Plasmonic Nonlocal Response Effects on Dipole Decay Dynamics in

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Plasmonic Nonlocal Response Effects on Dipole Decay Dynamics in the Weak and Strong-Coupling Regimes Radoslaw Jurga, Stefania D'Agostino, Fabio Della Sala, and Cristian Ciraci J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b07462 • Publication Date (Web): 14 Sep 2017 Downloaded from http://pubs.acs.org on September 20, 2017

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The Journal of Physical Chemistry

Plasmonic Nonlocal Response Effects on Dipole Decay Dynamics in the Weak and Strong-Coupling Regimes Radoslaw Jurga,†,‡ Stefania D’Agostino,†,¶ Fabio Della Sala,†,§ and Cristian Ciracì∗,† †Istituto Italiano di Tecnologia (IIT), Center for Biomolecular Nanotechnologies, Via Barsanti, 73010 Arnesano, Italy ‡Dipartimento di Matematica e Fisica "E. De Giorgi", Università del Salento, 73100 Lecce, Italy ¶Institute for Materials Science and Max Bergmann Center of Biomaterials, TU Dresden, 01062 Dresden, Germany §Istitute for Microelectronics and Microsystems (IMM-CNR), Via Monteroni, Campus Unisalento, 73100 Lecce, Italy E-mail: [email protected]

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Abstract The largest increases in spontaneous decay rates of quantum emitters can be achieved using plasmonic structures that are characterized by closely spaced metallic elements. These systems can give rise to the smallest optical cavities attainable, offering a viable solution to achieve single molecule light-matter strong-coupling. On the other hand, their optical response might be strongly affected by nonlocal and quantum effects of the metal electron gas. In this work, we analyze the impact of nonlocal effects on the emission properties of a single quantum emitter coupled to a plasmonic system characterized by deeply sub-wavelength gap regions, in both the weak and the strongcoupling regimes. We find that the presence of nonlocality imposes strict limits to the achievability of strong-coupling with single molecules, in apparent contrast to recent experiments, suggesting that a more refined theory might be required. These limits are even larger if a k-dependent absorption is included in the calculations. These results place boundaries to the applicability of hydrodynamic methods.

Introduction A quantum photon emitter immersed in a plasmonic environment can experience a dramatic modification of its optical emission properties. The emitted light field can be scattered through the environment back on the emitter, which ultimately will experience a self-driving field. When the emitter is placed in a resonant optical cavity, a mutual exchange of energy between the emitter and the surroundings can be established. If the energy is irreversibly lost to the environment before any coherent exchange can occur – weak-coupling regime – the interaction manifests itself as a modification of the spontaneous decay rate of the emitter. 1 The enhancement of the spontaneous emission rate is crucial to a variety of applications, such as bioimaging 2 and single-photon sources 3 for quantum information. 4 The largest increases in spontaneous decay rates can be achieved using plasmonic systems that are characterized by closely spaced metallic elements, such as bowtie nanoantennas 5 or film-coupled nanocubes. 6,7 2


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If the emitter is strongly coupled to the surrounding cavity, a reversible exchange of energy, known as Rabi oscillations, can be established between the emitter and the environment. 8–10 The resulting system is described as a new modified emitter whose eigenstates, part light, part matter, show a degeneracy splitting in comparison to the emitter in free space. The strong-coupled states could enable the creation of ultralow-power all-optical switches 11 and nanolasers. 12 This cavity quantum electrodynamics typically requires low temperatures and high-finesse optical resonators. 13,14 However, several studies have shown that plasmonic systems are ideal candidates to observe light-matter strong-coupling interactions at room temperature, 15–25 and very recently the first experimental observation of single-emitter strong-coupling has been carried out using a film-coupled nanoparticle system. 26 Plasmonic systems that are characterized by a deeply sub-wavelength dielectric region, as small as a fraction of a nanometer, sandwiched by two metallic elements, and can give rise to the smallest optical cavities achievable. When metallic elements are brought in such a close proximity nonlocal and quantum effects can strongly affect the system response to the optical fields. A series of experimental works have shown that local theories fail to correctly predict plasmonic resonances of such systems 27–31 and several strategies have been adopted in order to correctly describe these observations including effective methods, 32,33 time-dependent density functional theory (TD-DFT) 28,34,35 and the hydrodynamic theory (HT). 27,36–38 The HT is a semi-classical model that takes into account the nonlocal behavior of the electron response by including the Thomas-Fermi (TF) electron pressure, which prevents the charges to be crushed into an infinitesimally thin layer at the surface of the metal. The HT allows to study plasmonic systems of large dimensions that are computationally prohibitive for ab initio calculations, and it has proven to give an accurate description of the optical response of film-coupled nanoparticle systems 27,30 and ultra-small gold nanoparticle arrays. 31 Size-dependent broadening can also be taken into account by introducing a phenomenological

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imaginary component to the TF pressure term. 39 This model is then referred to as the generalized nonlocal optical response (GNOR) model. The HT and the GNOR model are usually combined with the assumption of a constant equilibrium density and hard-wall boundary conditions, however, a spatial dependence of the electron density, and hence spill-out effects, can be included by considering nonlocal contributions to the free-electron gas kinetic energy, in place of the simple TF kinetic energy. 40–43 Analogously to the study of the scattering properties of metal particles that are brought in close proximity, nonlocal effects are also expected to be important when a photon emitter is placed only a few nanometers from the metallic surface, and even more dramatic effects are expected when the emitter is placed in the nano-cavity between two metal nanoparticles. The role of nonlocality has been recently investigated analytically 44 in the context of spherically symmetric systems in the weak-coupling regime and numerically for very small particles of only a few nanometers in size 45 for which the scattering is negligible. In this work, we first investigate the fluorescence properties of a quantum optical emitter located in the gap between two metal nanoparticles (NPs) whose size is typically used in experimental setups. Our implementation allows us to consider nonlocal effects in systems of NPs that exceed several tens of nanometers in size and whose scattered energy becomes comparable to their absorption. More importantly we analyze the impact of nonlocal effects in the strong-coupling regime. We find that the presence of nonlocality imposes strict limits to the achievability of strong-coupling with single molecules, in apparent contrast to recent experiments, 26 suggesting that a more refined theory is required. These limits are even larger if the GNOR model is considered. These results place boundaries to the applicability of HT methods.

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Theory Emission properties of a quantum emitter embedded in a plasmonic system can be retrieved if the dyadic Green’s function G of the systems is known. In the context of the HT, the Green’s function can be formally obtained by solving the following system of equations: ∇ × ∇ × G − εbound k02 G − µ0 ω 2 Pfree = Iδ(r), 2

2

ξ ∇ (∇ · Pfree ) + (ω + iγω) Pfree =

(1)

−ε0 ωp2 G,

where k0 , ε0 and µ0 are as usual the wavenumber, the permittivity and the permeability in vacuum, εbound (ω) is the relative permittivity associated to the interband transitions and core electron response in the metal, Pfree is the dyad associated with the free-electron response, ωp and γ are the plasma frequency and the damping coefficient respectively (note that a time dependence of the form e−iωt is implied), and I is the identity operator and δ is the Dirac delta function. The first term in the second of Eqs. (1) is the nonlocal correction, where the parameter ξ is defined such that ξ 2 = β 2 − iωD, with β = (3/5)1/2 vF in the high-frequency limit (vF being the Fermi velocity) and D an ad hoc diffusion constant that provides additional line broadening. 39 By changing the values of ξ we can then switch between all the different models. Namely, if ξ = β = D = 0, the system of Eq. (1) is reduced to the local model; if β 6= 0 and D = 0, we have that ξ = β and we obtain the TF-HT; finally if β 6= 0 and D 6= 0, we have the GNOR model. Numerical values of these parameters are reported in Table 1. Although these semi-classical models take into account the nonlocal response of metals, a rigorous quantum mechanical treatment would also include quantum features associated with the emitter itself. Such approaches, as the TD-DFT, are computationally expensive making them unsuitable for the study of plasmonic systems of sizes of more than a few nanometres. Therefore the scope of this work focuses on the study of the nonlocal response of metals using the presented semi-classical methods, with the emitter modelled as a classical point dipole with dipole moment pc . 5



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Dipole decay dynamics in the weak-coupling regime A weak interaction of the emitter with its environment leads to a perturbation of the spontaneous decay rate γsp that can be obtained through Fermi’s golden rule as: 46,47

γsp =

2ω 2 |p|2 [ˆ np · Im {G(rd , rd )} · n ˆp ] , h ¯ ε0 c 2

(2)

where p = pc /2 is the transition dipole moment, n ˆ p is its unit vector, and rd is the location of the dipole. This spontaneous total decay rate is composed by the radiative decay rate γr corresponding to photons actually emitted and the non-radiative decay rate γnr corresponding to photons that are absorbed by the environment, such that γsp = γr +γnr . The non-radiative decay rate can be computed by integrating the losses (see Eq. (13) in the Numerical methods section) inside the metal. The fraction of the decay rate resulting in emitted photons, i.e. the probability that during a dipole transition a photon is radiated rather than absorbed by the environment, is thus given by the quantum efficiency q = γr /γsp . The enhancement of the total decay rate with respect to the vacuum, γsp /γ0 is the Purcell factor. In this paper, we assume the quantum efficiency in free space to be q0 = 1, that is to say we ignore any decay rate channels internal to the emitter. Another key quantity that is useful to define is the fluorescence enhancement ηem . In the weak-coupling regime, the energy levels of the emitter are negligibly depleted, so that the excitation rate γex is proportional to the incident field intensity and the fluorescence rate enhancement can be defined as:

ηem

|E(rd ) · n ˆ p |2 q , = |E0 (rd ) · n ˆ p | 2 q0

(3)

where E is the electric field evaluated at the emitter location; parameters with a subscript 0 indicate values in free space. The plasmonic response of the metallic system can produce a strong field that increases

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the excitation rate but that may simultaneously lower the quantum efficiency through ohmic losses. The competition between the two effects depends on the specific system, its geometry and the distance between the emitter and the metallic structure. 46 Moreover, the dipole nearfield of the emitter contains a wide spectrum of large k vectors associated with the evanescent fields that might be strongly affected by the nonlocal response of the metal.

Dipole decay dynamics in the strong-coupling regime In order to study this regime, we need to take into account non-perturbative dynamical effects. The information about these effects is contained in the far-field power spectrum of the emitter, which can be calculated if the decay rate and Lamb shift are known. 48 In the rotating wave approximation, the averaged power spectrum emitted to the far-field by a point-like dipole can be calculated as S(ω) = h ¯ ωq(ω)γsp (ω)S ′ (ω)/ (2π) where the dipole spectrum S ′ (ω) is given by the expression: 48 2 1 , S ′ (ω) = i (ω0 − ω − δω) + 21 γsp

(4)

in which ω0 is the dipole transition frequency and δω is the photonic anomalous Lamb shift defined as: 1 δω(ω) = P π¯hε0 c2

Z

∞ 0

ω ′2 |p|2 [ˆ np · Im {Gsc (rd , rd )} · n ˆ p ] dω ′ , ω′ − ω

(5)

and where the scattering dyadic Green’s function is defined as Gsc = G − G0 , with G0 being the free-space Green’s function. By extending the lower bound of integration to −∞ the integral can be calculated by using the Kramers-Kronig relations, resulting in

δω(ω) ≃

ω2 |p|2 [ˆ np · Re {Gsc (rd , rd )} · n ˆp ] . h ¯ ε0 c 2

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(6)


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Both the total decay rate γsp and the Lamb shift δω are proportional to the square of the transition dipole moment |p|2 which itself is proportional to the dipole oscillator strength:

f=

2me ω0 |p|2 , 3e2 h ¯

(7)

where me and e are respectively the mass and charge of the electron. To enter the strongcoupling regime, a sufficiently large dipole moment or oscillator strength is needed. An estimation for a lower bound required to the onset of the vacuum Rabi splitting can be obtained with the assumption that the decay rate has a Lorentzian shape:

γsp (ω) = γsp (ω0 )

1 Γ 2 2

(ω − ω0 ) +

2 . 1 Γ 2

(8)

Then, the threshold oscillator strength to enter the strong-coupling regime is fth = Γ/ (4γsp (ω0 )). 15

Numerical methods We consider metallic systems of relative permittivity ε embedded in air. The relative permittivity ε can be considered as composed by two contributions, one associated with free electrons εfree and the other accounting for the inner electrons polarizability εbound such that ε = εfree + εbound . In the local response approximation, εfree is well described by the Drude model as 49 εfree (ω) = −

ωp2 , ω 2 + iγω

(9)

where ωp is the plasma frequency and γ the damping coefficient. In order to describe realistic metals, the bound electron contribution εbound is obtained by fitting it to the experimentally measured relative permittivity εexp . Hence it is obtained by subtracting the Drude contribution from the relative permittivity such that εbound = εexp − εfree, Drude . This background contribution remains as calculated for the local case through this work and it is to be understood as a local contribution. Only the behavior of free electrons is described with nonlocality,

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through the second of Eqs. (1), where numerical values for the parameters used are shown in Table 1. Table 1: Parameter values for the description of metals Gold Silver Olmon (single-crystal) Palik 8.1 eV 8.99 eV 0.047 eV 0.025 eV 1.40 × 106 m s−1 1.39 × 106 m s−1 1.90 × 10−4 m2 s−1 3.61 × 10−4 m2 s−1 8.62 × 10−4 m2 s−1 9.62 × 10−4 m2 s−1

50,51

εexp h ¯ ωp 50 h ¯ γ 50,52 vF 53 D (A = 0.5) 54 D (A = 1) 54

Because the nonlocal effects are typically introducing spatial variations of the order of tenths of nanometers while the wavelength of light is of the order of hundreds of nanometers, the problem is strongly multiscale. Therefore we use the finite-element method (FEM) for simulations as it allows a variable mesh size. Furthermore the numerical solutions to Maxwell’s equations computed with the FEM fully take into account the electromagnetic behavior of the system, including higher-order plasmon modes. The simulations are carried out with Comsol Multiphysics. 55 Moreover, the investigated geometries feature an axial symmetry that allows to reduce the computational cost of the problem with a so-called 2.5D implementation. Axis-symmetric quantities such as the field E or the polarization P are written as a sum of cylindrical harmonics

Eρ,z,φ (ρ, z, φ) =

X

(m)

Eρ,z,φ (ρ, z) e−imφ .

(10)

m

This leads to the following weak form for the polarization equation:

X m

2π

Z " Ω

(m)

β

2

Pρ ρ

(m)

∂Pρ + ∂ρ

! ! (m) (m) (m) (m) im (m) ∂Pz ∂ P˜ρ im ˜ (m) ∂ P˜z P˜ρ + P + − P + + ρ φ ∂z ρ ∂ρ ρ φ ∂z i (m) ˜ ρdρdz = 0, (11) − (ω 2 + iγω)P(m) + ε0 ωp2 E(m) · P 9



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˜ is a test function with a cylindrical harmonic decomposition as in Eq. (10) but with where P the sign of the exponential argument being positive, and vectors with the superscript (m) (m) (m) (m) are defined as v(m) ≡ vρ ρˆ + vz zˆ + vφ φˆ with hats denoting unit vectors. Because each of

the integrals in the sum must be equal to zero, the weak forms can be solved for each value of m separately, the full 3D problem is then reduced to a set of 2D problems independent of φ, one for each value of the index. We solve the problem in the scattered field formulation, and for the emission study the incident field is given by the field associated to a radiating dipole in free space. In particular, for the field of a dipole p = pˆ z oriented along z used to model a quantum optical emitter, the decomposition into cylindrical harmonics only leaves the m = 0 term of the sum: |pc |eik0 r E= 4πε0

where r =

k 2 3ik0 3 − 0− 2 + 3 r r r

cos(θ) sin(θ) ρˆ

ik0 1 k02 sin2 (θ) 2 zˆ , (12) + 3 cos (θ) − 1 − 2 + 3 + r r r

p ρ2 + z 2 and θ = arctan(ρ/z). Similar expressions can be found for a dipole

oriented along the x direction, although in this case the field components contain cylindrical harmonics with azimuthal number m = ±1 (see Supporting Information). The fluorescence properties of the quantum emitter can be thus computed from the results of the FEM simulations. Among these properties, the non-radiative decay rate can be computed by integrating the absorbed power over the environment as: 1 γ0 γnr = 2 W0

Z

Re {J∗ · E} dV,

(13)

Ω

where γ0 = ω 3 |p|2 / (3¯hπε0 c3 ) and W0 = ω 4 |pc |2 / (12πε0 c3 ) are respectively the decay rate and the radiated power in free space. Then the radiative decay rate can be easily obtained by difference, γr = γsp − γnr . FEM calculations were validated using exact solutions for the local case of a metallic

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sphere (see Supporting Information).

Results and discussion Nanoparticle dimer in the weak-coupling regime As a first case of study, let us consider a dimer of metal NPs closely spaced surrounded by air. This system has been extensively investigated both theoretically 56–60 and experimentally 61,62 in various contexts. Despite its simplicity, in fact, this geometry allows to generate the maximum field enhancements achievable in plasmonic structures by concentrating light in deeply sub-wavelength volumes. In particular we consider gold (Au) NPs that are 60 nm in diameter, a size commonly encountered in experimental setups, and a single quantum emitter placed at the center between the NPs. Because the electric field in this region is mostly orthogonal to the metal surface we consider the dipole to be oriented along the dimer axis, as shown in Fig. 1a. By solving the system in (1) and using the relations (2) and (3) we calculate the Purcell factor γsp /γ0 , the quantum efficiency q and the fluorescence rate enhancement ηem as a function of the emitter frequency and the distance d between the NPs. Results for the local, the TF-HT and the GNOR models, respectively, are shown in Fig. 1. Some trends as a function of the gap size are common for all considered models. As the gap becomes smaller, the total decay rate grows broadly with no quenching occurring down to 0.1 nm, the smallest gap of the performed simulations.The quantum efficiency follows the trajectory of the peak resonance of the radiative decay rate, and at gaps below ∼ 2 nm its peak is noticeably shifted towards lower energies. A similar trend can be observed in the fluorescence enhancement. At these distances, however, nonlocal effects become important. As the gap shrinks, the total decay rates are lower for the TF-HT and GNOR models with respect to the local case. This is due to the fact that field enhancement is reduced with respect to the local case, and so are the losses, particularly for the TF-HT model. The quantum efficiency in this case is in fact higher (along the peak) down to a gap of ∼ 0.5 nm, although 11



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Strong-coupling in silver systems Let us start by considering silver (Ag) nano-structures. The higher interband losses in Au systems are in fact very detrimental to strong-coupling interactions and the presence of a dielectric material surrounding the metal is needed in order to push the resonances away from the d-band absorption. This will be analyzed in section . Moreover, in Ag systems, the total decay rate is closer to a Lorentzian function compared to Au systems, and consequently it allows to obtain a more accurate estimation of the oscillator strength threshold fth . In particular, we consider the geometries depicted in Fig. 2. The dipole is oriented along the rotational symmetry axis of the structures and is always located at 2 nm from the surface of the metal, such that the gaps of the dimers are d = 4 nm. The case of the single NP has been previously analyzed by D’Agostino et al. within the local response approximation 15 and it is in agreement with our results. In Fig. 2 is shown the dipole spectrum S ′ as a function of the oscillator strength of the dipole and its emission frequency for each model of the metal optical response. Two cases are included for the GNOR model using different values of the parameter D corresponding to different values of the constant A of the Kreibig model of size-dependent damping 63 for spherical NPs. To obtain the clearest picture of Rabi splitting, the dipole transition frequency ω0 is set to be that at which the maximum value of the total decay rate occurs. This maximum value depends on the model since nonlocal effects can cause a shift of the plasmon resonance, and therefore ω0 is different in each case. The figure also shows the threshold oscillator strength fth (dashed vertical line). In addition, a figure included in the Supporting Information shows the dipole spectrum as a function of ω0 in which anti-crossing can be clearly observed. For each geometry, the oscillator strength threshold increases as we move from a local response to a nonlocal one. For a single NP, it is respectively 2.1, 3.7 and 6.1 times larger than in the local case for the TF-HT and GNOR model with A = 0.5 (D = 3.61 × 10−4 m2 /s) and A = 1 (D = 9.62 × 10−4 m2 /s). This change is caused by the effect of nonlocality 13



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(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n

(o)

(p)

Figure 2: Dipole spectrum of an emitter in the vicinity of different Ag NP systems shown on the top (sizes are in nanometer), calculated using the local model (a-d), TF-HT model (e-h), the GNOR model with A = 0.51 (i-l) and the A = 1 (m-p), respectively. The dipole transition frequency is chosen to be at the peak of the total decay rate. The dashed vertical lines in the map plots indicate the oscillator strength threshold.

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the TF-HT. Because this geometry has sharp tips, the effects of plasmon damping with the GNOR model become very important due to the high surface to volume ratio of the structure. This leads to a stark difference relative to the local and TF-HT models, both neglecting this phenomenon. For the cone dimer, the oscillator strength threshold with GNOR (A = 0.5) is respectively 7.1 and 6.3 times higher than with the local model and the TF-HT, a bigger jump than for the film-coupled NP for which it is only 3.3 and 1.6 times higher. The considerable difference between local and nonlocal models shows that an accurate nonlocal description of the optical response of metals is essential for the investigation of strong coupling effects with plasmonic structures.

Strong-coupling in gold systems We now consider Au nanostructures. Despite having oscillator strength thresholds estimations similar to Ag systems, the upper branch of the Rabi split is faint in the dipole spectra unless we use the most drastic geometries such as the cone dimer as shown in the Supporting Information. Chikkaraddy et al. have however reported 26 the experimental observation of Rabi splitting in a film-coupled NP system. The difficulty of observing in our calculations the upper branch of the split in Au systems might be due to the higher absorption in the high energy region of the spectrum. If the permittivity of the surrounding dielectric medium is increased such as to shift the total decay rate peak toward lower energies, the upper branch of the Rabi split can become visible albeit still not with nonlocal models. Fig. 4 shows the dipole spectrum and the normalized total decay rate both as a function of the background permittivity for a fixed oscillator strength f = 1, and the dipole spectrum as a function of the oscillator strength for a fixed permittivity ε = 6 for the geometry used in Ref. 26 The dipole transition frequency is set to h ¯ ω0 = 2.2 eV. The results are shown for the local, TF-HT and GNOR (A = 1) models. The total decay rate has several resonances which are redshifted as the background permittivity increases. The main – largest and broadest – resonance yields the two main 16


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the emitter, all of which might turn out to be relevant for the strong-coupling regime in geometries with small gaps such as the ones considered here.

Conclusions In this paper we reported the effects of nonlocality on the fluorescence of a quantum emitter located between a dimer of gold nanospheres, and on the strong coupling of an emitter close to different metal nanostructures. We found that the fluorescence enhancement of a dipole by a dimer of gold NPs can be as much as 7 times lower with the GNOR model than in the case of a local response model when the gap is of the order of 1 nm. Additional insights could be obtained by analyzing the fluorescence of an emitter with an arbitrary position rather than restricted to the axis of symmetry of the dimer. This requires the development of new numerical approaches but can lead to an understanding of fluorescence in more general configurations. We have shown that nonlocality increases the oscillator strength threshold at which Rabi splitting can occur. The increase is caused by a total decay rate that is lower and broader. This demonstrates the importance of taking nonlocality into account when engineering plasmonic structures in the strong-coupling regime. However the difficulty to replicate the experimentally observed single molecule strong coupling with gold nanostructures suggests that we might have reached the limits of what the TF-HT and the GNOR model can be used for. The analysis of nonlocal effects for plasmonic structures in the strong-coupling regime might then require a more accurate description that takes into account the quantum nature of the microscopic behavior of the electrons.

Supporting Information • Anti-crossing in strong-coupling regime, strong-coupling with gold structures, dipole filed for a dipole oriented along x ˆ in cylindrical coordinates, classical decay rates near 18


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a metal sphere

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Plasmonic Nonlocal Response Effects on Dipole Decay Dynamics in

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