Coupling to one eigenmode: Weak and strong coupling regimes

In this section, we describe the coupling between a resonant scatterer and the environment in the case where the electromagnetic response at point rs is dominated by one eigenmode of

the electromagnetic field. This encompasses the case of engineered optical cavities [48, 15] or multiple scattering systems in the localized regime [4, 33].

Hybrid eigenmodes

In a weakly lossy system, the regularized Green function around eigenfrequency ωM correspond-

ing to an eigenmode eM reads (see Appendix B)

Greg(r, r′, ω) = c 2 2ωM e∗_M(r′) eM(r) ωM − ω − iΓM/2 . (1.44)

As derived in Appendix B, this expression is non-singular and corresponds to the regularized Green function. Denoting by u the direction of the electric field eM(rs) at the position of the

scatterer, one can use Eq. (1.44) to express the scattered regularized Green function as Sreg(rs, rs, ω) = c2 2ωM ρMuu ωM − ω − iΓM/2− ik 6πI, (1.45)

where ρM = |eM(rs)|2. Let us consider an isotropic resonant scatterer, with polarizability αs(ω)

in vacuum, given by Eq. (1.35). An eigenmode of the coupled system {scatterer+electromagnetic field} is characterized by a pole in the dressed polarizability given by Eq. (1.43). Since the dressed polarizability is a dyadic, the equation satisfied by the coupled eigenfrequencies depends on the direction. The coupled eigenmodes corresponding to a resonance of the scatterer in direction u are associated to poles of the coefficient u · α(ω)u. These eigenfrequencies thus satisfy the coupling equation

1 = k2αs(ω)u · Sreg(rs, rs, ω)u. (1.46)

Let us introduce the classical coupling constant, defined as g2_c = 3

4Γ

R

sΓMFP. (1.47)

gc is the classical analog of the coupling constant introduced in cavity QED to describe the

interaction between a quantum emitter and an optical cavity [13, 14, 49]. Using the Purcell factor introduced in section 1.1.2, and introducing the variable ∆ω = ω − ωM, Eq. (1.46)

transforms into ∆ω2+ i∆ω 2 ΓM + Γ NR s
− ΓMΓ NR s 4 + g 2 c  = 0. (1.48)

As the result of the coupling between the scatterer and the field, two hybrid eigenmodes, with complex eigenfrequencies ωM + ∆ω+and ωM + ∆ω− appear [where ∆ω+and ∆ω− are the two

solutions of Eq. (1.48)]. Depending on the parameters of both the scatterer and the eigenmode, the solutions of this equation are imaginary or real, giving rise respectively to the weak and strong coupling regimes.

Weak coupling regime

When the eigenfrequencies ωM+ ∆ω± are imaginary, the coupling between the eigenmode and

the scatterer only results in a change of the linewidth of both systems. In the weak-coupling regime, the losses out of the environment are considered much higher than those of the scatterer

ΓM ≫ Γs. (1.49)

The picture in this case is the following: as soon as a photon is emitted by the scatterer to its environment, the latter is immediately lost (i.e. radiated to the far field or absorbed in the environment). Hence, an emitted photon will never come back from the environment to the scatterer. Solving Eq. (1.48), one can show that, to the first order of Γs/ΓM,

∆ω+_{= −}i 2 Γ NR s + 3FPΓRs
(1.50) ∆ω−_{≈ −i}ΓM 2 . (1.51)

The eigenfrequency ∆ω−corresponds to the non-perturbed mode of the electric field, that keeps its resonant frequency ωM and linewidth ΓM. The eigenfrequency ∆ω+ corresponds to the

perturbed scatterer, which resonant frequency remains ωs= ωM, but which radiative linewidth

has become ΓR ΓR s = 3 FP= 3 ρ(rs, ω) ρ0 . (1.52)

We recover the well-known expression of the enhancement of the spontaneous decay rate driven by the Purcell factor. The factor 3 is due to the average over transition dipole orientation in our definition of the Purcell factor11_{. Note that the internal non-radiative linewidth is not affected}

by the coupling to the environment. Strong coupling regime

The strong coupling regime occurs when the eigenfrequencies ωM+ ∆ω± are real, meaning that

the two eigenmodes of the coupled system are no longer degenerate. The condition to reach this 11_{Let u be the orientation of the electric field at r}

s. Let v and w be two unit vectors that form an orthonormal

basis joint with u, the orientation averaged decay rate reads hΓi = (Γu+ Γv+ Γw) /3 = Γu/3 since Γv= Γw= 0

regime reads12 g_c2 _≥ Γ NR s − ΓM
2 16 . (1.53)

For a quantum two-level system, ΓNR

s = 0 and the condition is simply gc ≥ ΓM/4, which is

consistent with the usual criterion in cavity-QED [15]. In the usual formulation, the explicit use of the transition dipole of the two-level system makes the criterion slightly different (but equiva- lent). In our formalism the transition dipole is implicitly in the coupling constant gc through the

radiative linewidth ΓR_s (because of energy conservation, as commented in section 1.2). The eigen- frequencies of the hybrid eigenmodes of the coupled system {electromagnetic field + scatterer} then read ∆ω±_{= ±} " g_c2₋ Γ NR s − ΓM
2 16 #1/2 − iΓM + Γ NR s 4 . (1.54)

The resonant frequencies are splitted symmetrically around ωM and are separated by the Rabi

frequency, defined as ΩR= 1 2 " g2_{c −} Γ NR s − ΓM
2 16 #1/2 (1.55) The linewidth of the hybrid eigenmodes read

Γ = ΓM + Γ

NR s

2 . (1.56)

Note that ΓR

s is not implied in this linewidth, since this term corresponds to the radiation of

the scatterer towards the eigenmode, and hence does not correspond to losses out of the coupled system. Finally, let us stress that satisfying Eq. (1.53) is not sufficient to observe the splitting in the coupled system spectrum or to observe temporal Rabi oscillations. For such an observation, the Rabi frequency has to overcome the linewidth of the hybrid eigenmodes, i.e.

2ΩR≥ Γ. (1.57)

This condition reads

g_c2 _≥ Γ

NR s + ΓM

2

8 . (1.58)

To reach the strong coupling regime, the coupling constant needs to overcome the intrinsic losses of each uncoupled system.

Graphical criterion

The graphical interpretation of the coupling condition presented here results from a very inspir- ing conversation with Juan-Jos´e S´aenz (Universidad Aut´onoma de Madrid, Spain). The coupling

12_{The discriminant of Eq. (1.48) reads ∆ =}h_16g2

c− ΓM− ΓNRs

2i /4.

condition Eq. (1.46) can be written 1 αs(ω)

= k2u.Sreg(rs, rs, ω)u. (1.59)

The real part of this equation drives the eigenfrequencies of the hybrid eigenmodes, while its imaginary part drives their linewidth. Here, we focus on the eigenfrequencies, independently on the linewidths. Let us consider a resonant scatterer with a resonant frequency ωs, described by

Eq. (1.35). The real part of the left term of Eq. (1.59) reads Re  1 αs(ω)  = k 3 3πΓR s (ωs− ω) . (1.60)

Let us consider an eigenmode with eigenfrequency ωM. The regularized scattered Green function

is given by Eq. (1.45), and the real part of the right term of Eq. (1.59) reads Rek2u.Sreg(rs, rs, ω)u=

ωMρM

2

ωM− ω

(ωM− ω)2+ Γ2M/4

. (1.61)

The eigenfrequencies of the coupled system are found when Eq. (1.60) equals Eq. (1.61). We represent both expressions versus ∆ω = ω − ωM in Fig. 1.1, for two different sets of parameters

corresponding respectively to the weak and the strong coupling regime. The crossing between the

∆ω = ω − ωM

Re

k

2

_u

·S

reg

(r

s

, r

s

, ω)u

Re  α−1_s (ω) Re  α−1 s (ω) non-degenerate eigenmodes: strong coupling degenerate eigenmodes: weak coupling

Figure 1.1: Graphical representation of the weak and strong coupling regimes. (Green) Eq. (1.61) plotted as a function of ∆ω (Red and blue) Eq. (1.60) plotted as a function of ∆ω for two different set of parameters corresponding respectively to the weak and strong-coupling regimes.

obtained in the weak-coupling regime. Varying the slope of the red curve to reach the blue curve, two new intersections appear with the green curve. They correspond to the two eigenmodes with eigenfrequencies ωM±∆ω obtained in the strong-coupling regime. This graphical representation

is very helpful for a qualitative understanding of the coupled system {scatterer+electromagnetic field}. For example, when one increases ΓR

s, the slope of Eq. (1.60) decreases in absolute value,

and one tends to the strong coupling regime. This could have been intuited, since ΓR s is the

spontaneous decay rate of the emitter in free space. However, it can be directly deduced from this method13. Last but not least, this graphical representation could be useful to get insight on regimes where the analytical calculations become heavy, e.g. when the resonant frequencies of the scatterer and the eigenmode are shifted. This last idea is an open question that we have not addressed in the present thesis.

In document Emission, scattering and localization of light in complex structures: From nanoantennas to disordered media (Page 34-38)