Abstract

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

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 [14]. 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 Γbc, Γba, and Γca the spontaneous decay rates of each level. In practice, this three-level model can be used to describe a two-level atom (by taking Γbc=0), or a three-level system with a high decay rate towards the auxiliary level (ΓbcΓba) that provides the simplest model of a fluorescent molecule.

 

Fig. 1. Jablonski diagram of a three-level system. For Γbc=0, the system reduced to the model of a two-level atom. For ΓbcΓba, the three-level system is the simplest relevant model of a fluorescent molecule. In this case, Γbc corresponds to a fast non-radiative decay towards state |c, and Γca corresponds to the radiative transition.

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The state of the system is conveniently described by a density operator ρ^. 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 H^ according to [20]

dρ^dt=1i[H^,ρ^].
Using this equation is equivalent to using the Schrödinger equation for an arbitrary state |ψ(t) of the system, with the advantage of providing a straightforward description of mixed states. Since we are interested in the interaction between the three-level system and an external electric field, it is convenient to write H^=H^0+H^1, where H^0 is the unperturbed Hamiltonian (describing the emitter in absence of electric field) and H^1 is the interaction Hamiltonian (describing the coupling with the field). Following the procedure commonly used for two-level systems [2], we can construct the unperturbed Hamiltonian for three-level systems, which is
H^0=ωabσ^ab+σ^ab+ωacσ^ac+σ^ac,
where ωij is the Bohr frequency associated with the transition ij, and σ^ij+=|ji| and σ^ij=|ij| are the atomic raising and lowering operators, respectively. In order to describe the excitation of the emitter, we use a semi-classical description and assume that it interacts with a classical quasi-monochromatic electric field tuned to the transition ab. In the electric-dipole approximation, we can express the interaction Hamiltonian as [1,21]
H^1=dab·E(t)(σ^ab++σ^ab),
where E(t) is the electric field at the position of the emitter and dab=a|D^|b=b|D^|a is the dipole matrix element (or transition dipole). At this stage we did not consider the effects of spontaneous emission and other interactions with the environment (such as collisions with other molecules in a gas, with phonons in a solid, or with internal degrees of freedom in the emitter itself). These processes, assumed to be independent of the external exciting field, affect the populations and the coherences, and need to be included in Eq. (1). This leads to the master equation
dρ^dt=1i[H^0+H^1,ρ^]+{dρ^dt}relax
in which the last term accounts for the decay of populations and coherences due to spontaneous emission and dephasing processes (that contribute to the relaxation of the coherences). The form of the relaxation terms can be found by considering the three-level system in the absence of an external driving field. In this case we know that the populations of states |b and |c decay spontaneously, with the rates indicated in Fig. 1, allowing us to write dρbb/dt=(Γba+Γbc)ρbb and dρcc/dt=Γcaρcc+Γbcρbb. The coherences ρba and ρab=ρba*, which will be needed to compute the polarizability in the next sections, also decay according to dρba/dt=(γab/2)ρba, with a damping rate γab satisfying γabΓba+Γbc (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 L^d(ρ^) known as a Lindblad superoperator, which is sometimes used to explicitly include the relaxation terms in the master equation [22].

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 ρaa, ρbb, and ρcc, as well as the coherences ρab and ρba. 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 ρaa+ρbb+ρcc=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 (|ωωab|ωab) and the slowly varying envelope approximation (γabωab) [2]. This leads to the optical Bloch equations [1,8,21]:

dρbbdt=(Γba+Γbc)ρbb+2Im[ρabΩ(+)(t)],
dρccdt=Γcaρcc+Γbcρbb,
dρbadt=γab2ρbaiωabρba+i(ρbbρaa)Ω(+)(t),
where we have introduced the time-dependent Rabi frequency Ω(t)=[dab·E(t)]/, and its positive frequency component defined with the following convention:
Ω(+)(t)=0+Ω(ω)eiωtdω.
The time-dependent Rabi frequency characterizes the coupling strength between the three-level emitter and the electric field. In the absence of an external field, Ω(+)(t)=0 and the system spontaneously decays towards its lower energy state |a due to the damping rates of populations and coherences. In contrast, in the presence of an electric field, the terms in Ω(+)(t) couple the equations driving the populations and the coherences. In particular, Eq. (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 ab, we need to solve the coupled Bloch equations and find the expression of the coherences ρab and ρba.

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

ρba(ω)=Ω(+)(ω)ωabωiγab/2(11+s),
where s is the saturation parameter given by
s=2(2Γca+Γbc)Γca(Γba+Γbc)+Im[|Ω(+)(ω)|2ωabωiγab/2]dω.
Equation (9) can be used to compute the expectation value of the dipole moment operator defined as d=Tr(ρ^D^). More precisely, in order to define a polarizability matching the classical convention for monochromatic fields with a time dependence exp(iωt), we will need the positive frequency part of the expectation value that is given by d(+)(ω)=ρba(ω)dab. For a weak exciting field, we can neglect saturation effects (s1), and we obtain
d(+)(ω)=1(1ωabωiγab/2)[dab·E(+)(ω)]dab.
By definition of the polarizability αab(ω), we also have
d(+)(ω)=αab(ω)ε0E(+)(ω).
These two equations readily lead to the following expression of the polarizability characterizing the excitation of the three-level emitter:
αab(ω)=3πc3ωab3Γbaspωabωiγab/2uu.
In this expression, we have introduced the unit vector u characterizing the orientation of the transition dipole, such that dab=dabu, and denotes the tensor product. We have also introduced the spontaneous emission rate (or Einstein A coefficient) [2]
Γbasp=ωab3dab23πε0c3.
Note that very often ΓbaΓbasp since additional non-radiative processes can contribute to the decay of the excited state |b towards the ground state |a. From the expression of the polarizability αab(ω)=αab(ω)uu, we can deduce the expressions of the extinction and scattering cross sections σe(ω)=(ω/c)Im[αab(ω)] and σs(ω)=[ω04/(6πc4)]|αab(ω)|2 [23]. For quasi-resonant excitation (ωωab), we have
σe(ω)=3πc22ωab2γabΓbasp(ωabω)2+γab2/4,
σs(ω)=3πc22ωab2(Γbasp)2(ωabω)2+γab2/4.
Note that when the damping rate of the coherence equals the spontaneous emission rate (γab=Γbasp), the extinction cross section equals the scattering cross section. In this limit, light is scattered without absorption.

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 G (also denoted by field susceptibility). The tensor (electric) Green function is defined as the solution of the vector Helmoltz equation [5,24]

××G(r,r,ω)ω2c2ε(r,ω)G(r,r,ω)=δ(rr)I
satisfying the outgoing condition when |rr| (one also refers to it as the retarded Green function). In this equation, δ() is the Dirac delta function, I is the unit tensor, and ε(r,ω) is the space- and frequency-dependent dielectric function of the medium. Physically, the Green function connects a monochromatic electric-dipole source d(ω) located at a position r to the radiated electric field at a position r in the medium through the relation [25]
E(r,ω)=μ0ω2G(r,r,ω)d(ω).
Note that this relation holds for both classical dipoles and fields, and for quantum operators (the Green function is the same in classical and quantum electrodynamics). The Green function contains the electrodynamic response of the environment, and can be used to relate one or several dipole sources to the electric field in arbitrary geometries such as a cavity, an antenna, an interface supporting surface plasmons, or a more complex medium, which can all be treated formally on the same footing. It will be convenient to decompose the Green function as follows:
G(r,r,ω)=G0(r,r,ω)+S(r,r,ω),
where G0 is the free-space Green function and S is the change in the Green function due to the structured environment. Given the response of the dipole emitter (polarizability) and of the environment (Green function), we will now see that the coupling between them can be studied formally based on a picture borrowed from scattering theory [26,27].

4. DIPOLE EMITTER INTERACTING WITH AN ENVIRONMENT

In this section, we consider a two-level dipole emitter located at a position rs, with a fixed orientation of its transition dipole (defined by unit vector u), and characterized by its free-space polarizability α0(ω)=α0(ω)uu, with

α0(ω)=3πc3ω03Γ0ω0ωiγ0/2.
In this expression we assume ωω0, and we can use γ0Γ0 to account for non-radiative dephasing processes. We stress that this expression of the polarizability can also describe classical resonant scatterers [28].

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 Eexc generated by scattering of the incident field Einc 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

d(+)(ω)=α0(ω)ε0Eexc(+)(rs,ω)+α0(ω)k02S(rs,rs,ω)d(+)(ω),
where k0=ω/c. Note that the interaction with the environment is described by the modification of the Green function S=GG0 since the interaction of the emitter with itself through the vacuum field is already included in α0(ω). We can also define the dressed polarizability α(ω) such that
d(+)(ω)=α(ω)ε0Eexc(+)(rs,ω).
From Eqs. (21) and (22), we obtain
α(ω)1=α0(ω)1k02S(rs,rs,ω),
which gives a general expression of the dressed polarizability. Eigenmodes of the coupled system can be defined as poles in α(ω), or zeros of α(ω)1. This leads to the following general coupling equation:
ω2c2S(rs,rs,ω)α0(ω)=I.
Projecting on the direction u of the dipole, this can be rewritten as
ω2c2[u·S(rs,rs,ω)u]α0(ω)=1,
which is a scalar equation. The solutions of this coupling equation, considered as an implicit equation in ω, define the eigenfrequencies of the coupled system. For a two-level system, introducing Eq. (20) into Eq. (25), we find that the complex eigenfrequencies ϖp are solutions of
S(ϖp)=ω0ϖpiγ0/2,
where we use the notation
S(ω)=3πcΓ0ω0[u·S(rs,rs,ω)u].
Note that we can use ω=ω0 in prefactors since we already assumed ωω0 in Eq. (20). Solving Eq. (26) allows one to find the complex eigenfrequencies ϖp=ωpiγp/2, defining the central frequencies ωp and the linewidth γp of the eigenmodes of the coupled emitter-field system [3]. This leads to a simple description of different interaction regimes and their main features.

 

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 γ0. We can assume S(ω)S(ω0), and the solution to Eq. (26) simply becomes

ϖp=ω0i2γ0S(ω0).
Both the resonance frequency and the linewidth of the emitter are affected by the coupling, and are respectively given by
ωp=ω0Re[S(ω0)],
γp=γ0+2Im[S(ω0)].
We can see that the coupling induces a (classical) frequency shift δω=ωpω0 that scales with the real part of the Green function due to the environment. The linewidth is also modified, the change scaling with the imaginary part of the Green function. In the absence of non-radiative dephasing processes (γ0=Γ0), the change in the linewidth (or, equivalently, in the spontaneous decay rate) can be rewritten as
γpΓ0=1+2Im[S(ω0)]Γ0.
Introducing the partial (or projected) local density of states (LDOS), which is defined by ρu(r,ω)=2ω/(πc2)Im[u·G(r,r,ω)u] [25], we find
γpΓ0=ρu(rs,ω)ρu,0,
where ρu,0=ω2/(3π2c3) is the partial LDOS in vacuum. We recover the well-known fact that in the weak-coupling regime, the spontaneous decay rate is modified according to the change in the LDOS, which is known as the Purcell effect (the original paper by Purcell considers the particular case of a single-mode cavity with weak losses [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 |ωωm|ωm and γmωm, where ωm and γm are, respectively, the central frequency and linewidth of the mode, we can use the following single-mode expansion of the Green function:

G(r,r,ω)=c22ωmem(r)em*(r)ωmωiγm/2,
where em(r) is the normalized complex amplitude of the mode [25]. The change in the Green function S can be deduced from Eq. (33) by subtracting the contribution of the vacuum Green function G0. Only the imaginary part has to be subtracted, since the singular real part of G0 is not included in expression (33). For a discussion of this point, see [29,30]. This leads to
S(ω)=FmΓ0γm/4ωmωiγm/2iΓ0/2,
where we have introduced the Purcell factor of the mode defined by [25]
Fm=6πc3ωm2γm|em(rs)·u|2.
Note that in this definition, the factor |em(rs)·u|2, whose inverse defines the mode volume, depends on the emitter location and orientation (the Purcell factor is often defined using the maximum value of |em(rs)·u|2, a convention that we do not use here). Introducing Eq. (34) into the coupling Eq. (26), we find that the complex eigenfrequencies ϖp of the coupled system must satisfy
1=FmΓ0γm/4(ω0ϖpiγ0/2)(ωmϖpiγm/2)iΓ0/2ω0ϖpiγ0/2.
Solving this second-order equation, we obtain two solutions ϖp+ and ϖp given by
ϖp±=ϖ0+ϖm2±(ϖmϖ02)2+g2,
where g=FmΓ0γm/4 is the coupling constant, ϖm=ωmiγm/2 is the complex frequency of the mode, and ϖ0=ω0i(γ0Γ0)/2 characterizes the emitter. For 4g2|ϖmϖ0|2, developing the square-root term to first order, we would find two slightly modified eigenmodes (compared to the decoupled emitter and field mode), with a small frequency shift and a broadening, thus recovering the features of the weak-coupling regime. In contrast, for 4g2|ϖmϖ0|2 corresponding to the strong-coupling regime, the central frequency and the linewidth of the eigenmodes become
ωp±=ω0+ωm2±FmΓ0γm4,
γp±=(γ0Γ0)+γm2.
Equation (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 4g2|ϖmϖ0|2 often ensures that the frequency splitting can be experimentally observed, but is not always sufficient (for instance, when ϖmϖ0).

For the sake of illustration, let us consider a dipole emitter characterized by a central frequency ω0=2370  meV, a radiative linewidth Γ0=0.004  meV, and a total linewidth γ0=140  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 ωm=2220  meV and γm=40  meV. By increasing the Purcell factor Fm 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 105.

 

Fig. 3. (a) Evolution of the eigenfrequencies of the coupled system in the complex plane when increasing the Purcell factor Fm of the cavity. (b) Normalized frequency shift of the two eigenmodes versus the Purcell factor Fm. Error bars represent intervals bounded by ωp±γp/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 αi(ω)=αi(ω)uiui, the unit vector ui defining the fixed orientation of the transition dipole, with

αi(ω)=3πc3ω03Γiωiωiγi/2for  i=1,2.
Since we are considering the quasi-resonant regime with ωω1ω2, we use the average resonance frequency ω0=(ω1+ω2)/2 in all prefactors. We also introduce the following notations:
Sii(ω)=3πcΓiω0[ui·S(ri,ri,ω)ui],
Gij(ω)=3πcΓiΓjω0[ui·G(ri,rj,ω)uj].
While Sii(ω) describes the influence of each emitter on itself through the environment, Gij(ω) describes the interaction between them. Following the scattering picture used in Section 4, the relation between the induced dipoles d1=d1u1 and d2=d2u2 in each emitter and the excitation field are conveniently expressed in a matrix form MX=Y, where
X=(d1(+)(ω)d2(+)(ω)),
Y=(α01(ω)ε0u1·Eexc(+)(r1,ω)α02(ω)ε0u2·Eexc(+)(r2,ω)),
M=(1S11(ω)ω1ωiγ1/2G12(ω)ω1ωiγ1/2G21(ω)ω2ωiγ2/21S22(ω)ω2ωiγ2/2).
Note that reciprocity imposes that the Green function satisfies G12(ω)=G21(ω). The eigenfrequencies of the coupled system are found by solving det[M(ω)]=0. This leads to the following equation satisfied by the complex eigenfrequencies ϖp:
0=1S11(ϖp)ω1ϖpiγ1/2S22(ϖp)ω2ϖpiγ2/2+S11(ϖp)S22(ϖp)G122(ϖp)(ω1ϖpiγ1/2)(ω2ϖpiγ2/2).
This equation is a convenient starting point to discuss different interaction regimes.

B. Weak Coupling to the Environment

If the medium has a smooth dependence on frequency (no resonance), we can write Sii(ω)Sii(ω0) and Gii(ω)Gii(ω0). The two eigenfrequency solutions of Eq. (46) are then given by

ϖp±=ϖ1+ϖ22±(ϖ2ϖ12)2+G12(ω0)2,
where we have introduced ϖi=ωiiγi/2Sii(ω0), which corresponds to the eigenfrequency of each emitter considered alone in the environment [see Eq. (28)]. For strong dipole–dipole coupling between the emitters (4G12(ω0)2|ϖ2ϖ1|2) the central frequency and the linewidth of the two eigenmodes become
ωp±=ω1+ω22Re[S11(ω0)+S22(ω0)2]±Re[G12(ω0)],
γp±=γ1+γ22+2Im[S11(ω0)+S22(ω0)2]2Im[G12(ω0)].
We observe two eigenmodes characterized by a frequency splitting that scales with Re[G12(ω0)], i.e., with the strength of the electrodynamic coupling between the two dipoles. This is a feature of a strong-coupling regime between the emitters. The linewidths show the appearance of both a broadened (or superradiant) mode and a narrowed (or subradiant) mode.

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 ω1=ω2=2370  meV, radiative linewidth Γ1=Γ2=0.004  meV, and total linewidth γ1=γ2=140  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 d3  nm, the emitters can be considered independent. Also note that the condition 4G12(ω0)2|ϖ2ϖ1|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 ωp±γp/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 (4G12(ω0)2|ϖ2ϖ1|2). In this limit, we can perform a first-order expansion of the square root in Eq. (47), yielding

ϖp±=ϖ1+ϖ2±(ϖ2ϖ1)2±G12(ω0)2ϖ2ϖ1.
We see that the eigenmode + (resp. –) correponds to the modifications in frequency and linewidth of emitter 2 (resp. 1). It is interesting to show that this expression can describe irreversible energy transfer between two emitters (usually referred to as donor and acceptor), which at short distance is known as Förster resonant energy transfer (FRET) [31]. FRET has been widely used in biology as a mechanism to detect molecular interaction [32]. Short-distance energy transfer is also involved in the process of photosynthesis [33]. 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 γ1=Γ1), 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 γ2(Γ1,Γ2,2Im[S11(ω0)],2Im[S22(ω0)]) is assumed to be satisfied. Note that, as described in Section 2, the polarizability α2(ω) describes the excitation of emitter 2 only (subsequent fluorescent emission at a different frequency is implicit). We also assume ω1=ω2=ω0, 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
γp+=γ2,
γp=γ1+2Im[S11(ω0)]+4Re[G12(ω0)2]γ2.
As expected, eigenmode + (corresponding to emitter 2 or acceptor) has negligible broadening due to coupling to both the donor and the environment (this follows directly from the condition of a large γ2). More interestingly, the linewidth associated to eigenmode – (corresponding to emitter 1 or donor) is modified by the surrounding medium [second term on the right-hand side in Eq. (52)] and by the presence of the acceptor [third term on the right-hand side in Eq. (52), which will be denoted by Γinter]. We can observe that the presence of the acceptor can either increase or decrease the linewidth of the donor, depending on the sign of Re[G12(ω0)2]. 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 [34]. For distances much smaller than the wavelength λ0=2πc/ω0, we can assume Re[G12(ω0)2]|G12(ω0)|2. (This can be easily verified in free space, as long as the interdistance d<λ0/4.) 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 σe(ω0) and σs(ω0), which are, respectively,
σe(ω0)=6πc2ω02Γ2γ2,
σs(ω0)=6πc2ω02(Γ2γ2)2.
In the regime γ2Γ2, the scattering cross section is negligible, and we can assume that the absorption cross section σa(ω0) equals the extinction cross section. Then, the last term on the right-hand side in Eq. (52), usually referred to as the energy transfer rate Γet, can be written
Γet=6πΓ1σa(ω0)|u1·G(r1,r2,ω0)u2|2,
where we have used Eq. (42) and the assumption Re[G12(ω0)2]|G12(ω0)|2. This expression takes the usual form of the energy transfer rate in dipole–dipole interaction [35] (see also [36,37] for a full QED treatment). The main difference between Γet and Γinter 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 u1·G0(r1,r2,ω0)u2|r1r2|3. The free-space energy transfer rate therefore scales as Γet|r1r2|6, which is a feature of FRET, as initially derived by Förster [31]. 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 ω1=2370  meV and radiative linewidth γ1=Γ1=0.004  meV, and an acceptor (three-level molecule) with absorption frequency ω2=ω1 and total linewidth γ2=140  meV. We show in Fig. 5 the energy transfer rate Γet, 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 (Γinter). We see that both expressions coincide for d<100  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 106).

 

Fig. 5. Solid line: normalized modification in the donor linewidth due to the presence of the acceptor Γinter/Γ1 versus the distance d between donor and acceptor. Dashed line: normalized energy transfer rate Γet/Γ1 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

Fi=6πc3ωm2γm|em(ri)·ui|2for  i=1,2.
It follows that
Sii(ω)=FiΓiγm/4ωmωiγm/2iΓi/2,
Gij(ω)2=FiΓiFjΓjγm2/16(ωmωiγm/2)2.
The eigenfrequencies of the coupled system can then be found by inserting these expressions into Eq. (46), resulting in a third-order equation in the complex frequency ϖp whose roots can be found analytically. Different behaviors can be observed depending on the relative values of the two coupling constant g1 and g2, defined as
gi=FiΓiγm4for  i=1,2.
Indeed, if g1 and g2 are substantially different, the emitter with the larger coupling dictates the features of two split eigenmodes (resulting from strong coupling between this emitter and the field mode), while the third eigenmode is associated with the other uncoupled emitter. The situation is more complicated when g1g2 since in this case the features of the three eigenmodes depend on both emitters. As an example, let us consider two emitters with features matching those of fluorescent molecules at room temperature, characterized by different resonant frequencies (ω1=2370  meV and ω2=2070  meV), by a radiative linewidth Γ1=Γ2=0.004  meV and by a total linewidth γ1=γ2=140  meV. We assume the emitters coupled to a single-mode cavity with ωm=2220  meV and γm=40  meV. Moreover, we set the Purcell factor seen by emitter 1 to F1=3×106 so that this emitter is strongly coupled to the mode (see Fig. 3), while F2 is left as a free parameter allowing us to tune the coupling strength of emitter 2. The behavior of the central frequency ωp and linewidth γp of the three eigenmodes is shown in Fig. 6. For F2F1, 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 F2, this third eigenfrequency progressively changes to the free-space resonance frequency of emitter 1. In the regime F2F1, 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.

 

Fig. 6. (a) Evolution of the eigenfrequencies in the complex plane when increasing the Purcell factor F2 of emitter 2. (b) Normalized frequency shift of the three eigenmodes versus the Purcell factor F2. Error bars represent intervals bounded by ωp±γp/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 α0(ω) 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 Sii(ω)=S(ω) and Gij(ω)=G(ω). Finding the eigenfrequencies of the coupled system amounts to solving det(M)=0, where M is now a N×N matrix. This leads to the following equation for the complex eigenfrequencies ϖp:

(1S(ϖp)G(ϖp)ω0ϖpiγ0/2)N1
×(1S(ϖp)+(N1)G(ϖp)ω0ϖpiγ0/2)=0.
Whenever the surrounding medium can be considered as weakly resonant, the N eigenfrequencies are given by
ϖp=ω0i2γ0S(ω0)(N1)G(ω0),
ϖp+=ω0i2γ0S(ω0)+G(ω0),
where the solution ϖp+ has a multipicity N1. In contrast, if the surrounding medium is strongly resonant, and the emitters are quasi-resonant with one specific eigenmode m of the field, we obtain very different collective behavior. Using the notations ϖ0=ω0i(γ0Γ0)/2 and ϖm=ωmiγm/2, and introducing the coupling constant g=Γ0γmFm/4 with Fm the Purcell factor of the mode, we find N+1 eigenfrequencies given by
ϖp±=ϖ0+ϖm2±(ϖmϖ02)2+Ng2,
ϖp=ϖ0,
where the solution ϖp has a multiplicity N1. The eigenfrequencies given by Eq. (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 N identical emitters is simply Ng, where g 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 N, as one would obtain with the theory of Dicke superradiance in the weak-excitation limit [17]. In contrast, when the environment is strongly resonant, the splitting scales with N and the linewidth does not depend on N, in agreement with results obtained with the Jaynes–Cumming Hamiltonian [39]. The simple coupled-dipole model introduced in this tutorial therefore contains the main ingredients to describe collective interactions between quantum emitters under external excitation.

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].

Funding

Université Paris Sciences et Lettres (PSL) (ANR-10-IDEX-0001-02); LABEX WIFI (ANR-10-LABX-24).

Acknowledgments

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.

REFERENCES

1. C. Cohen-Tannoudji, J. Dunpont-Roc, and G. Grynberg, Atom–Photon Interactions: Basic Processes and Applications (Wiley, 2010).

2. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

3. S. Haroche, “Cavity quantum electrodynamics,” in Fundamental Systems in Quantum Optics, J. Dalibard, J. Raimond, and J. Zinn-Justin, eds. (Elsevier, 1992), pp. 767–940.

4. S. Haroche and J.-M. Raimond, Exploring the Quantum: Atoms, Cavities, and Photons (Oxford University, 2013).

5. L. Novotny and B. Hecht, Principles of Nano-Optics, 2nd ed. (Cambridge University, 2012).

6. A. F. Koenderink, A. Alù, and A. Polman, “Nanophotonics: shrinking light-based technology,” Science 348, 516–521 (2015). [CrossRef]  

7. E. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

8. L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, 1987).

9. B. Barnes, F. G. Vidal, and J. Aizpurua, “Special issue on strong coupling of molecules to cavities,” ACS Photon. 5, 1 (2018). [CrossRef]  

10. L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics 5, 83–90 (2011). [CrossRef]  

11. C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, “Observation of the coupled exciton-photon mode splitting in a semiconductor quantum microcavity,” Phys. Rev. Lett. 69, 3314–3317 (1992). [CrossRef]  

12. A. Kavokin and G. Malpuech, Cavity Polaritons (Elsevier, 2003).

13. T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004). [CrossRef]  

14. R. Chikkaraddy, B. de Nijs, F. Benz, S. J. Barrow, O. A. Scherman, E. Rosta, A. Demetriadou, P. Fox, O. Hess, and J. J. Baumberg, “Single-molecule strong coupling at room temperature in plasmonic nanocavities,” Nature 535, 127–130 (2016). [CrossRef]  

15. I. Medintz and N. Hildebrandt, FRET—Förster Resonance Energy Transfer: From Theory to Applications (Wiley, 2013).

16. G. S. Agarwal, Quantum Optics: Quantum Statistical Theories of Spontaneous Emission and Their Relation to Other Approaches (Springer, 1974).

17. M. Gross and S. Haroche, “Superradiance: an essay on the theory of collective spontaneous emission,” Phys. Rep. 93, 301–396 (1982). [CrossRef]  

18. D. Bouchet, D. Cao, R. Carminati, Y. De Wilde, and V. Krachmalnicoff, “Long-range plasmon-assisted energy transfer between fluorescent emitters,” Phys. Rev. Lett. 116, 037401 (2016). [CrossRef]  

19. A. Sipahigil, R. E. Evans, D. D. Sukachev, M. J. Burek, J. Borregaard, M. K. Bhaskar, C. T. Nguyen, J. L. Pacheco, H. A. Atikian, C. Meuwly, R. M. Camacho, F. Jelezko, E. Bielejec, H. Park, M. Lončar, and M. D. Lukin, “An integrated diamond nanophotonics platform for quantum-optical networks,” Science 354, 847–850 (2016). [CrossRef]  

20. C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics (Wiley, 1991), Vol. 1.

21. G. Grynberg, A. Aspect, and C. Fabre, Introduction to Quantum Optics (Cambridge University, 2010).

22. H. J. Carmichael, Statistical Methods in Quantum Optics 1: Master Equations and Fokker-Planck Equations, Theoretical and Mathematical Physics (Springer, 1999).

23. C. F. Bohren and R. D. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 2008).

24. P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part II, 1st ed. (McGraw-Hill, 1953).

25. R. Carminati, A. Cazé, D. Cao, F. Peragut, V. Krachmalnicoff, R. Pierrat, and Y. De Wilde, “Electromagnetic density of states in complex plasmonic systems,” Surf. Sci. Rep. 70, 1–41 (2015). [CrossRef]  

26. L. L. Foldy, “The multiple scattering of waves. I. General theory of isotropic scattering by randomly distributed scatterers,” Phys. Rev. 67, 107–119 (1945). [CrossRef]  

27. M. Lax, “Multiple scattering of waves,” Rev. Mod. Phys. 23, 287–310 (1951). [CrossRef]  

28. P. de Vries, D. V. van Coevorden, and A. Lagendijk, “Point scatterers for classical waves,” Rev. Mod. Phys. 70, 447–466 (1998). [CrossRef]  

29. M. Wubs, L. G. Suttorp, and A. Lagendijk, “Multiple-scattering approach to interatomic interactions and superradiance in inhomogeneous dielectrics,” Phys. Rev. A 70, 053823 (2004). [CrossRef]  

30. D. P. Fussell, S. Hughes, and M. M. Dignam, “Influence of fabrication disorder on the optical properties of coupled-cavity photonic crystal waveguides,” Phys. Rev. B 78, 144201 (2008). [CrossRef]  

31. T. Förster, “Zwischenmolekulare Energiewanderung und Fluoreszenz,” Ann. Phys. 437, 55–75 (1948). [CrossRef]  

32. P. R. Selvin, “The renaissance of fluorescence resonance energy transfer,” Nat. Struct. Mol. Biol. 7, 730–734 (2000). [CrossRef]  

33. G. D. Scholes, G. R. Fleming, A. Olaya-Castro, and R. van Grondelle, “Lessons from nature about solar light harvesting,” Nat. Chem. 3, 763–774 (2011). [CrossRef]  

34. K. Drexhage, “Influence of a dielectric interface on fluorescence decay time,” J. Lumin. 1, 693–701 (1970). [CrossRef]  

35. L. Novotny and B. Hecht, “Dipole-dipole interactions and energy transfer,” in Principles of Nano-Optics, 2nd ed. (Cambridge University, 2012), pp. 256–264. See Eqs. (8.159) and (8.160) for the expression of the energy transfer rate.

36. H. T. Dung, L. Knöll, and D.-G. Welsch, “Resonant dipole-dipole interaction in the presence of dispersing and absorbing surroundings,” Phys. Rev. A 66, 063810 (2002). [CrossRef]  

37. H. T. Dung, L. Knöll, and D.-G. Welsch, “Intermolecular energy transfer in the presence of dispersing and absorbing media,” Phys. Rev. A 65, 043813 (2002). [CrossRef]  

38. R. Vincent and R. Carminati, “Magneto-optical control of Förster energy transfer,” Phys. Rev. B 83, 165426 (2011). [CrossRef]  

39. G. S. Agarwal, “Vacuum-field Rabi splittings in microwave absorption by Rydberg atoms in a cavity,” Phys. Rev. Lett. 53, 1732–1734 (1984). [CrossRef]  

40. P. Berman, Cavity Quantum Electrodynamics (Academic, 1994).

References

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  • |
  • |

  1. C. Cohen-Tannoudji, J. Dunpont-Roc, and G. Grynberg, Atom–Photon Interactions: Basic Processes and Applications (Wiley, 2010).
  2. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  3. S. Haroche, “Cavity quantum electrodynamics,” in Fundamental Systems in Quantum Optics, J. Dalibard, J. Raimond, and J. Zinn-Justin, eds. (Elsevier, 1992), pp. 767–940.
  4. S. Haroche and J.-M. Raimond, Exploring the Quantum: Atoms, Cavities, and Photons (Oxford University, 2013).
  5. L. Novotny and B. Hecht, Principles of Nano-Optics, 2nd ed. (Cambridge University, 2012).
  6. A. F. Koenderink, A. Alù, and A. Polman, “Nanophotonics: shrinking light-based technology,” Science 348, 516–521 (2015).
    [Crossref]
  7. E. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).
  8. L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, 1987).
  9. B. Barnes, F. G. Vidal, and J. Aizpurua, “Special issue on strong coupling of molecules to cavities,” ACS Photon. 5, 1 (2018).
    [Crossref]
  10. L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics 5, 83–90 (2011).
    [Crossref]
  11. C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, “Observation of the coupled exciton-photon mode splitting in a semiconductor quantum microcavity,” Phys. Rev. Lett. 69, 3314–3317 (1992).
    [Crossref]
  12. A. Kavokin and G. Malpuech, Cavity Polaritons (Elsevier, 2003).
  13. T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
    [Crossref]
  14. R. Chikkaraddy, B. de Nijs, F. Benz, S. J. Barrow, O. A. Scherman, E. Rosta, A. Demetriadou, P. Fox, O. Hess, and J. J. Baumberg, “Single-molecule strong coupling at room temperature in plasmonic nanocavities,” Nature 535, 127–130 (2016).
    [Crossref]
  15. I. Medintz and N. Hildebrandt, FRET—Förster Resonance Energy Transfer: From Theory to Applications (Wiley, 2013).
  16. G. S. Agarwal, Quantum Optics: Quantum Statistical Theories of Spontaneous Emission and Their Relation to Other Approaches (Springer, 1974).
  17. M. Gross and S. Haroche, “Superradiance: an essay on the theory of collective spontaneous emission,” Phys. Rep. 93, 301–396 (1982).
    [Crossref]
  18. D. Bouchet, D. Cao, R. Carminati, Y. De Wilde, and V. Krachmalnicoff, “Long-range plasmon-assisted energy transfer between fluorescent emitters,” Phys. Rev. Lett. 116, 037401 (2016).
    [Crossref]
  19. A. Sipahigil, R. E. Evans, D. D. Sukachev, M. J. Burek, J. Borregaard, M. K. Bhaskar, C. T. Nguyen, J. L. Pacheco, H. A. Atikian, C. Meuwly, R. M. Camacho, F. Jelezko, E. Bielejec, H. Park, M. Lončar, and M. D. Lukin, “An integrated diamond nanophotonics platform for quantum-optical networks,” Science 354, 847–850 (2016).
    [Crossref]
  20. C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics (Wiley, 1991), Vol. 1.
  21. G. Grynberg, A. Aspect, and C. Fabre, Introduction to Quantum Optics (Cambridge University, 2010).
  22. H. J. Carmichael, Statistical Methods in Quantum Optics 1: Master Equations and Fokker-Planck Equations, Theoretical and Mathematical Physics (Springer, 1999).
  23. C. F. Bohren and R. D. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 2008).
  24. P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part II, 1st ed. (McGraw-Hill, 1953).
  25. R. Carminati, A. Cazé, D. Cao, F. Peragut, V. Krachmalnicoff, R. Pierrat, and Y. De Wilde, “Electromagnetic density of states in complex plasmonic systems,” Surf. Sci. Rep. 70, 1–41 (2015).
    [Crossref]
  26. L. L. Foldy, “The multiple scattering of waves. I. General theory of isotropic scattering by randomly distributed scatterers,” Phys. Rev. 67, 107–119 (1945).
    [Crossref]
  27. M. Lax, “Multiple scattering of waves,” Rev. Mod. Phys. 23, 287–310 (1951).
    [Crossref]
  28. P. de Vries, D. V. van Coevorden, and A. Lagendijk, “Point scatterers for classical waves,” Rev. Mod. Phys. 70, 447–466 (1998).
    [Crossref]
  29. M. Wubs, L. G. Suttorp, and A. Lagendijk, “Multiple-scattering approach to interatomic interactions and superradiance in inhomogeneous dielectrics,” Phys. Rev. A 70, 053823 (2004).
    [Crossref]
  30. D. P. Fussell, S. Hughes, and M. M. Dignam, “Influence of fabrication disorder on the optical properties of coupled-cavity photonic crystal waveguides,” Phys. Rev. B 78, 144201 (2008).
    [Crossref]
  31. T. Förster, “Zwischenmolekulare Energiewanderung und Fluoreszenz,” Ann. Phys. 437, 55–75 (1948).
    [Crossref]
  32. P. R. Selvin, “The renaissance of fluorescence resonance energy transfer,” Nat. Struct. Mol. Biol. 7, 730–734 (2000).
    [Crossref]
  33. G. D. Scholes, G. R. Fleming, A. Olaya-Castro, and R. van Grondelle, “Lessons from nature about solar light harvesting,” Nat. Chem. 3, 763–774 (2011).
    [Crossref]
  34. K. Drexhage, “Influence of a dielectric interface on fluorescence decay time,” J. Lumin. 1, 693–701 (1970).
    [Crossref]
  35. L. Novotny and B. Hecht, “Dipole-dipole interactions and energy transfer,” in Principles of Nano-Optics, 2nd ed. (Cambridge University, 2012), pp. 256–264. See Eqs. (8.159) and (8.160) for the expression of the energy transfer rate.
  36. H. T. Dung, L. Knöll, and D.-G. Welsch, “Resonant dipole-dipole interaction in the presence of dispersing and absorbing surroundings,” Phys. Rev. A 66, 063810 (2002).
    [Crossref]
  37. H. T. Dung, L. Knöll, and D.-G. Welsch, “Intermolecular energy transfer in the presence of dispersing and absorbing media,” Phys. Rev. A 65, 043813 (2002).
    [Crossref]
  38. R. Vincent and R. Carminati, “Magneto-optical control of Förster energy transfer,” Phys. Rev. B 83, 165426 (2011).
    [Crossref]
  39. G. S. Agarwal, “Vacuum-field Rabi splittings in microwave absorption by Rydberg atoms in a cavity,” Phys. Rev. Lett. 53, 1732–1734 (1984).
    [Crossref]
  40. P. Berman, Cavity Quantum Electrodynamics (Academic, 1994).

2018 (1)

B. Barnes, F. G. Vidal, and J. Aizpurua, “Special issue on strong coupling of molecules to cavities,” ACS Photon. 5, 1 (2018).
[Crossref]

2016 (3)

D. Bouchet, D. Cao, R. Carminati, Y. De Wilde, and V. Krachmalnicoff, “Long-range plasmon-assisted energy transfer between fluorescent emitters,” Phys. Rev. Lett. 116, 037401 (2016).
[Crossref]

A. Sipahigil, R. E. Evans, D. D. Sukachev, M. J. Burek, J. Borregaard, M. K. Bhaskar, C. T. Nguyen, J. L. Pacheco, H. A. Atikian, C. Meuwly, R. M. Camacho, F. Jelezko, E. Bielejec, H. Park, M. Lončar, and M. D. Lukin, “An integrated diamond nanophotonics platform for quantum-optical networks,” Science 354, 847–850 (2016).
[Crossref]

R. Chikkaraddy, B. de Nijs, F. Benz, S. J. Barrow, O. A. Scherman, E. Rosta, A. Demetriadou, P. Fox, O. Hess, and J. J. Baumberg, “Single-molecule strong coupling at room temperature in plasmonic nanocavities,” Nature 535, 127–130 (2016).
[Crossref]

2015 (2)

R. Carminati, A. Cazé, D. Cao, F. Peragut, V. Krachmalnicoff, R. Pierrat, and Y. De Wilde, “Electromagnetic density of states in complex plasmonic systems,” Surf. Sci. Rep. 70, 1–41 (2015).
[Crossref]

A. F. Koenderink, A. Alù, and A. Polman, “Nanophotonics: shrinking light-based technology,” Science 348, 516–521 (2015).
[Crossref]

2011 (3)

L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics 5, 83–90 (2011).
[Crossref]

G. D. Scholes, G. R. Fleming, A. Olaya-Castro, and R. van Grondelle, “Lessons from nature about solar light harvesting,” Nat. Chem. 3, 763–774 (2011).
[Crossref]

R. Vincent and R. Carminati, “Magneto-optical control of Förster energy transfer,” Phys. Rev. B 83, 165426 (2011).
[Crossref]

2008 (1)

D. P. Fussell, S. Hughes, and M. M. Dignam, “Influence of fabrication disorder on the optical properties of coupled-cavity photonic crystal waveguides,” Phys. Rev. B 78, 144201 (2008).
[Crossref]

2004 (2)

M. Wubs, L. G. Suttorp, and A. Lagendijk, “Multiple-scattering approach to interatomic interactions and superradiance in inhomogeneous dielectrics,” Phys. Rev. A 70, 053823 (2004).
[Crossref]

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
[Crossref]

2002 (2)

H. T. Dung, L. Knöll, and D.-G. Welsch, “Resonant dipole-dipole interaction in the presence of dispersing and absorbing surroundings,” Phys. Rev. A 66, 063810 (2002).
[Crossref]

H. T. Dung, L. Knöll, and D.-G. Welsch, “Intermolecular energy transfer in the presence of dispersing and absorbing media,” Phys. Rev. A 65, 043813 (2002).
[Crossref]

2000 (1)

P. R. Selvin, “The renaissance of fluorescence resonance energy transfer,” Nat. Struct. Mol. Biol. 7, 730–734 (2000).
[Crossref]

1998 (1)

P. de Vries, D. V. van Coevorden, and A. Lagendijk, “Point scatterers for classical waves,” Rev. Mod. Phys. 70, 447–466 (1998).
[Crossref]

1992 (1)

C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, “Observation of the coupled exciton-photon mode splitting in a semiconductor quantum microcavity,” Phys. Rev. Lett. 69, 3314–3317 (1992).
[Crossref]

1984 (1)

G. S. Agarwal, “Vacuum-field Rabi splittings in microwave absorption by Rydberg atoms in a cavity,” Phys. Rev. Lett. 53, 1732–1734 (1984).
[Crossref]

1982 (1)

M. Gross and S. Haroche, “Superradiance: an essay on the theory of collective spontaneous emission,” Phys. Rep. 93, 301–396 (1982).
[Crossref]

1970 (1)

K. Drexhage, “Influence of a dielectric interface on fluorescence decay time,” J. Lumin. 1, 693–701 (1970).
[Crossref]

1951 (1)

M. Lax, “Multiple scattering of waves,” Rev. Mod. Phys. 23, 287–310 (1951).
[Crossref]

1948 (1)

T. Förster, “Zwischenmolekulare Energiewanderung und Fluoreszenz,” Ann. Phys. 437, 55–75 (1948).
[Crossref]

1946 (1)

E. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

1945 (1)

L. L. Foldy, “The multiple scattering of waves. I. General theory of isotropic scattering by randomly distributed scatterers,” Phys. Rev. 67, 107–119 (1945).
[Crossref]

Agarwal, G. S.

G. S. Agarwal, “Vacuum-field Rabi splittings in microwave absorption by Rydberg atoms in a cavity,” Phys. Rev. Lett. 53, 1732–1734 (1984).
[Crossref]

G. S. Agarwal, Quantum Optics: Quantum Statistical Theories of Spontaneous Emission and Their Relation to Other Approaches (Springer, 1974).

Aizpurua, J.

B. Barnes, F. G. Vidal, and J. Aizpurua, “Special issue on strong coupling of molecules to cavities,” ACS Photon. 5, 1 (2018).
[Crossref]

Allen, L.

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, 1987).

Alù, A.

A. F. Koenderink, A. Alù, and A. Polman, “Nanophotonics: shrinking light-based technology,” Science 348, 516–521 (2015).
[Crossref]

Arakawa, Y.

C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, “Observation of the coupled exciton-photon mode splitting in a semiconductor quantum microcavity,” Phys. Rev. Lett. 69, 3314–3317 (1992).
[Crossref]

Aspect, A.

G. Grynberg, A. Aspect, and C. Fabre, Introduction to Quantum Optics (Cambridge University, 2010).

Atikian, H. A.

A. Sipahigil, R. E. Evans, D. D. Sukachev, M. J. Burek, J. Borregaard, M. K. Bhaskar, C. T. Nguyen, J. L. Pacheco, H. A. Atikian, C. Meuwly, R. M. Camacho, F. Jelezko, E. Bielejec, H. Park, M. Lončar, and M. D. Lukin, “An integrated diamond nanophotonics platform for quantum-optical networks,” Science 354, 847–850 (2016).
[Crossref]

Barnes, B.

B. Barnes, F. G. Vidal, and J. Aizpurua, “Special issue on strong coupling of molecules to cavities,” ACS Photon. 5, 1 (2018).
[Crossref]

Barrow, S. J.

R. Chikkaraddy, B. de Nijs, F. Benz, S. J. Barrow, O. A. Scherman, E. Rosta, A. Demetriadou, P. Fox, O. Hess, and J. J. Baumberg, “Single-molecule strong coupling at room temperature in plasmonic nanocavities,” Nature 535, 127–130 (2016).
[Crossref]

Baumberg, J. J.

R. Chikkaraddy, B. de Nijs, F. Benz, S. J. Barrow, O. A. Scherman, E. Rosta, A. Demetriadou, P. Fox, O. Hess, and J. J. Baumberg, “Single-molecule strong coupling at room temperature in plasmonic nanocavities,” Nature 535, 127–130 (2016).
[Crossref]

Benz, F.

R. Chikkaraddy, B. de Nijs, F. Benz, S. J. Barrow, O. A. Scherman, E. Rosta, A. Demetriadou, P. Fox, O. Hess, and J. J. Baumberg, “Single-molecule strong coupling at room temperature in plasmonic nanocavities,” Nature 535, 127–130 (2016).
[Crossref]

Berman, P.

P. Berman, Cavity Quantum Electrodynamics (Academic, 1994).

Bhaskar, M. K.

A. Sipahigil, R. E. Evans, D. D. Sukachev, M. J. Burek, J. Borregaard, M. K. Bhaskar, C. T. Nguyen, J. L. Pacheco, H. A. Atikian, C. Meuwly, R. M. Camacho, F. Jelezko, E. Bielejec, H. Park, M. Lončar, and M. D. Lukin, “An integrated diamond nanophotonics platform for quantum-optical networks,” Science 354, 847–850 (2016).
[Crossref]

Bielejec, E.

A. Sipahigil, R. E. Evans, D. D. Sukachev, M. J. Burek, J. Borregaard, M. K. Bhaskar, C. T. Nguyen, J. L. Pacheco, H. A. Atikian, C. Meuwly, R. M. Camacho, F. Jelezko, E. Bielejec, H. Park, M. Lončar, and M. D. Lukin, “An integrated diamond nanophotonics platform for quantum-optical networks,” Science 354, 847–850 (2016).
[Crossref]

Bohren, C. F.

C. F. Bohren and R. D. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 2008).

Borregaard, J.

A. Sipahigil, R. E. Evans, D. D. Sukachev, M. J. Burek, J. Borregaard, M. K. Bhaskar, C. T. Nguyen, J. L. Pacheco, H. A. Atikian, C. Meuwly, R. M. Camacho, F. Jelezko, E. Bielejec, H. Park, M. Lončar, and M. D. Lukin, “An integrated diamond nanophotonics platform for quantum-optical networks,” Science 354, 847–850 (2016).
[Crossref]

Bouchet, D.

D. Bouchet, D. Cao, R. Carminati, Y. De Wilde, and V. Krachmalnicoff, “Long-range plasmon-assisted energy transfer between fluorescent emitters,” Phys. Rev. Lett. 116, 037401 (2016).
[Crossref]

Burek, M. J.

A. Sipahigil, R. E. Evans, D. D. Sukachev, M. J. Burek, J. Borregaard, M. K. Bhaskar, C. T. Nguyen, J. L. Pacheco, H. A. Atikian, C. Meuwly, R. M. Camacho, F. Jelezko, E. Bielejec, H. Park, M. Lončar, and M. D. Lukin, “An integrated diamond nanophotonics platform for quantum-optical networks,” Science 354, 847–850 (2016).
[Crossref]

Camacho, R. M.

A. Sipahigil, R. E. Evans, D. D. Sukachev, M. J. Burek, J. Borregaard, M. K. Bhaskar, C. T. Nguyen, J. L. Pacheco, H. A. Atikian, C. Meuwly, R. M. Camacho, F. Jelezko, E. Bielejec, H. Park, M. Lončar, and M. D. Lukin, “An integrated diamond nanophotonics platform for quantum-optical networks,” Science 354, 847–850 (2016).
[Crossref]

Cao, D.

D. Bouchet, D. Cao, R. Carminati, Y. De Wilde, and V. Krachmalnicoff, “Long-range plasmon-assisted energy transfer between fluorescent emitters,” Phys. Rev. Lett. 116, 037401 (2016).
[Crossref]

R. Carminati, A. Cazé, D. Cao, F. Peragut, V. Krachmalnicoff, R. Pierrat, and Y. De Wilde, “Electromagnetic density of states in complex plasmonic systems,” Surf. Sci. Rep. 70, 1–41 (2015).
[Crossref]

Carmichael, H. J.

H. J. Carmichael, Statistical Methods in Quantum Optics 1: Master Equations and Fokker-Planck Equations, Theoretical and Mathematical Physics (Springer, 1999).

Carminati, R.

D. Bouchet, D. Cao, R. Carminati, Y. De Wilde, and V. Krachmalnicoff, “Long-range plasmon-assisted energy transfer between fluorescent emitters,” Phys. Rev. Lett. 116, 037401 (2016).
[Crossref]

R. Carminati, A. Cazé, D. Cao, F. Peragut, V. Krachmalnicoff, R. Pierrat, and Y. De Wilde, “Electromagnetic density of states in complex plasmonic systems,” Surf. Sci. Rep. 70, 1–41 (2015).
[Crossref]

R. Vincent and R. Carminati, “Magneto-optical control of Förster energy transfer,” Phys. Rev. B 83, 165426 (2011).
[Crossref]

Cazé, A.

R. Carminati, A. Cazé, D. Cao, F. Peragut, V. Krachmalnicoff, R. Pierrat, and Y. De Wilde, “Electromagnetic density of states in complex plasmonic systems,” Surf. Sci. Rep. 70, 1–41 (2015).
[Crossref]

Chikkaraddy, R.

R. Chikkaraddy, B. de Nijs, F. Benz, S. J. Barrow, O. A. Scherman, E. Rosta, A. Demetriadou, P. Fox, O. Hess, and J. J. Baumberg, “Single-molecule strong coupling at room temperature in plasmonic nanocavities,” Nature 535, 127–130 (2016).
[Crossref]

Cohen-Tannoudji, C.

C. Cohen-Tannoudji, J. Dunpont-Roc, and G. Grynberg, Atom–Photon Interactions: Basic Processes and Applications (Wiley, 2010).

C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics (Wiley, 1991), Vol. 1.

de Nijs, B.

R. Chikkaraddy, B. de Nijs, F. Benz, S. J. Barrow, O. A. Scherman, E. Rosta, A. Demetriadou, P. Fox, O. Hess, and J. J. Baumberg, “Single-molecule strong coupling at room temperature in plasmonic nanocavities,” Nature 535, 127–130 (2016).
[Crossref]

de Vries, P.

P. de Vries, D. V. van Coevorden, and A. Lagendijk, “Point scatterers for classical waves,” Rev. Mod. Phys. 70, 447–466 (1998).
[Crossref]

De Wilde, Y.

D. Bouchet, D. Cao, R. Carminati, Y. De Wilde, and V. Krachmalnicoff, “Long-range plasmon-assisted energy transfer between fluorescent emitters,” Phys. Rev. Lett. 116, 037401 (2016).
[Crossref]

R. Carminati, A. Cazé, D. Cao, F. Peragut, V. Krachmalnicoff, R. Pierrat, and Y. De Wilde, “Electromagnetic density of states in complex plasmonic systems,” Surf. Sci. Rep. 70, 1–41 (2015).
[Crossref]

Demetriadou, A.

R. Chikkaraddy, B. de Nijs, F. Benz, S. J. Barrow, O. A. Scherman, E. Rosta, A. Demetriadou, P. Fox, O. Hess, and J. J. Baumberg, “Single-molecule strong coupling at room temperature in plasmonic nanocavities,” Nature 535, 127–130 (2016).
[Crossref]

Deppe, D. G.

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
[Crossref]

Dignam, M. M.

D. P. Fussell, S. Hughes, and M. M. Dignam, “Influence of fabrication disorder on the optical properties of coupled-cavity photonic crystal waveguides,” Phys. Rev. B 78, 144201 (2008).
[Crossref]

Diu, B.

C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics (Wiley, 1991), Vol. 1.

Drexhage, K.

K. Drexhage, “Influence of a dielectric interface on fluorescence decay time,” J. Lumin. 1, 693–701 (1970).
[Crossref]

Dung, H. T.

H. T. Dung, L. Knöll, and D.-G. Welsch, “Resonant dipole-dipole interaction in the presence of dispersing and absorbing surroundings,” Phys. Rev. A 66, 063810 (2002).
[Crossref]

H. T. Dung, L. Knöll, and D.-G. Welsch, “Intermolecular energy transfer in the presence of dispersing and absorbing media,” Phys. Rev. A 65, 043813 (2002).
[Crossref]

Dunpont-Roc, J.

C. Cohen-Tannoudji, J. Dunpont-Roc, and G. Grynberg, Atom–Photon Interactions: Basic Processes and Applications (Wiley, 2010).

Eberly, J. H.

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, 1987).

Ell, C.

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
[Crossref]

Evans, R. E.

A. Sipahigil, R. E. Evans, D. D. Sukachev, M. J. Burek, J. Borregaard, M. K. Bhaskar, C. T. Nguyen, J. L. Pacheco, H. A. Atikian, C. Meuwly, R. M. Camacho, F. Jelezko, E. Bielejec, H. Park, M. Lončar, and M. D. Lukin, “An integrated diamond nanophotonics platform for quantum-optical networks,” Science 354, 847–850 (2016).
[Crossref]

Fabre, C.

G. Grynberg, A. Aspect, and C. Fabre, Introduction to Quantum Optics (Cambridge University, 2010).

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part II, 1st ed. (McGraw-Hill, 1953).

Fleming, G. R.

G. D. Scholes, G. R. Fleming, A. Olaya-Castro, and R. van Grondelle, “Lessons from nature about solar light harvesting,” Nat. Chem. 3, 763–774 (2011).
[Crossref]

Foldy, L. L.

L. L. Foldy, “The multiple scattering of waves. I. General theory of isotropic scattering by randomly distributed scatterers,” Phys. Rev. 67, 107–119 (1945).
[Crossref]

Förster, T.

T. Förster, “Zwischenmolekulare Energiewanderung und Fluoreszenz,” Ann. Phys. 437, 55–75 (1948).
[Crossref]

Fox, P.

R. Chikkaraddy, B. de Nijs, F. Benz, S. J. Barrow, O. A. Scherman, E. Rosta, A. Demetriadou, P. Fox, O. Hess, and J. J. Baumberg, “Single-molecule strong coupling at room temperature in plasmonic nanocavities,” Nature 535, 127–130 (2016).
[Crossref]

Fussell, D. P.

D. P. Fussell, S. Hughes, and M. M. Dignam, “Influence of fabrication disorder on the optical properties of coupled-cavity photonic crystal waveguides,” Phys. Rev. B 78, 144201 (2008).
[Crossref]

Gibbs, H. M.

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
[Crossref]

Gross, M.

M. Gross and S. Haroche, “Superradiance: an essay on the theory of collective spontaneous emission,” Phys. Rep. 93, 301–396 (1982).
[Crossref]

Grynberg, G.

C. Cohen-Tannoudji, J. Dunpont-Roc, and G. Grynberg, Atom–Photon Interactions: Basic Processes and Applications (Wiley, 2010).

G. Grynberg, A. Aspect, and C. Fabre, Introduction to Quantum Optics (Cambridge University, 2010).

Haroche, S.

M. Gross and S. Haroche, “Superradiance: an essay on the theory of collective spontaneous emission,” Phys. Rep. 93, 301–396 (1982).
[Crossref]

S. Haroche, “Cavity quantum electrodynamics,” in Fundamental Systems in Quantum Optics, J. Dalibard, J. Raimond, and J. Zinn-Justin, eds. (Elsevier, 1992), pp. 767–940.

S. Haroche and J.-M. Raimond, Exploring the Quantum: Atoms, Cavities, and Photons (Oxford University, 2013).

Hecht, B.

L. Novotny and B. Hecht, Principles of Nano-Optics, 2nd ed. (Cambridge University, 2012).

L. Novotny and B. Hecht, “Dipole-dipole interactions and energy transfer,” in Principles of Nano-Optics, 2nd ed. (Cambridge University, 2012), pp. 256–264. See Eqs. (8.159) and (8.160) for the expression of the energy transfer rate.

Hendrickson, J.

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
[Crossref]

Hess, O.

R. Chikkaraddy, B. de Nijs, F. Benz, S. J. Barrow, O. A. Scherman, E. Rosta, A. Demetriadou, P. Fox, O. Hess, and J. J. Baumberg, “Single-molecule strong coupling at room temperature in plasmonic nanocavities,” Nature 535, 127–130 (2016).
[Crossref]

Hildebrandt, N.

I. Medintz and N. Hildebrandt, FRET—Förster Resonance Energy Transfer: From Theory to Applications (Wiley, 2013).

Huffman, R. D.

C. F. Bohren and R. D. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 2008).

Hughes, S.

D. P. Fussell, S. Hughes, and M. M. Dignam, “Influence of fabrication disorder on the optical properties of coupled-cavity photonic crystal waveguides,” Phys. Rev. B 78, 144201 (2008).
[Crossref]

Ishikawa, A.

C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, “Observation of the coupled exciton-photon mode splitting in a semiconductor quantum microcavity,” Phys. Rev. Lett. 69, 3314–3317 (1992).
[Crossref]

Jelezko, F.

A. Sipahigil, R. E. Evans, D. D. Sukachev, M. J. Burek, J. Borregaard, M. K. Bhaskar, C. T. Nguyen, J. L. Pacheco, H. A. Atikian, C. Meuwly, R. M. Camacho, F. Jelezko, E. Bielejec, H. Park, M. Lončar, and M. D. Lukin, “An integrated diamond nanophotonics platform for quantum-optical networks,” Science 354, 847–850 (2016).
[Crossref]

Kavokin, A.

A. Kavokin and G. Malpuech, Cavity Polaritons (Elsevier, 2003).

Khitrova, G.

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
[Crossref]

Knöll, L.

H. T. Dung, L. Knöll, and D.-G. Welsch, “Intermolecular energy transfer in the presence of dispersing and absorbing media,” Phys. Rev. A 65, 043813 (2002).
[Crossref]

H. T. Dung, L. Knöll, and D.-G. Welsch, “Resonant dipole-dipole interaction in the presence of dispersing and absorbing surroundings,” Phys. Rev. A 66, 063810 (2002).
[Crossref]

Koenderink, A. F.

A. F. Koenderink, A. Alù, and A. Polman, “Nanophotonics: shrinking light-based technology,” Science 348, 516–521 (2015).
[Crossref]

Krachmalnicoff, V.

D. Bouchet, D. Cao, R. Carminati, Y. De Wilde, and V. Krachmalnicoff, “Long-range plasmon-assisted energy transfer between fluorescent emitters,” Phys. Rev. Lett. 116, 037401 (2016).
[Crossref]

R. Carminati, A. Cazé, D. Cao, F. Peragut, V. Krachmalnicoff, R. Pierrat, and Y. De Wilde, “Electromagnetic density of states in complex plasmonic systems,” Surf. Sci. Rep. 70, 1–41 (2015).
[Crossref]

Lagendijk, A.

M. Wubs, L. G. Suttorp, and A. Lagendijk, “Multiple-scattering approach to interatomic interactions and superradiance in inhomogeneous dielectrics,” Phys. Rev. A 70, 053823 (2004).
[Crossref]

P. de Vries, D. V. van Coevorden, and A. Lagendijk, “Point scatterers for classical waves,” Rev. Mod. Phys. 70, 447–466 (1998).
[Crossref]

Laloe, F.

C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics (Wiley, 1991), Vol. 1.

Lax, M.

M. Lax, “Multiple scattering of waves,” Rev. Mod. Phys. 23, 287–310 (1951).
[Crossref]

Loncar, M.

A. Sipahigil, R. E. Evans, D. D. Sukachev, M. J. Burek, J. Borregaard, M. K. Bhaskar, C. T. Nguyen, J. L. Pacheco, H. A. Atikian, C. Meuwly, R. M. Camacho, F. Jelezko, E. Bielejec, H. Park, M. Lončar, and M. D. Lukin, “An integrated diamond nanophotonics platform for quantum-optical networks,” Science 354, 847–850 (2016).
[Crossref]

Lukin, M. D.

A. Sipahigil, R. E. Evans, D. D. Sukachev, M. J. Burek, J. Borregaard, M. K. Bhaskar, C. T. Nguyen, J. L. Pacheco, H. A. Atikian, C. Meuwly, R. M. Camacho, F. Jelezko, E. Bielejec, H. Park, M. Lončar, and M. D. Lukin, “An integrated diamond nanophotonics platform for quantum-optical networks,” Science 354, 847–850 (2016).
[Crossref]

Malpuech, G.

A. Kavokin and G. Malpuech, Cavity Polaritons (Elsevier, 2003).

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Medintz, I.

I. Medintz and N. Hildebrandt, FRET—Förster Resonance Energy Transfer: From Theory to Applications (Wiley, 2013).

Meuwly, C.

A. Sipahigil, R. E. Evans, D. D. Sukachev, M. J. Burek, J. Borregaard, M. K. Bhaskar, C. T. Nguyen, J. L. Pacheco, H. A. Atikian, C. Meuwly, R. M. Camacho, F. Jelezko, E. Bielejec, H. Park, M. Lončar, and M. D. Lukin, “An integrated diamond nanophotonics platform for quantum-optical networks,” Science 354, 847–850 (2016).
[Crossref]

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part II, 1st ed. (McGraw-Hill, 1953).

Nguyen, C. T.

A. Sipahigil, R. E. Evans, D. D. Sukachev, M. J. Burek, J. Borregaard, M. K. Bhaskar, C. T. Nguyen, J. L. Pacheco, H. A. Atikian, C. Meuwly, R. M. Camacho, F. Jelezko, E. Bielejec, H. Park, M. Lončar, and M. D. Lukin, “An integrated diamond nanophotonics platform for quantum-optical networks,” Science 354, 847–850 (2016).
[Crossref]

Nishioka, M.

C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, “Observation of the coupled exciton-photon mode splitting in a semiconductor quantum microcavity,” Phys. Rev. Lett. 69, 3314–3317 (1992).
[Crossref]

Novotny, L.

L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics 5, 83–90 (2011).
[Crossref]

L. Novotny and B. Hecht, Principles of Nano-Optics, 2nd ed. (Cambridge University, 2012).

L. Novotny and B. Hecht, “Dipole-dipole interactions and energy transfer,” in Principles of Nano-Optics, 2nd ed. (Cambridge University, 2012), pp. 256–264. See Eqs. (8.159) and (8.160) for the expression of the energy transfer rate.

Olaya-Castro, A.

G. D. Scholes, G. R. Fleming, A. Olaya-Castro, and R. van Grondelle, “Lessons from nature about solar light harvesting,” Nat. Chem. 3, 763–774 (2011).
[Crossref]

Pacheco, J. L.

A. Sipahigil, R. E. Evans, D. D. Sukachev, M. J. Burek, J. Borregaard, M. K. Bhaskar, C. T. Nguyen, J. L. Pacheco, H. A. Atikian, C. Meuwly, R. M. Camacho, F. Jelezko, E. Bielejec, H. Park, M. Lončar, and M. D. Lukin, “An integrated diamond nanophotonics platform for quantum-optical networks,” Science 354, 847–850 (2016).
[Crossref]

Park, H.

A. Sipahigil, R. E. Evans, D. D. Sukachev, M. J. Burek, J. Borregaard, M. K. Bhaskar, C. T. Nguyen, J. L. Pacheco, H. A. Atikian, C. Meuwly, R. M. Camacho, F. Jelezko, E. Bielejec, H. Park, M. Lončar, and M. D. Lukin, “An integrated diamond nanophotonics platform for quantum-optical networks,” Science 354, 847–850 (2016).
[Crossref]

Peragut, F.

R. Carminati, A. Cazé, D. Cao, F. Peragut, V. Krachmalnicoff, R. Pierrat, and Y. De Wilde, “Electromagnetic density of states in complex plasmonic systems,” Surf. Sci. Rep. 70, 1–41 (2015).
[Crossref]

Pierrat, R.

R. Carminati, A. Cazé, D. Cao, F. Peragut, V. Krachmalnicoff, R. Pierrat, and Y. De Wilde, “Electromagnetic density of states in complex plasmonic systems,” Surf. Sci. Rep. 70, 1–41 (2015).
[Crossref]

Polman, A.

A. F. Koenderink, A. Alù, and A. Polman, “Nanophotonics: shrinking light-based technology,” Science 348, 516–521 (2015).
[Crossref]

Purcell, E.

E. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

Raimond, J.-M.

S. Haroche and J.-M. Raimond, Exploring the Quantum: Atoms, Cavities, and Photons (Oxford University, 2013).

Rosta, E.

R. Chikkaraddy, B. de Nijs, F. Benz, S. J. Barrow, O. A. Scherman, E. Rosta, A. Demetriadou, P. Fox, O. Hess, and J. J. Baumberg, “Single-molecule strong coupling at room temperature in plasmonic nanocavities,” Nature 535, 127–130 (2016).
[Crossref]

Rupper, G.

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Figures (6)

Fig. 1.
Fig. 1. Jablonski diagram of a three-level system. For Γ b c = 0 , the system reduced to the model of a two-level atom. For Γ b c Γ b a , the three-level system is the simplest relevant model of a fluorescent molecule. In this case, Γ b c corresponds to a fast non-radiative decay towards state | c , and Γ c a corresponds to the radiative transition.
Fig. 2.
Fig. 2. Representation of the two scattering processes involved in the electrodynamic interaction between a dipole emitter and a structured environment.
Fig. 3.
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 ω p ± γ p / 2 .
Fig. 4.
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 ω p ± γ p / 2 .
Fig. 5.
Fig. 5. Solid line: normalized modification in the donor linewidth due to the presence of the acceptor Γ inter / Γ 1 versus the distance d between donor and acceptor. Dashed line: normalized energy transfer rate Γ e t / Γ 1 calculated using expression (55).
Fig. 6.
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 ω p ± γ p / 2 .

Equations (65)

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d ρ ^ d t = 1 i [ H ^ , ρ ^ ] .
H ^ 0 = ω a b σ ^ a b + σ ^ a b + ω a c σ ^ a c + σ ^ a c ,
H ^ 1 = d a b · E ( t ) ( σ ^ a b + + σ ^ a b ) ,
d ρ ^ d t = 1 i [ H ^ 0 + H ^ 1 , ρ ^ ] + { d ρ ^ d t } relax
d ρ b b d t = ( Γ b a + Γ b c ) ρ b b + 2 Im [ ρ a b Ω ( + ) ( t ) ] ,
d ρ c c d t = Γ c a ρ c c + Γ b c ρ b b ,
d ρ b a d t = γ a b 2 ρ b a i ω a b ρ b a + i ( ρ b b ρ a a ) Ω ( + ) ( t ) ,
Ω ( + ) ( t ) = 0 + Ω ( ω ) e i ω t d ω .
ρ b a ( ω ) = Ω ( + ) ( ω ) ω a b ω i γ a b / 2 ( 1 1 + s ) ,
s = 2 ( 2 Γ c a + Γ b c ) Γ c a ( Γ b a + Γ b c ) + Im [ | Ω ( + ) ( ω ) | 2 ω a b ω i γ a b / 2 ] d ω .
d ( + ) ( ω ) = 1 ( 1 ω a b ω i γ a b / 2 ) [ d a b · E ( + ) ( ω ) ] d a b .
d ( + ) ( ω ) = α a b ( ω ) ε 0 E ( + ) ( ω ) .
α a b ( ω ) = 3 π c 3 ω a b 3 Γ b a s p ω a b ω i γ a b / 2 u u .
Γ b a s p = ω a b 3 d a b 2 3 π ε 0 c 3 .
σ e ( ω ) = 3 π c 2 2 ω a b 2 γ a b Γ b a s p ( ω a b ω ) 2 + γ a b 2 / 4 ,
σ s ( ω ) = 3 π c 2 2 ω a b 2 ( Γ b a s p ) 2 ( ω a b ω ) 2 + γ a b 2 / 4 .
× × G ( r , r , ω ) ω 2 c 2 ε ( r , ω ) G ( r , r , ω ) = δ ( r r ) I
E ( r , ω ) = μ 0 ω 2 G ( r , r , ω ) d ( ω ) .
G ( r , r , ω ) = G 0 ( r , r , ω ) + S ( r , r , ω ) ,
α 0 ( ω ) = 3 π c 3 ω 0 3 Γ 0 ω 0 ω i γ 0 / 2 .
d ( + ) ( ω ) = α 0 ( ω ) ε 0 E exc ( + ) ( r s , ω ) + α 0 ( ω ) k 0 2 S ( r s , r s , ω ) d ( + ) ( ω ) ,
d ( + ) ( ω ) = α ( ω ) ε 0 E exc ( + ) ( r s , ω ) .
α ( ω ) 1 = α 0 ( ω ) 1 k 0 2 S ( r s , r s , ω ) ,
ω 2 c 2 S ( r s , r s , ω ) α 0 ( ω ) = I .
ω 2 c 2 [ u · S ( r s , r s , ω ) u ] α 0 ( ω ) = 1 ,
S ( ϖ p ) = ω 0 ϖ p i γ 0 / 2 ,
S ( ω ) = 3 π c Γ 0 ω 0 [ u · S ( r s , r s , ω ) u ] .
ϖ p = ω 0 i 2 γ 0 S ( ω 0 ) .
ω p = ω 0 Re [ S ( ω 0 ) ] ,
γ p = γ 0 + 2 Im [ S ( ω 0 ) ] .
γ p Γ 0 = 1 + 2 Im [ S ( ω 0 ) ] Γ 0 .
γ p Γ 0 = ρ u ( r s , ω ) ρ u , 0 ,
G ( r , r , ω ) = c 2 2 ω m e m ( r ) e m * ( r ) ω m ω i γ m / 2 ,
S ( ω ) = F m Γ 0 γ m / 4 ω m ω i γ m / 2 i Γ 0 / 2 ,
F m = 6 π c 3 ω m 2 γ m | e m ( r s ) · u | 2 .
1 = F m Γ 0 γ m / 4 ( ω 0 ϖ p i γ 0 / 2 ) ( ω m ϖ p i γ m / 2 ) i Γ 0 / 2 ω 0 ϖ p i γ 0 / 2 .
ϖ p ± = ϖ 0 + ϖ m 2 ± ( ϖ m ϖ 0 2 ) 2 + g 2 ,
ω p ± = ω 0 + ω m 2 ± F m Γ 0 γ m 4 ,
γ p ± = ( γ 0 Γ 0 ) + γ m 2 .
α i ( ω ) = 3 π c 3 ω 0 3 Γ i ω i ω i γ i / 2 for    i = 1 , 2 .
S i i ( ω ) = 3 π c Γ i ω 0 [ u i · S ( r i , r i , ω ) u i ] ,
G i j ( ω ) = 3 π c Γ i Γ j ω 0 [ u i · G ( r i , r j , ω ) u j ] .
X = ( d 1 ( + ) ( ω ) d 2 ( + ) ( ω ) ) ,
Y = ( α 01 ( ω ) ε 0 u 1 · E exc ( + ) ( r 1 , ω ) α 02 ( ω ) ε 0 u 2 · E exc ( + ) ( r 2 , ω ) ) ,
M = ( 1 S 11 ( ω ) ω 1 ω i γ 1 / 2 G 12 ( ω ) ω 1 ω i γ 1 / 2 G 21 ( ω ) ω 2 ω i γ 2 / 2 1 S 22 ( ω ) ω 2 ω i γ 2 / 2 ) .
0 = 1 S 11 ( ϖ p ) ω 1 ϖ p i γ 1 / 2 S 22 ( ϖ p ) ω 2 ϖ p i γ 2 / 2 + S 11 ( ϖ p ) S 22 ( ϖ p ) G 12 2 ( ϖ p ) ( ω 1 ϖ p i γ 1 / 2 ) ( ω 2 ϖ p i γ 2 / 2 ) .
ϖ p ± = ϖ 1 + ϖ 2 2 ± ( ϖ 2 ϖ 1 2 ) 2 + G 12 ( ω 0 ) 2 ,
ω p ± = ω 1 + ω 2 2 Re [ S 11 ( ω 0 ) + S 22 ( ω 0 ) 2 ] ± Re [ G 12 ( ω 0 ) ] ,
γ p ± = γ 1 + γ 2 2 + 2 Im [ S 11 ( ω 0 ) + S 22 ( ω 0 ) 2 ] 2 Im [ G 12 ( ω 0 ) ] .
ϖ p ± = ϖ 1 + ϖ 2 ± ( ϖ 2 ϖ 1 ) 2 ± G 12 ( ω 0 ) 2 ϖ 2 ϖ 1 .
γ p + = γ 2 ,
γ p = γ 1 + 2 Im [ S 11 ( ω 0 ) ] + 4 Re [ G 12 ( ω 0 ) 2 ] γ 2 .
σ e ( ω 0 ) = 6 π c 2 ω 0 2 Γ 2 γ 2 ,
σ s ( ω 0 ) = 6 π c 2 ω 0 2 ( Γ 2 γ 2 ) 2 .
Γ e t = 6 π Γ 1 σ a ( ω 0 ) | u 1 · G ( r 1 , r 2 , ω 0 ) u 2 | 2 ,
F i = 6 π c 3 ω m 2 γ m | e m ( r i ) · u i | 2 for    i = 1 , 2 .
S i i ( ω ) = F i Γ i γ m / 4 ω m ω i γ m / 2 i Γ i / 2 ,
G i j ( ω ) 2 = F i Γ i F j Γ j γ m 2 / 16 ( ω m ω i γ m / 2 ) 2 .
g i = F i Γ i γ m 4 for    i = 1 , 2 .
( 1 S ( ϖ p ) G ( ϖ p ) ω 0 ϖ p i γ 0 / 2 ) N 1
× ( 1 S ( ϖ p ) + ( N 1 ) G ( ϖ p ) ω 0 ϖ p i γ 0 / 2 ) = 0 .
ϖ p = ω 0 i 2 γ 0 S ( ω 0 ) ( N 1 ) G ( ω 0 ) ,
ϖ p + = ω 0 i 2 γ 0 S ( ω 0 ) + G ( ω 0 ) ,
ϖ p ± = ϖ 0 + ϖ m 2 ± ( ϖ m ϖ 0 2 ) 2 + N g 2 ,
ϖ p = ϖ 0 ,

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