1. INTRODUCTION

Many aspects of light–matter interaction can be understood from the coupling between dipole emitters (or absorbers) and the electromagnetic field in a structured medium. Indeed, the basic processes in molecular spectroscopy, light scattering from small particles or atoms, fluorescence, nonlinear optics, or cavity quantum electrodynamics (QED) are most of the time described using electric (or magnetic) dipoles interacting with the electromagnetic field [1–4]. With the advent of nanophotonics, structuring the environment at scales much smaller than the wavelength is used to modify and control the emission and absorption dynamics of quantum emitters (such as molecules or quantum dots). This has become an active area of research, with fundamental and applied perspectives [5,6].

Depending on the strength of the interaction, different regimes are observed. In the weak-coupling regime, spontaneous emission can be either accelerated or inhibited, a phenomenon referred to as the Purcell effect [7]. When the emitter strongly couples to a specific mode of the electromagnetic field, two new hybridized eigenmodes (polaritons) are created, characterized by a frequency splitting or the appearance of Rabi oscillations in the time domain [8,9]. Initially the realm of cavity QED, changes in the spontaneous emission dynamics in the weak- and strong-coupling regimes have been demonstrated in nanophotonics using optical antennas [10], microcavities [11,12], photonic crystal cavities [13], or plasmonic cavities [14]. The mutual interaction between several emitters in the presence of an electromagnetic field also gives rise to different phenomena, from energy transfer between two molecules in weak coupling [15] to coherent collective interactions leading to sub- and superradiance [16,17]. Here as well, confining the electromagnetic field allows one to act on the coupling strength. For example, the range of energy transfer can be modified using surface plamons [18], and collective interactions can be enhanced using photonic crystal cavities [19].

In this tutorial, we propose a simple and unified approach to deal with the interaction between a quantum emitter and the electromagnetic field in a structured medium, and we show how the same starting point allows one to describe many different regimes and phenomena in light–matter interaction. The emitter is described by an electric polarizability, and the field is described in terms of a Green function. Assuming an external excitation, we address the coupling as a semi-classical scattering process (by semi-classical we mean that the field is not explicitly quantized), and we derive a coupling equation from which the eigenfrequencies of the resulting eigenmodes can be deduced. By choosing the correct model for the Green function, which describes the response of the environment, the formalism naturally leads to a description of the weak- and strong-coupling regimes. The intuitive scattering approach is easily extended to the situation of two emitters coupled through a structured environment. Interestingly, beyond coherent mutual interactions leading to strong coupling, the model also includes a description of incoherent energy transfer between molecules in the weak-coupling regime. Finally, we show how a generalization to a set of $N$ emitters provides an appealing coupled-dipole model to describe collective interactions.

The tutorial is organized as follows. In Section 2, starting from the optical Bloch equations, we derive the polarizability model that allows us to describe either the full dynamics of a two-level atom or the excitation dynamics of a three-level molecule. In Section 3, we introduce the concept of the Green function, which is a useful tool to describe the electrodynamic response of an arbitrary environment. In Section 4, we derive the coupling equation that drives the dynamics of the coupled emitter-field system, based on an intuitive scattering approach. From this equation, we show how the weak- and strong-coupling regimes emerge. In Section 5, we extend the scattering approach to the situation of two emitters coupled through a structured environment, focusing the analysis on the regimes of weak and strong dipole–dipole interaction. In the weak-coupling regime, we show how irreversible energy transfer can be described using appropriate polarizability models. In Section 6, we briefly discuss the generalization of the model to the collective interaction between $N$ identical emitters, with $N$ arbitrarily large. Finally, Section 7 summarizes the main conclusions.

2. POLARIZABILITY OF A DIPOLE EMITTER

The electrodynamic response of a subwavelength resonant scatter can be described in the electric-dipole limit using a dynamic polarizability. The same description holds for an atom or a fluorescent molecule. The interaction between a two-level atom and a classical monochromatic electric field is a textbook problem that is usually treated by solving the optical Bloch equations [1,8]. Here we use this framework to describe the excitation of a three-level system by a quasi-monochromatic electric field. The three-level model includes the two-level atom as a particular case. It also encompasses the main features needed to describe the excitation of a fluorescent molecule.

A. Three-Level Model

We consider a three-level system characterized by three stationary and non-degenerate eigenstates $|a⟩$, $|b⟩$, and $|c⟩$, as represented in Fig. 1, with ${\mathrm{\Gamma }}_{bc}$, ${\mathrm{\Gamma }}_{ba}$, and ${\mathrm{\Gamma }}_{ca}$ the spontaneous decay rates of each level. In practice, this three-level model can be used to describe a two-level atom (by taking ${\mathrm{\Gamma }}_{bc}=0$), or a three-level system with a high decay rate towards the auxiliary level (${\mathrm{\Gamma }}_{bc}\gg {\mathrm{\Gamma }}_{ba}$) that provides the simplest model of a fluorescent molecule.

 

Fig. 1. Jablonski diagram of a three-level system. For ${\mathrm{\Gamma }}_{bc}=0$, the system reduced to the model of a two-level atom. For ${\mathrm{\Gamma }}_{bc}\gg {\mathrm{\Gamma }}_{ba}$, the three-level system is the simplest relevant model of a fluorescent molecule. In this case, ${\mathrm{\Gamma }}_{bc}$ corresponds to a fast non-radiative decay towards state $|c⟩$, and ${\mathrm{\Gamma }}_{ca}$ corresponds to the radiative transition.

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The state of the system is conveniently described by a density operator $\widehat{\rho }$. The diagonal elements of this operator, known as populations, give the probability for the system to be in one of its eigenstates. The off-diagonal elements, known as coherences, describe dynamic effects related to the coherent superpositions of eigenstates. They enter, as we shall see, the expression of the polarizability. The evolution of the density operator is driven by the Hamiltonian $\widehat{H}$ according to [20]

B. Optical Bloch Equations

Finding the solution to Eq. (4) requires us to solve a system of nine equations. For our purposes, we need to compute the excited-state populations ${\rho }_{aa}$, ${\rho }_{bb}$, and ${\rho }_{cc}$, as well as the coherences ${\rho }_{ab}$ and ${\rho }_{ba}$. The coherences will allow us to compute the expectation of the dipole moment operators associated with transition $ab$. Since the density operator is Hermitian and satisfies the condition ${\rho }_{aa}+{\rho }_{bb}+{\rho }_{cc}=1$, we can reduce the problem to a set of three equations. As we assume the external electric field to be quasi-resonant with transition $ab$, we can use the rotating wave approximation ($|\omega -{\omega }_{ab}|\ll {\omega }_{ab}$) and the slowly varying envelope approximation (${\gamma }_{ab}\ll {\omega }_{ab}$) [2]. This leads to the optical Bloch equations [1,8,21]:

C. Polarizability

Assuming an excitation by a stationnary external field, we focus on the steady-state behavior of the coupled emitter-field system. In this regime, the solution of the optical Bloch equations can be found analytically. Solving Eqs. (5)–(7) in the frequency domain yields

3. FIELD RESPONSE: GREEN’S FUNCTION

While the electrodynamic response of a dipole emitter (or scatterer) is described by its polarizability, the linear response of the environment is conveniently described using the electric Green function $\mathbf{G}$ (also denoted by field susceptibility). The tensor (electric) Green function is defined as the solution of the vector Helmoltz equation [5,24]

4. DIPOLE EMITTER INTERACTING WITH AN ENVIRONMENT

In this section, we consider a two-level dipole emitter located at a position ${\mathbf{r}}_s$, with a fixed orientation of its transition dipole (defined by unit vector $\mathbf{u}$), and characterized by its free-space polarizability ${\mathit{\alpha }}_0(\omega )={\alpha }_0(\omega )\mathbf{u}\otimes \mathbf{u}$, with

A. Coupling Equation

The response of the dipole emitter to an external field can be understood as a two-step scattering process. First, the emitter is excited by the field ${\mathbf{E}}_{\mathrm{exc}}$ generated by scattering of the incident field ${\mathbf{E}}_{\mathrm{inc}}$ by the environment. Second, the emitter is excited by its own field scattered back by the environment. These two processes are represented schematically in Fig. 2. With these two processes in mind, the induced dipole can be written as

 

Fig. 2. Representation of the two scattering processes involved in the electrodynamic interaction between a dipole emitter and a structured environment.

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B. Weak Coupling

Let us first consider the situation in which the environment has a smooth frequency dependence at the scale of the emitter linewidth ${\gamma }_0$. We can assume $\mathcal{S}(\omega )\simeq \mathcal{S}({\omega }_0)$, and the solution to Eq. (26) simply becomes

C. Strong Coupling

We now assume that the emitter is coupled to an environment exhibiting sharp resonances, and is resonant (or quasi-resonant) with a specific mode so that we can restrict the problem to the interaction with a single mode. Assuming $|\omega -{\omega }_m|\ll {\omega }_m$ and ${\gamma }_m\ll {\omega }_m$, where ${\omega }_m$ and ${\gamma }_m$ are, respectively, the central frequency and linewidth of the mode, we can use the following single-mode expansion of the Green function:

For the sake of illustration, let us consider a dipole emitter characterized by a central frequency ${\omega }_0=2370\mathrm{meV}$, a radiative linewidth ${\mathrm{\Gamma }}_0=0.004\mathrm{meV}$, and a total linewidth ${\gamma }_0=140\mathrm{meV}$ (these values are typical of a fluorescent molecule at room temperature). We assume the emitter to be coupled to a single-mode cavity characterized by ${\omega }_m=2220\mathrm{meV}$ and ${\gamma }_m=40\mathrm{meV}$. By increasing the Purcell factor $F_m$ of the cavity, we can follow the evolution of the eigenfrequencies in the complex plane, as shown in Fig. 3(a). Both the frequency splitting and the change in the linewidth are observed. The dependence of the frequency splitting on the Purcell factor (that changes the coupling constant) is shown in Fig. 3(b). In this example, the critical Purcell factor, which separates the weak- and strong-coupling regimes, is on the order of ${10}^5$.

 

Fig. 3. (a) Evolution of the eigenfrequencies of the coupled system in the complex plane when increasing the Purcell factor $F_m$ of the cavity. (b) Normalized frequency shift of the two eigenmodes versus the Purcell factor $F_m$. Error bars represent intervals bounded by ${\omega }_p\pm {\gamma }_p/2$.

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5. TWO EMITTERS IN A STRUCTURED MEDIUM

In this section we describe the interaction between two dipole emitters in an environment, and discuss the strong- and weak-coupling regimes. In the weak-coupling regime, we show that the formalism encompasses the process of irreversible energy transfer between a donor and an acceptor.

A. Coupling Equation

We consider two dipole emitters located in an arbitrary medium, and excited by an external field. The emitters are characterized by their free-space polarizability ${\mathit{\alpha }}_i(\omega )={\alpha }_i(\omega ){\mathbf{u}}_i\otimes {\mathbf{u}}_i$, the unit vector ${\mathbf{u}}_i$ defining the fixed orientation of the transition dipole, with

B. Weak Coupling to the Environment

If the medium has a smooth dependence on frequency (no resonance), we can write ${\mathcal{S}}_{ii}(\omega )\simeq {\mathcal{S}}_{ii}({\omega }_0)$ and ${\mathcal{G}}_{ii}(\omega )\simeq {\mathcal{G}}_{ii}({\omega }_0)$. The two eigenfrequency solutions of Eq. (46) are then given by

To get orders of magnitude, let us take the example of two emitters in free space, with the same parameters as in the previous section, which are typical for fluorescent molecules at room temperature (central frequency ${\omega }_1={\omega }_2=2370\mathrm{meV}$, radiative linewidth ${\mathrm{\Gamma }}_1={\mathrm{\Gamma }}_2=0.004\mathrm{meV}$, and total linewidth ${\gamma }_1={\gamma }_2=140\mathrm{meV}$). Let us assume that the transition dipoles are oriented along the $z$ axis, and separated by a distance $d$ along a perpendicular direction (the $x$ axis). In these conditions, the critical distance for the observation of frequency splitting is 3 nm, and the change in the linewidth is negligible (see Fig. 4). For $d\gg 3\mathrm{nm}$, the emitters can be considered independent. Also note that the condition $4{\mathcal{G}}_{12}{({\omega }_0)}^2\gg {|{\varpi }_2-{\varpi }_1|}^2$, which we used to define the strong dipole–dipole interaction regime, is not sufficient for the observation of frequency splitting (that has to be larger than the linewidth).

 

Fig. 4. (a) Evolution of the eigenfrequencies in the complex plane when decreasing the distance $d$ between the emitters in free space. (b) Normalized frequency shift of the two eigenmodes versus the distance $d$. Error bars represent intervals bounded by ${\omega }_p\pm {\gamma }_p/2$.

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C. Weak Dipole–Dipole Interaction

On top of the assumption of weak coupling to the environment, we now assume that the two emitters are weakly coupled to each other ($4{\mathcal{G}}_{12}{({\omega }_0)}^2\ll {|{\varpi }_2-{\varpi }_1|}^2$). In this limit, we can perform a first-order expansion of the square root in Eq. (47), yielding

As a didactic example, let us consider a donor (two-level system) with emission frequency ${\omega }_1=2370\mathrm{meV}$ and radiative linewidth ${\gamma }_1={\mathrm{\Gamma }}_1=0.004\mathrm{meV}$, and an acceptor (three-level molecule) with absorption frequency ${\omega }_2={\omega }_1$ and total linewidth ${\gamma }_2=140\mathrm{meV}$. We show in Fig. 5 the energy transfer rate ${\mathrm{\Gamma }}_{et}$, calculated using Eq. (55), versus the distance $d$ between donor and acceptor. For comparison, we also display the change in the donor linewidth due to the acceptor that includes the scattering back-action (${\mathrm{\Gamma }}_{\text{inter}}$). We see that both expressions coincide for $d<100\mathrm{nm}$. Note that for larger distances, the difference would remain difficult to observe since the energy transfer efficiency is very low in this regime (on the order of ${10}^{-6}$).

 

Fig. 5. Solid line: normalized modification in the donor linewidth due to the presence of the acceptor ${\mathrm{\Gamma }}_{\text{inter}}/{\mathrm{\Gamma }}_1$ versus the distance $d$ between donor and acceptor. Dashed line: normalized energy transfer rate ${\mathrm{\Gamma }}_{et}/{\mathrm{\Gamma }}_1$ calculated using expression (55).

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D. Strong Dipole–Field Interaction

We now examine the regime of strong coupling of the two emitters to a single electromagnetic mode. To proceed, we use the expansion of the Green function in Eq. (33). For convenience we introduce the Purcell factor experienced by each emitter, defined by

 

Fig. 6. (a) Evolution of the eigenfrequencies in the complex plane when increasing the Purcell factor $F_2$ of emitter 2. (b) Normalized frequency shift of the three eigenmodes versus the Purcell factor $F_2$. Error bars represent intervals bounded by ${\omega }_p\pm {\gamma }_p/2$.

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6. GENERALIZATION: $N$ IDENTICAL DIPOLE EMITTERS IN MUTUAL INTERACTION

The approach can be extended to $N$ dipole emitters coupled to a structured environment. In the simplest situation, we can assume $N$ identical emitters with a polarizability ${\mathit{\alpha }}_0(\omega )$ given by Eq. (20), all with the same orientation of their transition dipole. We can also assume that all emitters see the same environment, so that ${\mathcal{S}}_{ii}(\omega )=\mathcal{S}(\omega )$ and ${\mathcal{G}}_{ij}(\omega )=\mathcal{G}(\omega )$. Finding the eigenfrequencies of the coupled system amounts to solving $\det (\mathbf{M})=0$, where $\mathbf{M}$ is now a $N\times N$ matrix. This leads to the following equation for the complex eigenfrequencies ${\varpi }_p$:

7. CONCLUSION

In summary, we have presented a semi-classical description of the interaction between one or several quantum dipole emitters and a structured environment under weak external excitation. The approach is based on a self-consistent coupling equation resulting from a scattering picture. This coupling equation serves as a starting point to discuss many interaction regimes, covering weak and strong coupling between a single emitter and the electromagnetic field, collective interactions between several emitters leading, for example, to superradiance, as well as energy transfer between two emitters. This simple approach provides both a unified description and an intuitive understanding of the behavior of dipole emitters in (nano)structured environments. It can also serve as a foundation for more elaborate models, including an explicit quantization of the electromagnetic field and/or saturation effects in the emitter dynamics [2,4,16,40].