## Abstract

We present an overview of the framework of macroscopic quantum electrodynamics from a quantum nanophotonics perspective. Particularly, we focus our attention on three aspects of the theory that are crucial for the description of quantum optical phenomena in nanophotonic structures. First, we review the light–matter interaction Hamiltonian itself, with special emphasis on its gauge independence and the minimal and multipolar coupling schemes. Second, we discuss the treatment of the external pumping of quantum optical systems by classical electromagnetic fields. Third, we introduce an exact, complete, and minimal basis for the field quantization in multiemitter configurations, which is based on the so-called emitter-centered modes. Finally, we illustrate this quantization approach in a particular hybrid metallodielectric geometry: two quantum emitters placed in the vicinity of a dimer of Ag nanospheres embedded in a SiN microdisk.

## 1 Introduction

In principle, quantum electrodynamics (QED) provides an “exact” approach for treating electromagnetic (EM) fields, charged particles, and their interactions, within a full quantum field theory where both matter and light are second quantized (i.e., both photons and matter particles can be created and annihilated). However, this approach is not very useful for the treatment of many effects of interest in fields such as (nano)photonics and quantum optics, which take place at “low” energies (essentially, below the rest mass energy of electrons), where matter constituents are stable and neither created nor destroyed, and additionally, there are often “macroscopic” structures such as mirrors, photonic crystals, metallic nanoparticles etc. involved. Owing to the large number of material particles (on the order of the Avogadro constant, ≈6 × 10^{23}), it then becomes unthinkable to treat the electrons and nuclei in these structures individually. At the same time, a sufficiently accurate description of these structures is usually given by the macroscopic Maxwell equations, in which the material response is described by the constitutive relations of macroscopic electromagnetism. In many situations, it is then desired to describe the interactions between light and matter in a setup where there are one or a few microscopic “quantum emitters” (such as atoms, molecules, quantum dots, etc.) and additionally a “macroscopic” material structure whose linear response determines the local modes of the EM field interacting with the quantum emitter(s).

The quantization of the EM field in such arbitrary material environments, i.e., the construction of a second quantized basis for the medium-assisted EM field that takes into account the presence of the “macroscopic” material structure, is a longstanding problem in QED. The most immediate strategy is to calculate the (classical) EM modes of a structure and to quantize them by normalizing their stored energy to that of a single photon at the mode frequency, *ω*. Consequently, there are in general no truly bound EM modes, and what is normally thought of as an isolated “cavity mode” is more correctly described as a resonance embedded in the continuum, i.e., a quasi-bound state that decays over time through emission of radiation. An interesting exception here are guided modes in systems with translational invariance (i.e., where momentum in one or more dimensions is conserved), as modes lying outside the light cone *ω* =* ck* then do not couple to free-space radiation [2]. An additional exception is given by “bound states in the continuum” [3], which arise owing to destructive interference between different resonances.

As a further obstacle to a straightforward quantization strategy as described above, the response functions describing material structures are necessarily dissipative owing to causality (as encoded in, e.g., the Kramers–Kronig relations). When these losses cannot be neglected, quantization is complicated even further by the difficulty to define the energy density of the EM field inside the lossy material [4], [5], [6].

Given all the points above, it is not surprising that there are many different approaches to quantizing EM modes in lossy material systems [2], [7], [8], [9], [10], [11], [12], [13], [14], [15]. In the following, we give a concise overview of a particularly powerful formal approach that resolves these problems, called macroscopic QED [16], [17], [18], [19], [20], [21], [22], [23], [24]. While there are excellent reviews of this general framework available (e.g., [22], [23]), we focus on its application in the context of quantum nanophotonics and strong light–matter coupling. In particular, we discuss the implications and lessons that can be taken from this approach on gauge independence and, in particular, the role of the so-called dipole self-energy term in the light–matter interaction in the Power–Zienau–Woolley (PZW) gauge, which has been the subject of some recent controversy [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38]. We then review in detail a somewhat nonstandard formulation of macroscopic QED that allows one to construct a minimal quantized basis for the EM field interacting with a collection of multiple quantum emitters. This approach was first introduced by Buhmann and Welsch [39] and subsequently rediscovered independently by several other groups [40], [41], [42], [43], [44]. The fact that this very useful approach has been reinvented by different researchers over the past decade or so partially motivates the current article, which intends to give a concise and accessible overview, and presents some explicit relations that have (to our knowledge) not been published before. We also note that with “minimal basis,” we are here referring to a minimal *complete* basis for the medium-assisted EM field, i.e., this basis contains all the information about the material structure playing the role of the cavity or antenna, and no approximations are made in obtaining it. This then makes it appropriate to serve as a starting point either for numerical treatments [45], [46] or for deriving simpler models where, e.g., the full EM spectrum is described by a few lossy modes [47].

In the final part of the article, we then present an application of the formalism to a specific problem, a hybrid dielectric-plasmonic structure [14], [48], [49]. In particular, we consider a dimer of metallic nanospheres placed within a dielectric microdisk, a geometry that is similar to that considered by Doeleman et al. [50].

## 2 Theory

Macroscopic QED provides a recipe for quantizing the EM field in any geometry, including with lossy materials. One particularly appealing aspect is that the full information about the quantized EM field is finally encoded in the (classical) EM dyadic Green’s function

### 2.1 Minimal coupling

In the following, we represent a short overview of the general theory of light–matter interactions in the framework of macroscopic QED. Since full details can be found in the literature [22], [23], we do not attempt to make this a fully self-contained overview, but rather highlight and discuss some aspects that are not within the traditional focus of the theory, in particular in the context of quantum nanophotonics.

The first step in the application of macroscopic QED is the separation of all matter present in the system to be treated into two distinct groups: one is the macroscopic structure (e.g., a cavity, plasmonic nanoantenna, photonic crystal, …) that will be described through the constitutive relations of electromagnetism, while the other are the microscopic objects (atoms, molecules, quantum dots, …) that are described as a collection of charged particles. This separation constitutes the basic approximation inherent in the approach and relies on the assumptions that macroscopic electromagnetism is valid for the material structure (the medium) and its interaction with the charged particles. While this is often an excellent approximation, some care has to be taken for separations in the subnanometer range, where the atomic structure of the material can have a significant influence [56], [57], [58], [59], [60], [61]. One significant advantage of this approach is that the microscopic objects are governed by the “standard” Hamiltonian of charged particles interacting through the Coulomb force. They can thus be represented using standard approximations, e.g., using the methods of atomic and molecular physics and quantum chemistry to obtain few-level approximations, or also of solid-state physics to obtain effective descriptions of their band structure, although care has to be taken with gauge invariance when such approximations are performed [62].

For simplicity, we assume that the medium response is local and isotropic in space, such that it can be encoded in the position- and frequency-dependent scalar relative permittivity ^{[1]}. The extension of the quantization scheme to nonlocal response functions can be found in reference [22]. We directly give the Hamiltonian

Here, the “atomic” Hamiltonian

where **1** and **0** are the Cartesian (3 × 3) identity and zero tensors, respectively. The index

The light–matter interaction Hamiltonian can be simplified in the long-wavelength or dipole approximation, i.e., if we assume that the charged particles are sufficiently close to each other compared to the spatial scale of local field variations that a lowest order approximation of the positions of the charges relative to their center of mass position

where ^{[2]}. Equation (3) explicitly shows that in the presence of material bodies, the emitter-field interaction has two contributions, one from longitudinal (electrostatic) fields owing to the Coulomb interaction with charges in the macroscopic body and one from transverse fields (described by the vector potential). The relevant fields are given by

where the longitudinal and transverse components of a tensor

with

We note that in the derivation leading to the above expressions, it is assumed that **r**, i.e., that the materials are lossy everywhere in space. The limiting case of zero losses in some regions (e.g., in free space) is only taken at the very end of the calculation. We will see that in the reformulation in terms of emitter-centered modes, subsection 2.4,

As mentioned above, the electrostatic contribution is not present in free space, and in the literature, it is often assumed that any abstract “cavity mode” corresponds to a purely transverse EM field. This is a good approximation for emitters that are far enough away from the material, e.g., in “large” (typically dielectric) structures such as Fabry–Perot planar microcavities, photonic crystals, micropillar resonators, etc. [64], but can break down otherwise. In general, this happens for coupling to evanescent fields [65] and in particular when subwavelength confinement is used to generate extremely small effective mode volumes, such as in plasmonic [66], [67] or phonon-polaritonic systems [68], [69]. This observation is particularly relevant as such subwavelength confinement is the only possible strategy for obtaining large enough light–matter coupling strengths to approach the single-emitter strong coupling regime at room temperature [70], [71], [72], [73]. For subwavelength separations, it is well known that the Green’s function is dominated by longitudinal components, while transverse components can be neglected [74]. In this *quasistatic* approximation, we thus have ^{[3]}. Conversely, owing to the strongly subwavelength field confinement, the long-wavelength or dipole approximation is not necessarily appropriate, and an accurate description requires either the direct use of the expression in terms of the electrostatic potential [75] or the inclusion of higher order terms in the interaction [76]. In this context, it should be noted that within an *ab initio* description (i.e., when the emitters are not treated as few-level systems), the term

### 2.2 Multipolar coupling

We now discuss the PZW gauge transformation [77], [78], [79], which is used to switch from the minimal coupling scheme discussed up to now to the so-called multipolar coupling scheme, which will then in turn form the basis for the emitter-centered modes we discuss later. This scheme has several advantageous properties: it expresses all light–matter interactions through the fields **E** and **B** directly, without needing to distinguish between longitudinal and transverse fields and allows a systematic expansion of the field-emitter interactions in terms of multipole moments. Additionally, in the multiemitter case, it also removes direct Coulomb interactions between charges in different emitters, which instead become mediated through the EM fields. This property makes it easier to explicitly verify and guarantee that causality is not violated through faster-than-light interactions. We only point out and discuss some specific relevant results here and again refer the reader to the literature for full details [22], [23]. The PZW transformation is carried out by the unitary transformation operator

where we have explicitly grouped the charges into several emitters, i.e., distinct (nonoverlapping) collections of charges, labeled with index *i*. The polarization operator *i* is usually defined as

where *i* and *α* belonging to emitter *i*. We note that there is considerable freedom in choosing a definition for

Since Eq. (7) describes a unitary transformation, physical results are unaffected in principle, although the convergence behavior with respect to different approximations can be quite different [26], [36]. Applying the transformation gives the new operators

where all operators are their PZW-transformed (primed) versions, but we have not included explicit primes for simplicity. In the long-wavelength limit, Eq. (9c) becomes simply

i.e., all field-emitter interactions are expressed through the dipolar coupling term, with the electric field operator given explicitly by

We note that the form of the bare-emitter Hamiltonian

which makes explicit the fact that the emitter Hamiltonian in the multipolar gauge is equivalent to the emitter Hamiltonian in minimal coupling plus a term containing the transverse polarization only. In order to arrive at this form, we have used that the Coulomb interaction can be rewritten as an integral over the longitudinal polarization. The transverse part of the polarization in Eq. (12) corresponds to the so-called dipole self-energy term [28]. When a single- or few-mode approximation of the EM field is performed *before* doing the PZW transformation, this term depends on the square of the mode-emitter coupling strength, but not on any photonic operator. However, when all modes of the EM field are included, as implicitly done here and as motivated by the fact that the term is not mode selective (it does not depend on any EM field operator), it is seen directly that this term becomes completely independent of any characteristics of the surrounding material structure, i.e., it cannot be modified by changing the environment that the emitter is located in. Instead, the bare-emitter Hamiltonian in the multipolar approach is simply slightly different than under minimal coupling. This raises the question whether in a few-mode approximation, such a term should be included in simulations of strongly coupled light–matter systems, i.e., whether the few-mode approximation should be performed before the PZW transformation or after [38]. Including the term improves some mathematical properties of the dipole approximation, in particular in large computational boxes and/or for very large coupling strengths [28], [33], [81]. However, it should also be remembered here that in realistic cavities capable of reaching few-emitter strong coupling, the dominant interaction term is due to longitudinal fields, for which this term does not exist (see discussion in subsection 2.1). Furthermore, it should be mentioned that a similar term can arise if the environment-mediated electrostatic interactions are taken into account explicitly instead of through the quantized modes, and one (or some) of the EM modes is additionally treated explicitly. The action of these modes on the emitters then has to be subtracted from the electrostatic interaction to avoid double counting them [27].

### 2.3 External (classical) fields

Adapting an argument by Sánchez-Barquilla et al. [45], here we show that macroscopic QED also enables a straightforward treatment of external incoming EM fields, in particular for the experimentally most relevant case of a classical laser pulse. Assuming that the incoming laser field at the initial time *t* = 0 has not yet interacted with the emitters (i.e., it describes a pulse localized in space in a region far away from the emitters), it can simply be described by a product of coherent states of the EM modes for the initial wave function, *n* here runs over all indices of the EM basis (*λ*, **r**, *ω*), and the *α*_{n}(0) correspond to the classical amplitudes obtained when expressing the laser pulse in the basis defined by these modes. In order to avoid the explicit propagation of this classical field within a quantum calculation, the classical and the quantum field can be split in the Hamiltonian using a time-dependent displacement operator [82]

We note that the above properties imply that within this framework, the action of any incoming laser pulse on the *full* emitter-cavity system can be described purely by the action of the medium-supported classical electric field on the emitters, with no additional explicit driving of any EM modes. This is different to, e.g., standard input–output theory, where the EM field is split into modes inside the cavity and free-space modes outside, and external driving thus affects the cavity modes^{[4]}.

Importantly,

such that, e.g.,

### 2.4 Emitter-centered modes

Following references [39–44], we now look for a linear transformation of the bosonic modes *i* could simply be extended to include up to three separate orientations per emitter. Our goal can then be achieved by defining *emitter-centered* or *bright* (from the emitter perspective) EM modes *i*:

where

such that the overlap matrix

to obtain a compact result [22], [23]. The normalization factor

We note that the coupling strength *G*_{i}(*ω*) of the emitter-centered mode *i* is directly related to the EM spectral density *μ* [83]. In the regime of weak coupling, i.e., when the EM environment can be approximated as a Markovian bath, the spontaneous emission rate at an emitter frequency *ω*_{e} is then given by 2*πJ*_{i}(*ω*_{e}).

Since the overlap matrix *S*_{ij}(*ω*) of the modes associated with emitters *i* and *j* is determined by the imaginary part of the Green’s function between the two emitter positions, it follows that the modes

which also implies that *i* only couples to the first *i* photon continua. Another possibility is given by Löwdin orthogonalization, with **S**. This approach maximizes the overlap

Using the orthonormal set of operators *ω*) of the

where

where

We mention for completeness that if the dark modes are initially excited, including them might be necessary to fully describe the state of the system. We have now explicitly constructed a Hamiltonian with only *N* independent EM modes *ω* [39], [40], [41], [42], [43], [44]. This Hamiltonian corresponds to a set of *N* emitters coupled to *N* continua and can be treated using, e.g., a wide variety of methods developed in the context of open quantum systems [84], [85], [86], [87].

Furthermore, one can obtain an explicit expression for the electric field operator based on the modes

These relations show that we can form explicit photon modes in space at each frequency by using orthonormal superpositions of the emitter-centered EM modes *z*-oriented emitter at the origin in free space. The procedure used here then gives exactly the *l* = 1, *m* = 0 spherical Bessel waves, i.e., the only modes that couple to the emitter when quantizing the field using spherical coordinates. Furthermore, it should be noted that these modes contain the absorption of the EM field in the material, which is encoded in the Green’s function that solves the macroscopic Maxwell equations including losses. This can be understood by remembering that macroscopic QED can be seen as solving the dynamics of an infinite set of coupled harmonic oscillators corresponding to the EM modes and material excitations (including reservoir modes that describe dissipation), cf. subsection 2. Since the coupled system consists only of harmonic oscillators, this problem can be diagonalized without approximations (i.e., without using a master equation or similar formalisms), resulting in a description of the full system in terms of continua of oscillators. The electric field operator as described by Eq. (22) can then be seen as a projection of the full solution onto the field subspace [23]. The end effect of this is that although the Hamiltonian, Eq. (21), is Hermitian, dissipation in the material is fully included. Finally, we note that it can be verified easily that inserting Eq. (22) in Eq. (9) and simplifying leads exactly to Eq. (21).

## 3 Example

As an example, we now treat a complex metallodielectric structure, as shown in Figure 1. It is composed of a dielectric microdisk resonator supporting whispering gallery modes, with a metallic sphere dimer antenna placed within. The SiN (*J*-aggregates, respectively [89]). As shown in the theory section, the emitter dynamics are then fully determined by the Green’s function between the emitter positions (as also found in multiple scattering approaches [54]). In Figure 2, we show the relevant values *i* = 1, *j* = 2, has a significant structure and changes sign several times within the frequency interval.

We now study the dynamics for the Wigner–Weisskopf problem of spontaneous emission of emitter 1, i.e., for the case where emitter 1 is initially in the excited state, while emitter 2 is in the ground state and the EM field is in its vacuum state, such that

The direct access to the photonic modes in this approach provides interesting insight into, e.g., the photonic mode populations, which are shown in Figure 4 at the final time considered here, *t* = 1000 fs. As mentioned above, there is some freedom in choosing the orthonormalized continuum modes *C*_{j}(*ω*) as any linear superposition of modes at the same frequency is also an eigenmode of the EM field. We have here chosen the modes obtained through Gram–Schmidt orthogonalization, which, as discussed above, have the advantage that emitter *i* only couples to the first *i* photon continua. In particular, emitter 1 only couples to a single continuum,

Finally, we also evaluate the electric field in time at a third position (indicated as point 3 in Figure 1), as determined by Eq. (22). This is displayed in Figure 5 and shows a broad initial peak due to the fast initial decay of emitter 1 (filtered by propagation through the EM structure, with clear interference effects visible) and then a longer tail due to the longer-lived emission from both emitters, which is mostly due to emitter 2 (which is less strongly coupled to the EM field) and its backfeeding of the population to emitter 1.

## 4 Conclusions

In this article, we have presented a general overview of the application of the formalism of macroscopic QED in the context of quantum nanophotonics. Within this research field, it is often mandatory to describe from an *ab initio* perspective how a collection of quantum emitters interacts with a nanophotonic structure, which is usually accounted for by utilizing macroscopic Maxwell equations. Macroscopic QED then needs to combine tools taken from both quantum optics and classical electromagnetism. After the presentation of the general formalism and its approximations, we have reviewed in detail the steps to construct a minimal but complete basis set to analyze the interaction between an arbitrary dielectric structure and multiple quantum emitters. This minimal basis set is formed by the so-called emitter-centered modes, such that all the information regarding the EM environment is encoded into the EM dyadic Green’s function, which can be calculated using standard numerical tools capable of solving macroscopic Maxwell equations in complex nanophotonic structures. As a way of example and to show its full potential, in the final part of this article, we have applied this formalism to solve both the population dynamics and EM field generation associated with the coupling of two quantum emitters with a hybrid plasmodielectric structure composed of a dielectric microdisk within which a metallic nanosphere dimer is immersed. We emphasize that this formalism can be used not only to provide exact solutions to problems in quantum nanophotonics but also to serve as a starting point for deriving simpler models and/or approximated numerical treatments.

**Funding source: **European Research Council

**Award Identifier / Grant number: **ERC-2016-StG-714870

**Funding source: **Spanish Ministry for Science, Innovation, and Universities – Agencia Estatal de Investigación

**Award Identifier / Grant number: **RTI2018-099737-B-I00

**Award Identifier / Grant number: **PCI2018-093145

**Award Identifier / Grant number: **MDM-2014-0377

## Acknowledgments

We thank M. Ruggenthaler for interesting discussions.

**Author contribution**: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.**Research funding**: This work has been funded by the European Research Council (doi: 10.13039/501100000781) through grant ERC-2016-StG-714870 and by the Spanish Ministry for Science, Innovation, and Universities – Agencia Estatal de Investigación (doi: 10.13039/501100011033) through grants RTI2018-099737-B-I00, PCI2018-093145 (through the QuantERA program of the European Commission), and MDM-2014-0377 (through the María de Maeztu program for Units of Excellence in R&D).**Conflict of interest statement**: The authors declare no conflicts of interest regarding this article.

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**Received:**2020-08-05

**Accepted:**2020-09-18

**Published Online:**2020-10-14

© 2020 Johannes Feist et al., published by De Gruyter, Berlin/Boston

This work is licensed under the Creative Commons Attribution 4.0 International License.