-emitter configurations, which is based on the so-called emitter-centered modes. Finally, we illustrate thisquantization approach in a particular hybrid metallodi-electric geometry: two quantum emitters placed in thevicinity of a dimer of Ag nanospheres embedded in a SiNmicrodisk.

Keywords: Quantum Nanophotonics, Macroscopic Quan-tum Electrodynamics, Emitter-centered Modes, HybridCavities

1 IntroductionIn principle, quantum electrodynamics (QED) providesan “exact” approach for treating electromagnetic (EM)fields, charged particles, and their interactions, within afull quantum field theory where both matter and light aresecond quantized (i.e., both photons and matter particles

*Corresponding author: Johannes Feist, Departamento de FísicaTeórica de la Materia Condensada and Condensed Matter PhysicsCenter (IFIMAC), Universidad Autónoma de Madrid, E-28049Madrid, Spain, e-mail: johannes.feist@uam.esAntonio I. Fernández-Domínguez, Departamento de FísicaTeórica de la Materia Condensada and Condensed Matter PhysicsCenter (IFIMAC), Universidad Autónoma de Madrid, E-28049Madrid, Spain, e-mail: a.fernandez-dominguez@uam.es*Corresponding author: Francisco J. García-Vidal, Departamentode Física Teórica de la Materia Condensada and Condensed Mat-ter Physics Center (IFIMAC), Universidad Autónoma de Madrid,E-28049 Madrid, Spain, and Donostia International PhysicsCenter (DIPC), E-20018 Donostia/San Sebastian, Spain, email:fj.garcia@uam.es

can be created and annihilated). However, this approach isnot very useful for the treatment of many effects of inter-est in fields such as (nano)photonics and quantum optics,which take place at “low” energies (essentially, below therest mass energy of electrons), where matter constituentsare stable and neither created nor destroyed, and addi-tionally, there are often “macroscopic” structures suchas mirrors, photonic crystals, metallic nanoparticles etc.involved. Due to the large number of material particles(on the order of the Avogadro constant, ≈ 6 · 1023), it thenbecomes unthinkable to treat the electrons and nuclei inthese structures individually. At the same time, a suffi-ciently accurate description of these structures is usuallygiven by the macroscopic Maxwell equations, in which thematerial response is described by the constitutive relationsof macroscopic electromagnetism. In many situations, itis then desired to describe the interactions between lightand matter in a setup where there are one or a few mi-croscopic “quantum emitters” (such as atoms, molecules,quantum dots, etc.), and additionally a “macroscopic”material structure whose linear response determines thelocal modes of the EM field interacting with the quantumemitter(s).

The quantization of the EM field in such arbitrarymaterial environments, i.e., the construction of a secondquantized basis for the medium-assisted EM field thattakes into account the presence of the “macroscopic” ma-terial structure, is a longstanding problem in quantumelectrodynamics. The most immediate strategy is to cal-culate the (classical) EM modes of a structure and toquantize them by normalizing their stored energy to thatof a single photon at the mode frequency, ℏ𝜔 [1]. However,an important point to remember here is that, even forlossless materials, electromagnetic modes always form acontinuum in frequency, i.e., there exist modes at anypositive frequency 𝜔. Consequently, there are in generalno truly bound EM modes, and what is normally thoughtof as an isolated “cavity mode” is more correctly describedas a resonance embedded in the continuum, i.e., a quasi-bound state that decays over time through emission ofradiation. An interesting exception here are guided modesin systems with translational invariance (i.e., where mo-

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2 J. Feist et al., Macroscopic QED for quantum nanophotonics

mentum in one or more dimensions is conserved), as modeslying outside the light cone 𝜔 = 𝑐𝑘 then do not coupleto free-space radiation [2]. An additional exception are“bound states in the continuum” [3], which arise due todestructive interference between different resonances.

As a further obstacle to a straightforward quantiza-tion strategy as described above, the response functionsdescribing material structures are necessarily dissipativedue to causality (as encoded in, e.g., the Kramers-Kronigrelations). When these losses cannot be neglected, quan-tization is complicated even further by the difficulty todefine the energy density of the EM field inside the lossymaterial [4–6].

Given all the points above, it is not surprising thatthere are many different approaches to quantizing EMmodes in lossy material systems [2, 7–15]. In the following,we give a concise overview of a particularly powerful for-mal approach that resolves these problems, called macro-scopic quantum electrodynamics (macroscopic QED) [16–24]. While there are excellent reviews of this general frame-work available (e.g., [22, 23]), we focus on its applicationin the context of quantum nanophotonics and strong light-matter coupling. In particular, we discuss the implicationsand lessons that can be taken from this approach on gaugeindependence and, in particular, the role of the so-calleddipole self-energy term in the light-matter interactionin the Power-Zienau-Woolley gauge, which has been thesubject of some recent controversy [25–37]. We then re-view in detail a somewhat non-standard formulation ofmacroscopic QED that allows one to construct a minimalquantized basis for the EM field interacting with a col-lection of multiple quantum emitters. This approach wasfirst introduced in [38], and subsequently rediscovered in-dependently by several other groups [39–43]. The fact thatthis very useful approach has been reinvented by differentresearchers over the past decade or so partially motivatesthe current article, which intends to give a concise andaccessible overview, and presents some explicit relationsthat have (to our knowledge) not been published before.We also note that with “minimal basis” we are here refer-ring to a minimal complete basis for the medium-assistedEM field, i.e., this basis contains all the information aboutthe material structure playing the role of the cavity orantenna, and no approximations are made in obtainingit. This then makes it appropriate to serve as a startingpoint for either numerical treatments [44, 45], or for de-riving simpler models where, e.g., the full EM spectrumis described by a few lossy modes [46].

In the final part of the article, we then present anapplication of the formalism to a specific problem, a hybriddielectric-plasmonic structure [14, 47, 48]. Particularly,

we consider a dimer of metallic nanospheres placed withina dielectric microdisk, a geometry that is similar to thatconsidered in [49].

2 TheoryMacroscopic QED provides a recipe for quantizing theEM field in any geometry, including with lossy materials.One particularly appealing aspect is that the full infor-mation about the quantized EM field is finally encodedin the (classical) electromagnetic dyadic Green’s functionG(r, r′, 𝜔). While there are several ways to derive the gen-eral formulation (see, e.g., [22] for a discussion of variousapproaches), a conceptually simple way to understand theframework is to represent the material structures througha collection of fictitious harmonic oscillators coupled to thefree-space EM field (which itself corresponds to a collec-tion of harmonic oscillators [50]). Formally diagonalizingthis system of coupled harmonic oscillators leads to a formwhere the linear response of the medium can be expressedthrough the coupling between the material oscillators andthe EM field. The end result is that the fully quantizedmedium-supported EM field is represented by an infiniteset of bosonic modes (labelled with index 𝜆, see below),defined at each point in space and each frequency, f𝜆(r, 𝜔),which act as sources for the EM field through the classicalGreen’s function. These modes are called “polaritonic” asthey represent mixed light-matter excitations [17]. Whilethis is a completely general approach for quantizing theEM field in arbitrary structures, it cannot be used “di-rectly” in practice due to the extremely large numberof modes that describe the EM field (a vectorial-valued4-dimensional continuum). Most uses of macroscopic QEDthus apply this formalism to derive expressions wherethe explicit operators f𝜆(r, 𝜔) have been eliminated, e.g.,through adiabatic elimination, perturbation theory, or theuse of Laplace transform techniques [22–24, 51–54]. Inparticular, macroscopic QED has been widely used in thecontext of dispersion forces, such as described by Casimirand Casimir-Polder potentials [23, 24].

2.1 Minimal coupling

In the following, we represent a short overview of the gen-eral theory of light-matter interactions in the frameworkof macroscopic QED. Since full details can be found in theliterature [22, 23], we do not attempt to make this a fullyself-contained overview, but rather highlight and discuss

J. Feist et al., Macroscopic QED for quantum nanophotonics 3

some aspects that are not within the traditional focusof the theory, in particular in the context of quantumnanophotonics.

The first step in the application of macroscopic QEDis the separation of all matter present in the system to betreated into two distinct groups: one is the macroscopicstructure (e.g., a cavity, plasmonic nanoantenna, photoniccrystal, . . . ) that will be described through the constitutiverelations of electromagnetism, while the other are themicroscopic objects (atoms, molecules, quantum dots, . . . )that are described as a collection of charged particles(which can then be approximated at various levels, e.g., astwo-level systems). This separation constitutes the basicapproximation inherent in the approach, and relies onthe assumptions that macroscopic EM is valid for thematerial structure (the medium) and its interaction withthe charged particles. While this is often an excellentapproximation, some care has to be taken for separationsin the sub-nanometer range, where the atomic structureof the material can have a significant influence [55–60]

For simplicity, we assume that the medium responseis local and isotropic in space, such that it can be encodedin the position- and frequency-dependent scalar relativepermittivity 𝜖(r, 𝜔) and relative permeability 𝜇(r, 𝜔) thatdescribe the local matter polarization and magnetizationinduced by external EM fields1. The extension of thequantization scheme to nonlocal response functions can befound in [22]. We directly give the Hamiltonian ℋ = ℋ𝐴 +ℋ𝐹 + ℋ𝐴𝐹 within the minimal coupling scheme [22, 23]

ℋ𝐴 =∑

𝛼

p2𝛼

2𝑚𝛼+

∑𝛼

∑𝛽>𝛼

𝑞𝛼𝑞𝛽

4𝜋𝜀0|r𝛼 − r𝛽 |, (1a)

ℋ𝐹 =∑

𝜆

∞∫0

d𝜔∫

d3r ℏ𝜔 f†𝜆(r, 𝜔)f𝜆(r, 𝜔), (1b)

ℋ𝐴𝐹 =∑

𝛼

[𝑞𝛼𝜑(r𝛼) − 𝑞𝛼

𝑚𝛼p𝛼 · A(r𝛼) + 𝑞2

𝛼

2𝑚𝛼A2(r𝛼)

].

(1c)

Here, the “atomic” Hamiltonian ℋ𝐴 describes a (nonrel-ativistic) collection of point particles with position andmomentum operators r𝛼 and p𝛼, and charges and masses𝑞𝛼 and 𝑚𝛼. The field Hamiltonian ℋ𝐹 is expressed interms of the bosonic operators f𝜆(r, 𝜔) discussed above,

1 Note that since the response functions are considered time-independent, effects due to the motion of the structure, such asin cavity optomechanics [61], cannot be treated without furthermodifications.

which obey the commutation relations[f𝜆(r, 𝜔), f†

𝜆′(r′, 𝜔′)]

= 𝛿𝜆𝜆′𝛿(r − r′)𝛿(𝜔 − 𝜔′), (2a)[f𝜆(r, 𝜔), f𝜆′(r′, 𝜔′)

]=

[f†𝜆(r, 𝜔), f†

𝜆′(r′, 𝜔′)]

= 0, (2b)

where 𝛿(r−r′) = 𝛿(r−r′)1, and 1 and 0 are the Cartesian(3 × 3) identity and zero tensors, respectively. The index𝜆 ∈ {𝑒,𝑚} labels the electric and magnetic contributions,with the magnetic contribution disappearing if 𝜇(r, 𝜔) = 1everywhere in space. The particle-field interaction Hamil-tonian ℋ𝐴𝐹 contains the interaction of the charges bothwith the electrostatic potential 𝜑(r) and the vector po-tential A(r), both of which can be expressed through thefundamental operators f𝜆(r, 𝜔) [22, 23]. Note that in theabove, we have neglected magnetic interactions due toparticle spin. We explicitly point out that although wework in Coulomb gauge, ∇ · A(r) = 0, the electrostaticpotential 𝜑(r) is in general nonzero due to the presenceof the macroscopic material structure, which also impliesthat the name “p · A-gauge”, which is sometimes em-ployed for the minimal coupling scheme, is misleading inthe presence of material bodies.

The light-matter interaction Hamiltonian can be sim-plified in the long-wavelength or dipole approximation,i.e., if we assume that the charged particles are sufficientlyclose to each other compared to the spatial scale of localfield variations that a lowest-order approximation of thepositions of the charges relative to their center of massposition r𝑖 is valid. For an overall neutral collection ofcharges, this leads to

ℋ𝐴𝐹 ≈ −d·E‖(r𝑖)−∑

𝛼

𝑞𝛼

𝑚𝛼

^p𝛼 ·A(r𝑖)+∑

𝛼

𝑞2𝛼

2𝑚𝛼A2(r𝑖),

(3)where d =

∑𝛼 𝑞𝛼^r𝛼 is the electric dipole operator of the

collection of charges, while ^r𝛼 and ^p𝛼 are the positionand momentum operators in the center-of-mass frameof the charge collection2. Eq. (3) explicitly shows thatin the presence of material bodies, the emitter-field in-teraction has two contributions, one from longitudinal(electrostatic) fields due to the Coulomb interaction withcharges in the macroscopic body, and one from transversefields (described by the vector potential). The relevant

2 While we here assumed a single emitter (i.e., a collection ofclose-by charged particles), the extension of the above formula tomultiple emitters is trivial. One important aspect to note is thatthe electrostatic Coulomb interaction between charges (secondterm in Eq. (1a)) is still present, i.e., there are direct instan-taneous Coulomb interactions between the charges in differentemitters.

4 J. Feist et al., Macroscopic QED for quantum nanophotonics

fields are given by [23]

E‖(r) =∑

𝜆

∞∫0

d𝜔∫

d3r′ ‖G𝜆(r, r′, 𝜔) · f𝜆(r′, 𝜔) + H.c.,

(4a)

A(r) =∑

𝜆

∞∫0

d𝜔𝑖𝜔

∫d3r′ ⊥G𝜆(r, r′, 𝜔) · f𝜆(r′, 𝜔) + H.c.,

(4b)

where the longitudinal and transverse components of atensor T(r, r′) are given by

‖/⊥T(r, r′) =∫

d3s 𝛿‖/⊥(r − s)T(s, r′), (5)

with 𝛿‖/⊥(r − s) the standard longitudinal or transversedelta function in 3D space. The functions G𝜆(r, r′, 𝜔) arerelated to the dyadic Green’s function G(r, r′, 𝜔) through

G𝑒(r, r′, 𝜔) = 𝑖𝜔2

𝑐2

√ℏ𝜋𝜖0

Im 𝜖(r′, 𝜔)G(r, r′, 𝜔), (6a)

G𝑚(r, r′, 𝜔) = 𝑖𝜔

𝑐

√−ℏ𝜋𝜖0

Im𝜇−1(r′, 𝜔)[∇′ × G(r′, r, 𝜔)]𝑇 .

(6b)

We note that in the derivation leading to the above expres-sions, it is assumed that Im 𝜖(r, 𝜔) > 0 and Im𝜇(r, 𝜔) > 0for all r, i.e., that the materials are lossy everywhere inspace. The limiting case of zero losses in some regionsof space (e.g., in free space) is only taken at the veryend of the calculation. We will see that in the reformula-tion in terms of emitter-centered modes, subsection 2.4,G𝜆(r, r′, 𝜔) disappears from the formalism relatively early,and only the “normal” Green’s function G(r, r′, 𝜔) isneeded (for which the limit is straightforward).

As mentioned above, the electrostatic contribution isnot present in free space, and in the literature it is oftenassumed that any abstract “cavity mode” correspondsto a purely transverse EM field. This is a good approx-imation for emitters that are far enough away from thematerial, e.g., in “large” (typically dielectric) structuressuch as Fabry-Perot planar microcavities, photonic crys-tals, micropillar resonators, etc. [62], but can break downotherwise. In general, this happens for coupling to evanes-cent fields [63], and in particular when sub-wavelengthconfinement is used to generate extremely small effectivemode volumes, such as in plasmonic [64, 65] or phonon-polaritonic systems [66, 67]. This observation is particu-larly relevant as such sub-wavelength confinement is theonly possible strategy for obtaining large enough light-matter coupling strengths to approach the single-emitter

strong-coupling regime at room temperature [68–71]. Forsub-wavelength separations, it is well-known that theGreen’s function is dominated by longitudinal compo-nents, while transverse components can be neglected [72].In this quasistatic approximation, we thus have A ≈ 0,cf. Eq. (4), and the light-matter interactions are all dueto electrostatic (or Coulomb) interactions, even withinthe minimal coupling scheme [35]3. Conversely, due tothe strongly sub-wavelength field confinement, the long-wavelength or dipole approximation is not necessarilyappropriate and an accurate description requires eitherthe direct use of the expression in terms of the electrostaticpotential [73], or the inclusion of higher-order terms in theinteraction [74]. As we have previously pointed out [35],doing so also resolves the formal lack of a ground statewhen the computational box is made too large and thedipole approximation is used [28, 33].

2.2 Multipolar coupling

We now discuss the Power-Zienau-Woolley (PZW) gaugetransformation [75–77], which is used to switch from theminimal coupling scheme discussed up to now to the so-called multipolar coupling scheme, which will then in turnform the basis for the emitter-centered modes we discusslater. This scheme has several advantageous properties: itexpresses all light-matter interactions through the fieldsE and B directly, without needing to distinguish betweenlongitudinal and transverse fields, and allows a system-atic expansion of the field-emitter interactions in termsof multipole moments. Additionally, in the multi-emittercase, it also removes direct Coulomb interactions betweencharges in different emitters, which instead become me-diated through the EM fields. This property makes iteasier to explicitly verify and guarantee that causality isnot violated through faster-than-light interactions. Weonly point out and discuss some specific relevant resultshere, and again refer the reader to the literature for fulldetails [22, 23]. The PZW transformation is carried out

3 It could then be discussed what the field modes should becalled in the limit when they contain negligible contributions frompropagating photon modes. However, since all EM modes that arenot just freely propagating photons will always have a somewhatmixed light-matter character, and since these modes always solvethe macroscopic Maxwell equations, they are conventionallyreferred to as “light”, “EM”, or “photon” modes. It is thusimportant to remember that this does not imply that they aresimply modes of the transverse EM field.

J. Feist et al., Macroscopic QED for quantum nanophotonics 5

by the unitary transformation operator

�� = exp

[𝑖

ℏ

∫d3r

∑𝑖

P𝑖 · A

](7)

where P𝑖 =∑

𝛼∈𝑖 𝑞𝛼^r∫ 1

0 d𝜎𝛿(r − r𝑖 − 𝜎^r) is the polar-ization operator of emitter 𝑖, and we have here explicitlygrouped the charges into several emitters, i.e., distinct(non-overlapping) collections of charges. Since this is aunitary transformation, physical results are unaffected inprinciple, although the convergence behavior with respectto different approximations can be quite different [26, 37].

Applying the transformation above gives the new op-erators ��′ = �� ����†. Expressing the Hamiltonian Eq. (1)in terms of these new operators then gives the multipolarcoupling form. The effect can be summarized by notingthat the operators A and r𝛼 are unchanged, while theircanonically conjugate momenta Π and p𝛼 are not. Inlight of the discussion of longitudinal (electrostatic) ver-sus transverse interactions, it is interesting to point outthat the electrostatic potential 𝜑(r) is also unaffected, i.e.,in the quasistatic limit, the PZW transformation has noeffect on the emitter-field interaction and discussions ofgauge dependence for this specific case become somewhatirrelevant. However, both the bare-emitter and the bare-field Hamiltonian are changed, as f𝜆(r, 𝜔)′ = f𝜆(r, 𝜔), i.e.,the separation into emitter and field variables is differentthan in the minimal coupling scheme. We here directlyshow the multipolar coupling Hamiltonian after addition-ally neglecting interactions containing the magnetic field.This leads to

ℋ𝐴 =∑

𝑖

[∑𝛼∈𝑖

p2𝛼

2𝑚𝛼+ 1

2𝜀0

∫d3r P2

𝑖 (r)

], (8a)

ℋ𝐹 =∑

𝜆

∞∫0

d𝜔∫

d3r ℏ𝜔 f†𝜆(r, 𝜔)f𝜆(r, 𝜔), (8b)

ℋ𝐴𝐹 = −∑

𝑖

∫d3r P𝑖(r) · E(r), (8c)

where all operators are their PZW-transformed (primed)versions, but we have not included explicit primes forsimplicity. In the long-wavelength limit, Eq. (8c) becomessimply

ℋ𝐴𝐹 = −∑

𝑖

d𝑖 · E(r𝑖), (9)

i.e., all field-emitter interactions are expressed throughthe dipolar coupling term, with the electric field operatorgiven explicitly by

E(r) =∑

𝜆

∞∫0

d𝜔∫

d3r′ G𝜆(r, r′, 𝜔) · f𝜆(r′, 𝜔) + H.c..

(10)

We note that the form of the bare-emitter Hamiltonianℋ𝐴 =

∑𝑖 𝐻𝑖 [Eq. (8a)] under multipolar coupling is

changed compared to the minimal coupling picture, andin particular can be rewritten as

𝐻𝑖 =∑𝛼∈𝑖

p2𝛼

2𝑚𝛼+

∑𝛼,𝛽∈𝑖

𝑞𝛼𝑞𝛽

8𝜋𝜀0|r𝛼 − r𝛽 |

+ 12𝜀0

∫d3r

[P⊥

𝑖 (r)]2, (11)

which makes explicit the fact that the emitter Hamilto-nian in the multipolar gauge is equivalent to the emitterHamiltonian in minimal coupling plus a term containingthe transverse polarization only. In order to arrive at thisform, we have used that the Coulomb interaction can berewritten as an integral over the longitudinal polariza-tion. The transverse part of the polarization in Eq. (11)corresponds to the so-called dipole self-energy term [28].When a single- or few-mode approximation of the EMfield is performed before doing the PZW transformation,this term depends on the square of the mode-emitter cou-pling strength, but not on any photonic operator. However,when all modes of the EM field are included, as implicitlydone here and as motivated by the fact that the term is notmode-selective (it does not depend on any EM field opera-tor), it is seen directly that this term becomes completelyindependent of any characteristics of the surrounding ma-terial structure, i.e., it cannot be modified by changingthe environment that the emitter is located in. Instead,the bare-emitter Hamiltonian in the multipolar approachis simply slightly different than under minimal coupling.This calls into question claims that this term should beincluded in simulations of strongly coupled light-mattersystems [28, 33]. However, it should be mentioned thata similar term can arise as if the environment-mediatedelectrostatic interactions are taken into account explic-itly instead of through the quantized modes, and one (orsome) of the EM modes are additionally treated explicitly.The action of these modes on the emitters then has tobe subtracted from the electrostatic interaction to avoiddouble-counting them [27].

2.3 External (classical) fields

Adapting an argument from Ref. [44], here we show thatmacroscopic QED also enables a straightforward treatmentof external incoming electromagnetic fields, in particularfor the experimentally most relevant case of a classical laserpulse. Assuming that the incoming laser field at the initialtime 𝑡 = 0 has not yet interacted with the emitters (i.e.,it describes a pulse localized in space in a region far away

6 J. Feist et al., Macroscopic QED for quantum nanophotonics

from the emitters), it can simply be described by a productof coherent states of the EM modes for the initial wavefunction, |𝜓(0)⟩ =

∏𝑛 |𝛼𝑛(0)⟩ =

∏𝑛 𝑒

𝛼𝑛(0)𝑎†𝑛−𝛼*

𝑛(0)𝑎𝑛 |0⟩,where the index 𝑛 here runs over all indices of the EMbasis (𝜆, r, 𝜔), and the 𝛼𝑛(0) correspond to the classicalamplitudes obtained when expressing the laser pulse inthe basis defined by these modes. In order to avoid the ex-plicit propagation of this classical field within a quantumcalculation, the classical and the quantum field can besplit in the Hamiltonian using a time-dependent displace-ment operator [78] 𝑇 (𝑡) = 𝑒

∑𝑛

𝛼*𝑛(𝑡)𝑎𝑛−𝛼𝑛(𝑡)𝑎†

𝑛 , where𝛼𝑛(𝑡) = 𝛼𝑛(0)𝑒−𝑖𝜔𝑛𝑡. Applying this transformation tothe wavefunction, |𝜓′⟩ = 𝑇 (𝑡)|𝜓⟩ adds a (time-dependent)energy shift that does not affect the dynamics and splitsthe electric-field term in Eq. (9) into a classical and aquantum part, E(r)′ = E(r) + Ecl(r, 𝑡).

We note that the above properties imply that withinthis framework, the action of any incoming laser pulse onthe full emitter-cavity system can be described purely bythe action of the medium-supported classical electric fieldon the emitters, with no additional explicit driving of anyEM modes. This is different to, e.g., input-output theory,where the EM field is split into modes inside the cavityand free-space modes outside, and external driving thusaffects the cavity modes. Importantly, Ecl(r, 𝑡) is the fieldobtained at the position of the emitter upon propagation ofthe external laser pulse through the material structure, i.e.,it contains any field enhancement and temporal distortion.Since the Hamiltonian expressed by the field operatorsf𝜆(r, 𝜔) by construction solves Maxwell’s equations inthe presence of the material structure, Ecl(r, 𝑡) can beobtained by simply solving Maxwell’s equations using anyclassical EM solver without ever expressing the pulse in thebasis of the modes f𝜆(r, 𝜔). It is important to rememberthat EM field observables are also transformed accordingto

⟨𝜓|𝑂|𝜓⟩ = ⟨𝜓′|𝑇 (𝑡)𝑂𝑇 †(𝑡)|𝜓′⟩, (12)

such that, e.g., ⟨𝜓|𝑎𝑛|𝜓⟩ = ⟨𝜓′|𝑎𝑛 + 𝛼𝑛(𝑡)|𝜓′⟩. This takesinto account that the “quantum” field generated by thelaser-emitter interaction interferes with the classical pulsepropagating through the structure, and ensures a correctdescription of absorption, coherent scattering, and similareffects.

2.4 Emitter-centered modes

Following [38–43], we now look for a linear transformationof the bosonic modes f𝜆(r, 𝜔) at each frequency in such away that in the new basis, only a minimal number of EM

modes couples to the emitters. To this end, we start withthe macroscopic QED Hamiltonian within the multipolarapproach Eq. (8), with the emitter-field interaction treatedwithin the long-wavelength approximation Eq. (9). Forsimplicity of notation, we assume that the dipole operatorof each emitter only couples to a single field polarization,d𝑖 = ��𝑖n𝑖. Alternatively, the sum over 𝑖 could simplybe extended to include up to three separate orientationsper emitter. Our goal can then be achieved by definingemitter-centered or bright (from the emitter perspective)EM modes ��𝑖(𝜔) associated with each emitter 𝑖:

��𝑖(𝜔) =∑

𝜆

∫d3r 𝛽𝑖,𝜆(r, 𝜔) · f𝜆(r, 𝜔) (13a)

𝛽𝑖,𝜆(r, 𝜔) = n𝑖 · G𝜆(r𝑖, r, 𝜔)𝐺𝑖(𝜔) , (13b)

where 𝐺𝑖(𝜔) is a normalization factor. Using Eq. (2), thecommutation relations of the operators ��𝑖(𝜔) reduce tooverlap integrals of their components, [��𝑖(𝜔), ��†

𝑗 (𝜔′)] =𝑆𝑖𝑗(𝜔)𝛿(𝜔 − 𝜔′), with

𝑆𝑖𝑗(𝜔) =∑

𝜆

∫d3r 𝛽*

𝑖,𝜆(r, 𝜔) · 𝛽𝑗,𝜆(r, 𝜔)

= ℏ𝜔2

𝜋𝜖0𝑐2n𝑖 · Im G(r𝑖, r𝑗 , 𝜔) · n𝑗

𝐺𝑖(𝜔)𝐺𝑗(𝜔) , (14)

such that the overlap matrix 𝑆𝑖𝑗(𝜔) at each frequency isreal and symmetric (since G(r, r′, 𝜔) = G𝑇 (r′, r, 𝜔)). Inthe above derivation, we have used the Green’s functionidentity∑

𝜆

∫d3s G𝜆(r, s, 𝜔)·G*𝑇

𝜆 (r′, s, 𝜔) = ℏ𝜔2

𝜋𝜖0𝑐2 Im G(r, r′, 𝜔)

(15)to obtain a compact result [22, 23]. The normalizationfactor 𝐺𝑖(𝜔) is obtained by requiring that 𝑆𝑖𝑖(𝜔) = 1, so

𝐺𝑖(𝜔) =√

ℏ𝜔2

𝜋𝜖0𝑐2 n𝑖 · Im G(r𝑖, r𝑖, 𝜔) · n𝑖. (16)

We note that the coupling strength 𝐺𝑖(𝜔) of the emitter-centered mode ��𝑖(𝜔) to emitter 𝑖 is directly related to theEM spectral density 𝐽𝑖(𝜔) = [𝜇𝐺𝑖(𝜔)/ℏ]2 for transitiondipole moment 𝜇 [79]. In the regime of weak coupling,i.e., when the EM environment can be approximated asa Markovian bath, the spontaneous emission rate at anemitter frequency 𝜔𝑒 is then given by 2𝜋𝐽𝑖(𝜔𝑒).

Since the overlap matrix 𝑆𝑖𝑗(𝜔) of the modes associ-ated with emitters 𝑖 and 𝑗 is determined by the imaginarypart of the Green’s function between the two emitter posi-tions, it follows that the modes ��𝑖(𝜔), or equivalently, thecoefficient arrays 𝛽𝑖(r, 𝜔), are not orthogonal in general.

J. Feist et al., Macroscopic QED for quantum nanophotonics 7

This can be resolved by performing an explicit orthogonal-ization, which is possible as long as the modes are linearlyindependent. When this is not the case, linearly dependentmodes can be dropped from the basis until a minimal setis reached [41]. In the following, we thus assume linearindependence. We can then define new orthonormal modes𝐶𝑖(𝜔) as a linear superposition of the original modes (andvice versa)

𝐶𝑖(𝜔) =𝑁∑

𝑗=1𝑉𝑖𝑗(𝜔)��𝑗(𝜔), (17a)

𝜒𝑖,𝜆(r, 𝜔) =𝑁∑

𝑗=1𝑉𝑖𝑗(𝜔)𝛽𝑗,𝜆(r, 𝜔), (17b)

which also implies that ��𝑖(𝜔) =∑𝑁

𝑗=1 𝑊𝑖𝑗(𝜔)𝐶𝑗(𝜔),where W(𝜔) = V(𝜔)−1. In the above, the transforma-tion matrix V(𝜔) is chosen such that [𝐶𝑖(𝜔), 𝐶†

𝑗 (𝜔′)] =𝛿𝑖𝑗𝛿(𝜔 − 𝜔′), which implies V(𝜔)S(𝜔)V†(𝜔) = 1. Thecoefficient matrices V(𝜔) can be chosen in various ways,corresponding to different unitary transformations of thesame orthonormal basis. We mention two common ap-proaches here. One consists in performing a Choleskydecomposition of the overlap matrix, S(𝜔) = L(𝜔)L𝑇 (𝜔),with V(𝜔) = L(𝜔)−1, where L(𝜔) and L(𝜔)−1 are lowertriangular matrices. This is the result obtained from Gram-Schmidt orthogonalization, with the advantage that 𝐶𝑖(𝜔)only involves ��𝑗(𝜔) with 𝑗 ≤ 𝑖 and vice versa, such thatemitter 𝑖 only couples to the first 𝑖 photon continua. An-other possibility is given by Löwdin orthogonalization,with V(𝜔) = S(𝜔)−1/2, where the matrix power is definedas S𝑎 = UΛ𝑎U†, with U (Λ) a unitary (diagonal) ma-trix containing the eigenvectors (eigenvalues) of S. Thisapproach maximizes the overlap [𝐶𝑖(𝜔), ��†

𝑖 (𝜔)] while en-suring orthogonality, and can thus be seen as the “minimal”correction required to obtain an orthonormal basis.

Using the orthonormal set of operators 𝐶𝑖(𝜔), whichare themselves linear superpositions of the operatorsf𝜆(r, 𝜔), we can perform a unitary transformation (sep-arately for each frequency 𝜔) of the f𝜆(r, 𝜔) into a basisspanned by the emitter-centered (or bright) EM modesand an infinite number of “dark” modes ��𝑖(𝜔) that spanthe orthogonal subspace and do not couple to the emitters,such that

f𝜆(r, 𝜔) =𝑁∑

𝑖=1𝜒*

𝑖,𝜆(r, 𝜔)𝐶𝑖(𝜔) +∑

𝑗

𝑑*𝑗,𝜆(r, 𝜔)��𝑗(𝜔).

(18)

where∫

d3r 𝑑*𝑗 (r, 𝜔) · 𝜒𝑖(r, 𝜔) = 0. The Hamiltonian can

then be written as

ℋ =∞∫

0

d𝜔

[𝑁∑

𝑖=1ℏ𝜔𝐶†

𝑖 (𝜔)𝐶𝑖(𝜔) +∑

𝑗

ℏ𝜔��†𝑗 (𝜔)��𝑗(𝜔)

−𝑁∑

𝑖,𝑗=1��𝑖

(𝑔𝑖𝑗(𝜔)𝐶𝑗(𝜔) + H.c.

) ]+

𝑁∑𝑖=1

��𝑖 (19)

where 𝑔𝑖𝑗(𝜔) = 𝐺𝑖(𝜔)𝑊𝑖𝑗(𝜔). We note here that V(𝜔)and thus W(𝜔) and 𝑔𝑖𝑗(𝜔) can always be chosen real dueto the reality of 𝑆𝑖𝑗(𝜔), but we here treat the general casewith possibly complex coefficients. Since the dark modesare decoupled from the rest of the system, they do notaffect the dynamics and can be dropped, giving

ℋ =𝑁∑

𝑖=1��𝑖 +

∞∫0

d𝜔

[𝑁∑

𝑖=1ℏ𝜔𝐶†

𝑖 (𝜔)𝐶𝑖(𝜔)

−𝑁∑

𝑖,𝑗=1��𝑖

(𝑔𝑖𝑗(𝜔)𝐶𝑗(𝜔) + H.c.

) ]. (20)

We mention for completeness that if the dark modes areinitially excited, including them might be necessary to fullydescribe the state of the system. We have now explicitlyconstructed a Hamiltonian with only 𝑁 independent EMmodes 𝐶𝑖(𝜔) for each frequency 𝜔 [38–43].

Furthermore, one can obtain an explicit expressionfor the electric field operator based on the modes 𝐶𝑖(𝜔),which to the best of our knowledge has not been presentedpreviously in the literature. This is obtained by insertingEq. (18) in Eq. (10), again dropping the dark modes ��𝑖(𝜔),and again using the integral relation for Green’s functionsfrom Eq. (15), leading to

E(+)(r) =𝑁∑

𝑖=1

∞∫0

d𝜔E𝑖(r, 𝜔)𝐶𝑖(𝜔) (21a)

E𝑖(r, 𝜔) =𝑁∑

𝑗=1𝑉 *

𝑖𝑗(𝜔)ℰ𝑗(r, 𝜔) (21b)

ℰ𝑗(r, 𝜔) = ℏ𝜔2

𝜋𝜖0𝑐2Im G(r, r𝑗 , 𝜔) · n𝑗

𝐺𝑗(𝜔) (21c)

These relations show that we can form explicit photonmodes in space at each frequency by using orthonormalsuperpositions of the emitter-centered EM modes ℰ𝑗(r, 𝜔).We note that this also provides a formal construction forthe “emitter-centered” EM modes, i.e., the modes createdby the operators ��𝑖(𝜔), with a field profile correspondingto the imaginary part of the Green’s function associatedwith that emitter. These modes are well-behaved: they

8 J. Feist et al., Macroscopic QED for quantum nanophotonics

solve the source-free Maxwell equations, are real every-where in space, and do not diverge anywhere (here, itshould be remembered that the real part of the Green’sfunction diverges for r = r′, while the imaginary part doesnot). As a simple example, we can take a single 𝑧-orientedemitter at the origin in free space. The procedure usedhere then gives exactly the 𝑙 = 1, 𝑚 = 0 spherical Besselwaves, i.e., the only modes that couple to the emitter whenquantizing the field using spherical coordinates. Finally,we note that it can be verified easily that inserting Eq. (21)in Eq. (8) and simplifying leads exactly to Eq. (20).

3 ExampleAs an example, here we treat a complex metallodielectricstructure, as shown in Fig. 1. It is composed of a dielectricmicrodisk resonator supporting whispering-gallery modes,with a metallic sphere dimer antenna placed within. TheSiN (𝜖 = 4) disk is similar to that considered in [49], withradius 2.03 𝜇m and height 0.2 𝜇m. Two 40 nm diameterAg (with permittivity taken from [80]) nanospheres sepa-rated by a 2 nm gap are placed 1.68 𝜇m away from thedisk axis. Two point-dipole emitters modeled as two-levelsystems and oriented along the dimer axis are placed inthe central gap of the dimer antenna, and just next tothe antenna (labelled as points 1 and 2 in Fig. 1). Theirtransition frequencies are chosen as 𝜔𝑒,1 = 𝜔𝑒,2 = 2 eV,while the dipole transition moments are 𝜇1 = 0.1 e nmand 𝜇2 = 3 e nm (roughly corresponding to typical singleorganic molecules and 𝐽-aggregates, respectively [81]). Asshown in the theory section, the emitter dynamics arethen fully determined by the Green’s function betweenthe emitter positions (as also found in multiple-scatteringapproaches [53]). In Fig. 2, we show the relevant valuesn𝑖 · Im G(r𝑖, r𝑗 , 𝜔) · n𝑗 , which clearly reveals the relativelysharp Mie resonances of the dielectric disk, hybridizedwith the short-range plasmonic modes of the metallicnanosphere dimer. In addition, it also shows that the cou-pling between the two emitters, i.e., the offdiagonal termwith 𝑖 = 1, 𝑗 = 2, has significant structure and changessign several times within the frequency interval.

We now study the dynamics for the Wigner-Weisskopfproblem of spontaneous emission of emitter 1, i.e., for thecase where emitter 1 is initially in the excited state, whileemitter 2 is in the ground state and the EM field is inits vacuum state, i.e., |𝜓(𝑡 = 0)⟩ = 𝜎+

1 |0⟩, where |0⟩ isthe global vacuum without any excitations and 𝜎+

1 is aPauli matrix acting on emitter 1. We also treat the light-matter coupling within the rotating wave approximation,

Fig. 1: Sketch of the hybrid metallo-dielectric structure we treat:A dielectric microdisk resonator with a metallic dimer antennaplaced on top. The positions of the two emitters are indicatedby points 1 and 2, while the field evaluation point used later isindicated as point 3.

i.e., we use∑𝑁

𝑖,𝑗=1 𝜇𝑖

(𝑔𝑖𝑗(𝜔)𝜎+𝐶𝑗(𝜔) + H.c.

)as the light-

matter interaction term. The number of excitations is thenconserved, and the system can be solved easily withinthe single-excitation subspace by simply discretizing thephoton continua in frequency [50]. The resulting emitterdynamics, i.e., the population of the excited states of theemitters, are shown in Fig. 3. This reveals that the EMfield emitted by emitter 1 due to spontaneous emission isthen partially reabsorbed by emitter 2. Comparison withthe dynamics of emitter 1 when emitter 2 is not present(shown as a dashed blue line in Fig. 3) furthermore revealsthat there is also significant transfer of population backfrom emitter 2 to emitter 1.

The direct access to the photonic modes in this ap-proach provides interesting insight into, e.g., the photonicmode populations, which are shown in Fig. 4 at the finaltime considered here, 𝑡 = 1000 fs. As mentioned above,there is some freedom in choosing the orthonormalizedcontinuum modes 𝐶𝑗(𝜔), as any linear superposition ofmodes at the same frequency is also an eigenmode ofthe EM field. We have here chosen the modes obtainedthrough Gram-Schmidt orthogonalization, which, as dis-cussed above, have the advantage that emitter 𝑖 onlycouples to the first 𝑖 photon continua. In particular, emit-ter 1 only couples to a single continuum, 𝐶1(𝜔), whileemitter 2 couples to the same continuum, and additionallyto its “own” continuum 𝐶2(𝜔). This makes the comparisonbetween the case of having both emitters present or only

J. Feist et al., Macroscopic QED for quantum nanophotonics 9

1.00 1.25 1.50 1.75 2.00 2.25 2.50ω (eV)4002000

200400600800ni·I

mG

(ri,

r j,ω

)·n

j [1/nm] G1, 1

G2, 2 × 1000

G1, 2 × 100

Fig. 2: Green’s function factor n𝑖 · ImG(r𝑖, r𝑗 , 𝜔) · n𝑗 connectingthe two emitters in the geometry of Fig. 1.

0 200 400 600 800 1000t (fs)0.00.20.40.60.81.0

Pe(t

)

Emitter 1Emitter 2Emitter 1 alone

Fig. 3: Population of emitters 1 (black line) and 2 (orange line)for the Wigner-Weisskopf problem with emitter 1 initially excited.The blue dashed line shows the dynamics of emitter 1 if emitter 2is not present.

including emitter 1 quite direct, as can be observed inFig. 4. In particular, any population in the modes 𝐶2(𝜔)must come from emitter 2, after it has in turn been excitedby the photons emitted by emitter 1 into continuum 1.

Finally, we also evaluate the electric field in timeat a third position (indicated as point 3 in Fig. 1), asdetermined by Eq. (21). This is displayed Fig. 5 andshows a broad initial peak due to the fast initial decayof emitter 1 (filtered by propagation through the EMstructure, with clear interference effects visible), and thena longer tail due to the longer-lived emission from bothemitters, which is mostly due to emitter 2 (which is lessstrongly coupled to the EM field) and its back-feeding ofpopulation to emitter 1.

1.96 1.98 2.00 2.02 2.04ω (eV)0

204060

|Cj(ω

)|2

C1, em. 1 & 2C2, em. 1 & 2C1, only em. 1C2, only em. 1

Fig. 4: Population ⟨𝐶†𝑗 (𝜔)𝐶𝑗(𝜔)⟩ of EM modes at time 𝑡 =

1000 fs in the presence of both emitters (solid black and orangelines) and in the presence of only emitter 1 (dashed blue andgreen lines). Gram-Schmidt orthogonalization has been used here,so that emitter 1 only couples to continuum 1 (i.e., populationsfor 𝑗 = 2 are identically zero when emitter 2 is not present), whileemitter 2 couples to both continua.

0 200 400 600 800 1000t (fs)0.00.10.20.30.4

|E|2

(arb.u.)With emitter 2Without emitter 2

Fig. 5: Time-dependent electric field intensity |E|2 at position 3

in Fig. 1, for the case when both emitters are present (black solidline) and when only emitter 1 is present (orange dashed line).

4 ConclusionsIn this article we have presented a general overview ofthe application of the formalism of macroscopic quantumelectrodynamics in the context of quantum nanophotonics.Within this research field, it is often mandatory to describefrom an ab-initio perspective how a collection of quantumemitters interacts with a nanophotonic structure, whichis usually accounted for by utilizing macroscopic Maxwellequations. Macroscopic QED then needs to combine toolstaken from both quantum optics and classical electromag-netism. After the presentation of the general formalism andits approximations, we have reviewed in detail the stepsto construct a minimal but complete basis set to analyze

10 J. Feist et al., Macroscopic QED for quantum nanophotonics

the interaction between an arbitrary dielectric structureand multiple quantum emitters. This minimal basis setis formed by the so-called emitter-centered modes, suchthat all the information regarding the EM environmentis encoded into the EM dyadic Green’s function, whichcan be calculated using standard numerical tools capa-ble of solving macroscopic Maxwell equations in complexnanophotonic structures. As a way of example and to showits full potential, in the final part of this article, we haveapplied this formalism to solve both the population dy-namics and EM field generation associated to the couplingof two quantum emitters with a hybrid plasmo-dielectricstructure composed of a dielectric microdisk within whicha metallic nanosphere dimer is immersed. We emphasizethat this formalism can be used not only to provide ex-act solutions to problems in quantum nanophotonics, butcould also serve as a starting point for deriving simplermodels and/or approximated numerical treatments.

Funding: This work has been funded by the EuropeanResearch Council (doi:10.13039/501100000781) throughgrant ERC-2016-StG-714870 and by the Spanish Min-istry for Science, Innovation, and Universities – Agen-cia Estatal de Investigación (doi:10.13039/501100011033)through grants RTI2018-099737-B-I00, PCI2018-093145(through the QuantERA program of the European Com-mission), and MDM-2014-0377 (through the María deMaeztu program for Units of Excellence in R&D).

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