We provide a simple semi-classical formalism to describe the coupling between one or several quantum emitters and a structured environment. Describing the emitter by an electric polarizability, and the surrounding medium by a Green function, we show that an intuitive scattering picture allows one to derive a coupling equation from which the eigenfrequencies of the coupled system can be extracted. The model covers a variety of regimes observed in light–matter interaction, including weak and strong coupling, coherent collective interactions, and incoherent energy transfer. It provides a unified description of many processes, showing that different interaction regimes are actually rooted on the same ground. It can also serve as a basis for the development of more refined models in a full quantum electrodynamics framework.
© 2019 Optical Society of America
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 . 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 , microcavities [11,12], photonic crystal cavities , or plasmonic cavities . 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  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 , and collective interactions can be enhanced using photonic crystal cavities .
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 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 identical emitters, with 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 , , and , as represented in Fig. 1, with , , and the spontaneous decay rates of each level. In practice, this three-level model can be used to describe a two-level atom (by taking ), or a three-level system with a high decay rate towards the auxiliary level () that provides the simplest model of a fluorescent molecule.
The state of the system is conveniently described by a density operator . 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 according to 2], we can construct the unperturbed Hamiltonian for three-level systems, which is 1,21] 1). This leads to the master equation 1, allowing us to write and . The coherences and , which will be needed to compute the polarizability in the next sections, also decay according to , with a damping rate satisfying (the equality holding only when pure dephasing processes, which do not change the energy states, can be neglected). Note that, formally, the last term in Eq. (4) can be represented by an operator known as a Lindblad superoperator, which is sometimes used to explicitly include the relaxation terms in the master equation .
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 , , and , as well as the coherences and . The coherences will allow us to compute the expectation of the dipole moment operators associated with transition . Since the density operator is Hermitian and satisfies the condition , we can reduce the problem to a set of three equations. As we assume the external electric field to be quasi-resonant with transition , we can use the rotating wave approximation () and the slowly varying envelope approximation () . This leads to the optical Bloch equations [1,8,21]:5) shows that the evolution of the excited-state population depends on the phase difference between the coherences (related to the dipole moment operators) and the time-dependent Rabi frequency (related to the external field). In order to compute the polarizability associated with the transition , we need to solve the coupled Bloch equations and find the expression of the coherences and .
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 yields9) can be used to compute the expectation value of the dipole moment operator defined as . More precisely, in order to define a polarizability matching the classical convention for monochromatic fields with a time dependence , we will need the positive frequency part of the expectation value that is given by . For a weak exciting field, we can neglect saturation effects (), and we obtain 2] 23]. For quasi-resonant excitation (), we have
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 (also denoted by field susceptibility). The tensor (electric) Green function is defined as the solution of the vector Helmoltz equation [5,24]25] 26,27].
4. DIPOLE EMITTER INTERACTING WITH AN ENVIRONMENT
In this section, we consider a two-level dipole emitter located at a position , with a fixed orientation of its transition dipole (defined by unit vector ), and characterized by its free-space polarizability , with28].
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 generated by scattering of the incident field 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 as21) and (22), we obtain 20) into Eq. (25), we find that the complex eigenfrequencies are solutions of 20). Solving Eq. (26) allows one to find the complex eigenfrequencies , defining the central frequencies and the linewidth of the eigenmodes of the coupled emitter-field system . This leads to a simple description of different interaction regimes and their main features.
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 . We can assume , and the solution to Eq. (26) simply becomes25], we find 7], the change in the LDOS being given in this case by the so-called Purcell factor).
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 and , where and are, respectively, the central frequency and linewidth of the mode, we can use the following single-mode expansion of the Green function:25]. The change in the Green function can be deduced from Eq. (33) by subtracting the contribution of the vacuum Green function . Only the imaginary part has to be subtracted, since the singular real part of is not included in expression (33). For a discussion of this point, see [29,30]. This leads to 25] 34) into the coupling Eq. (26), we find that the complex eigenfrequencies of the coupled system must satisfy 38) shows the appearance of two new eigenmodes of the strongly coupled system, with resonance frequencies split around the average resonance frequency of the uncoupled systems. Frequency splitting is a feature of the strong-coupling regime, which can be experimentally observed when the splitting is larger than the linewidth of the new eigenmodes. Note that the strong-coupling condition often ensures that the frequency splitting can be experimentally observed, but is not always sufficient (for instance, when ).
For the sake of illustration, let us consider a dipole emitter characterized by a central frequency , a radiative linewidth , and a total linewidth (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 and . By increasing the Purcell factor 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 .
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 , the unit vector defining the fixed orientation of the transition dipole, with4, the relation between the induced dipoles and in each emitter and the excitation field are conveniently expressed in a matrix form , where
B. Weak Coupling to the Environment
If the medium has a smooth dependence on frequency (no resonance), we can write and . The two eigenfrequency solutions of Eq. (46) are then given by28)]. For strong dipole–dipole coupling between the emitters () the central frequency and the linewidth of the two eigenmodes become
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 , radiative linewidth , and total linewidth ). Let us assume that the transition dipoles are oriented along the axis, and separated by a distance along a perpendicular direction (the 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 , the emitters can be considered independent. Also note that the condition , 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).
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 (). In this limit, we can perform a first-order expansion of the square root in Eq. (47), yielding31]. FRET has been widely used in biology as a mechanism to detect molecular interaction . Short-distance energy transfer is also involved in the process of photosynthesis . To compute the eigenfrequencies in this regime, we need to specify the polarizability models for emitter 1 (donor) and emitter 2 (acceptor). We shall assume that the donor is an ideal two-level atom (with ), while the acceptor is a three-level system, with a large excited-state decay rate towards the auxiliary radiative level (as in a florescent molecule). This means that the condition is assumed to be satisfied. Note that, as described in Section 2, the polarizability describes the excitation of emitter 2 only (subsequent fluorescent emission at a different frequency is implicit). We also assume , meaning that the emission frequency of the donor matches the absorption frequency of the acceptor. Under these conditions, the linewidths of the eigenmodes of the coupled system are 52)] and by the presence of the acceptor [third term on the right-hand side in Eq. (52), which will be denoted by ]. We can observe that the presence of the acceptor can either increase or decrease the linewidth of the donor, depending on the sign of . This can be understood as the result of changes in the relative phase between the induced dipole in the donor and the field backscattered by the acceptor at the donor position, as in the process giving rise to oscillations in the fluorescence lifetime of an emitter in front of a reflective interface . For distances much smaller than the wavelength , we can assume . (This can be easily verified in free space, as long as the interdistance .) This means that at short distance the linewidth of the donor is always increased by the presence of the acceptor. Moreover, from Eqs. (15) and (16), one can deduce the on-resonance expressions of the extinction and scattering cross sections and , which are, respectively, 52), usually referred to as the energy transfer rate , can be written 42) and the assumption . This expression takes the usual form of the energy transfer rate in dipole–dipole interaction  (see also [36,37] for a full QED treatment). The main difference between and is that the latter includes back-action from the acceptor to the donor due to scattering, which disappears in the energy transfer regime. In free space, the Green function at short distance can be taken in the quasi-static limit, and follows the scaling . The free-space energy transfer rate therefore scales as , which is a feature of FRET, as initially derived by Förster . In more complex geometries, inserting the appropriate Green function into Eq. (55) allows one to compute the change in the FRET rate due to the environment (see, for example, Refs. [18,38]).
As a didactic example, let us consider a donor (two-level system) with emission frequency and radiative linewidth , and an acceptor (three-level molecule) with absorption frequency and total linewidth . We show in Fig. 5 the energy transfer rate , calculated using Eq. (55), versus the distance 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 (). We see that both expressions coincide for . 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 ).
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 by46), resulting in a third-order equation in the complex frequency whose roots can be found analytically. Different behaviors can be observed depending on the relative values of the two coupling constant and , defined as 3), while is left as a free parameter allowing us to tune the coupling strength of emitter 2. The behavior of the central frequency and linewidth of the three eigenmodes is shown in Fig. 6. For , we observe the two eigenfrequencies resulting from the strong coupling between the emitter 1 and the field mode, while the third eigenfrequency is associated with the unperturbed emitter 2. By increasing , this third eigenfrequency progressively changes to the free-space resonance frequency of emitter 1. In the regime , two eigenfrequencies characterizing strong coupling between emitter 2 and the field emerge, while the third eigenfrequency is associated with the unperturbed emitter 2. This behavior can be interpreted as follows: the emitter with the largest coupling constant strongly couples to the field mode, creating two frequency shifted new eigenmodes, leaving the other emitter out of resonance.
6. GENERALIZATION: IDENTICAL DIPOLE EMITTERS IN MUTUAL INTERACTION
The approach can be extended to dipole emitters coupled to a structured environment. In the simplest situation, we can assume identical emitters with a polarizability 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 and . Finding the eigenfrequencies of the coupled system amounts to solving , where is now a matrix. This leads to the following equation for the complex eigenfrequencies :64) can be compared to the solutions for one emitter strongly coupled to the field given by Eq. (37). These solutions only differ by a modification of the coupling constant, and the coupling constant for identical emitters is simply , where is the coupling constant for one emitter. Interestingly, the weak-coupling and strong-coupling situations lead to very different results. In particular, when the environment is weakly resonant, the splitting in frequency and the linewidth scale with , as one would obtain with the theory of Dicke superradiance in the weak-excitation limit . In contrast, when the environment is strongly resonant, the splitting scales with and the linewidth does not depend on , in agreement with results obtained with the Jaynes–Cumming Hamiltonian . The simple coupled-dipole model introduced in this tutorial therefore contains the main ingredients to describe collective interactions between quantum emitters under external excitation.
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].
Université Paris Sciences et Lettres (PSL) (ANR-10-IDEX-0001-02); LABEX WIFI (ANR-10-LABX-24).
We acknowledge helpful discussions with Arthur Goetschy. D. B. acknowledges ESPCI Paris for a three-month postdoctoral grant that permitted the achievement of this work.
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